- Formations :
EXTRAIT 1 :
Ground-coupled acoustic airwaves from Mount St. Helens provide constraints on the May 18, 1980 eruption Jeffrey B. Johnson and Stephen D. Malone Abstract The May 18, 1980 Mount St. Helens eruption perturbed the atmosphere and generated atmosphere-to-ground coupled airwaves, which were recorded on at least 35 seismometers operated by the Pacific Northwest Seismograph Network (PNSN). From 102 distinct travel time picks we identify coherent airwaves crossing Washington State primarily to the north and east of the volcano. The travel time curves provide evidence for both stratospheric refractions (at 200 to 300 km from the volcano) as well as probable thermospheric refractions (at 100 to 350 km). The very few first-hand reports of audible volcano sounds within about 80 km of the volcano coincide with a general absence of ground-coupled acoustic arrivals registered within about 100 km and are attributed to upward refraction of sound waves. From the coherent refracted airwave arrivals, we identify at least four distinct sources which we infer to originate 10 s, 114 s, 180 s and 319 s after the onset of an 8:32:11 PDT landslide. The first of these sources is attributed to resultant depressurization and explosion of the cryptodome. Most of the subsequent arrivals also appear to be coincident with a source located at or near the presumed volcanic conduit, but at least one of the later arrivals suggests an epicenter displaced about 9 km to the northwest of the vent. This dislocation is compatible with the direction of the sector collapse and lateral blast. We speculate that this concussion corresponds to a northern explosion event associated with hot cryptodome entering the Toutle River Valley. Keywords: acoustic airwaves ; ground-coupled seismicity; Mount St. Helens 1. Introduction Mount St. Helens (MSH) erupted spectacularly on the morning of May 18, 1980 following an 8:32:11 PDT magnitude Ml 5.1 earthquake and consequent large landslide/sector collapse onset, which was observed approximately 10 s later . This northward-directed avalanche induced an abrupt unloading of a pressurized magmatic system (e.g.,  and ), which led to the onset of a vertical eruption column at 8:32:47 PDT, northward-directed lateral blast at 8:32:56 PDT, and Plinian phase, which initiated at approximately 8:37:00 PDT . According to satellite imagery  convective plume rise then alternated with repeated column collapse and associated co-ignimbrite ash columns for more than 8 h. Because clouds and plume effectively obscured much of the vent and northern flanks of the volcano starting only a few tens of seconds after the initial eruption onset, visual identification of subsequent explosive pulses was inhibited . Nevertheless, based upon analysis of seismic records  and , and photographic and satellite imagery  and , there is evidence to support at least two explosive pulses within the first minutes of the initial 8:32:11 PDT earthquake. Due to the complex and extended-duration source processes and saturation of proximal seismographs the identification of individual eruptive phases has been somewhat difficult to constrain seismically. As a result, the time history of potential eruptive pulses at the onset of the MSH paroxysm is not well determined. In an attempt to better understand the eruptive chronology during the first 5 min of the eruption, we focus here on the analysis of acoustic airwaves recorded on regional seismometers 15 to 32 min after the initiation of the eruption. First-hand reports gathered from the general public offer some potential insight into the nature of airwaves produced on the morning of May 18, 1980. More than 1200 people responded to a poll that asked for their observations on the audibility of the climactic MSH eruption . With only a few exceptions, mapped reports of audible sound indicate a pronounced zone of silence that extended from as close as 10 km from the volcano to as far as 80 km. At greater offsets, observers cite variations on “a 15 minute barrage of sonic booms, thunder, and dynamiting.” . Other observers report “low-frequency concussions, ear-popping, and faint sonic booms” that suggest low-frequencies and/or near-infrasound (< 20 Hz) pressure disturbances. Though the eruption remained audible at distances as far as 750 km, well into Montana, California, and British Colombia, the greatest “zone of loudness” was reported at about 200 km with a possible second zone of loudness identified at farther offsets . The alternating zones of sound intensity are qualitative, but hint at first and second refracted arrivals from ray paths turning in the high-velocity stratosphere or thermosphere . These atmospheric refractions are common for acoustic waves ducted in the atmosphere and have been observed during high-energy explosive testing [e.g., ], as well as during other volcanic eruptions such as Krakatau, Pinatubo, and Pavlov (e.g., ,  and ). 2. Data: MSH airwaves recorded by seismometers Regional seismic data provided a comprehensive record of the May 18 MSH eruptive activity because they responded to the relatively energetic ground-coupled pressure perturbations radiated by the volcano into the atmosphere. In 1980 the Pacific Northwest Seismograph Network (PNSN) operated 72 seismographs distributed throughout Washington and into parts of northeastern Oregon. A subset of 35 of these stations, located primarily to the north and east of MSH, recorded clear airwaves associated with the May 1980 eruption (Fig. 1). As these recordings were made during the dawn of the digital seismic age, data was recorded on a mixture of digital and hard copy media. Due to the relatively slow propagation speed of sound in the atmosphere (relative to seismic waves in the ground), much of the ground-coupled acoustic arrivals reached the regional seismic network long after the primary seismic shaking associated with the Ml 5.1 earthquake had dissipated. This enabled us to identify robust ground-coupled airwave arrivals on seismograms at instruments located 67 to 340 km from the source (see examples in Fig. 2). Unfortunately, because the atmosphere-to-ground transmission is so complex, influenced by incidence angle, signal amplitude, site response, and acoustic frequency, we are reluctant to utilize seismic trace characteristics to recover details about the original sound intensity or frequency content of the airwave. It is not possible, for instance, to distinguish whether these records originate from infrasonic (< 20 Hz) or sonic waves impinging upon the earth, or alternatively, by mass flow along the ground/atmosphere interface (e.g., caused by abrupt barometric changes or wind), or potentially by other types of seismo-acoustic phases (e.g., ground-coupled Rayleigh waves ).
EXTRAIT 2 :
Our analysis of ground-coupled airwaves begins with handpicked arrival times identified on 35 seismographs located 67 to 340 km from the volcano and encompassing an azimuthal distribution of − 35° to + 103° (relative to north). Between 8:47:18 and 8:56:07 PDT a total of 102 distinct arrivals are identified from digital and paper records with as many as 7 arrivals picked from certain individual stations (e.g., BLN; see Fig. 2). After 8:56:07 PDT a few additional transients were identified at individual stations, but none of these were linked to arrivals at neighboring stations so their significance is unclear. Although we possess no evidence for airwaves arriving prior to 8:47:18 PDT it should be noted that this record could potentially be incomplete. In May 1980 some seismograms were recorded only on digital media following automated earthquake triggering, which may not have occurred for small or isolated ground-coupled airwaves. Although continuous film records were scanned for additional arrivals, only a subset of stations were available from this medium. Arrival picks may be viewed in a time–distance fashion to identify logically connected “coherent” arrivals propagating across the network. We interpret 91 of the picked arrivals as belonging to nine distinct acoustic travel time (or arrival) curves based upon a reasonable move-out (Fig. 3). The nine coherent airwave travel time curves are separated into three distinct families based upon their apparent velocity. Two arrivals, with apparent velocities of 501 and 520 m/s, are found at close offsets (100 to 200 km). Three arrivals, with very consistent apparent velocities of 334 to 337 m/s, are identified at distances greater than 180 km. And at least four arrivals, with apparent velocities between 371 and 452 m/s, are also found at these greater offsets. Finally, two additional isolated arrivals at extremely close offsets are identified at the seismic station LON, 67 km from MSH. Asterisks () indicate the picks used in arrival curve linear regression fits. Small circles (•) indicate additional picks not easily attributed to any coherent travel time move-out. Encircled asterisks in arrivals #1/2 indicate stations located primarily to the north of MSH. We are confident in the Fig. 3 travel times curves, which are identified by examining how individual arrivals belong to a single coherent move-out. A linear regression is then applied to the arrivals to establish a slope, which is inversely proportional to the apparent velocity. In a few cases arrivals occur in quick succession (within 20 s of one another; e.g., station BLN) and their inclusion or exclusion in a particular curve is a judgment decision based upon a best fit with arrivals at similar offsets. Single erroneous picks may modestly affect the apparent velocity of an arrival; for instance, incorporation of the second BLN arrival would change the apparent velocity of the corresponding curve by + 13 m/s. Overlapping travel time curves with different slopes are not identified in our data. In a few instances we have discounted an arrival that did not easily fit into any specific travel time curve. Examples of selective exclusion include the omission of the second arrival at station NAC or the second arrival at station SPW (refer to Fig. 2); some of these ‘mispicks’ could potentially be explained by ambient noise. In spite of these subjective decisions we are satisfied with the identification of the primary coherent arrivals and the estimation of their approximate apparent velocities. It is notable that all arrival time picks were identified prior to, and independent of, the subsequent ray path modeling presented below. 3. Modeling Coherent arrivals are analyzed in terms of their apparent velocities to determine their angle of incidence at the earth's surface and turning altitudes. We then employ forward ray path modeling to recover likely atmospheric propagation paths and transit time for refracted energy at different altitudes. The calculated transit times require that multiple source origin times (i.e., multiple events) be invoked to explain the multiple arrivals in the ground-couple seismic data. Based upon minimization of arrival time residuals we demonstrate that at least one of the later travel time curves should be attributed to a source displaced from the MSH vent region. 3.1. Propagation paths Recovered apparent velocities ca provide information about the angle of incidence i with which presumed MSH airwaves are impinging upon the ground (at z ≈ 0) in the vicinity of the seismographs. Here the angle of incidence is measured in the traditional sense, with respect to a vertical incidence and the intrinsic sound speed c(z) in the atmosphere is calculated from virtual acoustic temperature ( where T(z) at a specific height is measured in kelvin). To solve for incidence angle at ground level the temperature c(z ≈ 0) must be known or estimated. Table 1 displays calculated incidence angles for the nine travel time curves presented in Fig. 3 at a range of temperatures (T(0) = 0 to 20 °C; 273–293 K). These temperatures encompass a reasonable range of conditions in Washington State in May at elevations below 2.5 kilometer elevation and at 9:00 AM in the morning. The nine travel time curves have been separated into three different groupings, or families (i.e., A–C), based upon similar apparent velocities and incidence angles. These families suggest three distinct turning altitudes in a horizontally stratified atmosphere. Travel time curve number (as displayed in Fig. 3), associated apparent velocity, incidence angles for temperature range, and apparent velocity family grouping based upon similarities in incidence angle. Assuming that ray theory is appropriate for our propagating acoustic waves, a ray parameter p may be conserved throughout the ray trajectory in non-moving media . Here we consider that incidence angle, intrinsic sound speed and apparent velocity are functions of altitude z: where rEarth is the earth's radius at z = 0. A windless atmosphere is an incomplete approximation of the effective velocity structure. For an atmospheric structure that is radially stratified in terms of temperatures and horizontal winds, a modified ray parameter can be utilized [e.g., from ]: In this case u(z) is the horizontally wind speed in the direction of propagation. The ray path turning altitude is found for horizontal incidence angles (i.e., i = 90°). We can thus expect energy to return to earth from altitudes where the following condition is satisfied: At tropospheric to lower thermospheric altitudes (i.e.,< 150 km) the coefficient on the right side of Eq. (4) remains relatively constant and earth curvature effects, in terms of ray path modeling, are found to be relatively insignificant.
EXTRAIT 3 :
Because radiosonde data is limited to a few tens of kilometers altitude, we utilize a COSPAR 1986 International Reference Atmosphere (CIRA) model  and  to determine turning altitudes for airwaves propagating into the upper stratosphere and thermosphere. CIRA provides tabulated empirical data for monthly zonal winds and temperatures at 10 degree latitude increments. Fig. 4 shows sample temperature, wind, and calculated turning altitude profiles (both eastward and northward) for several latitudes (40° and 50°) during 3 months (April, May, and June), which are intended to bracket potential conditions for the May 18, 1980 MSH eruption. Because the CIRA data extend to only 120 km, we supplement the temperature profile for higher altitudes with modeled data for May 18th at 46.2° N from the Mass Spectrometer Incoherent Scatter (MSIS 90) model  and . Due to extreme and largely unconstrained, variability of wind data in the thermosphere above 120 km, we fix the zonal winds at 0 m/s and comment on the potential influence of high thermosphere winds in the Discussion section. The third and fourth profiles of Fig. 4 display an ‘effective velocity’ (i.e., the lefthand side of Eq. (4)) for both eastward and northward-directed acoustic waves. In the case of northward propagating sound we have set the traditionally less intense meridional winds to zero to highlight extreme variations that might be encountered in the atmosphere. We infer that radiated sound is capable of refracting back to earth where this effective velocity exceeds the various apparent velocities of the different families (C, B, and A) recovered directly from the seismic data. This figure shows the clear capabilities for sound turning in both the stratosphere/mesosphere and thermosphere under a range of atmospheric profiles. Using a predetermined atmospheric structure it is possible to estimate atmospheric propagation paths using ray tracing  and calculate transit times for a known acoustic source at pre-determined altitude [e.g., ,  and ]. It is important to consider that the forward modeling requires strong assumptions about the temperature and wind structure up into the thermosphere and that direct measurements in and above the stratosphere are not available near MSH on May 18, 1980. We are forced instead to rely on empirical models such as MSIS 1990 and COSPAR CIRA 1986 for estimates of horizontally stratified atmospheric conditions. These profiles give average atmospheric structures for a specific latitude and season, but in reality the conditions will vary day to day and during the course of a day. Short-term (hourly) variations may be especially pronounced for winds in the thermosphere above about 150 km . It is uncertain how precisely ray theory can be applied to the atmosphere for the wavelengths in question. Scattering is expected for acoustic waves due to localized wind shear and/or turbulence, which results in extreme temperature gradients over potentially very small distances. Although ray theory is thus not a satisfactory predictor of all the acoustic arrivals commonly observed [e.g., ], we apply ray path modeling here to obtain travel time estimates for expected stratospheric/mesospheric and thermospheric refractions. Toward this goal, ray tracing provides valuable insights into regional sound propagation and is computationally simple to perform compared to alternative methods, such as finite difference wave propagation models. Forward modeling is vital and is used in this study for comparison with observed ground-coupled airwave arrival times in order to deduce the source origin time(s) of the various events. We illustrate projected ray paths from a hypothetical MSH source at 2.5 kilometer elevation according to Eq. (3) (see Fig. 5a). Ray paths are shown for acoustic waves propagated in a zonal (easterly) and meridional (northerly) direction. The zonal propagation uses winds and temperatures from COSPAR CIRA 1986 tables taking the average of the 40 and 50° N May profiles. Above 120 km, MSIS 1990 modeled temperatures for May 18 at 46.2° N, 122° W are utilized. The meridional ray path modeling is performed using an atmosphere with the same temperature profile and no horizontal winds. Though meridional winds in the upper atmosphere are by no means stagnant, they are significantly less than zonal winds. The zero velocity wind field is useful for demonstrating a potential extreme scenario. The ray tracing is performed by conserving the wind-adjusted ray parameter (p; Eq. (3)) for a range of initial inclinations ranging from 0 to 90 degree incidence. Fig. 5a illustrates a range of conceivable propagation paths in the two orthogonal directions. It is interesting to note that when the ray tracing is performed according to classical ray theory in two dimensions, acoustic energy returns to earth only at very limited distances (< 20 km and > 250 km). This significant shadow zone offers a convenient explanation for the lack of audible sounds at intermediate distances from the volcano. It is also noteworthy that there is no predicted stratospheric refraction for the modeled eastward propagating acoustic energy. Scattering and three dimensional structure can likely explain how acoustic energy returns to the earth at a much wider range of azimuths and propagation distances than shown in this simplistic ray tracing model, and thus why the observed shadow zone is smaller than predicted. 3.2. Evidence for multiple sources and source locations Arrival times from rays with different ray parameters are used to construct synthetic travel time curves for both stratosphere and thermosphere refracted energy (Fig. 5b). These curves can then be compared to the observed arrivals #3–9 (families B and C in Fig. 3) to estimate source origin times for the different observed arrivals. As many as five distinct origin times (i.e., distinct events) are invoked to match the primary observed arrivals (Fig. 6). Multiple travel time curves with similar slopes and significantly different y intercepts point strongly to the existence of multiple sources following an 8:32:11 PDT earthquake. An alternative explanation, multipathing of rays, can not be entirely discounted considering our simplified two dimensional assumptions and modeling. However, we note that even the earliest family of arrivals (family C; arrivals #7–9), which occur in quickest succession, are separated by 75 s. If these arrivals were due to a single source time, the arrival time difference must be explained by transit distances, which would vary by about 25 km and which would require turning altitudes that vary by many tens of kilometers. For ray path modeling in a typically stratified atmosphere we would not expect turning altitudes to have such an exaggerated range for a single ray parameter value.
EXTRAIT 4 :
Assuming that arrivals at a specific offset and with similar observed apparent velocities are due to multiple sources, we identify three distinct arrivals corresponding to stratosphere refractions, two arrivals corresponding to thermosphere refractions at near-offsets (less than 200 km) and four distinct arrivals corresponding to thermosphere refractions at greater offsets (beyond 200 km). The earliest source associated with any of the travel time curves is identified as a thermosphere refraction (arrival #5), which points to an origin time shortly after 8:32:11 PDT. A corresponding stratosphere refraction that might be associated with this original event is notably absent in our data. Although an accompanying stratosphere refraction is not evident for the first thermosphere refraction, the second thermosphere refraction (arrival #3) appears associated with the first definitive stratosphere refraction (arrival #7). This source would correspond to an event occurring at 8:34:05 PDT approximately 114 s after the earthquake, and based upon analysis of travel time differences (see below), is likely associated with arrival #1, which is recorded at closer offsets. A subsequent thermosphere refraction (arrival #6) may be associated with either one (or both) of the stratosphere refractions #8/9, which occur in relatively rapid succession and are inferred to have source origin times at 8:34:50 PDT and 8:35:20 PDT (159 s and 189 s after the original earthquake). There is no clearly distinguishable stratosphere refracted travel time curve corresponding to the last thermosphere refraction (arrival #4), which is inferred to have an 8:37:30 source origin time (319 s after the original event). Arrival #4 appears as a continuation of, and is likely associated with, arrival #2 recorded at near offsets. Of the four thermosphere refracted travel time curves that are identified at further offsets, two of them (arrivals #3/4) appear to be associated with the travel time curves identified at closer offsets (arrivals #1/2). This conclusion is based upon the continuous nature of their line segments (refer to Fig. 3) and the consistent ΔT2–1 time differences, which are similar to the ΔT4–3 time differences (see Table 2). These two distinct sources appear to radiate coherent airwaves that can be traced from 110 to 340 km from the volcano. From Table 2, we further note evidence for consistent time differences between arrivals #5/6 (ΔT6–5; mean = 177 s) and the time difference (also 177 s) between the ground-coupled phases identified at seismograph LON (67 km from MSH). Furthermore, we justify our identification of three distinct stratosphere refractions as corresponding to three separate sources based upon consistent time delays (ΔT9–8 and ΔT9–7). A few other ground-coupled arrivals identified prior to arrival #7 hint at potential additional sources, but not enough stations are picked to clearly define additional travel time curves prior to arrival #7. For arrivals #1/2, the stations located to the north are highlighted in parentheses. Mean and standard deviation time differences are provided for each station grouping. For arrivals #1/2 we note a systematic difference between ΔT2–1 for stations located to the north (mean 209 s; standard deviation 3.4 s) and stations located to the east (mean 226.5 s; standard deviation 1.9 s). If atmospheric structure remains unchanged during this 3.5 minute interval, this systematic difference can be attributed to a displacement in source location for the second event relative to the first event. To first order, the second source should be 17 s closer to northern stations than to eastern stations. Utilizing the average apparent velocity of the two arrivals in family A (ca(A) = 510 m/s), it appears that the second source should be 9 km closer to the northerly stations than to the easterly stations. To more precisely constrain the source region responsible for arrival #2, we performed a 2-D grid search of possible epicenters assuming that the first source corresponds to the MSH vent/conduit/summit. We then attempted to minimize root mean squared time residuals (TRMS), which are based upon differences between the observed (ΔT2–1) and expected (ΔTExpected) arrival times differences at n = 8 stations: where for all stations: Here Dvent is the horizontal distance between the MSH vent and each seismic station, Dloc is the distance between grid search location and each seismic station, and ΔT2–1 is the observed time differences between arrivals #1/2. Each of these three values is station dependent. Calculations are made for a source zone that is assumed to be small (i.e., a point source) and neglects potential source elevation variations, which are a minor influence. In this manner the source location with smallest TRMS can be mapped (see Fig. 7). Assuming arrival #1 corresponds to the MSH vent, arrival #2 presents a very large residual for both a subsequent MSH vent source (TRMS = 9 s) as well as for a hypothesized Spirit Lake epicenter (TRMS = 8 s) [Moore and Rice, 1982]. The residual TRMS reaches a minimum of 2.6 s for an epicenter located 9 km to the northwest of the volcano. We are reluctant to precisely pinpoint this source location because of the strong assumptions about the acoustic source as a point location and the atmospheric structure, which is simplistically modeled here as stratified and static since the eruption onset. Nevertheless, a dispersed region to the NW of MSH appears to be indicated by consistently low residuals (TRMS < 3.0) over a large region. We feel confident that this is evidence for a subsequent acoustic event occurring in the vicinity of Johnston Ridge, Coldwater Ridge, or the Toutle River drainage. 4. Discussion The analyses of ground-coupled acoustic airwaves produced by MSH provide substantial constraints on its eruptive activity, but also present several important unresolved issues. We now focus briefly on two of the primary unresolved issues: The first is related to the eruptive chronology on the morning of May 18, 1980 and speculation about specific physical sources responsible for the multiple airwave observations (see Table 3). The second is a commentary on the suitability of ray theory for effective prediction of acoustic arrivals at regional distances.
4.1. Comments on eruptive chronology Our data indicate that four or five distinct acoustic sources occur in the vicinity of MSH vent/conduit within 319 s of the original 8:32:11 PDT earthquake. These inferred source times and their relation to seismic events, and other observed eruptive chronology, are highlighted in a comparative timeline (Fig. 8).
EXTRAIT 5 :
Based upon our transit time modeling, the occurrence of the first observed thermosphere refraction (arrival #5) coincides closely with the earthquake and/or landslide initiation. Time resolution of the forward ray-path modeling is such that an 8:32:11 PDT earthquake/landslide onset and subsequent 8:32:21 PDT cryptodome explosion event, as postulated by Brodsky et al. , would be virtually indistinguishable in our data. Although landslide and avalanche events are known to radiate substantial low frequency acoustic energy to regional distances (e.g., ), we suggest that explosive concussions would be a more natural mechanism for high-amplitude sound generation that also contains an audible component  and . As such, we propose that the first identified acoustic source is likely an explosion (or series of explosions) occurring at 8:32:21 PDT, which was induced by the mass movement unloading effect of the large landslide. The relatively sustained time duration of the ground-coupled airwaves for arrival #5 (see for example the 25 s waveform from station EUK in the Fig. 2 inset) suggests a potential extended-duration source that is characteristic of an extended-duration sequence of explosive pulses. These explosive pulses might correspond to vertical seismic forces identified by Kanamori et al.  as the cyptodome is incrementally depressurized (refer to Fig. 1 in Brodsky et al.  for example). The timing of the visible manifestation of the eruption onset, which includes a vertical plume rise at 8:32:47 PDT and lateral blast initiation at 8:32:56 PDT, is constrained by Voight  from the sequence of G. Rosenquist photos. However, we are unable to identify clear acoustic manifestation of these events as potential acoustic sources despite extensive transit time modeling under a range of conditions (utilizing May, June, and July profiles at 40° and 50° N). Variability in atmospheric structure and winds may account for propagation time uncertainties on the order of only a few tens of seconds (e.g., Fig. 5b), but these uncertainties are probably not sufficient to associate arrival #5 with the initial lateral blast. We largely discount the potential, and speculated, influence of supersonic transmission velocities, because even though the MSH airwaves might have originated as shock waves (i.e.,  and ), reasonable shocks would decay to sonic speeds within a few kilometers of the source. It simply appears as though the initial explosive eruption occurring at 8:32:47–56 PDT was not energetic or impulsive enough to be responsible for the acoustic airwaves recorded by the PNSN network. Following arrival #5, which we attribute to decompression-related explosive concussions, we hypothesize that the next source (8:34:05 PDT; arrivals #1/3/7) corresponds to a large explosive event from the vicinity of the depressurized conduit. According to Hoblitt  a large explosion followed the initial vertical column/lateral blast by 60–70 s, which would correspond to 8:33:47 to 8:33:57 PDT. Malone et al.  and Kanamori et al.  provide corroboration for the Hoblitt  source with seismic evidence for a second event occurring about 2 min after the initial 8:32:11 PDT earthquake. Such timing by both Hoblitt  and Malone et al.  coincides remarkably well with our inferred 8:34:05 PDT acoustic source. Seismic and remote sensing arguments are used by Hoblitt  to suggest that this second event was somewhat more powerful than prior event(s). The second event of Hoblitt  corresponds well to the 8:34:05 PDT inferred acoustic source (arrivals #1/3/7), which may be conjoint with a seismically identified vertical thrust force identified by Kanamori et al.  and modeled by Brodsky et al. . Although the timing of this vertical thrust is given a time of 8:34:35 ( 30 s after the inferred source for arrivals #1/3/7), at least two smaller vertical thrust forces, occurring at 8:35:00 PDT and 8:35:22 PDT , exhibit timing that is very close to that of our stratosphere refracted arrivals #8 (8:34:50 PDT) and #9 (8:35:20 PDT). Several earlier vertical seismic thrusts identified by Kanamori et al.  at 8:32:45, 8:33:10, and 8:33:45 PDT, are conspicuously absent in our data set and may be explained by poor coupling of these hypothesized events to the atmosphere. We attribute at least two of the thermosphere refracted arrivals, including the large 8:34:05 source, to potential large explosive blasts. It is possible that one or both of these sources, or a combination of explosive pulses, may also be responsible for the acoustic-gravity phases, which were produced by MSH and observed worldwide [e.g., , , ,  and ]. For example, the microbarograph located 925 km from MSH at Berkely, CA recorded a wavetrain 50 to 56 min after the 8:32:11 earthquake that includes two primary pulses of periods 5 and 6 min with amplitudes 350 and 220 Pa. These pulses were attributed to acoustic gravity waves generated by two distinct sources occurring approximately 6 min apart . Though none of our ground-coupled arrivals provide explicit validation for two distinct energetic pulses separated by 6 min, it is notable that two sources (8:34:05 and 8:37:30 PDT; corresponding to arrivals #1/3 and #2/4) are especially prominent and are clearly recorded across most of the PNSN network (ranging from 151 to 341 km). These arrivals, separated by 3.5 min, might be associated with the two significant pulses recorded at the Berkeley microbarograph, especially if the excitation of the second acoustic gravity wave was delayed relative to the first. We speculate that because gravity waves are generated by the injection of a large buoyant air mass (i.e., volcanic plume) into the atmosphere, a fast-rising column followed 3.5 min later (i.e., at 8:37:30 PDT) by a more slowly rising pulse could account for the timing discrepancy. Eyewitness accounts may provide some limited constraints on the sequence of events at the very onset of the May 18 MSH eruption, although the visual observations of the volcano were largely obscured by ash and clouds shortly after 8:33:00 PDT . A couple of observers comment on being able to observe a shock wave, similar to that produced by a “nuclear explosion” that occurred “shortly after the initiation of a vertical eruption cloud.” The timing of this event is uncertain and may or may not have been associated with a source of the recorded ground-coupled airwaves. An observer 17 km NE of the vent also reported seeing the horizontal blast (at 8:32:56 PDT) and a shockwave “shortly following” the vertical eruption.
EXTRAIT 6 :
This observer also cites “a clap of thunder” followed by a notable pressure change and is one of a very few people to report concussive noises within a few tens of kilometers of the volcano. Though exact timing is unclear, we speculate that this shock could be associated with an 8:34:05 PDT origin (arrivals #1/3/7), which is considered here to be the 8:33:46–56 PDT event of Hoblitt . In general, most first-hand audio reports within the zone of devastation primarily referred to ‘rumbling’ noises , but one observer 18 km north of MSH mentions three “rifle shots” at an unspecified time after the eruption, with an associated pressure change that “forced the observer to the ground.” It is possible that these ‘rifle shots’ could also be associated with the 8:34:05 PDT event, or subsequent eruptive pulses inferred by Kanimori et al. . It is also possible that many small shocks were produced during the first few minutes by the volcano and not propagated regionally. At closer offsets National Weather Service barometers within Washington State recorded atmospheric perturbations associated with the eruption. The closest barograph in Toledo, WA, 54 km from MSH, shows a 373 Pa spike followed after a short pause by a 13-minute 394 Pa decompression and then a second longer-duration compression. Reed  proposes that the decompression was associated with strong inflowing winds (towards the volcano) inducing a regional pressure low. In this scenario, inflowing winds are postulated to be a response to the MSH buoyant column rise. The secondary compression is hypothesized to result from mass injected into the atmosphere . Unfortunately, the low temporal resolution afforded by the meteorological barometers inhibits the identification of relatively high-frequency energy that may be associated with near-infrasound (1 to 20 Hz) and/or sonic disturbances, which are the probable excitation mechanisms of the majority of our recorded ground-coupled recordings. In other words, multiple airwaves arriving in quick succession would be indistinguishable on the Toledo, WA long-period barometric records. Many observers near to the volcano specifically reported strong inflowing winds headed towards the volcano about 5 min after the onset of the eruption. Several reports, from 25, 29, and 23 km N of the vent, comment on the northward-traveling blast cloud, which was suddenly “stood up” by vigorous winds (up to 80 miles per h) blowing south off Riffe Lake approximately 5 min after the eruption onset. The cloud was stood up NNW of the volcano and may coincide with observations of Moore and Rice , who claim that a cloud centered 12–14 km north of the volcano began to ascend at 8:36:00. They speculate that the origin of this cloud may be the collision of a gas-charged dacitic cryptodome with Johnston Ridge/Toutle River drainage generating a significant ‘northern explosion,’ which was responsible for a 25 km high column displaced to the north of MSH . Additional first-hand observations substantiate a displaced column to the north or northwest of the volcano. Photos taken by J. Christensen from near the summit of Mount Adams, 50 km E of MSH, clearly show the region around Spirit Lake and Coldwater Ridge were enveloped in a blast cloud and a convective cloud may be seen ascending in the north (see Fig. 9). In contrast, the region just to the south of the volcanic cone is entirely clear at this time. These observations have led to speculation that a hot portion of the MSH cryptodome could have slid into Spirit Lake and generated a significant (and time-delayed) phreatic blast from this vicinity ( and unreferenced information posted at http://www.answers.com/topic/1980-eruption-of-mount-st-helens). Others, including Hoblitt , conclude that this ‘northern explosion’ may have resulted from interaction of the pyroclastic density flow with rough topography in the Toutle River drainage, initiating a buoyant ash cloud. The study here of the acoustic airwaves, and specifically of arrival #2, supports a secondary northern source that is located substantially to the west of Spirit Lake (Fig. 7). Based upon our acoustic evidence, we maintain the possibility of a displaced ‘northwest source’ in the vicinity of Toutle River Drainage and/or Johnston Ridge that is unassociated with a postulated Spirit Lake event. 4.2. Comments on suitability of ray tracing Observational data coupled with forward ray tracing indicate MSH acoustic refraction from high velocity regions in both the stratosphere and thermosphere. Although sound absorption in the thermosphere can be severe [e.g., ], this study along with those of others provides evidence that acoustic perturbations can return to earth from altitudes well above 100 km [e.g.,  and ]. Although the ray tracing performed in this study is able to generally reproduce the PNSN-recorded arrival times, it is less effective at predicting arrivals closer than 200 km from MSH. For instance, arrivals #1/2 include 9 stations that lie within the predicted acoustic shadow zone. It is clear that arrivals #1/2 do not represent stratospheric refractions because their apparent velocities (501 and 520 m/s) are far too fast, but they do strongly suggest rays turning in the high velocity thermosphere, well above 120 km. If this energy is in fact reaching the thermosphere, the problem is that ray theory does not satisfactorily predict acoustic energy returning to earth at such close offsets. We are unable to model these near-offset ‘thermosphere refractions’ despite attempts to force extreme (post-eruption) changes to the atmospheric velocity structure in the vicinity of MSH. Thermosphere winds are the least well-constrained parameters in our forward ray path modeling and may offer one potential explanation for the observation of near-offset thermosphere refractions. Thermosphere winds are affected by solar activity and vary according to location, season, and most significantly to time of day . Because we have no empirical measurements of thermosphere winds for the morning of May 18, we modeled acoustic radiation for a dramatic range of conceivable wind velocities. Hedin et al.  indicate that longitudinally averaged annual zonal winds can vary from − 130 m/s to + 90 m/s at 6:00 AM and 6:00 PM respectively at 45° N latitude. We thus attempted to model travel time curves for exceptional wind conditions (+/− 150 m/s) above 120 kilometer altitude. We found that extreme winds in the thermosphere in the direction of acoustic propagation do facilitate downward refraction, however they only succeed in bringing the nearest offset to about 170 km.
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It is still puzzling, and observationally significant, that we see apparent thermosphere refracted energy closer than 120 km for both zonal and meridional propagation. We conclude that classical ray theory may be deficient at predicting arrivals at these close offsets. The MSH data appear to provide evidence for the prevalence of leaky atmospheric waveguides and/or the importance of dispersion and scattering during regional sound propagation (e.g., ). One last unresolved issue is related to the two ground-coupled airwaves recorded at the seismic station LON only 67 km from the MSH vent. Based upon the time difference between the two observed arrivals at LON (177 s; Table 2), it would appear as though the sources responsible for the LON arrivals are conjoint with the sources responsible for arrivals #5/6 originating at 8:32:21 and 8:35:20 PDT. However, if we assume that the initial uncorking of MSH is responsible for the 8:47:50 PDT arrival at LON, this would imply a net transit time of 15 min ( 900 s) for a horizontal propagation distance of only 67 km (i.e., a straight-line velocity of 74 m/s). For this arrival to be an acoustic wave (with average velocity in excess of 300 m/s), the propagation path would need to be 300 km and thus require an effective ‘reflection’ in the thermosphere at an altitude of 150 km. Because internal atmospheric sound reflections are not considered plausible, we conclude that the LON arrival(s) can not be caused by ground-coupled acoustic waves. Rather, the picked arrivals may reflect mass transport of the atmosphere due to abrupt regional barometric changes and potential associated winds. The LON ground-coupled seismic deflections could be associated with phenomena that were documented by Rosenbaum and Waitt  and that might have been induced by the massive buoyant column rise . 5. Summary Throughout Washington State, both people and seismometers ‘heard’ the paroxysmal eruption of MSH on the morning of May 18, 1980. Data corresponding to ground-coupled airwaves substantiate that the first 5 min of the eruption was complicated with multiple discrete events occurring during this time. Although the first ‘acoustic event’ likely corresponded to uncorking of the cryptodome following the initial landslide onset, subsequent events may be associated with other potentially diverse phenomena, such as the onset of vertical and lateral explosive pulses from the central vent, and convective plume rise originating from the northwest of the MSH edifice due to hot debris avalanche and/or cryptodome slamming into the Toutle River drainage. This displaced northwest source is substantiated by acoustic arrival time residuals recorded across the network of PNSN seismometers. Perturbations of the atmosphere during the MSH eruption produced high-intensity acoustic waves, both low-frequency and audible, which were heard by humans and simultaneously recorded by seismometers. Based upon acoustic arrival times across the PNSN seismic network we infer that much of this energy radiated into the stratosphere and thermosphere before refracting back to earth. Scattering of the acoustic energy facilitated acoustic energy returning to earth at a greater range of offsets than would generally be expected with ray theory. Nevertheless, a significant shadow zone (region of inaudibility) was preserved within a few tens of kilometers of the volcano. This shadow zone, which was noted by the general public, and has been similarly observed at other erupting volcanoes, is most easily attributed to the upward refraction of acoustic airwaves in a temperature stratified atmosphere. Instability and surge development in debris flows Barbara Zanuttigh & Alberto Lamberti  Debris flows are often described as a succession of surges, which are characterized by enhanced peak depth and velocity and therefore by a tremendous increase of their destructive power. For given characteristics of the base flow, if the channel is sufficiently long to allow an appreciable wave development, the linear stability analysis in shallow streams is shown to provide a reasonable prediction of the critical flow condition and of the instability growth rate. The one-dimensional (1-D) theory, however, does not allow the determination of the wave period of the fastest growing perturbations. Debris waves most frequently develop following a mechanism similar to water roll waves: Instabilities grow up becoming clearly distinguishable waves, and then waves overtake one another with increasing wave period and amplitude. The typical hydrograph of a multiple-peak event is shown to be composed of a first surge, which is usually characterized by the highest depth, the longest duration, the greatest erosive power, and the most symmetrical shape, and of secondary waves that burst on the flow tail in the recession phase. The characteristics of the first surge can be explained by two different mechanisms. All waves that rise up near the flood crest run faster than this first surge and coalesce into it, causing its high depth and great volume. Moreover, segregation during the flow induces the concentration of boulders at the fronts, contributing to its depth enhancement, erosive power, and symmetrical shape. When a debris surge impacts a structure, the force pattern can be interpreted as the superposition of the reflection of the bouldery front and the formation of a vertical muddy jet due to the impact of the front wedge. Wave reflection can be described by a 1-D mass and momentum balance across the front, whereas the pressure impulse, due to the incompressibility of the interstitial fluid, can be analyzed through inviscid formulations validated for the representation of tsunami forces.
 Debris flows occur when a mixture of poorly sorted sediment and water, in almost equal quantity, surges down a slope or a channel similarly to a fresh concrete mixture [Iverson, 1997a]. According to most researchers [Costa and Williams, 1984; Phillips and Davies, 1991], sediment concentration in the mixture should be at least 50% by volume or 70% by mass; below this limit the mixture is said to be a hyperconcentrated suspension. The feature of sediment being poorly sorted, which makes the flowing mass appear as a mixture of mud, cobbles, and boulders, is actually as important as the sediment concentration or the presence of water. The characteristic property of debris flows is the dynamic interaction among the boulders and the muddy matrix, deriving from the mixture of water and the finersediment fraction. In the absence of such interaction we can have a ‘‘rock avalanche’’ (when the fluid phase is dynamically irrelevant) or sediment transport by water (when the effect of the solid phase on the fluid one is dynamically irrelevant).
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According to this definition many other events identified as mudflows, lahars, etc. may be regarded as debris flows [Johnson, 1984]; that is, the definition adopted is not selective.  Debris flows are often described by eyewitnesses as a sequence of surges impacting on defense structures and destroying houses and fields. Scientists frequently related this succession of surges to flood events from subbasins or to failure of natural dams along the stream. These events, because of their sporadic character, are usually described by images of the resulting effects: erosion in the formation area, devastation along the channel, and deposition in the fan area. A precise documentation of the surges and the attribution of this phenomenon to instability of the regular flow are more rare but can be found both in prototype and in the laboratory. Despite the differences in space timescales and in the rheology of the flowing material, multiple surges developing in debris flows have characteristics similar to water roll waves [Takahashi, 1991]. The formation of multiple surges has the following implications. The flow at wave crests becomes significantly higher and faster than in the corresponding regular flow. As a consequence, overtopping of the channel banks occurs more easily, and the flow peak thrust on obstacles is amplified by more than 1 order of magnitude. Moreover, the repeated action of surges causes structure fatigue. In synthesis the destructive power of the flow dramatically increases. In the presence of surges the wood trunks and the boulders, which are normally lifted toward the free surface where velocity is higher, are segregated also toward the surge front. On the other side, water and/or the finer sediments, the muddy matrix, are segregated toward the bed and the surge tail. This finally results in the common appearance of stony convex fronts, pushed forward and laterally by the muddy tail of the surge and forming stony lateral levees. In the deposition area the single surges may follow different paths, causing the typical branching shapes of deposition areas.  The objectives of this paper are to examine the critical conditions for instability formation, to identify the most important parameters affecting the development of waves, and to analyze surge characteristics such as depth, velocity, shape, texture, and impact force. For the scope of the paper it is assumed that the mixture behaves as incompressible, either because it is saturated with a liquid or because it is so dense that particles have multiple contacts not allowing mixture compression. Dense granular flows are also considered because, as a consequence of their incompressibility, they behave similarly to debris flows as far as the formation of waves is concerned. In many descriptions of multiplesurge events the surge textural structure is apparently presented as the dynamic cause of the formation of waves: Surge fronts behave as a [Sharp and Nobles, 1953, p. 551] ‘‘bouldery dam . . . pushed along by the finer, more fluid debris impounded behind.’’ Even if the textural structure strongly affects surge dynamics and, during the surge cycle, bed erosion and deposition do often occur, this paper follows a different conceptual approach. The uniform flow instability of a homogeneous debris mass is regarded as the fundamental process causing the formation of waves even in a uniform material or over a nonerodible bed; surge texture as well as erosion/deposition patterns are related phenomena that show up when sediment is highly heterogeneous or the bed is mobile.  The structure of the paper is the following: Section 2 documents the formation of multiple surges in prototype by reviewing experiences in selected basins. Section 3 is dedicated to a similar description of laboratory cases, where it can be easily controlled that no other cause may induce the formation of waves except the inherent instability of the regular flow. Both progressive and regressive instabilities are considered. Section 4 examines criteria for determining unstable flow conditions over a nonerodible bed for different rheological models of the flowing material. Section 5 describes the propagation of a single surge, showing its typical profile and celerity, and the process of wave development. Section 6 analyses the segregation mechanism and the time evolution of surge texture. Section 7 summarizes the available knowledge on forces exerted by surges on obstacles. Conclusions are finally drawn in section 8.
2. PROTOTYPE EVIDENCE OF SPONTANEOUS SURGE FORMATION
 Several qualitative descriptions of moving surges and of multiple-surge events can be found in the literature: Singewald , Jahns , Sharp and Nobles , Morton and Campbell , Wasson , Pierson , Li et al. , Costa and Williams , Johnson , Davies et al. , Davies [1993, 1997], Zhang , Melis et al. , Iverson [1997a], Arattano and Moia , and Major . Besides qualitative observations several measurements document debris flow events as a sequence of surges; see, for instance, the hydrographs recorded in the Acquabona channel, northeastern Italy [Berti et al., 1999]; in the Illgraben torrent, southwestern Switzerland [Rickenmann and Weber, 2000; McArdell et al., 2003]; in the Rio Moscardo, northeastern Italy [Marchi et al., 2002]; and in the Jian-Jia Ravine, China [Li et al., 1983]. In the following sections some basins are presented in which multiple-peak events often occur; the first two cases, Fully and Mount Thomas, are only monitored, whereas the others are also equipped with measurement stations. For each catchment, information as complete as possible is reported, and a database of events is finally prepared in Table 1 to summarize the data of this whole section.
2.1. Fully, Switzerland
 The ‘‘Torrent du Bossay’’ catchment, in southwest Switzerland, has an extension of 2.9 km2 above the fan apex. The channel above the fan apex is 2.9 km long, with a mean slope of 45%. On 15 October 2000, numerous debris flow surges destroyed 6 ha of vineyards above the Fully village; fortunately, no buildings were damaged and no people were injured. The event, which is documented by two video recordings [Rickenmann, 2001; Arattano and Trebbo, 2000], was artificially triggered by the unintentional release from ‘‘Lac infe´rieur’’ of 150,000 m3 of water, which was discharged over approximately 20 hours onto the debris slope at ‘‘Les Garette,’’ causing several surges of similar intensity.
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The deposit area covered about 60,000 m2, with thickness around 7 m in the middle and 4 m in the lower section for a total deposit volume of 350,000–400,000 m3. Boulder size at the surge front was about 1 m. From video records, flow velocity reached 10 m/s with flow depths of about 1–1.5 m, and estimated maximum discharges were in the range of 250–500 m3/s depending on the considered surge.
2.2. Mount Thomas, New Zealand
 Debris flows occur in the steep, unvegetated ravine at the head of the south fork of Bullock Creek in Mount Thomas State Forest, North Canterbury, New Zealand. The catchment active fan has an extension of 0.46 km2, and the channel downstream of the fan apex is around 1.5 km long with mean slope of 11%.  Description of the 3-day debris flow that occurred in April 1978 is given by Pierson [1980, pp. 235–236]: The debris flow came down the entrenched channel as in successive pulses or surges, usually about 10 to 20 minutes apart. Between surges, the fluid could be characterized either as a very fluid slurry or as a very muddy water. This sediment-laden water, which had the consistency of a motor oil, flowed turbulently around borders and streambed irregularities, and developed numerous standing waves and a very irregular, agitated surface. . .. One very large surge front observed on 22 April was roughly 3 m high. This was immediately followed by a rapid increase in discharge, plus an accompanying gradual increase in the viscosity of the slurry (to the consistency of wet concrete). Standing waves and eddies would shrink and the agitated surface become smooth as turbulent flow yielded laminar flow. . .. As the debris surge passed, the discharge decreased, flow become laminar again, the level dropped and the flow surface slowed down.  Fluid density of the flowing material ranged from 1.59 g/cm3 between surges to 2.13 g/cm3 during surges. Solid volume concentration of 33% between surges and 66% during surges was obtained. Materials collected from surges had an average diameter of 16 mm, and gravel made up about 70% of the solid; about 20% was sand, 6% was silt, and 4% was clay. Between surges, solid material in the more watery fluid comprised roughly 20% gravel, 54% sand, 15% silt, and 11% clay, with the gravel moving as bed load rather than suspended load.  The events in Mount Thomas are considered by Davies  as the typical example of small-gully debris flows, which are characterized by several surges per event and occur every few years when intense rain falls on the already saturated catchment. Surges are around 1 m high, moving relatively slowly (1 m/s) and often jerkily; they are characterized by coarser boulders (as coarse as the front is high) than in the rest of the flow. Between surges a rapid and turbulent stream of low density flows down the gully. Davies  relates surge formation with slurry density (gs _ 1.6–1.8). Intense slope erosion causes input of fine material into the channel that forms a slurry; if the slurry is dense enough, surges are formed with accumulation of boulders at the front, forming a temporary dam; the slurry overtops this dam or causes it to slide into motion. This scheme will be recalled in section 3 to describe the conditions for development of regressive instabilities. If the slurry is not dense enough, the event will not progress beyond a normal high streamflow with coarse material moving as bed load at the base of the flow, in which no damming nor surging occurs.
2.3. Yellow River Basin, China
 The Yellow River in northern and northeastern China is the country’s second largest river, with a drainage area of about 795,000 km2. During the flooding season, suspended sediment concentration in the Yellow River is very large, usually around 40%, and in its tributaries may rise up to 50% [Yang et al., 1996]. During the declining period of a long flood wave, when the sediment concentration produced by the flood remains high, oscillations in surface level and discharge are observed [Engelund and Wan, 1984]. A typical hydrograph recorded during a flood [Qian, 1980] in a tributary, the Black River (Figure 1), shows the presence of a first surge followed after a certain time by waves, very similar to water or muddy roll waves. The time series indicates an overall decline in mean surface level, associated with flood recession, over a period of roughly 10 hours. Superimposed on this overall decline, oscillations are evident with relatively high frequency and period of approximately 10 min. The formation of instabilities and waves was also observed at Lijiahe hydrological station of Xiaolihe River on 17 June 1963. The event duration was 18.5 hours, and the flow was intermittent, with waves lasting from 2 to 4 min and reaching a depth of 0.5 m [Ying, 1996; Qian and Wan, 1983].
2.4. Mount Yakedake, Japan
 Mount Yakedake is located in the Japanese Alps, close to the Sakurajima volcano, and since 1970 the Kamikamihori Valley is monitored by means of video recordings, Doppler anemometers, and sediment collection. The mean channel length above fan apex is around 1.5 km with an average slope of 8%. Data collected for 46 debris flows show that in the middle channel reach, because of several retention works, the flow becomes quasi-uniform [Suwa and Okuda, 1983]. Maximum values of front velocity are in the range of 10–15 m/s in the upper reach (average slope 30%), decrease down to 2–7 m/s in the quasi-uniform reach, and finally drop to 1–3 m/s in the lower reach (slope 5%). According to Suwa  all debris flows have similar characteristics that can be summarized by the event of 21 July 1985. The debris event consisted of two main surges, composed of a coarse front and a muddy tail. The surge front has a lower flow velocity than the debris body as a result of two effects: the greater frictional forces between the front boulders and the valley bottom and the greater friction among boulders in the local absence of muddy slurry; that is, no fine matrix filled up the boulder interstices of the front edge, as observed also by Ishikawa  in the Name River, Japan, and by Pierson  on Mount St. Helen, United States.
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Peak hydraulic values generally occur in the following order: flow depth first, then flow rate, and finally, surface velocity in Suwa’s [1988, p. 156] opinion ‘‘as a result of this significant change in the mechanical state of flowing debris.’’ The second surge presents the same characteristic of the first one; between one surge and the following, secondary muddy waves usually show up.
2.5. Acquabona, Italy
 The Acquabona drainage basin is composed of a dolomitic rock headwater basin (above 1650 up to 2667 m above sea level) and of a deeply incised debris flow channel around 1300 m long, with slope ranging from 18% at the downstream end of the incised reach down to 10% at the measurement station 3. The erosion banks are up to 30 m high (in the middle part), and slope angles range from 40_ to 50_. Figure 2 presents the hydrographs measured at station 3, 1200 m downstream the channel inlet, during three events: 25 and 27 July and 17 August 1998 [Lamberti, 1999; Berti et al., 1999]. The events that occurred at the end of July 1998 can be considered small-scale events since they mobilized small volumes of material: 600–700 m3 and 400–500 m3 on 25 and 27 July, respectively. The rainstorms that triggered the events have similar intensity and duration (8–11 mm over 30 min), the major difference being due to antecedent precipitation: The 25 July rainstorm was preceded by 14 mm of rain over 7 hours, whereas the 27 July rainstorm occurred after 46 dry hours. The debris flows originated during the most intense precipitation period, though the water peak inflow did not arrive from the upper rock basin in the initiation area. It seems that these two events started in the middle of the channel. Both events are essentially single peaked (Figures 2a and 2b) and have short duration (4 and 2 min, respectively); the event of 25 July shows small instabilities in the recession phase, whereas a secondary wave seems more marked on 27 July. Front velocity of the main surges was 0.5 and 1.0 m/s, respectively. The rainstorm that triggered the event of 17 August was, once again, very intense: 25.4 mm of rain in 30 min. The volume of the deposits has been estimated to be about 8000–9000 m3; the overall duration of the event was approximately 38 min, and more than 20 different surges passed in the channel (Figure 2c). Surge celerity ranged from 1.8 to 7.7 m/s. In all events the bulk concentration was high, roughly estimated around 60%. The difference among the hydrograph shapes presented in Figure 2 can be explained by the very different Froude numbers Fr in the July events and in the August one; if we calculate Fr based on surge depths and velocities, we obtain Fr = 1 for 25 July, Fr = 0.3 for 27 July, and Fr = 1.7 for 17 August (by assuming the average values of surge depth and velocity in the second part of the event equal to 1.3 m and 6.0 m/s, respectively).
2.6. Illgraben, Switzerland
 The Illgraben torrent is part of the Illbach basin in southwest Switzerland and has an extension of 10.5 km2 above the fan apex. It contains an unusually large proportion of unvegetated, steep, unstable slopes with abundant colluvium derived from weak carbonate and greywacke bedrock. The channel downstream from the fan apex is 2.6 km long, has a mean slope of 16%, and is presently deeply incised. Several debris flows occur per year, generally following intense rainfall in the catchment [Rickenmann et al., 2001], and some of them present multiple-peak hydrographs [McArdell et al., 2003]. Two events are shown in Figure 3: The first, 28 June 2000 (Figure 3a), is a granular debris flow, characterized by a total volume of 100,000 m3, maximum discharge of 92 m3/s, maximum depth of 2.9 m, maximum velocity of 4.5 m/s, and 17 roll waves; the second, 28 June 2001 (Figure 3b), is a muddy debris flow, characterized by a total volume of 36,000 m3, maximum discharge of 92 m3/s, maximum depth of 1.6 m, maximum velocity of 6.3 m/s, and 40 roll waves.
2.7. Rio Moscardo, Italy
 The Moscardo torrent, a small stream in the eastern Italian Alps [Marchi et al., 2002], commonly displays at least one event per year. The basin area is around 4.1 km2; the main channel, well defined on the fan and not subject to serious avulsions, is 2.76 km long. Average basin slope is 63%; average channel slope is 37%. Initiation points can vary from event to event, generally being located at the head of the main channel; typical gradients in the area are of 20_–30_ for the main channel and of 30_–50_ for channel banks and hillslopes. Hydrographs were measured in the period 1989–1995 by means of two ultrasonic sensors installed in the mid-fan area on a channel stretch 300 m long, with an average slope of 10%; in 1995 a third sensor was placed farther upstream, extending the total length of the monitored channel to 370 m. Hydrographs acquired at the two (1990–1994) and at the three (1996–1997) measurement stations are reported in Figure 4; the scales for time and stage are the same for all plots. The recorded hydrographs show relevant differences from event to event. Sometimes the debris flow appears as a single, well-defined wave with a steep front followed by smaller waves that develop during the recession phase (13 August 1991, 30 September 1991, 11 July 1993, 20 July 1993, and 8 July 1996); in other cases the first surge is less steep, lasts longer, is associated with instabilities, and is followed by secondary surges (1 September 1992, 14 September 1993, and 22 June 1996). Because of the progressive widening of the channel and the abrupt decrease in slope at the downstream station due to the presence of a ford, there is sometimes a clear deposition in correspondence with this station as on 17 August 1990 and 18 July 1994. Aggradation can be also occasionally found in other stations, like at the upstream one in the event of 27 June 1997.
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2.8. Xiaojiang River, China
 The Xiaojiang River is located in northeast Yunnan Province, China, and is one of the tributaries of the Yangzi River. The valley has an extension of about 2000 km2 and is affected by over 150 mudflow gullies, the most dangerous being the Jiang-Jia [Li et al., 1983] and the Dachao rivers [Li and Luo, 1981]. It has to be stressed, however, that the scale of the mudflows and the damage provoked is far greater for Jiang-Jia than for Dachao River. On the basis of the investigations performed in this area, it can be concluded that the mudflow might burst 20–30 times each year, and sometimes it can last up to 12 hours. The maximum discharge can reach 2500 m3/s. Once in a while the mudflow can carry away more than 500,000 m3 of debris rock and mud as well as stones of up to 5 m in diameter. Following Li and Luo [1981, p. 472]: The structural mudflows always burst in repeated waves. There is not any fluid flow between the waves. There are usually 20– 30 waves in each event, but sometimes up to 100– 200 waves during bursting of the mudflow. Its frontal surge is 3 – 5 m high. The frontal surge has a maximum velocity of about 13– 15 m/s and a maximum density 2.37 g/cm3.  Jiang-Jia channel [Davies et al., 1992] is from 25 to 500 m wide, depending on location and time (i.e., whether or not the valley bottom is incised), and slopes at about 17% over a 5.5-km length to its confluence with the Xiaojiang River. The channel bed material is fluvially winnowed or washed old debris deposit ranging from 10_4 to 102 mm; between debris flow events the channel is occupied by a braided, gravel bed river of about 25 m overall width. Descriptions of flow behavior in this channel have outlined the following repetitive sequences of phenomena at the observation section [Li and Luo, 1981]. Following the onset of rain, streamflow first rises to about 5 m3/s and then suddenly decreases. A series of small surges occurs, initially of thin mud that lays a ‘‘muddy blanket’’ over the braided gravel bed, as each successive surge travels farther downstream. Following the mud waves, a series (10–100) of large (up to several meters) rapid (more or less 10 m/s) waves of dense (about 2000 kg/m3) slurry occurs having the appearance of wet concrete. Between waves the slurry flow becomes nearly stationary. The interval between waves is about 1–5 min and is usually remarkably constant within each event. As the rainstorm dissipates, the waves become smaller and less dense and are followed by reestablishment of muddy stream flow. According to the long time observations in the basin by Li et al.  the largest discharge was of 2420 m3/s, the longest duration of bursts was more than 12 hours, and the greatest number of successive waves was 310.
2.9. Synthesis of Available Data
 Eyewitnesses and measured hydrographs of either muddy or stony debris flow document that debris events are often formed by a first isolated surge followed by a sequence of surges or secondary muddy waves. Measurements of surge characteristics (height and period) during real events are rare; Table 1 summarizes all the events that occurred in the basins presented in sections 2.1–2.9, for which most of the relevant process parameters are available: event volume, maximum discharge, surge velocity, and depth. Sometimes comments on the nature of the debris are also present; unfortunately, information on size of boulders, bulk concentration, and yield stress are almost always lacking or presented only through pictures so that no precise data of this type can be included in Table 1.
3. SPONTANEOUS SURGE FORMATION IN LABORATORY EXPERIMENTS
 This section aims at providing experimental evidence of instability formation and description of the development process of progressive and regressive waves in a wide variety of flows under properly controlled conditions. Main characteristics and appearance of regressive and progressive instabilities can be synthesized through the description by Davies  of type ‘‘A’’ (also named ‘‘small gully’’) and type ‘‘B’’ (‘‘large valley’’) debris flows, respectively. Progressive instabilities arise from small undular surface waves in a stationary slurry material, amplify translating downstream, then break, and because of overtaking, continue growing in amplitude and length. Regressive instabilities are typically originated by the temporary blockage of material due to the bridging of clusters of large grains across the channel or between the bed and the surface. A stationary or slow moving dam forms, behind which the material still in motion builds up until the downstream force is sufficient to overcome the interparticle and particle-boundary friction, the dam breaks, and the jammed material is set in motion again. The structure of this section is as follows: Section 3.1 presents experiments aimed at studying roll waves both in Newtonian flows, essentially water in laminar and turbulent conditions, and in non-Newtonian flows, which consist of water mixtures and dry granular flows. The case of pure water is included because instabilities in debris flows have characteristics similar to water roll waves [Takahashi, 1991] as will be shown in section 5. Section 3.2 recalls a few experimental observations on regressive waves in clay suspensions and dry granular flows. In both sections 3.1 and 3.2 the available literature is recalled in a chronological order. Section 3.3 briefly discusses the relevance of scale effects.
3.1. Progressive Instabilities
3.1.1. Newtonian Flows
 In steep regular channels or slopes the spontaneous formation of waves in a water stream developing into bores has been known for a long time [Cornish, 1910, 1934; Jeffreys, 1925]. Waves move downstream as bores connected by gentle profiles, where depth regularly increases from a trough to the adjacent crest. Because of the characteristic presence of a roller at each bore, they are usually called roll waves.  The pioneering work by Brock [1967, 1969] provided a great amount of information on roll wave formation and development in water turbulent flows. The experimental setup consisted of a channel 0.11 m wide and 40 m long, whose slope varied in the range of 5–11%. Flow depths of the unperturbed flow were in the range of 5–8 mm with Froude numbers between 3 and 6.
EXTRAIT 12 :
Water level in time was measured at several sections along the channel by means of pressure gauges. From these experimental results, three different development phases of water waves can be recognized (Figure 5). The first is referred to as ‘‘initial phase’’ and is characterized by small symmetric waves with exponentially increasing amplitude: Development curves of wave crests h0max (flow depth at wave crest hmax scaled by the normal depth h0, h0max = hmax/h0) are initially concave upward. Mean wave period T0 I (mean wave period adimensionalized by channel slope i and normal depth, T0 I = TI i/(g/h0)1/2) is almost constant, and wave formation is essentially related to instability of the uniform flow. The second phase is a ‘‘transition phase’’ in which wave periods start to increase since some wave overtaking takes place. In this phase, wave troughs h0min (flow depth at trough hmin divided by normal depth, h0min = hmin/h0) rapidly reach the final constant value h0minF, and h0max increases with rate ah0max. The transition point can be defined as where the constant T0 I line and the final constant slope T line cross each other: In correspondence with this point we have the typical crest height at transition h0maxT. The third (and final) phase is characterized by frequent wave overtaking with steep fronts or shock waves: The number of waves decreases, and the mean period increases quasi-linearly. The mean trough elevation is practically constant, whereas crest elevation grows with progressively decreasing rate with increasing distance from the channel inlet (h0max development curves are eventually convex). Free surface profiles versus time show a Gaussian symmetric structure during instability and an evident nonsymmetric form during the final overtaking phase. Figure 6 shows depth profiles, derived by numerical simulations of Brock’s  experiments [Zanuttigh and Lamberti, 2002b], at different distances from the channel inlet and during the same time interval: Close to the channel inlet (x = 13 m, x0 = 15,000), waves are essentially due to instability, whereas at greater distance (x = 70 m, x0 = 140,000) coalescence prevails, reducing the number of waves and producing their strongly asymmetric shape, with peaked crests much higher than the normal depth and flat trough. Conditions for roll wave formation in a water laminar flow were examined by Julien and Hartley . The tests were conducted in a 0.21-wide by 9.75-m-long rectangular flume, whose slope was adjusted between 1.4 and 4%. The characteristic depth and Froude number of the base unperturbed flow were in the ranges of 1–2 mm and 0.9–2.3, respectively. Measurements of wave celerity and period were performed at the distance from the channel inlet at which waves were observable. Julien and Hartley highlighted that a sufficient length is required for unstable perturbation to grow up to an observable level [Montuori, 1961, 1963] and showed that such length scales proportionally to h/sin J with a proportionality constant approximately equal to 40, a value which is much lower than in a turbulent flow, where on the basis of Brock’s [1967, 1969] experiments it is about 100. Liu et al.  carried out experiments with water and glycerine-water solutions (50% by weight) to analyze two-dimensional (2-D) waves and verify theoretical results on the instability of laminar film flows. The fluid was pumped on a plane 2 m long and 0.50 m wide with inclination up to 10_; the unperturbed film thickness was about 0.002 times the plane width, and the velocity of the uniform flow was around 0.03 m/s. Small pressure variations with fixed frequency and amplitude were applied to the entrance manifold. Instabilities developed as isolated wave groups caused by perturbations traveling downstream with increasing intensity and asymmetry. The initial instability was a periodic perturbation strongly dependent on the noise frequency; however, with increasing distance from the source, intrinsic nonlinearity gradually dominated, and waves became statistically independent from the input noise. The growth rate and phase celerity of the forced perturbations were measured through methods based on laser beam deflection and fluorescence imaging. The critical Reynolds number Rec and its relation to neutrally stable frequency were also derived and confirmed the predictions obtained by the linear method due to Anshus and Goren .
3.1.2. Non-Newtonian Flows
 Davies [1988, 1990] used a laboratory moving bed flume to analyze the development, behavior, and characteristics of waves in a high-concentration grain-fluid mixture. In his apparatus the bed, 50 mm wide and 2 m long (referring to the working cross section), moves up channel at a constant speed between stationary perspex walls, while the flowing material (water or grains in water) remains statistically stationary with respect to the walls (bed velocity is equal and contrary to mean flow velocity relative to the bed). The solid grains were 4-mm-long PVC cylinder with diameter equal to 4 mm; a small number of similar 8-mm cylinders were used to study the behavior of large grains. Channel slope ranged from 5_ to 19_, while bed speed ranged from 0.25 m/s to 1.17 m/s. Flow characteristics and their change were observed as grains were gradually added to the water flow. With no grains and for channel slopes steeper than 5_, the flow intensity was sufficient to form and amplify roll waves moving downstream relative to the bed. Few grains had negligible effect on the flow; the grains dispersed throughout the channel length; grains were moved downstream by rolling crests and returned upstream again in the shallower and slower flow between crests. As the concentration of grains increased (keeping bed speed and slope constant), the roll wave amplitude increased: A roll wave tended to collect grains as it moved downstream. For moderate concentrations, grains remained close to the bed and were left behind the wave. With still some grains added, these grain accumulations tended to form stationary grain waves through which the roll waves moved. When 8-mmdiameter grains were present within a shearing body of 4-mm-diameter grains, the large grains were carried up to the surface of the flow, and hence forward to the front of the flow by the higher surface velocity, if the flow depth was less than about 25 mm. If the flow was deeper, this tendency was much weaker.
EXTRAIT 13 :
In conclusion, when channel steepness was greater than a certain threshold, waves in pure water or in grain-water mixtures spontaneously developed in absence of any external forcing; particle segregation toward the front was evident when the surge depth was smaller than 3 times the coarsest fraction diameter; otherwise, large clasts remained dispersed in the surge body.  Takahashi  examined the flow depth variation in time for an experimental debris flow over a rigid bed flume 20 cm wide and 18_ steep. A heterogeneous sediment was mixed with water and was poured into the flume with constant flow rate (in Figure 7a water discharge is 1.6 L/s; solid discharge is 0.6 L/s); two kinds of materials were used, the diameters of which were 1.2 and 4 mm (corresponding to open and solid circles, respectively, in Figures 7b–7e); volume average concentration of the grains in the mixture varied between 0.18 and 0.45. Figures 7b and 7e show that Frc is below 1 since waves are observed for Fr down to 0.8. Takahashi recognized that roll waves developed with characteristics similar to those of water flow since his experimental points were in good agreement with the theoretical curves obtained for water by Iwagaki .  Lanzoni and Tubino  observed the formation of instabilities close to water roll waves in their experiments on glass beads and water mixtures. Experiments were carried out in a 10-m-long, 0.2-m-wide and up to 31_ steep flume. The bottom surface was roughened by cementing one to two layers of the material used in the tests. The bed was initially covered with a 9-cm-thick layer of grains, and the debris flow was activated by feeding a fixed water discharge upstream of the flume. The materials adopted were glass beads and two types of gravel. Glass beads had average diameter ds = 3 mm, density 2.65 g/cm3, static friction angle 25_, and packing concentration 0.63. For this material, water discharge was 1.6 or 2.3 L/s, flume steepness varied in the range of 9_–12_, and mean solid volumetric concentration ranged from 0.31 to 0.54. During some experiments with glass beads, characterized by an average flow depth and velocity in the range of 1.9–5.9 cm and 2.4–6.9 m/s, respectively, waves characterized by a high degree of regularity showed up on the debris surface, with amplitude ranging about 2–3 times ds, wavelength of approximately 20–30 ds, and wave speed close to uniform flow velocity.  In his experiments on mudflows, Coussot  did not make a systematic study of unstable flow characteristics but only recorded the conditions for which instability appeared (Figure 8). The experimental setup consisted of a rectangular channel 8 m long, whose width and steepness were between 0.1 and 0.6 m and 3 and 27%, respectively. Channel walls (bottom and sides) were either rough (expanded metal with an equivalent roughness of 0.06 m) or smooth (plywood). The material adopted was a waterkaolinite mixture, with volume concentration varying in the range of 20–25%. The water discharge supplied at the top of the channel was varied between 0.27 and 3.85 L/s. Resulting flow depths were in the range of 0.013 – 1.070 times the channel width. Starting from a stable flow, keeping the discharge constant and increasing the slope, Coussot observed that the flow became systematically unstable above a certain slope, which was a function of the discharge, as for water roll waves. When no external disturbance was applied to the flow, small waves appeared at the free surface at some distance downstream. They moved down, growing progressively to become roll waves, characterized by greater depth at crest and higher velocity than the uniform corresponding flow (i.e., for the same discharge). The distance between each of these rolls seemed to fluctuate around a mean value depending on the discharge, slope, and fluid characteristics. In these cases, if a small perturbation was created upstream, it also rapidly degenerated into a specific roll wave.  Schonfeld  analyzed the development of roll waves in dry and saturated granular flows in two facilities, at McGill University and at the U.S. Geological Survey (USGS) flume, respectively. At McGill University, two flumes were used, a shorter one, which was around 2 m long and 0.6 m wide, and a longer one, which was around 6 m long and 0.1 m wide. In both channels all the experiments were carried out with 45 kg of spherical glass beads with a diameter equal to 0.5 mm. A typical run lasted less than 1 min in the shorter flume and from 3 to 5 min in the longer one. All available data came from the longer flume, since almost no wave was appreciable in the shorter one. The USGS flume is a reinforced concrete channel 95 m long, 2 m wide, and 1.2 m deep that slopes 31_, an angle typical of terrain where natural debris flows originate. Various combinations of gravel, sand, and loam were run down the chute after being saturated with water sprinkled from above and injected from below overnight. Most of the runs consisted of about 10 m3 of sediments and typically took less than 40 s to complete. In both cases of granular and saturated flows, waves spontaneously developed along the channels, and measurements of flow depth and speed, wave depth, speed, and period were performed. Figure 9 shows a representative record of flow depth, normal stress, and basal fluid pressure made at the USGS debris flow flume [Iverson, 1997a]. The flow depth registration is very similar to the ones for granular flows: Several sharp surges can be recognized together with an onset of instabilities of the flowing material particularly evident in the surge recession phase like in prototype cases (compare, for instance, Figure 9, top left, with Figures 2c and 6).  Wang  performed experiments aimed at studying the growth of instabilities in non-Newtonian laminar flows. The channel that he adopted was 26 m long, 0.6 m wide, and 0.5 m high, with glass-sided walls and a smooth steel bottom plate.
EXTRAIT 14 :
Channel steepness was varied in the range of 2–8%. Clay suspensions were recirculated through the channel. The clay material had a density of 2.68 g/cm3 and median diameter of 0.2 mm. The flow rate was controlled by an inlet valve and measured by a magnetic flowmeter. Clay and water were well mixed so that the suspensions behaved like a viscous liquid rather than a solid water mixture. Flow depth was in the range of 1–12 cm. Unstable conditions were forced by feeding into the flow 3–5 L of extra clay suspension from a tank placed at 2 m from the channel inlet. The larger the yield stress, the closer the depth to the critical one, the faster the wave grew up. The obtained wave height along the channel in Figure 10 resembles the development curve of water roll waves in Figure 5.  Forterre and Pouliquen  analyzed long-wave instability in dense granular flows. The experimental setup was composed of a plane, 2 m long and 0.35 m wide, and a reservoir with a gate that was instantaneously pulled up. The side and bottom plates of the channel were made by glass walls; the bottom plate was roughened by gluing one layer of the particles used for the flow. Two materials were tested, glass beads characterized by a mean diameter equal to 0.5 mm and sand with average diameter equal to 0.8 mm. The channel slope was varied between 24_ and 29_ for glass beads and between 32_ and 36_ for sand particles. Flow thickness was in the range of 2.5–6.5 mm. A perturbation of known frequency and amplitude was imposed on the free surface flow, as in the work of Liu et al. , by blowing a thin jet of air through three loudspeakers embedded in a 2-D nozzle with a 1-mm slit. The typical free surface deformation obtained with such a setup was about 0.25 mm high, frequency being in the range of 1–20 Hz. In the case of glass beads the channel length was insufficient to allow wave development, and thus this kind of forcing was always necessary, whereas for sand particles, waves developed for slow flows even without being forced. Liu et al. provided for both glass beads and sand particles: the wave spatial growth rate as a function of the perturbation frequency with varying flow thickness at a given inclination of the plate; the wave phase velocity, always close to the speed of the gravity waves, as a function of Fr for different angles of plate inclination; and the cutoff frequency as a function of Fr with varying flow thickness and inclination of the plate. For both glass beads and sand particles, instabilities showed up and were essentially controlled by Fr, the growth rate mechanism being similar. The first unstable waves were the longest downstream traveling ones; for any unstable condition, only waves with frequency below a certain cutoff value did actually grow. Waves developed along the channel with increasing depth and time interval and with a progressive shift of wave energy from high to low frequencies (see Figure 11) as in water roll waves [Zanuttigh and Lamberti, 2002b].
3.2. Regressive Instabilities
 Few works can be found in the literature that analyze regressive instabilities, which are essentially aimed at examining the ‘‘channel clogging’’ phenomenon (i.e., the stick-slip behavior observed in some natural rivers and basins like the Yellow River and Mount Thomas, see sections 2.3 and 2.2, respectively). Engelund and Wan  investigated the rheological behavior of bentonite water suspensions in a 10-m-long tilting flume 0.3 m wide. When the discharge was large, the flow remained steady, and no oscillation of water level was detected, whereas at small discharges, intermittent flow like ‘‘river clogging’’ appeared. An example of the measured surface level variations is shown in Figure 12. Waves cause a slow rising level of the free surface followed by a sudden drop. Mud is arrested and accumulates on the upstream wave slope, up to its crest, until the yield stress is exceeded at the crest base, and then the crest material slides down in the downstream trough, and the crest form moves upstream. In the same experimental setup already described in section 3.1.2, Wang  analyzed the critical conditions for river clogging in the laminar flow of clay suspensions. Tested conditions included five different clay concentrations and three bed slopes (0.4–1.2%). For given bed slope and clay concentration, after achieving stable and uniform flow, the dis- charge was progressively reduced till river clogging occurred. Only when the shear stress was lower than or close to the suspension yield stress, did the suspension stop moving and suddenly flow again after a while, because the continuous incoming discharge raised the surface slope and driving shear stress.  Stationary waves were observed by Zanuttigh and Lamberti [2002a] in a rotating drum partially filled by dry granular materials (Figure 13). The experimental equipment consisted of a cylinder with inner diameter of 390 mm and axial length of 131 mm, which was mounted on a pair of friction rollers and rotated around its axis at a constant speed in the range of 0–10 rpm. The drum front and back walls were made of 10-mm-thick glass to allow optical measurements; the internal surface of the cylinder was roughened with sandpaper to avoid the tendency of the whole bed to slide on the boundary as a solid body. Many tests with glass spheres and natural sand of irregular shape were performed at different rotation speeds in the range of 1–6 rpm. In Figure 13 the drum is rotating at 6 rpm and is partially filled with natural sand grains of 0.84- to 1.19-mm diameter. If the drum rotation speed was very low (_2 rpm), instabilities showed up as a succession of avalanches that started after a period of rest when, because of a rigid rotation of drum and grains, the material friction angle was exceeded. For greater rotation speeds (_3 rpm and Fr _ 1), waves propagating upstream with celerity similar to stream velocity developed. Wave amplitude and celerity were reconstructed experimentally, by means of optical measurements, and theoretically, based on the Takahashi  flow profile over an erodible bed.
EXTRAIT 15 :
3.3. Synthesis of Literature Data
 Laboratory experiments demonstrate the spontaneous formation of surges for water, grain-water mixtures, mudflows, debris flows, and dry granular flows, i.e., the formation of surges in flowing materials even in the absence of any forcing action. Researchers agree about the existence of some flow intensity threshold, the exceedance of which is necessary to generate roll waves, and in some cases the necessity of a sufficient length for instabilities to grow up to an appreciable level is highlighted.  In order to give the reader a more rapid overview of the quoted literature the main parameters and results of all the works are summarized in Table 2. When dealing with the experimental results described in sections 3.1 and 3.2, attention should be paid to scale effects: Cohesion and permeability are very important parameters that do not scale with Froude’s law. Moreover, the dimension of the experimental apparatus has significant effect on the flow, as stressed by Davies [1988, pp. 15–16] the pulsing nature of the flows requires that even at laboratory scale a very long channel would be needed to allow full development and measurement of the waves. Even then, obtaining data from the moving wave would be very difficult as would be correcting the data for the effect of the channel sidewalls, and also choosing the appropriate material mixture so that the small-scale flow behaves in the same way as a large field debris flow; scaling up the laboratory results to apply to the full-scale situation would also be at best a tentative procedure.  Nevertheless, Davies, while continuing the discussion on the implications for debris flows of his experiments on water–PVC particles, also enhances how the qualitative features of the experimental surges appear similar to proto- type surges, and he hypothesized that debris flows can be represented by a slurry, the nature of which is not crucial, provided only that its apparent viscosity is sufficient to cause the shearing of large grains to be macroviscous.
4. INSTABILITY OF UNIFORM DEBRIS FLOW AND ROLL WAVES FORMATION
 This section aims to provide criteria to identify the conditions that lead to the formation of progressive and regressive waves in homogeneous flows. The structure of the section is as follows. Section 4.1 recalls the fundamentals of the 1-D linear stability analysis that allows the determination of both the unstable flow conditions and the maximum celerity of developing instabilities. Section 4.2 deals with unstable conditions for progressive waves and considers both Newtonian flows, as water streams, and non- Newtonian flows, which are often adopted in common practice to approximate debris flow rheology. The case of pure water is included because, apart from similarities between water and debris waves, it has been deeply analyzed with similar methods under simpler conditions, providing undoubted conclusions. Section 4.3 describes the formation of regressive instabilities due to the ‘‘stick-slip’’ process.
4.1. Development of Instabilities 
Uniform flow instability on a sloping plane can be analyzed with a fully 2-D approach or with a 1-D vertically integrated approach. Vertical integration is usually performed under the following hypotheses: (1) long-wave approximation, i.e., the effect of neglecting vertical acceleration by assuming a hydrostatic total pressure distribution; (2) constant fluid density and viscosity; and (3) very long waves, i.e., distribution of shear stress and longitudinal velocity along the vertical as in a uniform flow. The resulting 1-D equations, generally used in open channel flows, are (see the full text in pdf) where h is flow depth, u is flow velocity, b is momentum correction factor, J is channel slope angle, subscripts t and x denote the derivative in time and long the channel, respectively, g is acceleration due to gravity, andt0 is bottom shear stress.  Savage and Hutter  relaxed the assumption of hydrostatic thrust by including in the pressure force in equation (2) a coefficient K, which represents the ratio of the normal stress in the longitudinal direction to the normal stress in the orthogonal direction. In equation (2), K = 1 is adopted as Forterre and Pouliquen  did on the basis of numerical simulations carried out for dense granular flows by Prochnow et al.  and Ertas et al. . Moreover, for a pure shearing of a fluid characterized by an isotropic stress-strain relation, K = 1. However, for nonuniform flows, K could depend on flow divergence [Gray et al., 1999; Wieland et al., 1999]. By accounting for equation (1), if the momentum coefficient b is not significantly different from 1, equation (2) divided by h becomes (see the full text in pdf)  Equation (2) is often called the (von Karman’s) momentum integral equation, and equation (3) is referred to as the head balance or approximated momentum equation. Equation (3) leads to a simpler analysis and is often preferred to equation (2), but it contains an additional approximation.  If h0, u0 is a uniform flow regime solution of equations (1) and (2), the following parameters are chosen to scale the others: (1) h0 for any length perpendicular to channel profile; (2) u0 for any longitudinal velocity, and (3) l0 = u0 2 /gi for any longitudinal length, with i = tanJ. We fix thus (see the full text in pdf) ; where, for the sake of simplicity, the scaled variables are named as the nonscaled ones in this section only. Equations (1) and (2) are in dimensionless form (see the full text in pdf).  After the substitution of the continuity in the momentum equation and after the linearization around the regime solution u = 1 and h = 1, equation (4) becomes in synthetic form (see the full text in pdf) where tu an th denote the derivative of the shear stress t with respect to u and h. U is the vector of the independent variables; C is the matrix of the coefficients of the source terms.
EXTRAIT 16 :
The matrix of the coefficients A on the left-hand side of equation (6) is the Jacobian matrix; eigenvalues l can be found by evaluating det jA _ lIj = 0, where I is the identity matrix, that gives l1;2 ¼ b _ (see the full text in pdf)  The eigenvector matrix K of equation (5) shall satisfy the relation AK = lK and thus is given by (see the full text in pdf):  The determinant of K is equal to _2D, and its inverse is (see the full text in pdf): The characteristics variables W are W (see the full text in pdf); and equation (5) can be rewritten in the characteristic form as (see the full text in pdf) where L is the diagonal matrix whose diagonal elements are the eigenvalues (7) and (see the full text in pdf)  In order to have the development of instabilities, at least one of the diagonal terms in (9) should be positive, so that even if the other mode is zero or decreases to zero, the considered one grows. As D is intrinsically positive, the instability conditions result by highlighting (see the full text in pdf)  Since b is intrinsically greater than 1, it decreases the critical value of a and thus increases the critical Froude number Frc. For a viscous fluid, tu = 1 and th = _1 in laminar flow, whereas in turbulent flow tu = 2 and th = _1/3 or th = 0 if a Manning-Strickler- or a Che´zy-type resistance law is assumed. Equation (10a) corresponds to the unstable condition for a progressive wave, which is discussed in section 4.2, whereas equation (10b) represents the instability of a regressive wave, which is responsible for stick-slip motion and is described in section 4.3. Equation (10b) cannot be satisfied in a Newtonian flow.
4.2. Progressive Instabilities
4.2.1. Newtonian Flows
 The instability of a uniform water stream leading to roll waves was studied by several authors. Jeffreys , Dressler and Pohle , Ishihara et al. , Yih [1955, 1963], Benjamin , Mayer , Escoffier and Boyds , and Liggett  theoretically analyzed instability and wave development in a laminar or turbulent stream of a Newtonian fluid. The main experimental results on the same topics obtained by Brock , Julien and Hartley , and Liu et al.  have already been recalled in section 3.1. All these works start from the standard equations for open channel flows already given by equations (1) and (2), but most adopt the hypothesis of uniform flow velocity (b = 1) or the head balance equation (3). For turbulent flow, on the basis of the assumption of constant Che´zy coefficient, Jeffreys  obtained that uniform flow is unstable whenever Fr is greater than 2. As a special case of a more widely applicable theory that is recalled in section 4.2.2, Trowbridge , by accounting for variation of the friction coefficient but assuming b = 1, obtained the following expressions for Frc: (see the full text in pdf) for smooth turbulent flow and (see the full text in pdf) for rough turbulent flow, where k is von Karman constant (ffi0.4) and f is the Darcy-Weissbach friction factor defined by t0 = fru2/8 and equal to O(0.03). The resulting Frc is somewhat below 2.  For a laminar flow, Ishihara et al. , on the basis of the asymptotic long-wave assumption, found from equation (2) with b = 6/5 that the flow becomes unstable above Frc = 0.577 (= 1/ see the full text in pdf). In the analysis presented by Trowbridge , on the basis of equation (3), Frc = 0.50. Benjamin  and Yih [1955, 1963], on the basis of a more rigorous analysis of the complete linearized 2-D equation (of the Orr-Sommerfeld type), obtained for low Reynolds numbers (Re = O(1)) and long waves that (see the full text in pdf) or, what is equivalent, the critical Reynolds number Rec is (see the full text in pdf)  The 1-D momentum balance equations (1) and (2) represent the asymptotic form of the vertically integrated 2-D equations; if the 2-D flow is stable, there is no unstable perturbation of the regime solution, and thus the simple intensification of the regime velocity distribution must also decay in time-space to the regime solution. The inequality of the values of Frc (Frc = 0.5, 0.577, 0.527, respectively) obtained by Trowbridge , Ishihara et al.  and Benjamin , and Yih [1955, 1963] is due to the type of analysis. In fact, Benjamin  and Yih [1955, 1963] solved the 2-D problem; Ishihara et al.  and Trowbridge  both performed a 1-D analysis, but the first work assumed a velocity distribution consistent with the 2-D problem. The difference in Frc is, however, moderate.  Smith  extended the analysis by Benjamin  to a thin liquid film flowing down a steep chute by including the effect of friction at the free surface. Two different mechanisms for generating 2-D perturbations are examined: the application of a fixed stress at the free surface (for instance, wind friction) and the presence of a surface layer of particles characterized by constant velocity. A linear stability analysis, such as the ones by Benjamin  and Yih [1955, 1963], is carried out to the second order, starting from the result that unstable waves are long waves and adopting a long-wave asymptotic expansion scheme. It is shown that Rec depends not only on the channel slope but also on the free surface stress. The derived dispersion relation does not provide information either on the most unstable mode or on the cutoff frequency.  It can be concluded that the condition for instability is Fr > Frc, where Frc is around 0.5 for laminar flow and somewhat less than 2 for turbulent flow; lower values of Frc are obtained by using the head balance rather than the longwave momentum balance equation. These conclusions are generally valid even if all these works (and all the experiments in section 3.1) do not account for variation in channel bed topography. In fact, the 1-D analysis performed by Balmorth and Mandre  proves that the effects due to sinusoidal perturbation of the channel bed on roll wave formation are modest. When bed perturbation amplitude to uniform flow depth ratio is equal to 0.1 and 0.2, the value of Frc decreases from 2 to 1.99 and to 1.95, respectively. In any case, Frc never drops to less than 1.5, since for high bed perturbation amplitude the formation of step and pool system stabilizes the stationary flow.
EXTRAIT 17 :
4.2.2. Non-Newtonian Flows 
The analysis of uniform flow stability for Newtonian fluids was extended by Trowbridge  to any flow where the bed shear stress is expressed as a single valued function of local vertical mean velocity and flow depth t0/r = t(h, u). Trowbridge’s analysis is based on equation (3), but all the main results remain valid with minor modifications when equation (2) is used as has been shown in section 4.1. Trowbridge examined the space-time evolution of a sinusoidal perturbation of the base uniform flow with height h0 and velocity u0 satisfying the flow resistance law for the given channel slope. By using equations (1) and (3) in a reference system moving with the unperturbed stream and accounting only for linear terms in the perturbation, Trowbridge obtained the following quadratic equation for the complex perturbation celerity c: (see the full text in pdf)  If the imaginary part of c is positive for any value of the wave number k, the corresponding perturbation grows in time, and the base flow is unstable.  For vanishing k the propagation speed of disturbances approaches the kinematic wave speed, whereas for very large k the propagation speed approaches the linear frictionless wave speed, but equation (3) is unable to adequately represent short waves in the presence of internal stress. When @t/@u is negative, one of the solutions for c has a positive imaginary part that indicates the formation of instabilities, which grow very quickly and generate finite oscillations of the stick-slip type, discussed in section 4.3. Trowbridge’s interest was focused on the instability of the base flow in the case of a positive value of (see the full text in pdf) that leads to the condition (see the full text in pdf) which coincides with equation (10a) when b = 1 and corresponds to ‘‘Craya criterion’’ [Craya, 1952]: The base flow is unstable when the kinematic wave speed exceeds the linear, frictionless wave speed. For unstable conditions of the base flow the temporal growth rate of the perturbation monotonically increases from zero at k ! 0 to a maximum finite value at k!1. The approach is therefore unable to determine the fastest growing mode that one might expect to characterize the spectral peak of developing waves. On the basis of the energy balance of the flow perturbation, Trowbridge  provided the following physical interpretation of equation (12). The total energy of the perturbation increases in time if the perturbation of work done by gravity exceeds the perturbation of work dissipated by bed shear stress, i.e., if (see the full text in pdf) ; where the angle brackets denote the spatial average over one wavelength and h and v are the disturbances of free surface elevation and velocity, respectively, with respect to the steady uniform flow conditions.  Lanzoni and Seminara  developed a linear stability analysis of the uniform solution for debris stream in order to interpret the formation of surface waves in the experiments by Lanzoni and Tubino  just described in section 3. The approach neglects the relative motion of the two phases but accounts for the longitudinal variation of the depth-averaged concentration and explains the formation of debris waves as a result of a classical instability mechanism similar to the one describing water roll waves. Water-grain flows appear to be unstable for Fr around 1.0, with a quite satisfactory agreement among theoretical and experimental stable/unstable conditions.  Ng and Mei  provided a 1-D theory for permanent roll waves on a shallow layer of a power law viscous fluid. They applied a linear stability analysis of the uniform base flow to equations (1) and (2) by representing a mudflow with values of the power law exponent in the range of 0.1–0.4, the lower values corresponding to the higher mud density. By means of a nonlinear analysis of periodic roll waves, Ng and Mei described the waves as smooth profiles with depth monotonically increasing in downstream direction (as given by Dressler ) and showed that only waves longer than a certain threshold cause positive energy loss at the shock. Waves having just the threshold wavelength, i.e., that develop initially with no energy dissipation at the shock, are named ‘‘minimal roll waves,’’ and Ng and Mei assume that this is the wavelength that actually develops. A similar analysis was performed for a Bingham fluid by Liu and Mei , who therefore highlighted the effect of a yield stress, and for a dilatant fluid by Longo , who proved that in this case the minimum roll wave theory does not apply.  For a Newtonian fluid the theory is verified by comparison with the measurements of Julien and Hartley ; in this case, no periodic finite amplitude waves are possible if the uniform flow is stable. For mud or debris flows, very long roll waves may still exist even if the corresponding uniform flow is stable; that is, roll waves generated in a steep reach of the channel may propagate with little or no attenuation at all on a milder lower reach. Coussot  applied Trowbridge’s  analysis to a steady, uniform free surface flow of a Herschel-Bulkley fluid on an infinitely wide inclined plane. For such fluid, Frc drops to 0.25 or less. The comparison between experimental and theoretical conditions showed that no unstable flow occurs below the theoretical limit; however, many stable flow points fall above the theoretical limit. Coussot  related this discrepancy to uncertainty in empirically determining flow instability because of the limited channel length, an argument that has been already highlighted for water flows in section 3.1.1.  Similarly to Trowbridge , Forterre and Pouliquen  performed a 1-D linear stability analysis by adopting the friction law proposed by Pouliquen , which accounts for the reduction in the internal friction angle of the flowing material due to the greater mobility of a multiple layer of particles with respect to a single layer. Theoretical results are compared by Pouliquen to experiments, which have been already presented in section 3.1.2. The model is able to quantitatively predict the stability threshold and the wave phase velocity but fails in representing the (decreasing) growth rate of high-frequency components beyond the maximum measured value. Forterre and Pouliquen  provide an empirical formulation to calculate the cutoff frequency, which is around twice the frequency of the measured fastest growing waves.
EXTRAIT 18 :
 The Trowbridge stability condition (12) was applied by Zanuttigh and Lamberti [2004a] to different rheological laws: collisional, Bingham (B), Herschel-Bulkley (HB), and generalized viscoplastic (GVP) fluid, within a realistic range of depths (0.2–3.0 m), velocities (0.5–10 m/s), and rheo- logical parameters. We recall here for convenience that the GVP fluid model is expressed as (see the full text in pdf) where rm is average mixture density, r is water density, h is total depth, y is local depth at which stress is estimated, f is the dynamic friction angle, m1 is the consistency index, g is the shear rate, and h is the model exponent. The sum of the first two terms on the right-hand side of (13) represents the yield stress ty, whereas the third term t0 is the stress due to shear rate. The GVP model [Iverson, 1985; Chen, 1988a, 1988b] contains as trivial particular cases the B (h = 1.0 and f = 0_), HB (h = 1/3 and f = 0_), and grain inertia (GI) (ty = 0 and h = 2.0) models. Figures 14a–14d show the critical Froude number Frc as function of the bed stress nondimensionalized with the yield stress for the GVP model, taking into account the variable ratio between the yield and bed stress or between the lower shearing and the upper blocked layer thicknesses. Cases are selected starting from a central configuration (m1/tc = 0.5 s3/kg2, tc/rm = 1.0 m2/s2, h = 1.0, and f = 2_) and varying each rheological parameter in excess and defect. Frc normally results in the range of 0.2–0.4 and mainly depends on the degree of mobility or yield to bed stress ratio ty/t0 and on the exponent of the rheological law: By increasing this exponent (1/3–2.0), Frc almost proportionally increases (0.25–0.67). The friction angle and the consistency index show a secondary influence; by increasing one of these parameters, Frc increases. When bed shear stress is much greater than the yield stress, Frc depends only on the model exponent: It is 0.25, 0.50, and 0.67 for the HB, B, and GI models, respectively. It must be remarked that when the bed shear stress is close to the yield stress, Frc drops to almost zero and almost all flows become unstable.  Davies [1990, 1997] observed that in small gullies the nonuniformity of debris flow behavior has its origin in the occurrence and instability of nondepositing macroviscous flow. Davies started from Bagnold’s criterion for the occurrence of macroviscous flow, used the work by Rickenmann , introduced values of rheological parameters derived from experiments [Wan, 1982; Rickenmann, 1990] and field surveys [Pierson, 1980; Li et al., 1983], and finally, provided the following criterion for the occurrence of pulses: ty _ 25 Pa: ð14 Equation (14) was tested by Davies  against Chinese [Li and Luo, 1981; Li et al., 1983; Zhang et al., 1985] and other field data [Pierson, 1980; Costa and Williams, 1984; Johnson, 1984]. Since there is no theoretical evidence that instability is peculiar of macroviscous flows, equation (14) can be interpreted as the expression that for high yield stress almost all flows develop instabilities, as said above.
4.3. Regressive Instabilities
 The essential mechanism of the formation of regressive perturbations is the increase of resistance to motion as velocity decreases. This can be due to the following causes: (1) the casual aggregation of the largest boulders in the debris mass, (2) the reduction of the friction angle from the static value to the residual or dynamic value as shear starts in a boulder mound, and (3) the breakdown due to shear of the weak links that characterize a cohesive material in the muddy matrix. Whenever the bed shear stress decreases with increasing stream velocity, a regressive instability develops, as given by equation (10a); the resulting celerity relative to the channel is normally very weak compared to progressive waves and is quite often upstream directed.  In the following the three initiation mechanisms listed above for the development of regressive instabilities are briefly explained. The first mechanism consists of the tendency of coarse grains to jam across the width or the depth of a channel at high concentrations. Bagnold , Savage and Sayed , and Walton  show this tendency to be important where the channel width or depth is less than few grain diameters, as it can be for debris flows carrying large boulders. Such a vertical jam in a region of locally high bulk concentration creates a temporarily stationary dam. When the dam reaches a certain height, the dam toe becomes unstable and slides over the resting bed; the fracture and the expansion wave move upstream along the sliding body; the toe flows down in the channel as a surge wave.  The second mechanism of instability occurs, for instance, in debris avalanches. In a static granular medium made up of frictionless grains, the contacts among particles and contact forces are oriented in any direction. Compression of the mixture intensifies every contact force proportionally to pressure, and the mean effective stress remains isotropic. When a shear stress is applied that can be resisted without shearing, forces through contacts oriented along extension lines are deactivated and forces oriented along contraction lines are intensified, causing an average orientation of contact forces or self-organization of contacts; as a consequence of this the granular material can resist shear to some extent. If grains show up friction at their contacts, satisfying, for instance, the Coulomb t-s proportionality relation, contact forces are oblique with respect to the microscopic contact surface. If grains are not too loose when shear starts up, the obliquity of the macroscopic contact force relative to the macroscopic shearing surface (or apparent friction angle) reaches a peak that is actually the sum of two angles: the dilation angle, representing the mean obliquity of contacts (forcing a dilation when shear starts), and the true or residual friction angle, representing the mean obliquity of forces relative to active contacts. When shear is intensified, grains start agitating and touching each other along random directions, causing the progressive disorganization of contacts and the reduction of the apparent static friction angle to the residual one.
EXTRAIT 19 :
The dilation angle, i.e., the difference between the peak and the residual angle, is strongly dependent on particle shape and packing density and varies from a minimum near 0_ up to 12_–15_ for tightly packed angular grains.  The third mechanism is common in mud streams, as, for instance, in the already described Yellow River [Qian, 1980], at the Jian-Jia Ravine [Davies, 1997], or at Wrightwood [Morton and Campbell, 1974], and it is the cause of what is usually named ‘‘river clogging,’’ i.e., the flow freezing when the shear stress falls below the minimum shearing stress. Debris flows are repeatedly said to ‘‘flow as fresh concrete’’ [Iverson, 1997a; Davies, 1997], and in concrete [Ukraincik, 1980] a behavior similar to bentonite suspensions (see the experiments by Engelund and Wan  noted in section 3) is normally observed.  In all these mechanisms the regressive expansive wave is accompanied by the formation of a downstream surge. The surge front exerts very high shear stress on the bed, and Bagnold  showed that in some circumstances the bed may be scoured to almost unlimited depth; see also section 5.1. Regressive erosions can be also caused by the interaction of a supercritical stream and an alluvial bed, whose extreme forms are rapids and waterfalls. In the case of debris flows these forms are usually named ‘‘nickpoints’’ [Pierson, 1980; Davies et al., 1991, 1992].
5. SURGE DEVELOPMENT
 This section aims to analyze the characteristics of surge propagation in the case of stable and unstable conditions and in particular to describe the process of debris wave development. The structure of the section is as follows. The propagation of a single surge, focusing on its celerity and shape, is the subject of section 5.1. Section 5.2 analyzes debris wave development in time and space through field, laboratory, and numerical data. First, it is examined if in debris flows as in water streams a minimum channel length is necessary for the formation of clearly recognizable waves. Then, wave development rate (depth and period) along the channel is compared for homogeneous and heterogeneous flows. Analytical and numerical descriptions of wave profile are provided in section 5.3. Section 5.4 briefly describes the changes in the flowing mass during the event due to channel erosion, deposition, or remobilization of the deposited material.
5.1. Surge Propagation
5.1.1. Surge Celerity  The surge front is an abrupt change in flow characteristics moving downstream as an almost stationary flow in a reference frame moving with the front. This approximation, probably proposed for the first time by Witham , is named a uniformly progressive flow by Chow . In the case where the front advances on a dry nonerodible bed, in the reference moving with the front there is no momentum flux both upstream, because the relative velocity is null, and downstream, because there is no flowing mass. The momentum balance near the surge front reduces therefore to the balance of pressure gradient, weight component, and bed friction, i.e., under the assumption of a longitudinal thrust due only to a gradient in the hydrostatic pressure: (see the full text in pdf)  When the downstream bed is dry, the basal stress and the pressure gradient grow indefinitely at the front, and neither of them can be neglected; on the contrary, when a stream is present before the front, the basal stress is usually ignored, and the front speed can be derived only by mass and momentum balance across the front. On the basis of video analysis at Jiang-Jia Ravine, Davies  suggested that the behavior of debris surges is similar to a moving surge in water: The surge depth decreases on entering a deeper pool of stationary material and increases again as it leaves the pool and enters shallower deposits downstream. On the basis of the theory for a uniformly progressive wave in still water by Chow , Davies considered a scheme composed of a surge, characterized by front celerity cf and depth hf, and a stationary material of thickness ht lying downstream of the moving surge. By neglecting the basal stress over the front length and assuming that the material is homogeneous, the following expression for cf can be derived: cf ¼ (see the full text in pdf).
5.1.2. Surge Profile
 The description of a single surge caused by a pointwise release of material is given at the surge scale and near the surge front by Hunt  for a Newtonian fluid and by Huang and Garcı´a [1997, 1998] for a Bingham plastic or a Herschel-Bulkley flow. Huang and Garcia essentially use a matched asymptotic expansion technique, which is shown to fit experimental results after a propagation distance from the release point that corresponds to a bed drop greater than few stream depths [Arattano and Savage, 1994].  This method combines a kinematic wave propagation model far from the surge front and a stationary flow approximation in a reference frame moving with the surge front. The first scheme neglects streamwise pressure gradient and acceleration compared to bed friction and weight; the second one neglects time/downstream variation of surge conditions. Depth and mean velocity upstream of the surge are assumed to satisfy the uniform flow condition (h0, u0). In the front zone the mean velocity along any vertical is equal to the front speed; in order to keep this mean velocity constant the weight component due to channel slope is sufficient at the surge crest (and behind it), whereas at the surge front a pressure gradient due to downstream decreasing depth is necessary.  Relations for surge depth and velocity for laminar and turbulent flow are reproduced in the following from Hunt . Symbols adopted by Hunt  and used in the equations below are clarified by Figure 15, which shows the composition of inner and outer solution for one specific test and a zoomed image of the region where inner and outer solution have to match. For a laminar (left side or top) or turbulent (right side or bottom) flow the mean surge velocity us according to the kinematic wave approximation is expressed by (see the full text in pdf) where n is the cinematic viscosity and C is a dimensionless constant related to the Darcy-Weisbach friction factor f and the Che´zy coefficient c through the expression C = c/ (see the full text in pdf) .
EXTRAIT 20 :
The outer solution h(x, t) due to a release in the origin (x = 0, t = 0) is obtained from the kinematic wave model, i.e., the assumption of a constant state along kinematic characteristic lines (x/t = dq/dh, q being the discharge per unit length), (see the full text in pdf) and has to be applied far from the surge front down to the surge position (see the full text in pdf) where A is the released volume per unit width. The resulting surge height is (see the full text in pdf)  The longitudinal shape of the outer solution is (see the full text in pdf) which strongly depends on the resistance law. The inner solution H(x; hs(t)), to be applied near the front, is obtained by introducing into equation (15) the appropriate basal stress, velocity, and depth relations (see the full text in pdf) is measured downstream from the instantaneous mean front position and (see the full text in pdf) is the x coordinate of the surge tip. In both cases of laminar and turbulent flow the composite solution hc is obtained by adding the inner H(x; hs(t)) and outer h(x, t) solutions and by subtracting their common matching term hs, i.e., by subtracting the cross-hatched area in Figure 15 from the outer solution extended as a constant downstream of the front: (see the full text in pdf)  All variables in the equations above are dimensional. Equation (22) shows that the front depth (as well as velocity, discharge, and momentum) rapidly decreases in the downstream direction; such decrease is more rapid in the case of a laminar flow, where the kinematic celerity is 3 times the average velocity, than in a turbulent flow where the celerity is only around 3/2 the average velocity.  In the case of mud the presence of yield stress lets some mass be deposited behind the surge, and the surge progressively loses volume as it moves downstream. Equations conceptually similar to the ones reported above are given by Huang and Garcı´a [1997, 1998], where the flowing material is represented as a Bingham and a Herschel-Bulkley fluid, respectively. The presence of the yield stress gives much more complicated expressions, some of which are in a nonclosed form. The shape of the tip for a Bingham fluid is described by an ordinary differential equation [Huang and Garcı´a, 1997], and it is shown in Figure 16 taken from Hungr . Symbols given by Hungr  do not correspond to definitions given by Hunt  and by Huang and Garcı´a ; in particular, the longitudinal coordinate x is upstream directed with origin at the tip toe, whereas x is downstream directed with the origin at the mean front position (dw _ x). The surge is in all cases assumed to be followed by uniform flow conditions. The rheological behavior appears to have only moderate effect on front shape. The extreme cases in Figure 16 are represented by equation (22) (left) for the laminar flow and by the following for a pure plastic material (t = constant): (see the full text in pdf)  In the case of a Coulomb-type frictional flow (t = rghi) the front is wedge-shaped, and there is no limit depth behind the surge, as observed by Hungr . The wedge rises from the bed at an angle approximately equal to the difference between friction and bed slope angles. By comparing these shapes with a picture of a real surge front (Figure 17) the prototype front appears steep and similar to the Newtonian profile in Figure 16.  Iverson [1997a, 1997b] and Hungr  analyzed the front profile for the case of a heterogeneous progressive surge. The shape of the coarser debris surges is controlled by their heterogeneity, which causes a higher crest than in a homogeneous flowing mass (Figure 18). Because of the material heterogeneity, uniform flow conditions cannot be reached, and the relation between bed shear stress and flow velocity and depth depends also on material composition, i.e., on position behind the surge front. It is assumed that a surge of viscous (Newtonian) slurry is upstream supplied at a rate sufficient to maintain a normal depth h0. The surge front contains a boulder accumulation that extends from the tip to a distance Lb behind it, measured along the slope. The concentration of large clasts, and thus the frictional component of flow resistance, decreases from a maximum at the tip to zero at the end of the frontal boulder accumulation. Such a variation in effective stress was measured in flume experiments and used in the unsteady flow model by Iverson [1997a, 1997b]: (see the full text in pdf) where tf is the frictional component, tn is the viscous component, m is the average mixture viscosity, ru is the pore pressure ratio, and the dynamic friction angle 8 is assumed equal to 32_. The values of ru range from 0 at the surge front to 1 at the end of the frontal accumulation. Figure 18 shows the resultant profile: The presence of a higher resistance to flow at the front is the cause of the increase in the surge height and the formation of a backwater profile upstream of the boulder accumulation zone. The free surface slope at the front equals the friction angle of boulders; therefore the intensification of peak surge height is greater the longer the accumulation zone is and the milder the channel slope is. In conclusion, the kinematic wave description of the whole surge body approximates reality only if and until roll waves do not develop, which is the subject of section 5.2.
5.2. Wave Development Process
 In some cases, debris waves do not show up in nature even if Frc is exceeded. Among basins in which waves are well documented from measurements and observations, the Moscardo torrent, already described in section 2.7, is considered an example. Some of the hydrographs measured along the channel (Figure 4) are reported at the left-hand side in Figure 19. Effects of the channel length can hardly be appreciated because the two hydrographs are measured in correspondence with two stations located at a distance of 300 m. In all cases a first surge is present, and secondary waves develop during recession and become more evident with increasing effective duration of the event (from top to bottom of Figure 19), independently of peak discharge.
EXTRAIT 21 :
 The shape and the intensity of surges differentiate during the events (Figures 2c and 19). The first surge (mainly composed by boulders and coarser fractions) has an almost symmetrical shape and lasts for a relevant time; depending on the event duration, subsequent surges can show up as small waves in the descending phase of this first surge or as distinct (muddy) secondary waves with asymmetrical shape, similar to water roll waves in the overtaking phase. The overtaking process of debris waves was first explained by Sawada and Suwa , who noticed that the interval between surges is smallest at the beginning stage and increases following flow down the course.  A detailed numerical analysis of a multiple-surge event was performed in Acquabona for the event of 17 August 1998 (section 2.5), with a 1-D model [Zanuttigh and Lamberti, 2004a, 2004b] based on shallow water equations and on the second-order accurate weighted average flux scheme [Toro, 1997]. The code was successfully validated against tests on mudflow dam breaks by Laigle and Coussot  and accurately reproduced water roll wave development by comparison with Brock’s  results [Zanuttigh and Lamberti, 2002b]. The HB, B, and GVP rheological laws are implemented in the code and have been calibrated to optimize the representation of wave statistics in the field. The best approximation for the selected event was obtained by the GVP model with m1/tc = 0.3 s3/kg2, tc/rm = 1.5 m2 /s2, h = 1.0, and f = 1.5_. The simulated wave amplitude along the channel in time is shown in Figure 20. Since the beginning of the event, waves form at the rising part of the front starting from the very upper part of the channel; the development of high waves at the flood tail is evident around 500 m from the channel inlet. At the beginning and at the end of the event, in fact, when the condition of incipient flow occurs, Frc is almost zero (i.e., instabilities always develop). Around 500 m downstream of the inlet, wave grouping clearly appears from the flow peaks and from the modulation in wave amplitude with secondary waves that overtake one another in secondary fronts.  On the basis of the observations in the Moscardo torrent and on the numerical results along Acquabona channel the interpretation sketched at the right-hand side in Figure 19 has been provided. Debris flows tend almost always to develop in theoretically unstable conditions, but debris waves are not always observed. In fact, the flood surge moves with the velocity uf of the masses of which it is composed (black line in the right plot in Figure 19), whereas the unstable perturbations (dash-dotted shaded lines in the right plot in Figure 19) are associated with celerities dx/dt = (u + c). Just after the peak of the event, perturbations run faster than the front and thus coalesce in the first main surge (solid shaded lines in the right plot in Figure 19). When flow depth is decreased so that celerity of the perturbations equals uf,, instabilities can run over the tail of the first surge without ever reaching the front and therefore form recognizable waves, with intensity and celerity similar to the first wave; see Figures 2c and 19. This interpretation is confirmed by observations in the Illgraben torrent, where the coupling of video recordings with radar measurements allowed the production of a detailed picture of the event of 28 June 2000 [Zanuttigh and McArdell, 2004]. The flow consisted of a single coarse granular surge moving over an initially dry bed carrying boulders of more than 2 m in diameter. The tail of the debris flow was muddy with an onset of 17 low-amplitude roll waves. Figure 21 shows flow depth in time at the radar (downstream in the channel) together with particle velocity nondimensionalized by front velocity. Wood logs, boulders, and roll waves were identified from video recordings. It is interesting to note that after the first 75 s all particles move downstream slower than the front, with the exception of one wood log, which appeared to be rolled on the flow surface. Roll waves are visible only around 500 s after the front passage and have more or less the same velocity as the front. A more in-depth analysis of the conditions for debris wave formation and development is presented in the following based on the laboratory and field data reported in Table 3. Experimental values are derived from Brock [1967, 1969] in the case of a rough channel inlet, fromJulien and Hartley , from Schonfeld , and from Forterre and Pouliquen , whereas field values are calculated from measured hydrographs during events already described in Table 1. Table 3 includes, for each case, the channel length L and the average slope i from the inlet to the measurement station, the depth h0 and velocity u0 of the base unperturbed flow, the average depth at wave crest Hm, and the average surge period Tm. The values taken from Forterre and Pouliquen  for sand particles and from Julien and Hartley  and Brock  for water correspond to the critical stability condition; it is worthwhile to recall that the data of Julien and Hartley  were obtained for laminar flows. Laboratory tests with glass beads from Schonfeld  and prototype conditions (with the exception of the July 1998 events in Acquabona) are characterized by appreciable instabilities. To allow the reader an easier comprehension of Figures 22–24, the symbols have been chosen based on the following two criteria: a fixed shape for each different channel (e.g., a circle for all the data collected in the Moscardo torrent) and solid or open symbols for events characterized by clear multiple-peak and singlepeak events, respectively, with eventual development of low-amplitude roll waves.  Figure 22 presents all the data in Table 3 in the plane given by the Froude number of the base unperturbed flow Fr0 and by the channel length L nondimensionalized with i and u0, a scale which is suggested in section 4.1 and is commonly used in representing water roll waves [Montuori, 1961; Julien and Hartley, 1986]. Only the data of two debris events (the July 1998 events in Acquabona, Figures 2a and 2b) correspond to stable conditions.
EXTRAIT 22 :
With increasing Fr0 the length necessary for wave formation decreases; for nonlaminar flows a rough estimate of the relation between the base flow and the scaled channel length is given by Fr0 = 2.8[log(Lgi/u0 2 )]_1.5. Figures 23 and 24 analyze wave development along the channel, in terms of depth Hm and period Tm, respectively, nondimensionalized following Brock’s [1967, 1969] usual scale for water roll waves. With respect to Table 3, additional data for water waves in the case of rough channel inlet [Brock, 1967] are included. Figures 23 and 24 show that the channel length L necessary for the development of appreciable waves strongly depends on rheology and on turbulence: In fact, for water flows, L is much lower in laminar (ffi50h0/i) than in turbulent (ffi100h0/i) conditions, and L for glass beads is much greater (ffi1000h0/i) than for water. Even if there is some scatter, the growth rate of Hm for glass beads [Schonfeld, 1996] and water [Brock, 1967] is similar; data for mixtures of water and sand fall more or less between the two data clouds of water and dry particles, just above the limit for water turbulent flows. Field data collected in the Moscardo torrent and measurements at the USGS flume (all performed on a smooth channel) show both a linear tendency characterized by a much steeper growth rate, in particular the USGS data, than glass beads and water. The higher steepness can be explained by considering that prototype and USGS data correspond to heterogeneous flows: Surge dynamics is significantly affected by segregation that induces a great enhancement in flow depth at crests Hm because of the presence of boulders at the surge fronts.  Only data for water waves clearly show wave development for overtaking with the tendency of the curve Hm/h0 to be concave downward. The channel at McGill University was probably too short to allow the development of this final phase of wave coalescence (associated periods Tm in Figure 24 appear almost constant), whereas debris flows in Moscardo and in the USGS flume seem to grow indefinitely. Figure 24 shows the variation of average wave period Tm along the channel. Field values of Tm have been calculated based on clearly recognizable surges, and thus they are calculated only for multiple-peak events. The instability phase, characterized by an almost constant period, is present only in the data of water waves. Data obtained at McGill University and in the USGS flume fall on the same line, which is parallel to the growth rate for water when overtaking is dominant. Field data are insufficient to say something conclusive, but they have a much steeper linear growth rate, which is sign of a very efficient mechanism of wave coalescence.  In conclusion, the global picture of wave development in homogeneous and heterogeneous flows can be obtained by combining Figures 22, 23, and 24. The traveling distance necessary for waves to show up is related to flow rheology (Figure 22), and wave development rate depends both on flow rheology and on the position along the channel (Figures 23 and 24); for fixed rheology and channel length the stable or unstable flow conditions and the type of instabilities depend on Fr0 (Figure 22). Wave development in debris flows may have similar characters to water turbulent flows in the instability phase. This behavior can be associated with the similarity between the turbulent water flow rheology, represented by the mixing length model (see the full text in pdf); and the grain inertia regime for debris flows [Bagnold, 1955; Takahashi, 1991] (see the full text in pdf); where l ffi min(0.4y, 0.10h) is the mixing length, y being the distance from the bottom, ail2 = O(1), and ds is grain diameter. The distributions of l and ds along the vertical are similar since ds increases along the vertical because of segregation and is limited by flow depth h. The differences in wave development for water and debris flow seem to start when longitudinal segregation causes a significant intensification of the overtaking process (Li/h0 ffi 600).
5.3. Wave Profile
 On the basis of the work by Dressler  for periodic wave trains in water streams and on the minimum roll wave theory already presented in section 4.2.2, Ng and Mei  and Liu and Mei  derived the shape of periodic surges for a power law and a Bingham fluid, respectively. In both cases the stream is supposed to be shallow; therefore surge celerity satisfies equation (26). The minimum roll wave theory seems to provide a reasonable description of roll waves in the instability phase; then because of overtaking, wave period and length increase. In fact, waves in prototype (conditions well above threshold values in Figure 22) resemble much longer than minimum length waves evaluated by Ng and Mei ; see Figures 25 and 26. Figures 25 and 26 represent surge depth, velocity, and bed stress profile for a constant power law exponent (0.4) of the rheological law, with varying channel slope and Froude number. Figure 25 reproduces the profiles in the case of wavelength corresponding to minimum roll wave condition, whereas Figure 26 refers to a long roll wave. The power law exponent equal to 0.4 represents the case of basal stress well above the yield value; attention should be focused on the solid line, which corresponds to a typical prototype slope condition. All variables are scaled with the corresponding uniform flow value. It is worth mentioning that Longo  found that this theory cannot be applied to a homogeneous dilatant fluid.  Several authors attempted to numerically describe the propagation of irregular waves based on shallow water equations (sharp fronts): Among others, Liu and Mei  used a constant parameters Bingham rheological model; Zanuttigh and Lamberti [2004a, 2004b] and Zanuttigh and McArdell  calibrated different rheological laws to reproduce wave statistics derived from field data. Iverson [1997a, 1997b], Denlinger and Iverson , and Iverson and Denlinger  represented debris flows as deforming masses of granular solids variably liquefied by a viscous pore fluid. The model simulates the flow through de Saint Venant equations and considers the basal pore pressure as an additional state variable, which is governed by a diffusion equation and affects the apparent rheology of the granular mass. The effect of flow heterogeneity on surge shape, discussed more in depth in section 6.4, is represented in the simulations by Hungr  and Savage and Iverson .
EXTRAIT 23 :
5.4. Mass and Material Changes During Flow
 Because of bed and banks erosion, mobilization of deposited material, or lateral/rear deposition in the channel, the flowing mass along the channel varies. Deposits do not have the same average composition of the flowing mass and therefore cause modification of its properties. In any case, the huge shearing stress and strain within a mass flowing in a channel of variable section and the energy dissipation due to the drop, often exceeding 1000 m, cause breakage of the greatest aggregates, i.e., reduction of the characteristic boulder size, and a significant production of fine materials by abrasion. This increase of fine fractions, often combined with some flowing water, causes a substantial transformation from granular behavior in the upper slopes to muddy behavior in the lowest slopes. For instance, at Ontake-san, Japan, a stony debris flow occurred in 1984 and was characterized by a volume of 34 Mm3, a drop of 1550 m, a run out of 11.6 km, a decrease of the kinematic viscosity from 500 to 5 m2/s, and a decrease of the apparent friction in a sliding block model from 0.20–0.25 to 0.05–0.06 [Voight and Sousa, 1994]. The major crests of an event may flow over the banks and originate secondary flows, which are usually quite dry and soon stop, producing a deposit of the bouldery crests as lateral levees. The deposition of boulders on lateral levees (Figure 27) deprives the flowing mass of the contrasting fraction and thus allows the muddy fraction to proceed further. Therefore this selective deposition process on banks is also one cause of the frequently observed downstream thinning of the flow.
6. SURGE TEXTURE
 During debris flow propagation the coarser material tends to accumulate at the surge front, enhancing its depth, and the finer materials accumulate at its tail [Johnson, 1984]. This segregation mechanism is very significant for the applications, since it produces an increase of the debris erosive power, of its run out distance, and of its impact force.  The aim of this section is to synthesize what is known so far about the segregation process based on physical models, the characteristics of this process in the field, and its consequences and, finally, to examine how it is possible to reproduce segregation effects while modeling debris events. The structure of the section is as follows. Section 6.1 analyses the segregation mechanism trough a series of experimental works that are recalled in a chronological sequence. Section 6.2 shows in particular how to predict the distance from the channel inlet at which segregation is completed. Section 6.3 presents prototype evidence and the most relevant effects of segregation for technical purposes. Finally, section 6.4 briefly points out how numerical models simulate debris flow events by also accounting for segregation of the flowing material.
6.1. Experimental Analysis of the Segregation Mechanism
 The first interpretation of segregation dynamics was given by Bagnold . In a grain mixture flowing in the grain inertia regime (i.e., grain impacts dominate stresses), since stresses under the same strain and concentration are proportional to the squared grain size, the coarser fractions dilate more and concentrate at the surface, whereas the finer fractions concentrate near the bottom. This argument is apparently based on the additive character of partial pressure among the different size fractions, which is questionable.  Bridgwater  identified in the kinetic sieving the dominant mechanism driving particle size segregation in shallow granular surface flows. As the grains avalanche downslope, there are fluctuations in the void spaces among the particles; when a void opens up under a layer of grains, the small particles are more likely to fit into the available space. The fines therefore percolate to the flow bottom, and mass conservation dictates that there is the corresponding reverse flow of large particles toward the free surface.  Several studies on the segregation of dry granular mixtures refer to the kinetic theory for multicomponent mixtures given by De Haro et al. . In particular, a series of analyses and experiments considering binary mixtures of different-sized spheres can be found in the literature, on the basis of which some models were developed that associate the segregation, limited just to percolation, to diffusive phenomena fundamentally due to three independent causes: the number fraction gradient, the granular temperature gradient, and the pressure gradient [see, e.g., Brown, 1978; Okuda et al., 1980].  Takahashi  clearly described the connection between vertical and longitudinal segregation. As soon as a debris flow originated, larger particles move upward while the smaller ones move downward; therefore if a debris flow stops, an inverse grading in the deposited layer is evident. If the flow continues, because of the higher velocity of the upper layers, larger particles move ahead accumulating at the front. In fact, when a particle reaches the front, it tumbles down to the bottom and is buried in the flow, but if it is larger than the surrounding particles, it appears soon again on the top of the flow and moves ahead. Takahashi thus explains the accumulation of boulders at the debris flow front as a consequence of the repetition of such mechanism along the travel distance.  Suwa  claims that the big boulders observed at the front often have a size nearly equal to the flow depth, so they cannot be lifted only by dispersive forces, and the mechanism of their accumulation toward the front is due rather to their higher velocity with respect to the smaller boulders and the surrounding fluid. An empiric statistical approach was proposed by Savage and Lun , who experimentally analyzed a binary mixture of PVC spheres flowing down an inclined chute. They defined a probabilistic distribution function of void spaces and a characteristic diameter, whose value represents the possibility of the single particle being captured by the lower layers, and statistically solved the problem by means of the maximum entropy approach [Jaynes, 1963; Brown, 1978].  Takahashi  carried out experiments whose purpose was to quantify the segregation process both along and across the flow of a complex water-grain mixture.
EXTRAIT 24 :
The experimental setup consisted of an inclined chute with erodible bed and variable length and a collecting bin divided by sectors where the mixture freely fell at the chute outlet. The explanations of the experimental setup and of the results are, however, insufficient to allow the repetition of these experiments and a complete analysis of the data. The deducible qualitative results are that (1) the solid concentration and the content of the coarser fraction at the front increase with increasing space left to the debris for flowing and (2) even for the longest channel length (equal to 4 m) the front presents a component of fine-medium particles.  A qualitative contribution on the segregation process of a grain-water mixture was given by Davies  in his experiments aimed at analyzing in detail the debris surge formation and composition (see section 3.1.2). The increase of granular concentration in a water flow over a nonerodible bottom, moving at a constant speed, induces a gradual change from a dispersed to a heterogeneous granular distribution inside the fluid matrix. Local accumulations of grains progressively develop into stationary granular waves, which are formed by a dry front, a central uniformly deep body, and a fluid tail.  Vallance and Savage  performed experiments with dry particles of different sizes and densities and different particle-liquid mixtures flowing down a chute. The setup adopted is essentially the same as that of Takahashi . In uniform, steady experiments with binary mixtures of small and large particles the small ones fall downward, and the large migrate upward. In slow, dry, frictional flows the downstream segregation is so efficient that zones composed of 100% small and large particles, separated by a concentration jump, occur. In rapid, dry, collisional flows, segregation is less efficient because of diffusive mixing that smoothes the vertical concentration profiles so that concentration jumps blur or disappear. The presence of a viscous fluid shows an inhibitive action with respect to size segregation; when the interstitial fluid has the same density as the particles, little or no segregation occurs. In steady, uniform flows, segregation of particles by density differs fundamentally from particle segregation by size. In dry granular flows, dense particles efficiently move downward in upper parts of the flow but do not efficiently display light particles in the lower part of the flow. Concentration jumps, as observed with particles of different size, do not occur because neither dense nor light particles percolate preferentially; instead, imbalances in contact forces, which favor the dense ones to move downward when dense and light particles encounter each other, are the sole cause of segregation. A conclusion similar to that of Vallance and Savage  regarding the effect of the interstitial fluid on segregation process was achieved by Samadani and Kudrolli . The images of the pile that results after bidisperse color-coded particles are poured into a silo show that segregation is sharply reduced and preferential clumping of small particles is observed when a small volume fraction of fluid is added to dry mixtures.  Recently, Zanuttigh and Di Paolo  performed experiments on continuously dispersed dry granular mixtures using a setup very similar to the one adopted by Takahashi  and Vallance and Savage . Complex dry granular mixtures segregate rapidly and efficiently after having flowed about 40 times the normal depth. No effect on segregation velocity seems due to the grain size composition since the segregation appears more marked not more rapid in the presence of a coarser matrix. The elaboration of results allowed the reconstruction of longitudinal grain size composition profiles, which show boulders localized at the front and on the debris surface, whereas medium fraction prevails inside the surge body and fine fraction prevails at its tail.
6.2. Evaluation of the Segregation Speed
 The experimental works recalled so far deal with qualitative rather than quantitative aspects of the segregation process. Few works can be found in the literature that analytically derive the distance at which binary mixtures are completely segregated, and they are here recalled for the sake of completeness. In prototype the segregation of debris flows usually happens, based on videos and eyewitnesses, in an negligible time and space with respect to event duration and channel length. By combining the kinetic sieving and the ‘‘squeeze’’ expulsion mechanisms, Savage and Lun  obtained the net percolation velocity of each particle size. The mass conservation equation for fines was then solved by the method of characteristics to derive the development of concentration profiles with downstream distance. By assuming a linear velocity distribution along the vertical, Savage and Lun derived the expressions for the distances, along the channel xsr and from the bed ysr, where the complete separation of the small and large particles occurs. Vallance and Savage  extended this solution to a more general power law velocity distribution. Timespace variation of xsr and ysr can be obtained by using the model developed by Gray and Thornton  for dry granular flows and extended by Gray and Hogg  with the inclusion of an interstitial fluid; details of the model are recalled in section 6.4. The model was validated against the concentration jumps observed by Savage and Lun  and Vallance and Savage . Gray and Hogg  highlighted the dependence of xsr on two parameters: (see the full text in pdf) where Ls is the typical surge length, ^r is the relative density difference, B is a nondimensional factor that determines the magnitude of the pressure perturbations from the hydrostatic that drives the flow, D is an interparticle drag coefficient, and zsr is the height of the sharp inflow interface at x = 0.  The distance at which segregation is completed is thus directly related to the density of the interstitial fluid. As the density of the interstitial fluid increases, ^r and thus xsr increase; that is, complete segregation occurs farther downstream. If the density of the fluid and the grains match, the segregation by kinetic sieving is inhibited as in the experiments by Vallance and Savage . If ^r changes sign, so that the particles are buoyant, the direction of segregation is reversed, and normally graded layers will be formed sufficiently downstream.
6.3. Prototype Evidence and Major Consequences of the Segregation Process
EXTRAIT 25 :
 The segregation process can be appreciated in several records of prototype events, and it significantly affects surge shape and dynamics, as has been discussed in section 5.2. Figure 28 shows the same surge moving down channel at Wrightwood, California, on 20 May 1969 at the same instant in correspondence with two different sections: Boulders appear mainly concentrated at the front, while the tail is composed of fine-medium fractions and behaves as a muddy flow. Variation of grain size, percentage of gravel, discharge, flow depth, and surface velocity are available for the event of 5 September 1983 at Mount Yakedake (see section 2.4). The front is composed almost only of gravel for very few seconds (more or less 5 s); after 10 s the gravel percentage dropped to 50%. The change in flow composition from coarse to fine fractions during the event, with boulders that do not disappear from flow surface immediately after the front passage, is evident, for instance, in the surface images of the events of 5 September 1983 at Mount Yakedake [Suwa and Okuda, 1985] and of 1 January 1994 at Acquabona [Lamberti, 1999] and in some close views of the 19 January 2002 debris flow front in the Curah Lengking River [Lavigne et al., 2003]. In multiple-peak events the first surge is typically the coarsest one, followed by a medium-finer tail; then, secondary waves may reproduce the first surge, with boulders concentrated at the front and on the surface and a progressively finer matrix, or they may consist of muddy unstable waves.  Figure 29 shows an idealized representation of debris waves and deposits by Johnson . Figure 30 presents some video pictures of the debris flow of 8 July 1996 in Rio Moscardo [Marchi et al. 2002]. The selected event was preceded by a precursory surge, which consisted of a sediment-water mixture, characterized by strong turbulence. This first surge carried many logs and trees in its frontal portion (Figure 30a); coarse woody debris was supplied by shallow landslides affecting forested slopes along the torrent. About 1 min after this initial surge the debris flow peak arrived with a number of large boulders (up to 2–3 m in diameter) in an abundant muddy matrix (Figure 30b). Neither rigid plug nor laminar behavior can be identified at the debris flow front; behind the front the presence of boulders became sporadic with a progressive decrease in the concentration of solid particles. A few minutes after the passage of the front peak, further secondary waves of smaller size were observed (Figure 30c).  The reconstructed wave profile and composition in time for the event that occurred in Acquabona on 17 August 1998 are shown in Figure 31 [Genevois et al., 2001]. Surface velocities from geophone logs and from image analysis confirm that during this event velocities vary in a wide range (2.3 _ 8.4 m/s, see section 2.5), depending essentially on the solid concentration of the flow. Such variations are observable both between successive surges and, in a single surge, between front and tail. This behavior can be explained by dynamic processes acting at the front of the flow that produce the typical segregation of coarsest clasts and by consistent variations of water content between different surges. The inferred decreasing trend in the solid concentration of the flow from the beginning to the end of the event leads to the hypothesis of the presence of limits in the availability of effectively mobilizable loose material along the channel and in the initiation area. This could explain very fluid surges that cannot reach the solid concentration at equilibrium because of the lack of material to entrain. Turbulence in the flow was observed only rarely between successive surges and was observed more often toward the end of the event, when the flow depth and the solid concentration significantly dropped.  Debris flow often appears also as a single or multiple muddy waves (Figure 32) in which segregation is not as evident as noted above: Boulders are partially concentrated at the front and partially well mixed inside a matrix of finer particles and sporadically show up on flow surface; refer, for instance, to the events of 28 June 2000 and 2001 at Illgraben (Figure 33), the event of 20 May 1995 at Dorfbach, Switzerland [Rickenmann, 2001], the event of 27 August 1991 at the Belvedere Glacial, Italian Alps, and the event of 13 August 1995 at Marderello, Italy [Arattano and Trebbo, 2000].  The segregation process is relevant in technical applications, as is briefly pointed out in the following. Segregation and debris erosive power are strictly related, as was observed by, among others, Li and Luo , Okuda et al. , Pierson , and Davies et al.  and as was experimentally investigated by Rickenmann et al. . The erosive power may be relevant during the mobilization and the development phase, when the front becomes deeper and picks up boulders, and it is very reduced in the following propagation phase, when the flow is completely segregated and solid interaction forces dominate at the front and fluid resistance forces prevail at the tail [Iverson, 1997a, 1997b].  The local particle size distribution can also have a subtle feedback onto the bulk flow. When the large particles are rougher than the small ones and there is a strong shear through the avalanche depth, the large particles tend to concentrate at the front and resist the motion. This leads to an instability in which the large particles are pushed to the sides to form stationary lateral levees that channel the flow and lead to significantly longer debris flow run-out distances [Iverson and Vallance, 2001].  Finally, the impact force of a mudflow can be increased by a factor of 6 or so if boulders are present at the surge front [Watanabe and Ikeya, 1981]. The estimation of impact forces is the subject of section 7.
6.4. Segregation in Numerical Modeling of Debris Flows
 So far, a numerical model able to account for the segregation process during debris surge propagation without simplified assumptions and/or parameters to be calibrated on the basis of experimental/field data has not been developed.  Iverson and Denlinger  and Savage and Iverson  derived a one-dimensional model of pore pressure evolution in debris flows that reproduces the influence of segregation on surge propagation in terms of surge shape and velocity.
EXTRAIT 26 :
The model is based on the observation that because of their different permeability and compressibility, coarse rubble snouts are characterized by great frictional resistance and fluid pressure dissipation, whereas finer-debris slurries behind the snouts sustain high pore fluid pressure. In the work of Iverson and Denlinger  the pore pressure evolution obeys a homogeneous advective diffusion equation, unforced by changes in flow geometry. Savage and Iverson  extend this approach to forced pore pressure diffusion by merging well-established theories for sediment consolidation and granular avalanche motion. In this model it is necessary to introduce the values of a characteristic length and height, which have been chosen on the basis of the results from experiments carried out at the USGS debris flume.  Gray and Thornton  and Gray and Hogg  used the mixture theory to derive a model for particle size segregation by kinetic sieving in shallow gravity-driven granular-free surface flows. In the first work a two-phase theory, dealing with dry binary flows composed of small and large particles, was developed; the latter work extends this theory to a material composed of large particles, small particles, and a passive interstitial fluid that allows buoyancy effects to be incorporated into the final segregation equation. By assuming a linear velocity drag among the particles and that the fines carry less of the overburden pressure as they fall down through the gaps, Gray and Thornton  derived an expression for the segregation flux in terms of the volume fraction of small particles. This representation of particle segregation velocity was included in a total-variation-diminishing Lax-Wendroff scheme that Gray and Hogg  suggested would be possible to incorporate in a numerical model like the one by Iverson and Denlinger .
7. FORCES ON OBSTACLES
 Avalanches and debris flows exert enormous impact forces on obstacles in their path, such as bridge piers, defense walls, and buildings. Field measurements performed at Mount Yakedake [Suwa and Okuda, 1983] show that impact forces consist of two distinct parts: fluid dynamic pressure up to 10 kN/m2 and collisional forces of single boulders up to 102–104 kN/m2. From the more than 70 impact force graphs collected by Zhang  since 1982, fluid dynamic pressures as high as 5 _ 103 kN/ m2 and impulse forces of individual boulders as great as 3 _ 103 kN were found. Large-scale model tests of flexible barriers by DeNatale et al. [1997, 1999] showed that forces in the anchor cables exceeded the load cell measure range of 45 kN (square-mesh wire-rope netting with 30 cm, 20 cm, and 15 cm openings as well as 30-cm-diameter interlocking steel rings). Measurements also indicated that the peak loads in the tie back anchors were highly transient and occurred at the time of maximum momentum against the net.  As field monitoring and measurements in mountain areas are generally quite difficult and expensive because of the particular environment and the nature of debris events, experiments were often adopted to improve the knowledge of debris forces and thus the design of defense works. Until the present the modelization has been focused on mixture impacts against dams with height equal or higher than flow depth, which is, according to Hungr et al. , the most common case in the field: Forces on walls were analyzed for mixtures of water and polyvinylchloride or anionic-resin mixtures by Armanini and Scotton , for grain-water steady surges by Scotton and Deganutti , for sandy mudflows by Daido , for broad size range of dry granular mixtures by Zanuttigh and Lamberti , and for similar saturated mixtures by Ghilardi et al. .  Several empirical formulae can be found in the literature to express the design force per unit length F, and they can be divided into two main types. The first type uses as reference force the static pressure term adequately amplified by a multiplying coefficient k [see, e.g., Lichtenhan,1973; Armanini and Scotton, 1993]: F ¼ krmg h2 2 : ð28Þ  The second type of formulae enlightens the momentum of the incoming flow [see Daido, 1993]: (see the full text in pdf)  The values of h or u in (28) and (29), depending on the authors, may correspond to normal conditions or to the maximum surge depth/velocity.  Armanini and Scotton  first observed two types of impact, corresponding to different global viscosity of the saturated mixture. At the impact, if the global flow viscosity is low, the current forms vertical jets, whereas if the global viscosity of the flow is dominant, the flow forms a reflected wave that propagates upstream.  Inside a saturated mass, neutral overpressures are present and persist during the impact process [Iverson 1997a]; vertical accelerations at the impact prevail when the interstitial fluid is incompressible. In fact, the formation of vertical jets at walls occurs also in the case of clear water [Partenscky, 1988; Ramsden 1996], whereas it does not show up in the presence of a dry granular flow that reflects more or less as a single back wave [Zanuttigh and Lamberti, 2006]. Both in water and in saturated mixtures down a steep chute [Ghilardi et al., 2006], pressure overshoots lasting less than 0.1 s were recorded at the impact on vertical walls, with peak values up to 1 order of magnitude higher than the force corresponding to the hydrostatic front pressure (Figure 34).  The relevance of vertical accelerations in the impact process led Armanini and Scotton  to sketch two different schemes, which are reported in Figure 35. They computed the dynamic impact pressure by applying the global momentum balance to the control volume sketched in Figure 35 with reference to a fixed coordinate system 1 2 rmgy2 1 þ rmy1w2 ¼ 1 2 rmgy2 1 þ y1Dp þ @ @t Z 8c rmwd8; ð30Þ where, to be consistent with notations in Figure 35, here w is wave celerity, y1 is flow depth in front of the wall, and Dp is the dynamic impact pressure. The flow is produced by the instantaneous abatement of a retention dam on a dry channel, so that w = 2 (see the full text in pdf), with h0 being the undisturbed depth of the flow retained upstream of the dam.
EXTRAIT 27 :
In case of a reflected waveforms (Figure 35b), equation (30) yields (see the full text in pdf) whereas in the presence of vertical accelerations (Figure 35c), equation (30) gives Dp rmg ¼ kh0; ð32Þ with k = 4 for nonviscous fluid over a horizontal bottom.  Recent experiments both in dry and saturated mixtures [Ghilardi et al., 2006] suggest the use of formulae developed for reproducing the overshoot of a tsunami wave against a vertical wall [Cross, 1967]: (see the full text in pdf) where the coefficients CF and Cp do not require any calibration since they depend only on the angle formed between the free surface of the surge front and the channel bottom.  All the semiempirical existing equations for estimating the design forces exerted by debris flows on structures do not represent either the impact of single boulders or fatigue effects due to the repeated pulsation in time of the debris against the structure.  In the laboratory the grains-obstacles system is elastic, whereas in nature the system of debris flow and the defense/retain work is plastic, and deformation and breakage occur. See in Figure 36, for instance, the damage to the former bridge and the wall due to the action of single exceptionally massive rock unit and in Figure 37 the abrasion of the large check dam in the Illgraben torrent due to the repeated impact of rocks.  Some remarks on boulder effects on the impact process are available only in the work by Scotton and Deganutti , where they observed that some of the largest particles in the mixture left the debris surge because of agitation and showed an impact pressure (on the pressure gauge) up to twice the average surge impact pressure. The formation of multiple surges may induce, because of the increase in flow depth, an increase in impact forces of an order of magnitude with respect to mean values; effects can be seen, for instance, in the video recordings of the event of 10 October 2000 at Fully (Switzerland), in which a light pole at the channel fan collapses just at the passage of one of the debris surges.
 Two types of perturbations are frequently observed in debris flows: (1) progressive instabilities, with greatest velocity at wave crests, and (2) regressive instabilities, characterized by the greatest velocities at the front toe. Both progressive instabilities and unstable surge toe of regressive instabilities may rapidly reach the shape of roll waves, which are particularly relevant in practice since they are characterized by increasing depth and period while they travel downstream and can thus significantly build up the destructive power of the debris flow.  The formation of clearly appreciable waves requires two conditions: the instability of the base flow and a sufficient channel length. When these conditions are not satisfied, the flow is stable, and the matched asymptotic expansion method used by Hunt  and Huang and Garcı´a [1997, 1998] provides a good analytical description of the propagation and attenuation of a single homogeneous surge in a prismatic channel. The instability of the regular flow, according to a 1-D description, occurs when the kinematic wave speed exceeds the inertia wave speed: The flood wave runs over the small dynamic waves making discontinuities coalesce in big surges with steep fronts. This condition results in the Froude number of the base flow Fr0 being greater than a critical value Frc. The 1-D linear stability analysis provides a reasonable value of Frc, which mainly depends on the bed stress to yield stress ratio: It is just above zero if the bed stress is just above the yield value, whereas when the bed stress is well above the yield value, it depends on the exponent of the rheological law, varying from 1/4 for a Hershel-Bulkley fluid up to 2/3 for debris in the grain inertia regime.  The process of wave development in homogenous flows is composed of an initial instability phase where perturbations with almost constant period and increasing depth show up; then the overtaking process lets wavelengths and heights grow farther downstream. Wavelength or period in the initial unstable phase can be obtained from the minimal wave hypothesis by Ng and Mei , verified only for Newtonian flows and not applicable to dilatant fluids [Longo, 2003], or from the empirical law by Forterre and Pouliquen , developed for granular flows. An analytical description of wave profile is available only for periodic roll waves of a homogeneous fluid [Ng and Mei, 1994; Liu and Mei, 1994]. The channel length required for unstable perturbation to grow up to an observable level strongly depends on flow rheology and scales proportionally to uniform flow depth over channel slope [Montuori, 1961, 1963]. The proportionality constant is about 50 for water laminar flows [Julien and Hartley, 1986], 100 for water turbulent flows [Brock, 1967], and 1000 for glass beads [Schonfeld, 1996]. Available data for debris flows do not allow the estimation of the value of this constant.  Wave development, adequately scaled, is similar for debris flows and homogeneous turbulent flows. Since measurement stations are usually placed in the downstream channel reach, debris wave characteristics are available in prototype when overtaking has already occurred. The measured values of average crest height to unperturbed flow depth ratio show a growth tendency similar to turbulent flows but a steeper growth rate because of the strong effect of segregation on surge dynamics. Multiple-peak events are generally composed of a first isolated surge, which results from coalescence of all unstable perturbations around the peak, and of secondary waves, which develop during the recession phase and become more evident with increasing duration of the event. The first surge, which mainly consists of boulders, has an almost symmetrical shape and lasts for an appreciable time, whereas secondary waves are generally more muddy and are characterized by an asymmetrical shape, similarly to water roll waves in the overtaking phase.  Several numerical models reproduce with moderate success debris flows like a homogeneous fluid or a twophase mixture, since they do not describe the segregation process. Only the works by Iverson [1997a, 1997b], Iverson and Denlinger , and Savage and Iverson  reproduce the influence of segregation on surge propagation in terms of surge shape and velocity.
EXTRAIT 28 :
So far, few models have been calibrated to represent debris wave statistics through field data [e.g., Fraccarollo and Papa, 2000; Zanuttigh and Lamberti, 2004a, 2004b; Rickenmann et al., 2003; Zanuttigh and McArdell, 2004].  Debris waves are characterized by peak intensity of both flow depth and velocity approximately 3 times higher than the average values, so their effect on the debris flow devastating power must not be neglected. Several formulae are available in the literature for evaluating the design force exerted by a debris flow against a wall, either by considering the peak force due to the impulse of the muddy front wedge or the force due to the complete reflection of a perfect fluid surge. Most of these formulae are empirically based and thus are affected by the necessity of calibration. A promising method that does not require calibration [Ghilardi et al., 2006] consists of the use of formulae specifically developed for tsunami forces against walls [Cross, 1967]. So far, no formulation exists for estimating the effects on structures due to the repeating pulsation of debris surges or to the impact of single boulders.
Landslides in vibrating sand box: What controls types of slope failure and frequency magnitude relations?
Oded Katz and Einat Aharonov
Although landslides are a worldwide significant natural hazard, their physics is not well understood. Here, landslides were induced in a vibrated box filled with wet (practically cohesive) sand, simulating natural slope failure. The questions addressed were (a) what controls the type of slope failure and (b) what controls frequency magnitude relations of landslides.
In the experiments, two end-member slope failure types were obtained: during application of only horizontal acceleration, a failure plane rapidly developed, followed by a box-sized slump. In contrast, under application of only vertical acceleration, mode I fractures formed slowly, dissecting the slope into blocks. The fractures caused a strength heterogeneity and were followed by block-sliding. In the vertical shaking experiments, a power law size distribution of slide-blocks was measured, controlled by the fracture distribution.
The experiments suggest that heterogeneity may be a major control on the size distribution of natural landslide inventories: In a homogeneous environment, the landslide will have a characteristic size of the whole system. In a heterogeneous slope, sizes of landslides will reflect the heterogeneity.
Following the above experimental observations our hypothesis is that natural landslides may be divided into two groups small and large. The processes controlling their formation are different:
I. The smaller natural landslides occur as slumps within the unconsolidated, rather homogeneous, sediments typical of the upper few meters close to the surface. The size of these slumps is determined by the dependence between failure depth (constrained to be the depth of the unconsolidated sediments) and area.
II. In contrast to the homogeneous upper layer, rock mass below the unconsolidated sediment is always heterogeneous due to fractures, layers and bedding. This preexisting heterogeneity is the source of the power law decay observed for the large landslide portion in natural landslide distributions.
Finally, in light of these insights, in hazard evaluation, it is advised to fit a regression line to the power law decay of the large landslide inventory only and consider the characteristic landslide separately.
Keywords: landslides ; frequency magnitude; power law; characteristic size; sand box; self-organized criticality
Landslide is a general term for unstable downwards movement of slope material that occurs when available gravitational forces, increased pore pressures and seismically induced shear stresses overcome the slope strength. Field observations divide landslides into few main generic types (e.g.  and ); relevant to this study is a division based on the mechanics of failure: The terms rock-slides (addressed hereafter as slides) and rock-falls define failures that occur in a layered sequence or in pre-fractured solid rock, respectively. Slumps, on the other hand, develop in continuous, unconsolidated or even earth-like geological material . Landslides are hazardous phenomena and are therefore often analyzed, using an engineering approach, to ensure slope stability and reduce life and property loss (e.g. ). In parallel to the study of landslides as isolated slope failure events, the statistical characteristics of large populations of landslides and avalanches have also become a recent focus of study in the physics and geophysics communities: Natural landslides, similar to other natural hazardous phenomena such as earthquakes and forest fires, often follow power law frequency magnitude relations  and . Naturally occurring landslide populations (mainly consisting of shallow disrupted landslides), both seismically and hydrologically triggered, show non-cumulative power law frequency magnitude (area) relations with a slope of 2.3–3.3 for the large landslides part of the population (Fig. 1), and a flattening-out/roll-over in the small landslides part of the population  and . One possible explanation for the roll-over is technical, where small size landslides on the edge of the mapping resolution are under-mapped and therefore missing from the inventory . In contrast, others  have claimed that the roll-over observed in the distribution does not reflect a non-completeness of the data sets, as it takes place in landslides with areas about an order of magnitude larger than the observational limit. Rock-fall inventories also exhibit a frequency magnitude power law, but for this type of slope failures, the frequency magnitude slope is shallower than that observed for slumps and slides, with a slope of 1, and also no roll-over is observed . The source of the observed power law behavior and the roll-over are still not well understood, though several explanations have been offered (e.g. ), and will be reviewed in the discussion. Understanding the physics behind the different observed size distributions is important as a basic physics questions and also for practical reasons—assessing erosion rates and providing landslide hazard analysis (e.g. ). One of the major hazard analysis tasks at hand is assessing probabilities of landslide-induced tsunamis, which in turn rests upon knowledge of the probabilities of the inducing landslide event size and speed.
(sse the full text in pdf) is inverse gamma probability), probability density (p) is, where N is the total number of landslides in the inventory and δNL is the number of landslides with area between AL and AL + δAL (reproduced from ).
EXTRAIT 29 :
The work presented here studies the physics of landslides induced in a vibrating box filled with wet sand, simulating natural slope failure. Two main questions are addressed in this study: first, what controls the type of slope failure and, second, what controls the observed frequency magnitude relations. In answer to these questions, our work shows: 1. In contrast to the common belief, that the different types of landslides, i.e. slumps and slides, reflect only the geotechnical character of the slope (i.e. ), our experimental results show that both slumps and slides may occur on a single slope, corresponding to different applied acceleration directions. 2. Two parallel processes were observed to occur during shaking: (a) development of a slope instability along a circular failure plane which encompasses the whole slope and (b) development of fragmentation and material heterogeneity. We suggest that, while the two processes occur in parallel, the process that is more significant of the two controls the type of slope failure, as well as the final size distribution. Although the shaking sand box is a simplified setting, the mechanisms of slope failure exhibited serve as an analog for larger-scale natural slope failures. The question of what controls slope stability and frequency magnitude relations is both basic and applied, and investigation of the underlying physics in different settings will enable, in the end, a landslide hazard analyses using a probabilistic approach, similar to the approach used for earthquake hazard analysis (e.g. ). 2. Experimental setting The material used in the experiments is quartz-sand with 1% wt. tap water. Grain diameter is about 0.5 mm (30–40 mesh) and grain roundness and sphericity varies from low to high (Fig. 2). The addition of water induces an apparent cohesion (negative pore pressure), due to the capillary forces that act between grains. Mixing of sand and water took place immediately before starting each run. An amount of 1% wt. of added water results in an intriguing slope failure picture, while adding less (0.5% wt.) or more (2% wt.) water results in slope failure by surface grain flows only, or in stable slopes (under the experimental slope angles and acceleration range), respectively. Due to mixing, water distribution in our sand box is expected to be practically homogenous during each experiment. As the relative amount of water was small, pores were mostly full with air, and the negative pore pressure that formed can be shown to be a few orders of magnitude larger than the hydraulic forces driving flow. Thus, we expect no flow and no change from the initial homogeneous and well-mixed state. Experiments were highly reproducible with 1% weight of water. Our box is made of Plexiglas with 1.2 cm thick walls and a 2 cm thick base. Box dimensions are 28 cm2 base and 28 cm height. Sand is set in a pile-like formation with the crest resting on the top of one inner face and the foot on the opposite face base, which is only 4 cm high (Fig. 2). The box is bolted to an industrial shaker. Slope angle was set initially under static conditions, at an angle of θ 40° from the horizon, where θ θr (θr is the angle of repose); when vibrations are applied, the slope at θ becomes unstable. Vibrating frequency was set constant at 10 Hz and individual test durations lasted up to a few minutes. Three different acceleration directions were tested: (1) vertical, (2) horizontal, slope-parallel (parallel to the slope strike), and (3) horizontal, slope-normal (in a direction normal to the slope) (Fig. 2). Acceleration magnitudes ranged from 0.2 to 1.1 g. In addition, control experiments with lower initial slope angle and vibration frequency of 5 Hz were also performed. Slope performances were continuously recorded using a digital camera (Pulnix, PC-640CL) and a frame grabber with 30 frame/s rate. The experimental accelerations may be scaled to natural slopes in the following manner:
where a is the applied acceleration, f is the frequency and l is a typical length scale (subscript m and p refer to box model and field prototype, respectively). From Eq. (1), the relationship between the applied frequency and the field frequency may be deduced: fp = fm × (lm/lp)0.5. Model box dimension (lm) is 0.28 m and applied frequency (fm) is 10 Hz. We refer to a field length scale (lp) of 10 m – 100 m and the calculated scaled field frequency (fp) is 1.67 Hz–0.53 Hz, within the expected lower range for earthquakes.
Reported here are 22 individual experiments performed under the above described initial slope angle and vibration frequency, and under increasing accelerations ranging from 0.2 to 1.1 g. Summary of the results is provided in Table 1 and Fig. 3, and a detailed description is given in the following section.
3.1. Types of slope failure
Under horizontal shaking and low to moderate accelerations (Table 1: E118-14–16, 18–21) slopes remained stable with minor sporadic short termed (about a second) surface grain flows, mainly from the uppermost part of the slope. Above a threshold acceleration (0.6 g and 0.8 g for shaking normal and parallel to slope direction, respectively; Table 1: E118-12, 13, 17 and E118-22–23, respectively), surface flow intensified and was followed by a box-wide slump, initiated after about three seconds at the upper half of the slope (Fig. 4a–b). The slump first remained coherent and then progressively disintegrated (Fig. 4b–c). Under continued slope-normal accelerations, post-slumping shaking left the slope apparently stable with a smooth surface morphology. Under slope-parallel accelerations, post-slumping shaking induced block-slides (block-slides are defined here as large rectangular chunks of sand that remained coherent for significant duration of their sliding), which initiated from a step-like scarp that migrated up the slope (Fig. 4c). Surface morphology remained blocky.
Under weak-medium vertical shaking (Table 1: E267-4–6), the slope remains stable with sporadic and short termed low-volume surface grain flows, mainly from the uppermost slope crest. Strong vertical shaking (Table 1: E267-7, 10, 11, 14, 17 and 18) yielded different behavior than that observed in horizontal shaking: here vibrations larger than a 1.0 g threshold induced mainly a few centimeters wide and about 1 cm high block-slides from a step-like scarp that migrated up the slope (Fig. 5a–c).
EXTRAIT 30 :
Block-slides started a few tens of seconds after the initiation of acceleration. Prior to each block-slide, a tension crack initiated and slowly propagated a few centimeters above the scarp (Fig. 5 and Fig. 6). This crack determined the block dimension in the slope dip direction, while its intersection with the scarp determined its dimension in the slope strike direction. Block-sliding rate was approximately one block every few seconds to tens seconds. Final slope cross-section is S shaped with normal faulting at its flattened crest. Final slope angle was about 35°.
Relative estimated sliding velocity was highest under slope-normal horizontal acceleration, lower under slope-parallel horizontal acceleration and significantly lower under vertical acceleration. When vertical and horizontal acceleration experiments were performed on slopes with an angle lower than θr, only surface grain flows were observed under the above specified acceleration range. An increase of acceleration magnitudes above the slope failure thresholds (Table 1: E267-19), yielded instability, whole box flow and convective-like behavior, probably as slope angle approaches zero.
3.2. Frequency magnitude (area) relations
The upper surface area of 64 block-slides induced under vertical acceleration of 1.0 g (tests E267-7 and E267-11) were measured to enable characterization of frequency size relations. For each block, the exact digital-camera frame showing the first increment of noticeable down-slope motion was chosen. The frame then was forwarded to a GIS environment and the block upper surface digitized as a polygon with GIS calculated area. Block surface areas ranged over three orders of magnitudes from 0.1 cm2 to 10.1 cm2.
Experimental block areas were plotted on a frequency (probability density, p) magnitude (measured block surface area, AL) diagram, following ,
where N is the total number of blocks in the inventory and δNL is the number of blocks with area between AL and AL + δAL. The area scale in the experiments is related to natural landslides populations via the chosen scaling lp/lm. Assuming lp/lm = 300, a 3 cm × 3 cm experimental slide corresponds to a natural slide of area 100 m2.
Experimental frequency magnitude relations show power law behavior with a slope of − 1.2 over the entire measured block surface area scale range (Fig. 7). This slope is smaller than the slope of the field-observed landslide inventory , which is not surprising, since it is has been recently understood and excepted that rock-falls have different distributions than landslides  and , and our blocks correspond to rock-falls rather than to landslides. After converting surface area distributions to volume distributions, using Eqs. (19), (39) and (40) of , our block volume probability distribution follows
where α equals − 1.13, close to the slope measured for natural rock-fall inventories, for which α equals − 1.07 (Fig. 7 of ) and very similar to that obtained for fragment size distributions produced by the fragmentation process of rocks and other material .
The discussion section consists of two main parts. The first part (Section 4.1) focuses on the acceleration direction as a control on the style of slope failure. This part is a direct result of the experimental observations. The second part (Section 4.2) focuses on the role of inhomogeneity in controlling the slope failure size distribution. This part is a consequence of insights gained from the experiments. First (Section 4.2.1), we review previous works that tried to determine what physical processes control the size distribution of slope failures. This is followed by a discussion of how our experimental observations (Section 4.2.2) as well as how natural landslides (Section 4.2.3) fit within the current understanding of controls on landslide size distribution. Finally, in the last part of the discussion (Section 4.3), the implications of this work to hazard analysis are discussed.
4.1. Acceleration direction as a control on the style of slope failure
The style of slope failure is usually related to the geological setting : slides are expected to occur on weak bedding planes in layered rock sequences or in pre-fractured rock mass, while slumps are expected to occur in soil or continuous weak rock formations, where failure planes develop through the rock material. In this work, the acceleration direction is identified as a previously unknown control on the style of slope failure: Under horizontal accelerations, box-sized slumps occurred. Under (horizontal) slope-parallel accelerations, slumping was followed by retrograde block-slides. Under (horizontal) slope-normal accelerations, no block-slides were ever observed. Vertical acceleration never showed slumping, but instead slope failure occurred via retrograde, variable-sized block-slides.
We speculate that the physical basis for this behavior is that different acceleration directions activate failure on planes with different orientations, followed by gravity-induced down-hill displacement. In the horizontal slope-normal acceleration, a spoon-like failure plane developed in the sandy material (as is often described in slumps) and the slope failed due to the fact that the sum of gravity force and down-slope component of seismic acceleration, caused a driving moment that exceeded the retaining moments (e.g. ). The slump disintegrated and continued to move down-slope with slope stabilization after a single failure event. In contrast, the horizontal slope-parallel acceleration does not induce a down-slope acceleration component and therefore does not contribute directly to the slope destabilizing moments balance. Instead, slope-parallel vibrations induce shearing parallel to the slope, so that finally a shear plane develops within the sand. The shear plane localizes to form a failure plane, and the slope then slumps under gravity. Since slopes under slope-normal vibrations have an acceleration component perpendicular to the slope, they experience slumping at a lower acceleration (0.6 g) than slopes under slope-parallel accelerations that lack this acceleration component, therefore becoming unstable at a higher acceleration (0.8 g).
As for vertical accelerations, in this case, only block-slides are observed and no slumps occur at all. It is easily calculated that no slumping is expected in vertical acceleration, because vertical acceleration on a slope which is stable at 45° does not change the factor of safety. On the other hand, our experimental results show that vertical accelerations do induce slope instability and block-slides. This, we suggest, occurs because vertical accelerations lead to the formation of fractures and a preferred weakened shear zone.
EXTRAIT 31 :
Two possible conceptual mechanisms are offered for the observed block-sliding under vertical acceleration: The first conceptual mechanism pictures tensional cracks opening at the surface due to vibration-induced extension (Fig. 8a). These cracks propagate (in both horizontal and vertical directions). A block-slide is initiated when a crack happens to intersect a potential failure plane at the subsurface. This potential failure plane is located where stresses couple with material strength properties to cause an instability: the factor of safety on this layer is smaller than 1 and sliding starts on this layer (Fig. 8b–c). The second conceptual model pictures block-slides as initiating on a failure plane in the substrate. This sliding causes extension in the rock layer (Fig. 8d) till a tension crack develops, enabling a block-slide (Fig. 8e–f). Sizes of block-slides are controlled in the first mechanism by pre-sliding crack distributions and in the second mechanism by the depth of weak substrate layers, the material tensile strength and the fragmentation process.
Tension cracks in the vertical vibration experiments were observed to form on the slope surface in a general slope-parallel direction for up to tens of seconds before block-slide initiated at the upward migrating step-like scarp. This observation agrees with the first mechanism for block-slide formation: the sliding process initiates at the surface with a propagating tension crack. At a depth of about 1 cm below the surface, the cracks intersect the slope-parallel potential failure plane (A–A′ in Fig. 8). This weak failure plane develops parallel to the slope upper surface due to the vertical accelerations. The blocks then slide on this weak plane due to gravity and the seismic acceleration slope-normal component. The depth of the slope-parallel plane is determined by a combination of material parameters , slope geometry and box size.
4.2. The role of inhomogeneity in controlling slope failure size
This section discusses the possible origin of the characteristic and power law behavior for the slumps and block-slides, respectively, observed in the experiments. This question relates to the general question of size distribution and power law behavior in landslides both in nature and in previous experiments and to scaling in other breakdown processes ,  and .
4.2.1. Controls on size distributions—review of previous work
One physical explanation for the emergence of power law distributions in landslides was offered even before landslides were known to exhibit power law behavior:  numerically modeled a sand-pile using a constant input of grains from above. Using simple system stability rules for slope failure the output of numerical grains was observed to occur in an avalanche style failure with a power law frequency magnitude (size) relation. This spontaneous emergence of power law avalanche sizes in a homogeneous system was termed by  as self-organized criticality (SOC). Following the introduction of the sand-pile and the SOC model, many works were subsequently devoted to the study of slope failure in granular materials. The different numerical and experimental studies obtained basically one of two classes of frequency size distributions: either a power law , , , , , ,  and  or a characteristic size , ,  and .
The power law distributions obtained above generated much interest in relation to natural landslide. However, the models that produce power law distributions, (cellular-automaton, forest-fires, spring-blocks) either use stability rules that are physically inconsistent with processes occurring in natural slope failures or fail to reproduce the slope of the observed power law size distributions .
The other mode of size distribution observed in experiments and models is characteristic landslide size. Characteristic landslides initiate when the slope angle θm exceeds the angle of repose θr by δ. Each landslide causes slumping of the whole slope, returning the pile to its metastable state at θr (e.g. ). In this case, in contrast to the prediction of the SOC model, no power law frequency size distribution is observed, only a single landslide size, with an area of the whole experimental system, and a depth correlating with δ. The physical conditions controlling the transition from a characteristic size distribution to power law are not clear (although see some suggestions in  and ).
The above studies were performed with dry and non-cohesive grains effectively simulate homogeneous material above the size of few grains. The addition of cohesion or heterogeneity, either by using a distribution of material strengths ,  and , by adding fluids  or by using magnetic particles , was found to produce a different and interesting size-frequency dependence: Either a characteristic landslide size with broad tails that may show a power law relation, or a power law slope that depends on the material properties. For example,  simulated slides using a fractal topography with a power law spatial distribution of soil moisture (which has an effect of modulating substrate strength). The simulations produced a realistic landslide distribution, though in order to recapture the natural roll-over and characteristic slide size , introduced a cutoff topography, below which failure occurs as a unit. These simulations stress the importance of material strength, heterogeneity and geometry in controlling the size-frequency dependence.
Frequency magnitude relations of seismically and hydrologically naturally triggered landslide inventories of  show a non-trivial size distribution: a maximum (a roll-over) at landslide area of about 102–103 m2 and a power law tail (this combined function was fitted by  with a gamma function: Fig. 1). The roll-over observed in natural landslide inventories indicates a characteristic avalanche size. The majority of landslides in the inventory of  are shallow (a few meters depth) and occur within young, poorly cemented, unconsolidated and even earth-like sediment. The observed characteristic landslide, which has an area of 102–103 m2, is expected to have a depth of a few meters (1–5 m), as deduced from dependence between the landslide depth (h) and area (A) : (see the full text in pdf) where ε is 0.05 ± 0.02. This relationship between the area of slumps and their depth is not completely understood, although it is both empirically observed  and may be analytically obtained.
The larger landslides are mainly deep-seated slides and slumps and are far less common in natural landslide populations than the characteristic landslides. They often reflect preferred geological conditions, like dip slope or weak geological formations. In contrast to unconsolidated sediments, inventories of rock-falls, which are the typical failure mode of hard carbonates and crystalline steep rock faces, do not show roll-overs and characteristic sizes. Instead, they show a power law and a very shallow exponent, close to 1 .
EXTRAIT 32 :
4.2.2. Controls on size distributions in the current experiments
In our experiments, there are two parallel processes that develop during shaking: (a) development of a slope instability along a circular failure plane (slump) which encompasses the whole slope and (b) development of discontinuities and material heterogeneity that dissect the slope into fragments. We suggest that both processes may occur in parallel, but the process that is more significant controls the final size distribution of landslides. When horizontal acceleration is applied, whole slope instability is the significant failure process and a characteristic box-sized slump occurs. When the applied acceleration is vertical, slope-normal components of gravitational and seismic accelerations act in opposite direction, slumping along a circular plane is not expected, and indeed not observed. Instead, the acceleration induces formation of vertical mode I fractures (Fig. 6), intersecting the subsurface failure plane (Fig. 8). These pre-sliding fractures become a source of heterogeneity. Based on our experimental photos, the size distribution of block-slides formed under vertical acceleration was apparently determined by the distribution of tensile cracks that developed in the sandy material (Fig. 6). The distribution of block sizes in our experiments (Fig. 7) follow a power law scaling very similar to that obtained for fragment size distributions produced by the fragmentation process of rocks and other material .
4.2.3. Controls on size distributions in natural slopes
It is argued here that, in nature, slopes consist of two different mechanical environments (Fig. 9a): The first is the upper few meters close to the slope surface. This environment is rather homogenous and consists of young, poorly cemented, unconsolidated and even earth-like sediments that have relatively low strength. This environment was modeled by our sand box in cases when the box stayed homogenous throughout the entire test and no major discontinuous developed till slope failure (Fig. 9b). The second mechanical environment in natural slopes is the rock mass below the unconsolidated sediment. Bedding, layers and fractures located within this environment make it heterogeneous. This environment was modeled by our sand box experiments in cases when pre-failure tensile fractures and discontinuities developed (Fig. 9c). In these cases, the sandy material lost its homogeneity and slope failure size was controlled by the heterogeneity.
Our experimental results, along with previous experimental studies and numerical simulations of landslides, show two end members of frequency size distributions: characteristic and power law.
a. A characteristic landslide (slump) size occurs when material is homogenous. The size of a slump in a homogeneous material will be the maximal possible. This is demonstrated by practical slope stability calculations, in which failure of a slump on a circular plane will take place at the maximum slump size, since the driving moments increase with slide size (e.g. ). However, the relationship between h and A must be maintained following Eq. (4) . Thus, in our experimental box, in cases for which material heterogeneity is small, the slide area A is driven towards maximum possible: the box size (a1 × b1, Fig. 9b), while h is determined by A. Given that our A is 10 cm × 28 cm, h is 1.6 cm (using Eq. (4)), where in our observations h is 1 cm. In contrast, for natural slopes, h is constrained to a few meters by the depth of the homogenous and weak sub-surface unconsolidated material (h1, Fig. 9a) and A is determined by h. The depth of the unconsolidated layer, h, does not vary too much in a given geological setting, and even in between sites , as it is controlled mainly by weather and hydrology of the area. Given that h ranges between 1 and 5 m, the area of natural slides from the homogenous unconsolidated layer, A, are thus constrained to range between 100 and 1000 m2 (using Eq. (4)). This range is the roll-over landslide size in the field observations of  that are presented here in Fig. 1.
b. A power law size distribution of landslides occurs when slope material is heterogeneous; the nature of the landslide size distribution is controlled by the heterogeneity. Sources for heterogeneity and power law landslides size distribution in the experimental box were the tensile fractures and discontinuities. In nature, heterogeneity arises due to pre-existing fractures , variable water content , variability in material properties  and, very importantly, from the natural variability in mechanical properties of sedimentary sequences. Fig. 10 shows an example where pre-failure fracture formed and will determine the size of the future slope failure. Similar pre-failure fractures have been claimed to be indicators for tsunami generating landslides .
Unlike the characteristic slides that span the whole available system size, the size of the power law slides is smaller than the system size. The system size in the experiments is the box (Fig. 9c), where in nature the system size is the entire slope (Fig. 9a). It should be noted that although heterogeneity is argued here to be the source for power law scaling, power law distributions may also occur even when material is homogeneous, as demonstrated in SOC models. However, as  and  already argued, the power law resulting from the SOC model is not physically relevant for landslides, as it arises from snow-ball or avalanche dynamics, that play only a minor role in natural landslides. Thus, we will not consider the SOC model any further, but instead will argue that preexisting heterogeneity is what produces deviations from characteristic landslide sizes and the emergence of a power law distribution.
The observed complex size distribution of natural landslides  is probably a result of the combined effect of the two processes (a and b), observed in our experiments and discussed above. The shallow small slides utilize the whole unconsolidated layer and appear on the characteristic (roll over) part of the size distribution, while the larger slides utilize preexisting heterogeneous bedding planes and fractures and appear on the power law tail of the size distribution.
Another indication that the roll-over is a result of the unconsolidated homogenous and uniform layer, and also an indication that this roll-over is not a measurement artifact as suggested by  is the fact that size distributions of rock-slides do not exhibit a characteristic size and a roll-over. In hard, consolidated rocks, avalanche size distributions are controlled only by bedding and tensile crack distributions, and lack a characteristic size dictated by the weathered soil depth.
EXTRAIT 33 :
The power law obtained in our vertical acceleration experiments is similar to that obtained in natural rock-fall inventories  and  and also to that observed in fragmentation processes . We conclude from this morphological similarity that a fragmentation process occurs in our box (under vertical accelerations), controlling the heterogeneity and size distributions. In nature, however, other heterogeneities in addition to fractures control the size distribution of slides. These include moisture and other strength variations; the most important of which is probably bedding plane strength variability.
The experimental results may be viewed as a process tree (Fig. 11). Homogeneity leads to characteristic slide sizes, while heterogeneity leads to a size distribution reflecting the heterogeneity distribution. In our experiments, we happened to induce heterogeneity by shaking; however, heterogeneity in nature is induced continuously by the many sources mentioned above, as well as by other sources. Thus, we view shaking only as a tool to induce natural variability by fracturing. This choice of heterogeneity led to a rock-fall distribution (α 1). Other heterogeneities might lead to other distributions.
A necessary additional layer of complexity must finally be added by considering that the failure process discussed above occurs on a distribution of slope sizes and shapes.  simulated this combined affect of topography and heterogeneity using a power law spatial distribution of soil moisture (which acted affectively as a heterogeneity modulating substrate strength) on top of a fractal topography. The simulations produced realistic landslide distributions, where in order to recapture the natural roll-over and characteristic slide sizes  introduced a cutoff topography, below which failure occurs as a single unit of minimal size. This artificial cutoff length is equivalent to the characteristic sediment depth discussed here. The simulations of  stress the importance of combining material strength heterogeneity, a characteristic slide size and variable slope geometry when attempting to obtain a realistic size-frequency dependence.
4.3. Implication to hazard analysis
The above discussion has an important implication to hazard analysis. Size distribution of landslides is used to evaluate landslide hazard and probability (e.g. ). According to the above discussion, the three parameter inverse gamma probability best fitted to the landslide frequency magnitude inventories of  actually follows two types of landslide distributions: characteristic and power law at the small and large landslide populations, respectively. It was shown here that each distribution is controlled by a different physical process and represents different geological agents that eventually provoke instability. As a consequence, we think that in hazard evaluation, when considering the probability of a given landslide size it is better to fit a regression line to the power law decay of the large landslide inventory only and take into account the characteristic landslide separately.
 clearly demonstrated (reproduced in Fig. 1) that landslide size distributions are trigger independent, showing similar distributions whether the trigger is seismic or meteorological. This can be explained from our results. In our experiments, the vertical shaking performed two roles nearly simultaneously: induced heterogeneity and triggered landslides. In contrast, on natural slopes the two processes, the slide triggering and the formation of heterogeneity, are not necessarily simultaneous or coupled. Fractures, topographic variations and other rock heterogeneities (especially mechanical variability due to bedding) are ubiquitous in rocks, and do not form only during earthquakes. While in our experiments the shaking first formed the heterogeneity and then blocks defined by this heterogeneity slid, natural deep-seated slides triggered either by earthquake or by storms utilize the preexisting heterogeneity. Once heterogeneity exists, the triggering mechanism itself is irrelevant as long as it drives the slope away from stability. For our horizontal acceleration experiments, the shaking triggered box-size instability. In this case, again, the system size and not the triggering mechanism determined the slide sizes. Since in both the homogenous and heterogeneous cases slide sizes were determined by purely geometrical parameters, we suggest that similar processes control meteorologically induced landslide inventories and seismically induced ones.
Natural slope failures were experimentally studied in an analog environment using a vibrating box filled with wet (practically cohesive) sand. The questions addressed in the study were (a) what controls the type of slope failure and (b) what controls frequency magnitude relations of landslides. These questions are both fundamental and applied, as size distribution of landslides are used in hazard evaluation to analyze the probability of given landslides sizes.
Experiments show that:
1. In contrast to the common idea, that the different types of landslides, i.e. slumps and slides, reflect the geotechnical character of the slope, our experimental results show that both slumps and slides may occur on a single slope, corresponding to different applied acceleration directions.
2. Size distributions of the landslides were different under horizontal and vertical acceleration. Specifically, under horizontal acceleration a characteristic landslide size was observed, while under vertical acceleration power law landslide size distribution was observed.
Two parallel processes were observed to occur in the experimental box during shaking: (a) development of a slope instability along a circular failure plane which encompasses the whole slope and (b) development of fragmentation and material heterogeneity. We suggest that, while the two processes occur in parallel, the process that is more significant of the two controls the type of slope failure, as well as the final size distribution. Under applied horizontal acceleration, slope instability is the significant failure process and a slump occurs. Failure repeatedly occurs where driving moments are maximum, creating a characteristic landslide that has the dimensions of the whole box. Under applied vertical acceleration, formation of vertical mode I fractures is the significant process. As a result, a power law size distribution of block-slides forms. These blocks detach from the now heterogeneous slope, and are the main type of slope failure.
It is suggested that the above two processes control size distributions of natural landslides inventories. The roll-over characteristic size found in the small landslides part of the inventories reflects the shallow slumps occurring within the unconsolidated rather homogeneous sediment, typical of the upper few meters close to the surface. In this environment, the slumping process utilizes the whole depth of the weak layer, creating a landslide size where driving moments are maximum, while material heterogeneity is not significant. At greater depths, heterogeneity due to fracturing and bedding becomes very important. This heterogeneity is a source for power law decay observed for the large landslide portion within natural landslide distributions. In this model, SOC dynamic is ruled out as the dominant physical process responsible for the power law decay observed in size distributions of natural landslides.
EXTRAIT 34 :
While in our experiments vibrations served to both induce heterogeneity and trigger landslides, we suggest that once mechanical heterogeneity exists it strongly controls landslide distributions independent of triggering mechanism. This agrees with the observation that landslides distributions are independent of inducing mechanisms . Our simple lab experiment do not reproduce (and cannot be expected to reproduce) the full natural heterogeneity, and thus cannot reproduce the natural observed exponent. Rather it was demonstrated that, in the presence of a heterogeneity, slide scaling follows this existing heterogeneity. It should be taken into account that the mechanisms discussed above occur on a distribution of slope sizes and shapes.  simulated this combined affect of topography and heterogeneity using a power law spatial distribution of soil moisture as a particular choice of heterogeneity on top of a fractal topography. In that work, a cutoff topography was used, below which failure occurs as a single unit of minimal size. We argue that this artificial cutoff length is the characteristic sediment depth, for the reasons discussed above. With such a combination of ingredients,  obtained realistic slide distributions. Thus, we suggest that our findings are relevant to natural landslides triggered by earthquakes as well as by meteorological events. In light of these findings, when performing hazard evaluation, it is advised to fit a regression line to the power law decay of the large landslide part only while taking into account the characteristic landslide separately. We believe that this procedure will allow a more accurate estimate of the probability of a given landslide size. Dust Devils On Earth And Mars Matt Balme and Ronald Greeley  Dust devils, particle-loaded vertical convective vortices found on both Earth and Mars, are characterized by high rotating wind speeds, significant electrostatic fields, and reduced pressure and enhanced temperature at their centers. On Earth they are subordinate to boundary layer winds in the dust cycle and, except possibly in arid regions, are only ‘‘nuisance-level’’ phenomena. On Mars, though, they seem to support the persistent background atmospheric haze, to influence the surface albedo through the formation of ‘‘tracks’’ on the surface, and to possibly endanger future exploration because of their high dust load and large potential gradients. High-resolution numerical simulations and thermophysical scaling models successfully describe dust devil–like vortices on Mars, but fitting dust devil action into the Martian global dust cycle is still problematic. Reliable parameterizations of their erosional abilities and solid temporal and spatial distribution data are still required to build and test a complete model of dust devil action. 1. INTRODUCTION  Dust devils are small whirlwinds made visible by entrained dust and sand. They are upward moving, spiraling flows caused by heating of near-surface air by insolation. The term ‘‘dust devil’’ is used to refer to sustained, particleloaded convective vortices to distinguish them from vortices that form in the same way but are too weak to pick up materials and become visible. They are common atmospheric phenomena on both Earth (Figure 1) and Mars (Figure 2) and have been observed for their general characteristics, measured in situ, and simulated both numerically and in the laboratory. They are distinct from tornadoes in that they are powered only by insolation, rather than release of latent heat, and form under clear skies with no association with thunderstorms.  Beginning with the descriptions of Baddeley , there has been more than a century of dust devil investigations. Although many of these studies were performed as adjuncts to other meteorological studies, some investigations focused specifically on dust devils [e.g., Sinclair, 1966; Ryan and Carroll, 1970; Fitzjarrald, 1973; Metzger, 1999; Renno et al., 2004], seeking to understand their role in convection, arid zone erosion, and sediment transport and their danger to light and unpowered aircraft. Although terrestrial dust devils have been studied in detail for decades, it is the discovery of their frequent occurrence on Mars in VO, MPF-IMP, MGS MOC NA/WA, and ODY THEMIS and, recently, MER and MEX HRSC (see Table 1 for acronym definitions) images that motivated this general review of their properties, mode of formation, and effects on the climate of both planets.  The next few decades will witness an unprecedented number of robotic missions to Mars and perhaps the first human missions. A sound understanding of the Martian environment is essential for planning such missions, and insight into dust devil processes is essential. Dust devils also affect scientific questions about climate, surfaceatmosphere interaction, and the cycles of erosion and sedimentation on Mars. Now is an ideal time to crystallize the current state of knowledge on dust devils, on both Earth and Mars, and to highlight future areas of work. 2. TERRESTRIAL DUST DEVILS: GENERAL CHARACTERISTICS 2.1. Geographical and Seasonal Occurrence  Dust devils usually occur in the summer in arid regions [Ives, 1947] such as (1) the southwest United States [see the full text in pdf], and elsewhere including the Middle East [Flower, 1936], China [Mattsson et al., 1993], and the Canadian sub-Arctic [Grant, 1949].  Sinclair  suggests that convective vortices and dust devils do not form solely because of ground heating by strong insolation but as a result of vertical instability in the atmosphere wherever there is a superadiabatic atmospheric lapse rate, a source of vorticity, and a supply of sand, dust or debris. Although these conditions commonly occur in hot, arid regions during the summer, they can also occur in winter or spring when cold air spreads over warmer ground or in the cold, dry conditions of the sub-Arctic.  The frequency of occurrence of dust devils is affected by many factors. The most active dust devil areas appear to be hot, flat surfaces [Mattsson et al., 1993] such as dry playas and riverbeds, especially those close to freshly ploughed and irrigated fields [Sinclair, 1969]. Gentle slopes favor dust devil formation; mountains and foothills do not [Brooks, 1960]. Although they tend not to form where there is extensive tree cover [Sinclair, 1969] or grass [Metzger, 1999], the existence of vegetation per se does not preclude dust devil formation [McGinnigle, 1966; Mattsson et al., 1993; Metzger, 1999]. Neither does moderate rock cover inhibit dust devil activity. For example, Metzger  found that areas in Nevada with rock cover >40% contained few dust devils, but areas with rock cover of 17–25% had many observable dust devils. In the Peruvian Andes, Metzger  observed boulder fields in volcanic terrain that acted as ‘‘breeding grounds’’ for thermal plumes and produced thousands of dust devils per week.
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Control of dust devil activity by topography is sometimes observed, as suggested by Williams , McGinnigle , Hallett and Hoffer , and Hess and Spillane , who report lines of dust devils forming parallel to local ridges.  Ideal regional characteristics for dust devil breeding grounds are (1) frequent strong insolation, (2) arid terrain with some rock cover but few trees, buildings, or grassy areas, and (3) gently sloping topography. Sinclair  suggests that ideal local conditions for dust devils include a plentiful supply of loose surface material, (2) ‘‘hot spots’’ or areas with anomalously high soil temperature, (3) local impediments to wind flow that can produce wake eddies or otherwise concentrate local vorticity, and (4) boundaries between different types of terrain (such as irrigated fields and arid desert) where strong horizontal thermal gradients can occur. 2.2. Size and Shape  Most dust devils are at least 5 times higher than they are wide [Hess and Spillane, 1990], but they can be extremely tall and thin or wider than they are tall. They are most densely particle-loaded near the ground [Sinclair, 1973], and when a defined columnar core is present, it often tilts toward the direction of motion by about 10_ [McGinnigle, 1966; Sinclair, 1973; Mattsson et al., 1993] and can sometimes be crooked or sinuous because of wind shear.  Dust devils range in height from a few meters to over 1 km and are generally less than 100 m in diameter [Mattsson et al., 1993]. Data from Sinclair , Flower , and Williams  suggest that _12% of dust devils are <3 m high, _50% are 3–50 m high, _33% are 50–300 m high and that only _8% are >300 m in height. Bell , however, reports dust devils observed from the air that are as high as 1000–2500 m. The height of a dust devil is most likely governed by atmospheric conditions and the type of material entrained [Ives, 1947], the visible height being controlled by how much and how high the material loading the dust devil can be carried. Sinclair  split the vertical structure of a ‘‘typical’’ dust devil into three regions. Region 1, the surface interface region, is heavily particle loaded and comprises the ‘‘vortex boundary layer’’ in which turbulent inflow occurs toward the center of the dust devil. Region 2, the main part of the dust devil, is characterized by a near-vertical column of rotating dust, with little exchange of dust between the column and the surrounding air [Sinclair, 1966]. Region 3, at the top of the dust devil, is where the rotation decays and any dust is expelled into the ambient atmospheric flow.  Sinclair [1965, 1969], Ryan and Carroll , and Snow and McClelland  made detailed statistical measurements of diameter for large samples of dust devils. The results, shown in Figure 3, have a mean diameter of _7 m, with the distribution skewed toward the smaller sizes. Snow and McClelland  and Metzger , however, suggest that small dust devils are underreported in ‘‘spotting’’ surveys because of the distances from which they are observed. The exception is the study of Carroll and Ryan  shown in Figure 3d that used only a 500 _ 300 m study area. Because their study area was small, the spotting data are extremely reliable and suggest that the frequency of occurrence is inversely related to size. Renno and Bluestein  suggest that available vertical atmospheric vorticity controls the diameter of dust devils, as discussed in section 5.  Dust devils vary widely in morphology (Figure 4) from columnar to inverted cones to disordered, rotating dust clouds [Metzger, 1999]. Metzger  found that _95% of dust devils observed in the Eldorado Valley region were V-shaped, only _4% being sharply defined columns. Less frequently, he observed broad rotating masses of dust with little structure but containing short-lived, dynamic ‘‘ropes.’’ The lower structure of a ‘‘typical’’ dust devil has been described as an ‘‘inverted cone with the apex touching or near the ground’’ [Ives, 1947; McGinnigle, 1966] or as ‘‘convex’’ [Sinclair, 1973] trending into a more cylindrical shape at some point above the ground (e.g., Figure 1c). In some studies [Sinclair, 1973; Balme et al., 2003a], dust-free cores are present in most of the observed dust devils, but in others, dust-free cores are rare [Metzger, 1999]. Metzger  suggests that different shapes of dust devils might occur in different terrain: columnar vortices being slightly more common over smooth playas and V-shaped ones found more frequently over rougher, shrubby alluvial plains. This suggests that, aside from the intensity and rotation of dust devils, the availability of different materials with different particle sizes or densities adds to the variety of morphologies observed.  Finally, dust devils frequently contain subvortices [Williams, 1948; Sinclair, 1973; Ryan and Carroll, 1970; Hallett and Hoffer, 1971; Metzger, 1999; Balme et al., 2003a] or have parasitic swirls trailing in their wake [Williams, 1948]. Hallett and Hoffer  describe subvortices disappearing, splitting apart, and reforming. Metzger  notes that individual dust devils can change in shape as they move, especially when they move into or over areas of different terrain, and sometimes virtually disappear before reforming again. 2.3. Sense of Rotation  Whether dust devils have a preferred sense of rotation has been a controversial issue [Durward, 1931; Flower, 1936; Williams, 1948; Sinclair, 1965]. Table 2 summarizes measurements of rotation sense and shows that cyclonic and anticyclonic flows are equally likely, although there is a suggestion that the largest dust devils (diameter >25 m) tend toward cyclonic rotation (65% spinning cyclonically [Sinclair, 1965]). Brooks  found that of 100 dust devils observed, all had cyclonic rotation but noted that it was often difficult for observers to distinguish the sense. To overcome observational problems, Sinclair  included only close-up measurements in which two independent observers had agreed. His data show no clear preference for rotation sense, and Brooks’ data remain anomalous. The conclusion that dust devils have no tendency toward a sense of rotation agrees well with theory; estimated ratios of inertial to Coriolis effects for even the largest dust devils show that they are too small to be affected by the Earth’s spin [Morton, 1966].
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Finally, and mysteriously, there have been several observations of dust devils completely reversing their sense of rotation [Williams, 1948]. 2.4. Diurnal Formation Rate  Dust devils form most frequently in the late morning and the early afternoon [Flower, 1936; Williams, 1948; Sinclair, 1969; Hallett and Hoffer, 1971; Snow and McClelland, 1990; Mattsson et al., 1993; Metzger, 1999]. Dust devils seldom form before 1000 LT or after 1730 LT [Sinclair, 1969; Snow and McClelland, 1990; Mattsson et al., 1993; Metzger, 1999]. Sinclair  and Metzger  note that dust devil sizes are not constant throughout the day. Sinclair  found that small dust devils peak in activity earlier than large ones and suggests that this reflects the time taken for a superadiabatic temperature profile to form through a deep layer of the atmosphere. However, Metzger  reports that the tallest dust devils occurred around 1100 LT and that later in the day the height stabilizes at _150 m.  There is some evidence that dust devil formation is ‘‘bursty’’ and that an hour or so of intense activity is frequently followed by a more quiescent period [Sinclair, 1969; Snow and McClelland, 1990]. Carroll and Ryan  note similar behavior but on a shorter timescale (5–15 min) and interpret the data to signify that the timescales of atmospheric convection govern dust devil activity. Sinclair  suggests that periods of particularly intense dust devil activity ‘‘stir up’’ the superadiabatic boundary layer to such an extent that it suppresses dust devil formation and requires some time to reestablish itself. 2.5. Lifetime and Frequency of Occurrence  Terrestrial dust devils are transient events and most last for only a few minutes [Idso, 1974], although Snow and McClelland  and Metzger  observe that lifetimes might be underestimated, especially for smaller dust devils that can grow or shrink as they travel. Metzger , Ives , and Mattsson et al.  report rare occurrences of large dust devils with lifetimes of 30 min to several hours. Ives  reports a large, stationary dust devil that lasted over 4 hours and large migratory dust devils in Utah with lifetimes >7 hours that traveled _60 km. Ives , Sinclair , and Metzger  found that large dust devils are longer-lived than smaller ones, Ives  suggesting an empirical relation of 1 hour of duration for every 300 m of height.  The frequency of occurrence is highly dependent on the season, time of day, and location. Most studies are not representative of the wider region because, of necessity, investigations have focused on areas where dust devils form frequently. The number of dust devils observed per day depends upon the size of the study area as illustrated in Table 3, reinforcing the fact that small dust devils are often ignored. Carroll and Ryan  found that >750 dust devils can occur per square kilometer per day. 3. SPECIFIC MEASUREMENTS OF TERRESTRIAL DUST DEVILS  Detailed wind speed, pressure, temperature, and dust load measurements of dust devils can be made in situ or using remote sensing. To date, most data have been obtained in situ because current remote sensing techniques have insufficient resolution. While in situ measurements have the advantage of allowing several parameters to be sampled simultaneously, they must contend with technical challenges such as a hostile environment that can damage sensitive equipment and the short lifetimes and unpredictable nature of the phenomena; in situ measurements require robust yet mobile sampling systems [Sinclair, 1973; Metzger, 1999; Tratt et al., 2003; Metzger et al., 2004a]. Choice of study area is also essential; it must have frequent, observable dust devil activity, easy vehicle access, and a surface that allows for rapid movement of the sampling system. Playas and their surrounding terrain are ideal study areas. Two playas in particular, Eldorado Valley in Nevada [Metzger and Lancaster, 1995; Metzger, 1999; Balme et al., 2003a; Metzger et al., 2004a, 2004b; Towner et al., 2004] and another in the Mojave Desert, southern California [Ryan and Carroll, 1970; Carroll and Ryan, 1970; Ryan, 1972; Fitzjarrald, 1973], have been the site of several studies. 3.1. Wind Speed Structure of Dust Devils  Wind speed measurements are either generalizations of many measurements or detailed studies of a few dust devils. Wind speeds are usually quoted as cylindrical components relative to the central point of the dust devil and include tangential velocity U, radial velocity V, and vertical velocity W. The magnitude of the total horizontal wind speed, Vh = (V2 + U2)1/2, is also frequently quoted as no directional measurements are required. Commonly, for studies of multiple dust devils, only the peak values of the components are reported. Most measurements within dust devils have been made at _2 m height following Sinclair [1964, 1973], although some measurements have been made very close to the ground [Balme et al., 2003a; Metzger et al., 2004b] and at heights up to _23 m [Kaimal and Bussinger, 1970].  Table 4 summarizes the ‘‘general’’ wind speed measurements. V is usually 5–10 m s_1, with peak values up to _20 m s_1. Vh values up to _25 m s_1 have been measured in situ in approximate agreement with remote sensing measurements for Vh made using lidar of 11 m s_1 [Bluestein and Pazmany, 2000] and 22 m s_1 [Schwiesow and Cupp, 1975]. (Italicized terms are defined in the glossary, after the main text.) Vertical wind speeds are generally about a quarter of the peak rotational wind speed; only Sinclair  and the qualitative estimates of Ives  and Hallett and Hoffer  suggest greater values for W. Typically, horizontal wind speeds within dust devils are <25 m s_1, and vertical wind speed is <10 m s_1.  Ryan and Carroll  provide a large, selfconsistent data set (>80 encounters with dust devils made at the same study area with simultaneous measurements at 2 m height of V and W and estimates of diameters). Their results suggest that larger dust devils have greater rotational wind speeds and that dust devils with greater rotational winds also tend to have greater vertical winds (Figure 5).
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 Detailed data for the velocity structure within dust devils are limited because of the difficulties of making highresolution in situ measurements. Horizontal profiles of wind speeds through dust devils [Sinclair, 1964, 1973; Kaimal and Bussinger, 1970; Fitzjarrald, 1973; Metzger, 1999; Balme et al., 2003a; Tratt et al., 2003] show that nearsurface horizontal wind speed has a minimum at the center of the dust devil and a peak at a radius concurrent with the visible dust-laden region and falls to zero away from the dust devil until there is no rotation. This is particularly obvious in Figure 6 [Metzger et al., 2004a], which shows wind speed measurements made in a vertical section through a dust devil. In general, the horizontal wind speed profiles approximate a Rankine vortex (Figure 7). Sinclair  finds good agreement of dust devil data with the Rankine model at heights of _2 m and _10 m, but outside the solidly rotating central region, recent measurements [Tratt et al., 2003] show that the wind speed profile is closer to an r_1/2 distribution than r_1, probably because of nonconservation of angular momentum caused by frictional losses near the surface. It is likely that the Rankine structure is applicable higher up in the dust devil where surface effects are negligible.  There is almost no systematic radial flow within the dust devil core [Sinclair, 1966], radial inflow instead occurs near the ground, with radial wind speeds greatest just outside of the dust column [Sinclair, 1966, 1973]. Inflow occurs both in front of and behind the dust devil as it moves across the surface. The visible dust devil column appears to be embedded within a larger region of radial inflow. Balme et al. [2003a] found an approximately linear increase in horizontal wind speed with the logarithm of heights from 0.05 to 1.90 m, suggesting that the radial inflow layer was at least 2 m deep.  Some researchers have found central downdrafts within dust devils [Kaimal and Bussinger, 1970; Sinclair, 1973]. Downdrafts are less intense (or not present) near the ground than at height within the dust devil devils [Kaimal and Bussinger, 1970; Sinclair, 1973]. Metzger  reports that at 2 m height most dust devils have no central downdraft. This suggests a stagnation point in the vortex and reversal in the direction of vertical flow at height from zero to a few meters above the ground (Figure 8).  Subvortices, ambient winds, and local gusts add to the variable nature of dust devils. However, stable, simple dust devils are characterized by (1) radial inflow near the surface (with peak inflow speeds just outside the dust column), (2) upward flow within the dust column (with possible downward flow at the center), and (3) tangential wind speeds that approximate a Rankine vortex and that peak at about the same radius as the visible dust column. At the center of the dust devil, vertical flow dominates; within the dust column, rotation and vertical flow dominate; just outside the column, inflow and rotation dominate. The most distant areas affected by the dust devils differ from the ambient winds only by weak inflow toward the dust devil. This structure is summarized in Figure 9.  The higher reaches of dust devils can only be sampled remotely or from aircraft. Sinclair  measured vertical wind speeds in ‘‘thermals’’ above large dust devils using an instrumented sailplane. At altitudes of 2000– 4000 m, warm upwellings a few tenths of a degree above ambient with vertical wind speeds of _2–4 m s_1 covered an area of 1–5 km in diameter above large dust devils. The vertical wind speeds were often reduced at the center of the flow, and in some cases, there was weak evidence for central downdrafts. Sinclair  also noted that surrounding these upwellings were regions of downward flow and that this structure was stable with time. This suggests that large dust devils are linked to a much larger continuous upward flow of air that extends to several kilometers height and expands to a few kilometers in diameter before returning downward. It is unknown whether this return flow can also be through the center of the thermal to link with downward flows measured near the ground at the center of some dust devils. 3.2. Temperature and Pressure Excursions Within Dust Devils  Dust devil cores commonly have small, positive temperature excursions [Sinclair, 1964, 1973; Fitzjarrald, 1973; Metzger, 1999; Tratt et al., 2003]. A summary of these measurements is given in Table 5. Temperature excursions <_10_C are found consistently [Sinclair, 1969, 1973; Tratt et al., 2003], but measurements with an order of magnitude higher sampling rate [Metzger, 1999] show temperature excursions as great as 20_C. The temperature excursion seems to be fairly stable to heights of _3 m [Tratt et al., 2003], but it weakens farther up in the core [Kaimal and Bussinger, 1970]. A cooler ring of air surrounding the warm cores has been reported [Ives, 1947; Ringrose, 2003], but available data are too poor to resolve detailed temperature structure.  In addition to the positive temperature excursion, negative pressure excursions or ‘‘pressure wells’’ are common at the center of dust devils as first noted by Ives  and summarized in Table 6. Ringrose  measured pressure wells at heights of 0.04, 1.0, and 1.8 m above the ground but found no correlation between maximum pressure drop and height. Most measurements of pressure wells in dust devils are only a few millibars from ambient, but both Metzger  and Ringrose  measured some pressure wells about an order of magnitude larger. These might represent a small population of dust devils with exceptionally deep pressure wells (Ringrose  suggests the tightest vortices have the deepest pressure wells), or they might represent a confined region of low pressure present in most dust devils but only rarely sampled, even in apparently central penetrations. 3.3. Electrical and Magnetic Structure  Dust storms can generate significant electrostatic fields because of contact between grains and between grains and the surface.
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This process is known as the triboelectric effect and has been observed in dust devils for several decades [Freier, 1960; Crozier, 1964, 1970; Farrell et al., 2003, 2004]. Table 7 summarizes the electrical measurements made in and near dust devils. Dust devils always appear to have negative electric fields, and charge densities of 105– 107 electrons cm_3 are not uncommon. Farrell et al.  suggest that the negative gradient is due to particle-sizedependent stratification caused by the tendency of small particles to become negatively charged during charging [Ette, 1971]. Thus, because the net flow in a dust devil preferentially transfers smaller particles upward compared to larger sand-grade material, a negative potential gradient is observed. Farrell et al.  estimated the potential difference over one particular dust devil as being as large as 0.8 MV.  In addition to electrostatic fields, Houser et al.  measured AC magnetic fields around and within dust devils. They measured ultralow-frequency (3–30 Hz) emissions as a dust devil approached their instruments and noted a peak in intensity as it passed over the sensors. Interestingly, the intensity remained high for about 12 s after the dust devil had passed before decreasing to ambient levels about 30 s after the encounter. Houser et al.  attributed this behavior to the entire dust devil radiating ULF emissions. The discovery of ULF emissions might be used in the future for remote sensing of dust devil activity or might give an indication of the ‘‘dustiness’’ of a vortex detected using other sensors. 3.4. Entrainment of Surface Material by Dust Devils  Dust devils are erosional agents: The simple fact that they are visible means that they remove material from the surface. For example, satellite images revealed tracks over sand dunes left by the passage of dust devils [Rossi, 2002]; these dunes have bimodal particle size distributions, and it is thought that the removal of the finer sands changes the albedo compared with undisturbed areas. However, the transport of sands in dust devils occurs only locally (typical small dust devils do not travel great distances and sand is lifted within the dust devil but returns to the surface a few tens of meters from the core). However, the transport and suspension of smaller particles (<25 _m) by dust devils is important for climate, air quality, and particle transport considerations. Dust devils efficiently transport dust vertically where it can be transported in suspension by regional winds for hours or days [Gillette and Sinclair, 1990]. Mattsson et al.  suggest that dust devils in North Africa might be a mechanism for dust injection into the atmosphere and transport into Europe.  Dust flux in dust devils has been estimated by aircraft measurements of vertical velocity and particle loading in dust devils. Fluxes up to _3 _ 10_3 kg m_2 s_1 at heights of _140 m were measured for very large dust devils [Gillette and Sinclair, 1990], but fine particles (<25 _m) made up only about 5% of this figure (_1.6 _ 10_4 kg m_2 s_1). Smaller dust devils were found to lift orders of magnitude less material. Lidar measurements of dust concentrations have also been used to estimate dust fluxes; for example, Renno et al.  estimated a particle flux of _1 _10_3 kg m_2 s_1 100 m above the surface, and Metzger  measured flux of _0.6 _ 10_3 to 4.4 _ 10_3 kg m_2 s_1 in the lower regions of dust devils. These values are similar and suggest that large, long-lived dust devils can remove hundreds or even thousands of kilograms of material from the surface during their lifetime. For the contiguous United States, Gillette and Sinclair  estimated that dust devils might be responsible for as much as two thirds of the total windblown dust for particle sizes of <25 _m and that, particularly in the southwest United States and other arid regions, dust devils could be a significant cause of poor air quality. Efforts to improve regional or global models to include dust devil processes are hampered by the fact that they fall below the resolution of most models. Cakmur et al.  initiated studies to parameterize dust lifting by local circulation into global climate models, but much remains to be done.  To complement field studies of dust devils, laboratory simulations and experiments are also conducted under controlled conditions. Vortex simulators have been used to model the dynamics of tornadoes and dust devils for several decades and have mainly focused on initiation and dynamics of the flows, but there have been some studies focused on particle lifting as well. Work using a vortex generator [Hsu and Fattahi, 1976] reported by Greeley et al.  and Greeley and Iversen  suggested that the horizontal shear stresses caused by swirling winds might be assisted in lifting particles by a ‘‘vacuum cleaner’’ effect caused by the low-pressure core associated with dust devil vortices. Later, using an apparatus specifically designed to simulate the particle-lifting action of terrestrial and Martian dust devils, Greeley et al.  confirmed that for dust size particles, dust devil vortices are more efficient at entraining material than their wind speeds alone account for. They suggest that the pressure well effect (referred to as the DP effect) is the probable cause. Neakrase et al.  have used the same apparatus to estimate the rates that these laboratory vortices remove dust from the surface and find excellent agreement with the field data reported above (0.2 _ 10_3 to 5 _ 10_3 kg m_2 s_1 in the laboratory compared with Metzger’s  field measurements of 0.6 _ 10_3 to 4.4 _ 10_3 kg m_2 s_1). This work is ongoing but reinforces fieldwork results that dust devils can play a dominant role in transporting dust into the atmosphere and reducing air quality in arid regions. 4. DUST DEVILS ON MARS 4.1. Background  Dust devils were first identified on Mars in VO images as small bright clouds with long tapered shadows [Thomas and Gierasch, 1985], although their existence had been hypothesized previously [Neubauer, 1966; Gierasch and Goody, 1973]. Many dozens of dust devils were found in VO images, but when high-resolution MOC images became available, many more dust devils on Mars were identified [Edgett and Malin, 2000; Malin and Edgett, 2001] (Figures 2c and 2d).
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Dust devils have also been observed in MEX HRSC images [Stanzel et al., 2005]. In addition to active dust devils, ‘‘tornado tracks’’ [Grant and Schultz, 1987], later shown to be dust devil tracks [Edgett and Malin, 2000], are seen in huge numbers in MOC NA images. Dust devils were also imaged directly from the surface by MPF IMP [Metzger et al., 1999] (Figure 2a) and the MER Spirit (Figure 2b), and meteorological data were used to infer their passage over the Viking [Ryan and Lucich, 1983; Ringrose et al., 2003] and MPF [Schofield et al., 1997; Murphy and Nelli, 2002] landers. Because of the difficulty of obtaining in situ data, techniques such as laboratory and numerical simulations have also been extensively employed to understand particle-lifting and formation mechanisms of Martian dust devils. 4.2. General Appearance and Size  Mars orbiter observations of active dust devils show that they are frequently a few kilometers high and hundreds of meters in diameter and tend to have narrow bases and broader tops [Thomas and Gierasch, 1985]. Up to 10 dust devils have been observed in a single MOC WA frame [Edgett and Malin, 2000; Malin and Edgett, 2001]. Table 8 shows that Martian dust devils can be an order of magnitude larger than terrestrial ones but that there are also many smaller examples that can probably only be detected from the surface. Recent images from MER (e.g., NASA MER Spirit press release, 19 August 2005, http://marsrovers.jpl. nasa.gov/gallery/press/spirit/20050819a.html) have confirmed earlier observations from IMP [Metzger et al., 1999, 2000] that Martian and terrestrial dust devils are similar in morphology (compare Figure 2b and Figures 1a–1c) and can be extremely common.  Dust devil tracks have been used to estimate dust devil diameter. Edgett and Malin , Malin and Edgett , and Balme et al. [2003b] note that most dust devil tracks are a few to tens of meters wide and that the diameters of the dust devils that formed them are presumably similar. The largest tracks observed are up to a few hundreds of meters in diameter, in agreement with images of active dust devils. 4.3. Seasonal Dependence, Diurnal Activity, and Geographic Distribution  The long lifetimes of the Viking landers and orbiters and MGS missions allow multiyear observation and measurement of active dust devils from orbit and in situ. These studies show that dust devil activity follows the season of maximum insolation [Ryan and Lucich, 1983; Thomas and Gierasch, 1985; Cantor and Edgett, 2002]. Most dust devil tracks are seen in images taken during regional spring and summer [Balme et al., 2003b]; these observations also show that the tracks ‘‘fade’’ on a timescale much shorter than one Martian year.  Analysis of _80 convective vortices recorded by MPF [Murphy and Nelli, 2002] shows a clear trend in diurnal activity: Most vortices occur between 1200 and 1300 local time, as seen for terrestrial dust devils. Analysis of Viking Lander 2 data by Ringrose et al.  shows a less clear pattern, although the peak is still 1200 LT. Moreover, these data show ‘‘bursts’’ of dust devil formation with fewer events in the half hour after a period of intense activity as seen on Earth.  Determining where dust devils form most frequently on Mars is challenging because of the sheer volume of data. Over 100,000 MOC images of suitable resolution have been taken, and still more are being acquired. With only a very small percentage containing active dust devils, searching the whole set is an enormous task, although progress is being made to automate the process [Gibbons et al., 2005]. Surveys of dust devil tracks have been made as a proxy for active dust devils, but data volume still limits these to only regional studies [e.g., Balme et al., 2003b; Fisher et al., 2005]. Some particularly active dust devil regions that have been identified include northern hemisphere low-lying regions such as Amazonis Planitia (_30_N, _190_E [Edgett and Malin, 2000; Fisher et al., 2005; Cantor and Edgett, 2002]), Casius (_40_N, _90_E [Fisher et al., 2005]), and the large impact basin in the southern hemisphere, Argyre Planitia (_50_S, _340_E [Balme et al., 2003b]). Fisher et al.  observed many active dust devils in Amazonis but relatively few dust devil tracks and many dust devil tracks but no active dust devils in Casius, perhaps suggesting that dust devil tracks are not a good proxy for dust devil activity. Geissler  found many more dust devil tracks between 45_ and 60_N in the dark terrain of Nilosyrtis (_45_N, _85_E) than in either bright or dark terrain to the south and a similar increase in the number of dust devil tracks between 40_ and 60_S in Phaethontis (_50_S, _210_E). Grant and Schultz  and Balme et al. [2003b] also found dust devil tracks to be most abundant between 50_ and 60_S.  Balme et al. [2003b] suggest that dust abundance on the surface (using albedo as a proxy) might control the formation of dust devil tracks; where there is more dust, more tracks will occur. However, taking all the regional studies performed to date together, there is no clear correlation with albedo [e.g., Geissler, 2005], and a more global, latitudinal control seems more likely. While these data are not exhaustive, they suggest enhanced dust devil erosion at latitudes of between 30_ and 65_ in both hemispheres but also that dust devil activity is regionally highly variable. 4.4. In situ Measurements of Wind Speed, Sense of Rotation, Pressure, and Temperature  Excursions in meteorological data made by the Viking 1 and 2 and MPF landers remain the only in situ measurements of active Martian dust devils; the recent MERs did not carry any dedicated meteorology instruments. There is a paucity of wind speed data in particular because of calibration problems with the MPF wind sensor [Schofield et al., 1997]. Also, it is generally unknown whether each detection represents a convective vortex or a particle-laden dust devil because it is difficult to infer if the vortex is dust-loaded (although one encounter with MPF was associated with a drop in power to the solar cells and was thus was assumed to be a dust-loaded vortex [Schofield et al., 1997]).
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Maximum wind speeds of up to 42 m s_1 at 1.6 m height were calculated for convective vortices passing over the Viking 1 and 2 landers from meteorology data by Ryan and Lucich . They estimated that wind speeds of >30 m s_1 were required to entrain surface material and therefore that seven of the detected vortices were dust devils. Peculiarly, most of the highest wind speed measurements were made during winter. Reexamining these data, Ringrose et al.  found seven events in which a vortex had passed over the Viking 2 lander and developed an algorithm to search wind speed and direction data excursions for ‘‘near misses’’ by vortices. Wind speeds of up to 46 m s_1 at 1.6 m height were calculated for vortices that passed directly over the instruments, but wind speeds of up to _100 m s_1 were inferred (using the Rankine vortex approximation as described in section 3.1) for vortices that passed within about five core radii of the sensors. Ringrose et al.  used a friction wind speed threshold criterion to determine whether the vortices were dust-laden and found that only a few inferred ‘‘near-miss’’ examples were sufficiently vigorous to entrain material.  Rotation sense was inferred using patterns of wind direction data. Neither Ryan and Lucich  nor Ringrose et al.  found any preference for rotation sense despite the fact that the larger size of Martian dust devils suggests they would be more influenced by planetary rotation than terrestrial ones.  In contrast to the wind speed instrumentation the MPF pressure sensors were more suited to detecting vortices than the Viking instruments, which had too slow a sample rate for detection of vortices [Ryan and Lucich, 1983]. Murphy and Nelli  identified 79 possible convective vortices from MPF pressure data and recorded pressure drops from _0.5 to _5 Pa (_0.075 to _0.75%). Over half of these encounters had pressure drops less than 1 Pa with relatively few ‘‘large’’ or intense (possibly dust loaded) vortices.  Positive temperature excursions within vortices measured by the Viking and MPF landers had maximum values of 5–6 K. These values are similar to terrestrial measurements. However, most of the measurements had low sample rates, and it is possible that higher sampling rates would give higher peak temperature excursions, as has been the case for Earth. 4.5. Entrainment of Surface Material by Dust Devils on Mars  As on Earth, observations of active dust devils and tracks indicate that they entrain surface material. Albedo decreases of at least 15% have been recorded for regions where dust devil tracks cover _50% of the surface [Geissler, 2005]. Another indicator that dust devils inject significant material into the atmosphere locally is the close match of diurnal variations in dust opacity observed by MPF [Smith and Lemmon, 1999] with the times when dust devil activity is greatest (midday through midafternoon).  It is difficult to make quantitative estimates of how much material Martian dust devils can entrain as there have been no in situ measurements of dust/sand loading. It is also unknown whether devils tracks indicate complete removal of a dust layer or represent ‘‘jostling’’ and infiltration of dust into a sandy surface [Greeley et al., 2005], making estimates from observations of tracks difficult. Nevertheless, optical depth measurements of dust columns were made from orbit and surface observations and used to estimate their particle load. Thomas and Gierasch  estimated optical depths of 0.3–0.5 along the path of illumination for dust devils in Viking orbiter images and calculated the dust loading to be 3 _ 10_5 kg m_3, assuming that the particles were 10 _m and the occluded path length was 250 m. Using a similar technique for MPF IMP images, Metzger et al.  found that dust devil columns were _3–4% darker than the sky. They estimate that the dust load was _10_5 to 10_4 kg m_3, similar to the results of Thomas and Gierasch .  Extrapolating these measurements of dust load to a reliable estimate of flux is complicated by several uncertainties: (1) Estimates of vertical wind velocity within Martian dust devils can only be based on terrestrial analogues (_7 m s_1 [Metzger et al., 1999]) or first-order modeling (_20 m s_1 [Renno et al., 2000; Ferri et al., 2003; Renno et al., 2004]). (2) It is unknown if the entire observed dust column is moving upward or if a downwelling central core is sometimes present as for Earth. (3) It is unknown how much material removed from the surface is expelled from the top of the dust devil and how much is ‘‘recycled’’ within the column and immediately redeposited. (4) It is unknown what area beneath the dust column is actively entraining material. Therefore, while these data can be used to indicate dust removal flux for single Martian dust devils, a conservative estimate of the uncertainty on the measurements is approximately 2–3 orders of magnitude.  Even larger uncertainties exist when trying to estimate flux from measurements of dust devil tracks. To convert observations of area and frequency of formation of tracks to a removal flux requires in situ measurements of how much material is removed per track coupled with measurement of the length of time it took to be emplaced. These data are unavailable for dust devils on both Earth and Mars. Recent MER Spirit Microscopic Imager observations have shown that sand particles within a dark linear feature (possibly a dust devil track) appear to have been cleaned of fine dust particles compared with the surface outside of the dark linear feature [Greeley et al., 2005]. Metzger  estimates that _50% of the dust cover was removed from a rock by a dust devil at the MER Gusev site but acknowledges the difficulty of estimating the total mass of material actually removed by the dust devil and in what time period.  Laboratory modeling of dust lifting using a vortex generator apparatus has been extended to Martian surface pressures [Greeley et al., 2003].
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Similar to the simulations of terrestrial atmospheric conditions, Greeley et al.  found that the particle-lifting ability of vortices does not diminish as rapidly for grain sizes >100 _m as it does for boundary layer winds, implying that vortices are the more efficient mechanism for lifting dust. Further experiments using this apparatus at Martian pressures seek to measure dust removal flux by laboratory vortices. Preliminary results suggest suspension loads of _1 _ 10_4 kg m_3 are obtainable for vortices with DP values of 0.7% of ambient pressure (L. D. V. Neakrase, personal communication, 2005), similar to estimates from observations made on Mars. 5. DUST DEVIL FORMATION 5.1. Overview  Dust devils form when surface insolation leads to a superadiabatic lapse rate, causing an unstably stratified atmosphere and strong convection. Dust devils appear to get their energy only from this insolation in contrast to tornadoes, which are powered in part by the release of latent heat within the column. In particular, the strength of the superadiabatic lapse rate in the region _0.3 to 10 m above the surface seems to control the frequency and size of dust devils formed [Ryan and Carroll, 1970; Carroll and Ryan, 1970], stronger superadiabatic lapse rates being associated with more and larger dust devils. ‘‘Burstiness’’ in formation rates [Sinclair, 1969; Carroll and Ryan, 1970; Snow and McClelland, 1990] suggests that intense convection temporarily inhibits dust devil formation because of overmixing of the adiabatic layer. Dust devils do not appear to be isolated convective phenomena and instead form a part of the local convective system [Sinclair, 1966; Kaimal and Bussinger, 1970; Ryan and Carroll, 1970; Hess et al., 1988]. Observations of thermal plumes several kilometers above large dust devils [Sinclair, 1966] suggest that a dust devil is the near-surface expression of a convective plume that has been somehow ‘‘spun-up,’’ larger examples probably extending over the depth of the whole convective boundary layer. However, it is unclear what governs the size, wind speed, pressure, and temperature excursions and frequency of formation of dust devils and why these particular convective elements form concentrated vortices when others form thermal plumes with little or no rotation.  Recent advances in numerical simulations of atmospheric dynamics, both for the Earth and Mars, have allowed investigation of convective phenomena at previously unprecedented spatial and temporal resolution. Mesoscale LES atmospheric models for Earth [Kanak et al., 2000; Kanak, 2005] and Mars [Rafkin et al., 2001; Michaels and Rafkin, 2001; Toigo and Richardson, 2002; Toigo et al., 2003] have begun to utilize resolutions sufficiently fine that they spontaneously generate convective vortices on similar scales to dust devils (although these models cannot deduce if the vortex would be dust loaded or not). The model vortices agree well with field measurements, showing similar pressure wells and diurnal behavior and velocity structure similar to real life dust devils. Models have the advantage that all vortex properties are instantly accessible, and it is likely that future work on dust devil initiation will rely heavily on such numerical simulations. 5.2. Vorticity Source  While terrain features are undoubtedly responsible for the rotation of some dust devils [Sinclair, 1969; Hallett and Hoffer, 1971], many form in flat regions with weak ambient winds [Mattsson et al., 1993], and thus another source of vorticity is required. Such sources might include concentration of vorticity from planetary rotation, mesoscale eddies, or kilometer-scale swirls or tipping of horizontal vorticity (i.e., horizontal boundary layer vortices) into the vertical plane. Because dust devils do not appear to show a preference for rotational direction, it is unlikely that the planetary rotation is the source of vorticity for dust devils as shown by considerations of Rossby number, Ro, the ratio of inertial and Coriolis forces for a flow system [Morton, 1966] given by Ro ¼ V=2WL; ð1Þ where V is a flow speed, L is a length characteristic of the flow, and W is the vertical component of the angular velocity of the planet’s rotation. Table 9 shows estimated values of Rossby number for terrestrial and Martian dust devils. Even for very large dust devils the Rossby numbers are orders of magnitude >1, implying that Coriolis forces are insignificant and that vorticity does not come directly from planetary rotation.  In a field study correlating local vorticity with observations of frequency of dust devil formation and sense of rotation, Carroll and Ryan  and Fitzjarrald  found that the horizontal scale of vorticity variations were of the order of hundreds of meters. Carroll and Ryan  also found that groups of dust devils with the same sense of rotation occurred often and that for larger ambient wind speeds, sense of dust devil rotation and measured vorticity were frequently in agreement. Dust devils were noted to form in areas with no local topographic obstacles or observed mesoscale phenomena [Carroll and Ryan, 1970], indicating that dust devils form from local sources of vorticity that change sign and amplitude with temporal scales of minutes and spatial scales of hundreds of meters. Observations that dust devils frequently occur near the boundary of irrigated fields [Sinclair, 1969] led Renno et al.  to suggest that horizontal atmospheric vortices formed from opposition of cold and warm air currents that were then twisted into the vertical by convection might be a vorticity source for dust devils. Similarly, the importance of convective tipping of horizontal vorticity in dust devil formation is demonstrated in extremely high resolution numerical LES simulations [Kanak et al., 2000; Kanak, 2005] that show vortices forming within convergent branches of convective cells. These models simulate environments with no mean winds, wind shears, or topography; vortices of similar scale and structure to dust devils are generated purely through the action of convection. Dust devil–like vortices were observed in LES models of the Martian atmosphere [Rafkin et al., 2001; Toigo et al., 2003], and tilting of horizontal vorticity into the vertical plane by convection appears to be the preferred formation mechanism for Martian dust devils [Toigo et al., 2003].  Although there is likely a variety of vorticity sources for dust devils, these results show that those that form in flat terrain with little mean wind are unlikely to be caused by large-scale (mesoscale atmospheric circulation or planetary rotation) or small-scale (spin-off from obstacles) vorticity sources.
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Instead, medium-scale tilting of horizontal vorticity by convection is the more probable mechanism. 5.3. Thermodynamics and Energy Balance of Dust Devils  The thermophysical ‘‘Renno’’ model [Renno et al., 1998; Renno and Bluestein, 2001; Renno et al., 2004] describes a dust devil as a heat engine. Steady state vortices in cyclostrophic balance are modeled assuming that heat input is from sensible heat flux at the surface, that heat output is from thermal radiation of air parcels subsiding outside of the vortex, and that losses are due to mechanical friction at the surface. Thus the intensity of the vortex can be described by its thermodynamic efficiency (the fraction of the heat input converted into work (_) and the fraction of the total mechanical energy consumed by friction near the ground (_)) and the thermal properties of the atmosphere. For the complete derivation readers should refer to Renno et al. , but the important points are summarized here as this model has been applied to both Earth and Mars. Renno et al.  state that the bulk pressure drop across a convective circulation is (see the full text in pdf) where p0 is the surface pressure at the center of the convective circulation, T1 and P1 are the temperature and pressure away from the influence of the circulation, R is the appropriate gas constant for the atmosphere, cp is the specific heat capacity at constant pressure for the atmosphere, and DT is the temperature perturbation for convective plumes over homogeneous surfaces given by Renno and Ingersoll  as (see the full text in pdf) where Fin is the surface heat flux, “ is the atmosphere’s emissivity, _R is the Stephan-Boltzmann constant, g is the acceleration due to gravity, H is the depth of the convective layer, and Tc is the temperature at the tropopause (the height at which the upward traveling warm air is assumed to be ejected from the convective system).  Thus, from (2), if _ _ 1 and Dp/p1 _ 1 (as would be expected for a typical convective plume or vortex), then the pressure drop can be approximated by (see the full text in pdf) and so, if the vortex is in cyclostrophic balance, (see the full text in pdf) where Vmax is the peak tangential wind speed.  Thus the wind speed and peak pressure excursion of a dust devil depend only upon the thermodynamics of its heat engine, which is governed by ambient conditions. Vortex size, according to Renno and Bluestein , is proportional to the background vorticity and must be accounted for separately.  This model is powerful in that it is simple, applicable to almost all environments, and describes a scaling relationship between key measurable parameters for individual dust devils. Also, measurable ambient parameters can be used to predict some properties of local dust devils from equations (3), (4), and (5). Predictions from (5) agree well with preliminary measurements of actual dust devils [Tratt et al., 2003], but a statistically valid number of reliable in situ measurements has not been made, nor have detailed ambient measurements been made temporally and spatially close enough to sampled dust devils to test the validity of (3). Nevertheless, this model has been successfully extended to Mars: Renno et al.  show that this model generates realistic temperature excursions and wind speeds when applied to measurements of pressure excursions by MPF, and Toigo et al.  show that pressure excursions predicted by this model agree well with numerical models (although they note that even better agreement is obtained using a lower estimate of mixing depth of 5–6 km rather than _45 km as used by Renno et al. ).  A recent observation by Lorenz and Myers  using thermal imaging suggests that material within the dust devil column is strongly heated by insolation and likely warms the air that supports it. If this is the case, then insolation of the lowest, most particle-laden parts of dust devils provides an important, and hitherto poorly recognized, contribution to the energy budget of the system. Also, this mechanism could serve as a positive feedback system (the more intense the dust devil, the dustier it becomes, therefore absorbing more solar energy and becoming even more intense) that might explain the long lifetimes of particularly large and dusty dust devils. 6. EFFECTS OF DUST DEVILS ON THE MARTIAN CLIMATE  The Martian atmosphere is thinner than Earth’s with a surface pressure of _5.2 mbar [Young, 1971] compared to _1000 mbar for Earth, so much higher wind speeds are required to pick up sand or dust on Mars. Wind tunnel studies [Greeley et al., 1976, 1981; Iversen and White, 1982] have shown that, like Earth, particles with diameter 80–100 _m (fine sand) are the easiest to move, having the lowest static threshold friction velocity, and that larger and smaller particles require stronger winds to entrain them into the flow (Figure 10). However, much of Mars’ atmospheric dust load is very small (_2 _m [Pollack et al., 1979, 1995; Smith and Mars Pathfinder Team, 1997; Tomasko et al., 1999; Lemmon et al., 2004]), and the boundary layer wind speeds required to entrain such fine material are in excess of those measured on the surface [Hess et al., 1977; Schofield et al., 1997; Magalhaes et al., 1999] or predicted by climate models [Haberle et al., 1999]. Nevertheless, fine dust is somehow being injected into the atmosphere to support the observed haze and to supply local [Cantor et al., 2001] and global (reviewed by Kahn et al.  and Zurek et al. ) dust storms. Greeley et al.  reviewed alternatives to direct lifting by boundary layer winds to raise dust, of which saltation impact (of easily moved sand into more difficult to move dust) has, until recently, been the prime candidate and has been used as the dust-lifting scheme in GCMs [Newman et al., 2002].  Given that dust devils were observed to be efficient transporters of fine material on Earth, they were proposed as a dust-lifting mechanism and possible triggers for global dust storms on Mars [Neubauer, 1966; Gierasch and Goody, 1973] even before they were identified in Viking orbiter images by Thomas and Gierasch . However, the increase in the number of observations of dust devils from MGS has led to renewed interest in dust devils and how they might affect the Martian climate.
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Also, recent observations of the high interannual repeatability of Martian atmospheric temperatures [Clancy et al., 2000; Richardson, 1998; Liu et al., 2003; Smith, 2004] seem to preclude slow fallout of dust from global dust storms as the source of the haze [Basu et al., 2004], suggesting that dust devils or small convective dust storms might instead play a role in maintaining the background dustiness.  Approximately 2 _ 10_2 kg m_2 yr_1 of dust must be removed from the Martian surface to support the observed atmospheric haze [Pollack et al., 1979], a value confirmed by dust settling rates found at the MPF landing site [Rover Team, 1997], and so the following question is posed: Can dust devils account for this amount of dust lifting? Balme et al. [2003b] using dust devil track densities observed in Argyre Planitia and Hellas Basin as an estimate for the mean of the whole of the Martian surface found that dust devils alone could not account for this flux but stressed that it is unknown what percentage of dust devils leave tracks, an observation supported by a lack of consistency between areas of high dust devil track density and areas observed to have frequent active dust devils [Fisher et al., 2005]. Ferri et al.  estimate that the local dust devil removal flux from the MPF site was an order of magnitude larger than required to support the background haze, and Fisher et al.  estimate that the dust devils flux in Amazonis was an order of magnitude higher still.  The differences in these studies highlight the problem of making global estimates from limited or local data sets, and it seems that an estimate of the total dust flux can only be made from orbiter observations of tracks or active dust devils. The two approaches can be summarized by equation (6) for active dust devils and equation (7) for dust devil tracks. For observations of active dust devils, D _ NF L A; ð6Þ where D is the total dust devil dust removal per year, N is the global number of dust devils that occur per Martian year, F is the mean removal flux for dust devils on Mars, L is the mean lifetime for a dust devil on Mars, and A is the mean instantaneous area each dust devil acts upon. The main advantages of this scheme are that it uses observations of active dust devils and that the number of dust devils observed in images from orbiters can be calibrated by lander images from the same sites. Another advantage is that it might be possible to estimate F by combining the Renno model with results from threshold and flux experiments and data on the availability of surface dust at a given site. Disadvantages of this method include the poor resolution and limited temporal and spatial coverage of images, possibly leading to poor statistics, and a failure to include the smaller, and perhaps most common, dust devils. Also, it is difficult to estimate flux per dust devil from simulations or from lander observations without dedicated sampling systems on Mars. For observations of dust devil tracks, (see the full text in pdf) where _ is the fraction of all dust devils that leave tracks, Nt is the number of tracks formed on Mars per year, At is the area of the average dust devil track, and mt is the mean mass of material removed per unit area to form a track. The biggest challenges this technique faces are estimating how much removed material a typical track represents and estimating _ for given locations and seasons. Again, lander observations are likely to be vital here (recent observations have provided some preliminary data [Metzger, 2005]), but laboratory simulations of the effects of vortices on analogue surfaces will also be important.  It is likely that empirical measurements will not prove sufficiently accurate to answer the question, and another technique must be employed. Recently, schemes to model dust devil flux within GCMs were developed [Newman et al., 2002; Basu et al., 2004]. In these schemes, dust flux by dust devil lifting is calculated at scales below the resolution of the climate model as a function of atmospheric parameters determined in the GCM. Alongside the dust devil parameterization is a boundary layer scheme that relies on saltation impact to trigger dust lifting.  Both Newman et al.  and Basu et al.  used the Renno thermodynamic model to derive a value of dust devil activity, L, based only on the sensible surface heat flux (from insolation), the depth of the boundary layer, and a tunable ‘‘efficiency parameter.’’ Newman et al.  used two parameterizations, one in which the dust devil injection flux was simply proportional to L and another that used a threshold criterion from early laboratory dust devil threshold experiments [Greeley and Iversen, 1985]. Basu et al.  did not use a specific dust-devil-lifting threshold criterion. Newman et al.  tuned their scheme to match observations of opacity and presented their results in arbitrary units of flux, whereas Basu et al.  tuned their free parameter to match year-round air temperature and used the ‘‘best fit’’ value to determine L.  Newman et al.  found that dust devil activity was greatest in a broad band at _±30_ latitude in each hemisphere’s summer and that a dustier atmosphere led to less dust devil activity. Very little lifting occurred poleward of 40_ latitude in either hemisphere. Both Newman et al.  and Basu et al.  found Amazonis to be an area of particularly high dust devil erosion in agreement with observations. Basu et al.  also found peak activity at middle/low latitudes in summer. In addition, the average northern hemisphere dust flux required to verify that their model corresponds well with the measurements of dust devil flux from MPF [Ferri et al., 2003]. Basu et al.  note that their dust devil scheme alone cannot initiate dust storms but that their boundary layer scheme cannot initiate dust storms and at the same time maintain the haze. Recent observations of active dust devils and tracks [Cantor and Edgett, 2002; Balme et al., 2003b] also show no evidence for dust devils triggering dust storms. The modeling tends to confirm observations that dust devils are not triggers for global dust storms but probably are responsible for maintaining the haze.
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 The latitudinal distributions of dust devil activity found in both models do not agree with the observed distribution of dust devil tracks discussed in section 4.3. Whether this is due to a lack of observational data or a flaw in the models is unknown. However, neither model accounts for the actual availability of dust at the surface, and factoring in this parameter perhaps from the dust cover index of Ruff and Christensen  might enhance the agreement. The combination of observations and modeling suggests that dust devils are the dominant process for maintaining the Martian haze, although confirmation awaits a global study of active dust devils or tracks. 7. ARE DUST DEVILS HAZARDS TO THE EXPLORATION OF MARS?  On Earth, dust devils do not form significant hazards to humanity. There are isolated reports of dust devils damaging temporary or half-built buildings [Idso, 1974], weather stations, and outdoor storage yards [Ives, 1947], and certainly, they can be a hazard to light aircraft during takeoff or landing [Hess and Spillane, 1990], but, in general, their main threat is to air quality in arid regions [Gillette and Sinclair, 1990; Mattsson et al., 1993].  On Mars, dust devils are often very large but are unlikely to pose a great physical threat. Although their wind speeds are poorly constrained, they are likely _100 m s_1, and while such speeds would be devastating on Earth, the thin Martian atmosphere means they likely will be no more harmful than dust devils on Earth. Close observations of MER images of dust devils might be searched for evidence of larger clods or pebbles in the flow as this might indicate stronger winds than exist on Earth and could suggest dust devils being more dangerous. The high particle content in dust devils was thought to pose a degradation hazard to solar panels on landing craft, but MER results have shown that the passage of a dust devil or wind gust over the lander actually cleared air-deposited dust from the solar panels, thus improving their output. However, the high particle density in dust devils might pose another risk: electrical damage through triboelectric charging. Farrell et al.  have shown that terrestrial dust devils can have huge potential gradients, and if the same is true on Mars, this could be a significant source of electrical hazard to landers. Even though no reports of damage caused by passage of dust devils over either MPF or MER have been made, making detailed measurements is still worthwhile. Another statistic that mitigates against dust devils as a hazard is that locally they are relatively uncommon; Ferri et al.  estimate that the instantaneous fractional area coverage of dust devil activity is only _2 _ 10_4. However, as some MOC images show regions with many orders of magnitude more dust devils than this, dust devil activity must be taken into account when selecting sites for future exploration. 8. CONCLUSIONS AND FUTURE WORK  Dust devils are widespread and common phenomena on Earth, occurring throughout the world especially in arid areas. They are efficient erosional agents and can lift substantial amounts of dust-grade particles even when the ambient wind speeds are below the predicted threshold velocity for a given region. Few attempts have been made to quantify the effect of dust devils on climate, although preliminary results have shown that dust devils could form a significant part of the dust transport cycle.  On Mars, dust devils are also widespread and common, and more effort has been made to integrate their effects into the global dust cycle than on Earth because of the absence of competing mechanisms to replenish the background dust haze. Recent modeling agrees with many of the observations from orbit and the surface and suggests that dust devils are the main mechanism for day-to-day dust injection. Dust devils are also responsible for local and regional changes in surface albedo, which might have a longer-term effect on climate through changing the rate of surface heating. There is no evidence that dust devils are responsible for triggering global dust storms. Their potential as hazards to robotic and human exploration has not yet been fully assessed. A key conclusion from this work is that terrestrial and Martian dust devils are alike in many ways. They have similar morphologies and similar pressure and temperature excursions (relative to the ambient atmospheric conditions). Numerical and laboratory modeling shows that dust devils on both planets form part of the larger convective system and have similar strong erosional effects on the surface.  The Renno thermodynamic model has been used to describe individual dust devils on Earth and Mars and as a basis for dust devil dust-lifting schemes in GCMs. The model suggests that the intensity of a dust devil is a function of the surface heat flux from insolation and the depth of the planetary boundary layer. One of the primary tasks of future dust devil fieldwork on Earth must be to test this theory with thorough observations of ambient meteorological conditions together with detailed measurements of pressure and velocity within mature dust devils, specifically testing equations (3), (4), and (5). Other important terrestrial investigations include (1) fieldwork to obtain horizontal profiles of flux within the dust column at some height above the ‘‘sand skirt’’ to estimate the dust transport properties of dust devils; (2) more measurements of surface shear stress and entrained particle sizes within the lowest levels of dust devils to better constrain the dynamics of particle lifting within dust devils; (3) further laboratory tests of flux and threshold at scales as appropriate to reality as possible to support measurements outlined in item 2; and (4) fieldwork in Niger (the only known terrestrial example of dust devil tracks) to determine how dust devil tracks are formed and what amount of material is removed to create visible tracks.  Future work on dust devils on Mars includes (1) a global study to measure the distribution of dust devils to obtain data from observations of active dust devils and supported by observations of dust devil tracks that will be essential for validating GCM models and important in hazard assessment for a given location; (2) more in situ measurements on Mars to determine electrical and dust hazard potential and provide ‘‘calibration’’ of numbers, sizes, and diurnal formation rates for orbiter images of the same region; and (3) measurements of wind speeds, pressure wells, and temperatures both of ambient conditions and within dust devils to test the Renno model and to constrain the particle-lifting abilities of dust devils on Mars.
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Because of the difficulties involved with making in situ measurements, developing remote sensing techniques and instruments that can be deployed on Mars is a priority.  Finally, it is likely that only high-resolution numerical models will allow a full understanding of dust devil formation, and as such they must be integrated into GCM models. Bridging the gap in resolution between local LES models and GCM models is not likely to be accomplished in the near term simply by using faster computers; some degree of parameterization of activity is needed. Similarly, empirical laboratory results for flux and threshold for particle lifting by vortices seem to be the only available option for developing a reliable dust-lifting scheme, and they must also be integrated into numerical models. A modern GCM that includes parameterization of dust devil formation and dust-lifting ability, together with good remote sensing and in situ data on the type and availability of surface materials, will provide a powerful tool in understanding the global climate and surface interactions of dust devils on Mars and on Earth. GLOSSARY Rankine vortex: A simple two-dimensional model of swirling flow in which the tangential velocity increases linearly with radius until a characteristic radius, at which point the tangential velocity then decreases as the inverse of radius. This means that vorticity is constant within the characteristic radius and zero outside it. Superadiabatic lapse rate: A lapse rate (vertical change in temperature) steeper than the dry adiabat (the lapse rate at which a dry parcel of air rising in the atmosphere cools without exchanging energy to the surroundings). Superadiabatic lapse rates usually only occur near the surface as a result of insolation of dry soil under clear skies and windless conditions. Cyclostrophic: In a swirling flow a case in which the pressure gradient and centripetal forces are balanced. Lidar: Specifically Doppler lidar. Used to remotely determine the velocity of particles in a dust devil. A laser beam is directed at the dust devil, and the wavelength change of light reflected from the entrained dust is measured. This allows precise measurements of the wind speed, but the temporal resolution can be poor.
Extensive valley glacier deposits in the northern mid-latitudes of Mars: Evidence for Late Amazonian obliquity-driven climate change
J.W. Head, D.R. Marchant, M.C. Agnew, C.I. Fassett and M.A. Kreslavsky
Understanding spin orbital parameter-driven climate change on Mars prior to 20 Ma ago requires geological evidence because numerical solutions for that period are chaotic and non-unique. We show geological evidence that lineated valley fill at low mid-latitudes in the northern hemisphere of Mars ( 37.5° N) originated through regional snow and ice accumulation and underwent glacial-like flow. Breached upland craters and theater-headed valleys reveal features typical of erosion in association with terrestrial glaciers. Parallel, converging and chevron-like lineations in potentially ice-rich deposits on valley floors indicate that flow occurred through constrictions and converged from different directions at different velocities. Together, these Martian deposits and erosional landforms resemble those of intermontaine glacial systems on Earth, particularly in their major morphology, topographic shape, planform and detailed surface features. An inferred Late Amazonian age, combined with predictions of climate models, suggest that the obliquity of Mars exceeded a mean of 45° for a sustained period. During this time, significant transfer of ice occurred from ice-rich regions (e.g., the poles) to mid-latitudes, causing prolonged snow and ice accumulation there and forming an extensive system of valley glaciers.
Keywords: Mars ; glaciation; lineated valley fill; climate change; deuteronilus Mensae
Recently, new spacecraft data for Mars Global Surveyor and Mars Odyssey, and insight into the nature of glaciation in Mars-like hyperarid cold polar deserts on Earth (e.g., ) have supported earlier hypotheses  that glaciation might have occurred on Mars in non-polar regions. New solutions and predictions for historical variations in the spin orbital parameters of Mars  show that the obliquity of Mars varied significantly from its present unusually low value (25.19°). These new solutions permit robust predictions of parameter variations over the last 20 Ma, but prior to that time the solutions are chaotic and non-unique. The solutions predict that the maximum obliquity over the history of Mars may have reached 82°; the standard model of insolation parameters  over 4 Gyr predicts a most probable obliquity value of 46°. Due to the non-uniqueness of solutions for orbital parameter variation during this time, one must rely on the geological record to provide evidence for the nature of climate change. Furthermore, atmospheric general circulation models taking into account the redistribution of polar volatiles and water vapor in the atmosphere equator-ward during periods of high obliquity predict deposition and retention of ice and snow at lower latitudes during these times , ,  and . Thus, it is appropriate to examine the geological record for evidence of the former presence of water and ice at mid-latitudes as one key indicator of volatile transport, climate change, and orbital parameter variation. Here we examine mid-latitude lineated valley fill to assess the presence and state of water in its formation and evolution in order to address the issue of past climate change.
Among the hallmark morphologies of the highland–lowland boundary region in the northern mid-latitude Deuteronilus–Protonilus Mensae area (30–50° N, 10–45° E) (Fig. 1) is the fretted terrain , consisting of (1) debris aprons that surround many of the massifs and valley walls and (2) lineated valley fill that occurs on the floors of many of the valleys , , , , , , , , , ,  and . The ages of these deposits are typically much younger than the adjacent plateau terrain or its breakup and the formation of the valleys themselves (e.g.,  and ). The margins of the debris aprons consist of rounded and convex upward topography, and at Viking resolution the debris aprons and the valley fill can appear smooth and relatively homogeneous or, in contrast, can be characterized by closely spaced parallel ridges and grooves a few to several tens of meters high. Some workers (e.g., ) argue that the valley fill lineations form mostly normal to the direction of flow due to converging flow from debris aprons on opposite sides of valleys or mesas, while others (e.g., ) argue that bending of ridges and grooves entering valleys from a side tributary supports flow in the direction along the valley.
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Recent analysis also reveals that lineated valley fill displays topographic slope reversals, interpreted to mean that along-valley flow was minimal . All agree that the materials represent some sort of viscous flow processes, but opinions differ on the details of the mechanism; most authors call on processes of gravity-driven debris flow, assisted by ice or water in the debris interstices, derived from either groundwater or diffusive water vapor exchange with the atmosphere (e.g., see , ,  and ). Some liken the process to rock-glacial flow (e.g.,  and ) with the source of the major deforming agent being ice from atmospheric frost deposition and diffusion into rock debris pore spaces  or, alternatively, mobilized interstitial ground ice . In this contribution, we assess whether glaciers, derived from snowfall, may have been a factor in the formation of lineated valley fill.
What criteria for the recognition of glaciers, past and present, can be employed in order to examine this possibility? Four types of observations can provide insight into the distribution of glacial-erosional landforms and deposits : (1) presence of the major morphologies and zones typical of glaciers (e.g., accumulation and ablation zones), (2) topographic variations (convex-upward profile and generally sloped in a down-flow direction), (3) planform (e.g., source regions, tributaries, convergence zones, main trunk valleys, etc.), and (4) surface features (e.g., lineations, crevasses, lateral, medial and terminal moraines, evidence of converging flow and surface deformation, etc.). We analyzed new THEMIS (Mars Odyssey), MOLA and MOC (Mars Global Surveyor) data to assess the nature and origin of lineated valley fill, and found evidence that glaciation (accumulation of snow and ice to sufficient thickness to cause its local and regional flow) has played a significant regional role in its formation.
1.1. New observations
We have analyzed numerous areas along the dichotomy boundary north of 30°N latitude and present in detail the results from an area in southern Deuteronilus Mensae, where a T-shaped valley occurs just south of a large depression (Fig. 1). The walls of the large depression are characterized by debris aprons and there is a break in the southern rim of the depression that leads to the top of the T-shaped valley about 100 km across. A topographic profile (Fig. 1) from the floor of the large depression across the southern wall, across the top of the T and along the meridional part of the T shows (1) the flat floor of the large depression, (2) a pole-facing convex-upward slope of the debris apron, (3) the elevated floor of the generally WNW-trending top of the T, and (4) the convex-upward slope of the generally NS-trending meridional part of the T. Note that the floor of the T-shaped valley along the top of the T lies at elevations as high as − 2600 m, almost 1 km above the large depression floor. The bottom of the valley along the vertical part of the T lies 500 m below the valley floor at the top of the T, implying a thickness of the valley fill in excess of hundreds of meters.
THEMIS image data (Fig. 2A) superposed on MOLA altimetry and viewed perspectively show lineations typical of valley fill (Fig. 2B), their characteristics and their directions of flow; details of the lineations are shown in MOC images and compared to terrestrial glacial features (Fig. 3). Examination of Fig. 1 and Fig. 2 shows a 15-km-diameter breached crater south of the eastern arm of the T (Fig. 2B, arrow A). At the southern interior wall of the crater, material is banked up against the wall and shows evidence of flow lineations extending away from the wall. Elongated slab-like features lie parallel with the wall and are separated by linear troughs (Fig. 3A). These features are very similar to bergschrunds (fractures separating flowing from stagnant ice) and seracs (isolated ice blocks) seen in terrestrial accumulation zones  (Fig. 3B). The presence of these characteristic features here, but the lack of smooth deposits typically seen in terrestrial ice accumulation zones, suggests that snow, if present at one time, no longer covers the surface, most likely having sublimed away. Downslope, the elongated slabs give way to numerous lineations on the floor of the crater (Fig. 2B, arrow C); these trend away from the southern crater wall, converge (Fig. 3C,D), and can be traced through the gap in the crater wall, joining lineations beginning at the eastern edge of the T (Fig. 2 and Fig. 3). This configuration is typical of constricted flow and convergence between ice masses moving at different velocities, causing chevron-like shear patterns  (Fig. 3D–F). Similar lineated valley fill extends from the mouth of a north-trending valley in the lower western part of the T (Fig. 2), is deformed by lineations from valley fill moving from the higher terrain to the west, and then joins the general lineated valley fill just to the east (Fig. 2B arrow G; Fig. 3G and H).
Along the southern margin of the western part of the T (Fig. 1 and Fig. 2), a portion of the lineated valley fill lies at the edge of the valley floor, but topographically above the central lineated valley fill (Fig. 2B, just below arrow G). The floor lineations are seen to be ridges interpreted to be debris-rich medial moraines (Fig. 3G and H). The terrace-like feature is typical of marginal stagnant portions of glacial flow on Earth which form and are isolated by changing flow dynamics (Fig. 3H, arrow). In terrestrial glaciers, this detachment and isolation usually occurs during the lowering of the central glacial surface, shear separation, and isolation of the marginal terrace due to the establishment of a new flow margin .
At the western part of the top of the T, additional lineated valley fill begins at the base of the broad amphitheater and converges at the T-junction with the lineated valley fill coming from the west and the east (Fig. 2A and B). From here, these three major flow lineation directions, together with several smaller ones (Fig. 2C), converge and extend down the vertical part of the T, with many of the lineations contorted within the area of the convergence (Fig. 2B, arrow I; Fig. 3I and J). At the end of the major lobe, the topography is broadly convex upward (Fig. 1) and the perspective view (Fig. 2B) shows the distinctive lobe-like nature of the valley fill as it extends into the adjacent low-lying terrain, displaying morphological relationships similar to distal glacier deposits in terrestrial environments (Fig. 3K and L).
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1.2. Origin of Lineated Valley Fill
What processes are responsible for the valley fill? We see evidence in the lineated valley fill for features typical of terrestrial valley glaciers  (e.g., crevasses at accumulation zones and between ice moving at different velocities, tributaries converging toward main trunk valleys and generating lineated debris bands, some of which show surface convolutions suggestive of flow deformation, convergence zones, converging flow and surface deformation, and lateral and medial moraines). Thus, the lineated valley fill and its complex patterns (Fig. 2C) resemble flow lines in glacial ice on Earth, particularly where glacial ice converges from different directions at different velocities and deforms into complex patterns (Fig. 3C, E, and I). Detailed analysis of the MOC, THEMIS and MOLA data suggests that changing environments and local topographic conditions (such as the crater walls and the narrow valleys) favored accumulation and preservation of snow and ice, and its glacial-like flow down into surrounding areas for distances approaching 70 km (Fig. 2B and C). Such local ice accumulation zones are typical of debris-covered glacial flow in the Antarctic Dry Valleys , a cold polar desert analogous to the environment on Mars , and of regional valley glaciers and plateau icefield landsystems in Baffin and Ellesmere Islands  and  in the Canadian Arctic. For example, the regional valley glaciers associated with the Ellesmere Island plateau icefield landsystem show very similar relationships of accumulation zones, valley fill and down-valley flow, and convergence and ice flow deformation (compare Fig. 2 and Fig. 4).
Alternative hypotheses for the origin of lineated valley fill focus on transport inward from the valley walls to form the lineations, and minimal subsequent down-valley movement (e.g.,  and ). The evidence documented here in the higher resolution THEMIS and MOC data strongly suggests an integrated flow network; although flow may in places start at crater and valley walls and move toward valley centers, the predominant flow is along-valley. The variable and undulating along-valley topographic profiles typical of lineated valley fill (reversing gradients) have been cited as evidence that flow is more likely to be locally across-valley rather than systematically along-valley . The integrated pattern of flow lineations documented in the new data (Fig. 2B and C), however, strongly supports along-valley flow, but also show that local accumulation zones and trunk valley convergence can produce along-valley topographic undulations. Furthermore, subsequent loss of volatiles and variation in the thickness of protective sublimation tills will enhance along-valley topographic variations.
Additional arguments against along-valley flow have included observations that lineated valley fill is found in valleys enclosed at both ends, and that floor textures of lineated valley fill often mimic the shape of valley walls . Both of these observations are consistent with the glacial hypothesis. For example, enclosed valleys can contain multiple accumulation zones and the resulting lateral flow in the enclosed depression is accommodated by sublimation and volatile mass loss. Furthermore, accumulation zones can occur along valley walls if the geometry is conducive to preferential accumulation of snow and ice; lineated deposits that mimic wall topography can then be produced as ice moves onto the valley floor. In addition, along-valley flow will produce lineations that reflect the basic topographic characteristics of the valley walls.
Most alternative theories to glaciation for the origin of lineated valley fill call on lubrication and flow of rock debris and focus predominantly on groundwater, ground ice or ice from atmospheric water vapor diffusion as the origin of the lubricant (e.g., , , , ,  and ). While such processes are likely to occur (for example, near-surface groundwater flow was probably more important in the Noachian, prior to about 3.7 Gyr ago, when the geothermal heat flux was much higher), we believe that the great lateral extent, continuity and direction of flow lineations, and their complex interactions consistent with glacial-like flow, are all evidence that supports glacial-like flow of debris-containing ice as the dominant process in formation of lineated valley fill in this region, rather than local flow of debris with an icy component in pore spaces originating from simple atmospheric water vapor diffusion.
The relationship between the lineated valley fill and the lobate debris aprons is not yet firmly established; however, the contiguous nature of many examples of lineated valley fill and debris aprons (Fig. 1) suggests that if the glacial interpretation of the valley fill is supported by further observations, then glacial ice may play more of a role in the formation of debris aprons than previously suspected (e.g., ).
2. Discussion and implications
Breached upland craters and theater-headed valleys reveal features typical of terrestrial intermontaine glacial settings, parallel lineations on valley floors resemble flow lines in glacial ice, converging lineations resemble ice flow through constrictions, and complex chevron-like flow patterns occur where lineated valley fill converges from different directions (Fig. 2 and Fig. 3). This example in the Deuteronilus Mensae region shows an integrated pattern (Fig. 2B and C) interpreted to represent snow and ice accumulation and along-valley flow for 70 km. These patterns resemble valley glacial systems on Earth in major morphology, topographic shape, planform and detailed surface features (compare Fig. 2, Fig. 3 and Fig. 4).
The current atmosphere and insolation conditions on Mars are not conducive to snow and ice accumulation at these latitudes . Recent developments in the modeling of the orbital parameter variations thought to be the drivers of climate change on Mars  have shown that robust predictions can be made only back to 20 million years before the present. Therefore, geological evidence is required to help distinguish among the family of plausible orbital parameter histories. The data presented here imply that, earlier in the Late Amazonian period (which extends from 400 million years ago to the present), significant climate change occurred which caused sustained snow and ice accumulation and flow at mid-latitudes to form a regional system of valley glaciers. Due to protective sublimation tills, some of these features may still be cored with glacial ice.
The age of similar deposits elsewhere in this region have been estimated to be Late Amazonian ( 300 Ma, with some of the deposits as young as 10 Ma) . This suggests that there may have been periods during the Amazonian when mid-latitude glaciation was extensive.
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What might the mid-latitudes of Mars have looked like during these periods? In the Deuteronilus region (Fig. 1 and Fig. 2) local accumulation zones in breached craters and tributary valleys clearly fed the glaciers. In some areas of the lineated valley fill in this latitude range, evidence has recently been presented that valleys containing lineated valley fill were once filled with ice which overflowed and extended out onto the marginal plateaus, leaving deposits after they receded back down into the valleys . The distribution of ice on the Ellesmere Island Agassiz Ice Cap, a plateau icefield landsystem , is an instructive example (Fig. 4). Accumulation zones include large expanses of adjacent high plateaus characterized by ice caps which then flow down into preexisting valleys into a marginal series of converging valley glaciers, that ultimately sublime and melt or sometimes extend out onto surrounding plains to produce local piedmont glaciers. Furthermore, on Mars, recent modeling of glacial flow in lineated valley fill suggests that the distribution of snow and ice would be much more likely to include regional ice deposits feeding valley fill and that the current valley fill configuration may be the remnants of a more extensive regional ice sheet .
What would cause the formation of regional snow and ice deposits at these latitudes during the Late Amazonian? Recent advances in global climate modeling have permitted the examination of the fate of volatiles such as H2O in response to changes in global patterns of orbital parameter-driven insolation. These studies , ,  and  show that at obliquities of 45° or more, water vapor mobilized from the polar regions is redistributed at mid-latitudes, is stable there as ice, and will accumulate if obliquity remains at these values. Although robust prediction of obliquity beyond 20 Ma is not possible, statistical studies of the possible behavior of obliquity over the last 250 Ma have been performed  and the range of solutions include scenarios from about 50 to 250 Ma with obliquity values from 5° to 65°. The geological evidence reported here supports orbital parameter scenarios where mean obliquity exceeds 45° during the Late Amazonian for a sufficiently long period to cause the observed ice accumulation and glacial flow. Tropical mountain glacier deposits have been documented in the equatorial regions of Mars  and  and glacial flow simulations  suggest significant atmospheric precipitation and accumulation. Furthermore, climate models  suggest that upwelling of water-rich air on the western flanks of Tharsis and the Tharsis Montes causes adiabatic cooling, snow precipitation and rapid accumulation sufficient to produce sustained snow and ice cover and glacial flow . We hypothesize that similar conditions occurred in the Deuteronilus–Protonilus region (Fig 1) when, during periods of higher obliquity, water-rich polar air encountered the several-kilometer-high dichotomy boundary scarp, rose and underwent adiabatic cooling, causing snow precipitation, regional ice accumulation, and valley glacial systems here and elsewhere (e.g., ) along the dichotomy boundary at this latitude range.
The detailed preservation of textures typical of glacial surface and near-surface features (e.g., lineations, folds, craters) suggests that the current surface of this deposit is close to that of the original surface, and that downwasting due to the loss of ice from underneath a surface sublimation till has been retarded. This interpretation is supported by the topography of the deposit (Fig. 1) which indicates that the deposit is many hundreds of meters thick. These data suggest that the debris till overlying the glacier deposits may have severely retarded the sublimation rate of the ice beneath the till. Geological evidence for ancient ice (millions of years old) has been described in the Antarctic Dry Valleys . If the lineated valley fill surface is indeed ancient and underlying ice is still preserved, this ice could provide a record of ancient climate change at latitudes readily accessible to future automated and human exploration.
Forward modeling of ice topography on Mars to infer basal shear stress conditions M. E. Banks and J. D. Pelletier  Understanding the history of ice caps on Mars could reveal important information about Martian geologic and climatic history. To do this, an ice reconstruction model is needed that operates over complex topography and can be constrained with a limited number of free parameters. In this study we developed a threshold-sliding model for ice cap morphology based on the classic model of Nye later incorporated into the models of Reeh and colleagues. We have updated the Nye-Reeh model with a new numerical algorithm. Although the model was originally developed to model perfectly plastic deformation, it is applicable to any ice body that deforms when a threshold basal shear stress is exceeded. The model requires three inputs: a digital elevation model of bed topography, a ‘‘mask’’ grid that defines the position of the ice terminus, and a function defining the threshold basal shear stress. To test the robustness of the model, the morphology of the Greenland ice sheet is reconstructed using an empirical equation between threshold basal shear stress and ice surface slope. The model is then used to reconstruct the morphology of ice draping impact craters on the margins of the south polar layered deposits using an inferred constant basal shear stress of _0.6 bar for the majority of the examples. This inferred basal shear stress value is almost 1/3 of the average basal shear stress calculated for the Greenland ice sheet. What causes this lower basal shear stress value on Mars is unclear but could involve the strain-weakening behavior of ice. 1. Introduction  The Martian polar regions have accumulated extensive mantles of ice and dust with volumes of _106 km3 [Smith et al., 1999]. The layered deposits in these regions are believed to preserve a record of seasonal and climatic cycling of atmospheric carbon dioxide, water, and dust over the past hundred million to perhaps even billion years [Pathare et al., 2005]. In addition to the polar caps, persuasive evidence of past glacial processes has been identified in areas such as the Argyre Basin, east of Hellas Basin, the northern fretted terrain, and among the Tharsis volcanoes [e.g., Head, 2000; Head and Marchant, 2003; Head et al., 2005, 2006; Moore and Davis, 1987; Kargel, 2004; Kargel and Strom, 1990, 1992]. Understanding more about the history of ice on Mars could reveal important information about Martian geologic and climatic history including variations in insolation, orbital parameters, volatile mass balance, large impacts, volcanic eruptions, catastrophic floods, and solar luminosity.
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To learn more about the past and present ice of Mars and its climatic and geologic implications, a model is needed that operates over complex topography capable of both ice reconstruction and forward modeling of erosion and deposition.  Previous approaches to the modeling of ice cap geometry and flow on Mars can be divided into simple ‘‘flow band’’ models that predict ice surface elevations along a one-dimensional (1-D) flow band in the x direction, and more sophisticated ‘‘map plane’’ models that predict surface elevations in the x and y directions. Models previously developed by Schmidt and Buchardt , Fountain et al. , Moore and Davis , and Hvidberg  have given valuable insight into possible mass balance processes, flow velocities, and the flow regime surrounding scarps and troughs of the Martian polar ice sheets. A wide range of information about the ice on Mars has also been provided by the more sophisticated models of Fisher , and Pathare et al. . Particularly important is the dynamic/thermodynamic ice sheet model SICOPOLIS [e.g., Greve, 2000; Greve et al., 2003, 2004; Greve and Mahajan, 2005; Mahajan et al., 2004]. Generally speaking, flow band models are advantageous in that they are more readily calibrated and interpreted. They are well suited for first-order ice sheet modeling but are of limited usefulness when applied to complex or steep bed topographies. Map plane models are more versatile and accurate in principle but require input parameters that may be difficult to constrain. A model is needed that incorporates the physical realism of the map plane models and their ability to work with complex bed topographies with the simplicity of flow band models. This becomes particularly important when reconstructing bodies on other planets, such as Mars, where parameter estimation may be far more challenging than on Earth.  In this study, we developed a threshold-sliding model based on the work of Nye [1951, 1952a, 1952b] and Reeh [1982, 1984] that is designed to capture the realism of map plane models with complex topography while minimizing the number of required input parameters. Nye’s  classic work on the geometry of perfectly plastic ice sheets and glaciers has long been used to estimate the thicknesses of former ice sheets. In his work, Nye developed a differential equation for the ice surface topography above an arbitrary or known bed if flow lines are known, which was later incorporated into the models of Reeh and colleagues [e.g., Reeh, 1982, 1984; Fisher et al., 1985]. The Nye-Reeh model was found to be applicable to ice sheets and glaciers that move when a threshold shear stress has been exceeded and was successfully used to reconstruct the Antarctic ice sheet [Reeh, 1982, 1984]. However, this model utilized a complicated computational approach.  In this paper we update the Nye-Reeh model with a new numerical algorithm. In this model, the geometry of present and past ice sheets and glaciers can be reconstructed using information about bed topography, ice margin positions, and threshold basal shear stresses. The advantage of this model, especially when applied to the surfaces of other planets, is that it requires constraints on only one physical parameter: the threshold basal shear stress. Research on Earth suggests that the value of the basal shear stress is constant on spatial scales of tens of kilometers and varies in a systematic way at larger spatial scales. While the threshold- sliding model works well in many cases, it has several well known limitations. For example, it cannot model the thermal effects on basal conditions nor can it represent surging and tidal forces when ice interacts with the ocean.  We test the model by simulating the ice thickness and morphology of the Greenland ice sheet. By adjusting the model to the appropriate surface conditions on Mars, we also demonstrate its potential in reconstructing the ice surface topography of modern and former ice sheets and glaciers on other planets. We also constrain modern basal shear stresses and discuss the implications for bed environmental conditions on Mars. To do this, we use the thresholdsliding model to reconstruct ice that drapes impact craters near the margins of the south polar layered deposits (SPLD). This application was chosen because it enables us to make relatively accurate predictions about the ice margins and the bed topography underneath the ice due to the radial symmetry of impact craters. The threshold-sliding model also provides a basis for a glacial erosion model that can be used to better understand the spatial distribution of glacial erosion for areas such as the Argyre basin on Mars. 2. Model Description 2.1. Previous Work  Field studies of modern ice sheets and glaciers on Earth have often shown that basal shear stress values lie between 0.5 and 1.5 bars [Nye, 1952a, 1952b], a surprisingly narrow range considering the spatial variability of observed basal conditions, including gradients in temperature, meltwater content, basal debris, till rheology, and other variables. Nye [1951, 1952b] developed a differential equation for the ice surface topography above an arbitrary or known bed if the flow lines are known, which was later incorporated into the models of Reeh and colleagues [e.g., Reeh, 1982, 1984; Fisher et al., 1985]. While his equation yields parabolic and logarithmic solutions for flat and inclined beds, solutions for more complex bed topography can be obtained numerically. The Nye-Reeh model is not only associated with perfectly plastic deformation of near-basal ice but is also applicable to complex basal mechanisms such as debris controlled frictional sliding and till deformation [Reeh, 1982, 1984]. It can be used with any ice sheets and glaciers that move when a prescribed threshold shear stress is exceeded whether relative motion occurs internally or by sliding over deforming till or directly over the bed [Reeh, 1982, 1984]. Threshold behavior may characterize these basal processes under many conditions, including the plastic behavior of deforming till [e.g., Tulaczyk et al., 2000] and the shear stress necessary to overcome kinetic friction during basal sliding [e.g., Lliboutry, 1979]. In this paper we use the term ‘‘threshold sliding’’ to refer to the class of ice sheets and glaciers that move when a threshold basal shear stress is exceeded.  Critical to Nye’s success in developing a single, independent equation for the geometry of ice sheets was his observation that the ice sheet geometry is independent of flow when the relative motion of the ice sheet and its bed is governed by a threshold shear stress.
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In these cases, the local ice thickness continually adjusts to the local ice surface slope to maintain a balance of forces independent of the pattern of accumulation and ablation. Accumulation and ablation rates affect the velocities of ice sheets and glaciers, but not their geometries as long as a threshold condition exists for motion and the margin is fixed or is a prescribed function of time. Used in conjunction with the Nye-Reeh model, information on the position of the ice margin enables the rates of accumulation and ablation to be eliminated from the reconstruction as long as the margin is static. 2.2. Basal Shear Stress  The basal shear stresses in ice sheets and glaciers are given by [Nye, 1951] (see the full text in pdf) where r is the density of ice, g is the gravitational acceleration, h is the ice thickness, and S is the ice surface slope using a small angle approximation in which sin(S) = S. Equation (1) assumes that gradients in longitudinal stresses are negligible, an accurate approximation for spatial scales several times the ice thickness [Hooke, 2005]. Field studies have long recognized that basal shear stresses calculated with equation (1) commonly range from about 0.5 to 1.5 bars [Nye, 1952a, 1952b]. Nye  used this observation to invert equation (1) to solve for ice thickness. Expressed as a function of ice thickness, h(x), and bed topography, b(x), S is given by S = jdh/dx + db/dxj. Using this expression, equation (1) becomes a nonlinear differential equation for h: (see the full text in pdf) Equation (2) can be solved if t and b(x) are known and proper boundary conditions are applied. The boundary conditions for equation (2) are h = 0 at the two ice margins on either side of the divide.  In addition to using a constant basal shear stress, the Nye-Reeh model can also be extended to include the spatial variability of basal shear stress observed in large modern ice sheets on Earth. Figure 1a is a gray scale image of the basal shear stresses in Greenland calculated from equation (1). This image clearly shows that basal shear stresses increase with distance from divides, with values ranging from 0.5 bar near divides to roughly 3 bars near margins, and have an average value of _1.41 bars.  One means of incorporating the observed spatial variability of basal shear stresses into the model is to consider them as a function of ice surface slope. In Figure 1b, the basal shear stresses of Greenland are plotted as a function of ice surface slope on a logarithmic scale. The plot illustrates that basal shear stresses increase from values as low as 0.5 bar at low slope values to higher, more variable values as slope values increase. A least squares power function fit to this data, (t = 15S0.55), is indicated by the solid line in Figure 1b. Certainly a higher-order fit would characterize the data more precisely because the data set has a significant convex curvature in its dependence on ice surface slope. However, t = 15S0.55 provides a useful firstorder correction to uniform basal shear stresses. Quantifying basal shear stresses as a function of distance from divides was also investigated but did not provide as clear a correlation. Power function fits such as t = 15S0.55, characterize most of the spatial variability in basal shear stress, but the limitations of the power function can be recognized in the reconstructions. In particular, divide regions in the model reconstructions have ice surface slopes that are noticeably steeper than in real ice sheets. However, these inaccuracies are limited to the divide region. Therefore, for large ice sheets, power function fits can be used as the basis for incorporating nonuniform basal shear stresses in the threshold- sliding model by replacing t with t(S) in equation (2). 2.3. Ice Sheet and Glacier Reconstructions in Three Dimensions  In three dimensions, equation (2) applies along the direction of flow lines. Since flow lines are parallel to the local ice surface gradient, the 3-D version of equation (2) is obtained by replacing S = jdh/dx + db/dxj with S = jrh + (see the full text in pdf) The basal shear stress may also be written as a function of S, t(S), to incorporate spatial variations as observed in Figure 1. As a solution to equation (3), we have developed a straightforward algorithm based on the accumulation of discrete ‘‘blocks’’ of ice on a grid. Our method mimics the accumulation of ice thickness and slope until a threshold is reached (illustrated schematically in Figure 2). Before the algorithm begins, grid points within the ice margin are identified. Ice will accumulate at these points while the other grid points remain ice free. The fundamental action item for each allowed grid point is simple: add a discrete unit of ice thickness if the resulting surface slope is less than a threshold value given by t/rgh. The ice surface slope is calculated as S = (Sx 2 + Sy 2)1/2 where Sx is the downhill slope in the x direction and Sy is the downhill slope in the y direction. During each ‘‘sweep’’ of the grid, the algorithm attempts to add ice to all of the allowed grid points. The grid is swept repeatedly until no additional blocks can be added. The end result is an exact solution to equation (3) for any bed topography. The value of t can be taken to be uniform in this model or it can be a function of the ice surface slope. Each sweep through the grid is from lowest to highest elevations to avoid oversteepening and to make the model more efficient. A coarse block size is used initially and is reduced in size by a constant factor once no more blocks can be added to the grid. This enables a coarse solution to be obtained quickly which is gradually smoothed as the block size is reduced. It should be emphasized that although this algorithm method is discrete, it is not a cellular automaton or otherwise an approximation to equation (3). Instead, it is an iterative numerical method that obtains an exact solution to equation (3).  It should be noted that there are different methods that have been applied to ice sheet modeling.
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Traditionally, ice sheet modeling has been based on prescribing climatic factors, such as the spatial distribution of net accumulation rate, over the potential ice-covered area and letting the ice sheet build up both vertically and laterally with free margins. This method utilizes inputs for climate or net accumulation together with ice rheology/threshold basal shear stress to model the ice sheet forward in time from some specified starting condition. However, if the ice margins are known, the ice sheet topography can also be reconstructed by starting the reconstruction method with the full lateral extent of the ice and growing the cap vertically until the threshold basal shear stress is reached. In this approach, only the ice margin and the threshold basal shear stress are needed as inputs. It is important to emphasize that this alternative method is static and does not reconstruct any of the intermediate stages in ice sheet growth. Instead, the algorithm only reconstructs the ice sheet topography for the ‘‘snapshot’’ of time at which the ice sheet occupies that specific margin. In this study we utilized this method because the spatial pattern of mass balance or net accumulation on Mars is unknown. A set of climatic conditions does exist that created the resulting ice morphology seen today. However, since these conditions are unknown, we use this method as a way to obtain the same end result. The efficiency of the model is demonstrated by comparing the reconstructed ice surface topographies with those observed. Of course, the threshold-sliding model can also be run using a spatial mass balance distribution instead of an ice terminus for instances in which the ice margin at a given time is unknown. When run this way, the model moves the ice sheet both upward and outward with the ice margin free to take on any shape in response to spatial variations in net accumulation and threshold basal shear stress. 3. Validation of the Threshold-Sliding Model to the Modern Greenland Ice Sheet  In this section we present a reconstruction of the ice surface topography of the modern Greenland ice sheet using our discrete algorithm method as a solution to equation (3). Greenland is an important case study because the bed topography is both well constrained and has a significant influence on the morphology of the ice sheet. Specifically, the principal divide in Greenland is offset from central Greenland by approximately 100 km (a flat bed would place the divide in the middle). This asymmetric profile could reflect a bed topographic control associated with higher bed elevations in eastern Greenland. Alternatively, the asymmetry of the ice sheet may be associated with an east-to-west accumulation gradient similar to that observed today [McConnell et al., 2000].  Figures 3a and 3b illustrate the bed and ice surface topography for modern Greenland [Bamber et al., 2001]. Areas below sea level are indicated in black (Figure 3a). This and all other reconstructions in the paper require three inputs: a DEM of bed topography, a ‘‘mask’’ grid defining the ice margin, and a function defining the basal shear stress. The bed topography was input directly into the model and the ice thickness data, derived from the difference between the ice surface and bed topography, was used to provide a binary ‘‘mask’’ grid defining the grid points that are allowed to accumulate ice. The mask grid has values of 1 where the ice thickness is greater than 0 (i.e., areas covered by ice), and values of 0 where the thickness is 0 (i.e., no ice coverage). For this reconstruction we used the shear stress relationship t = 15S0.55 observed in Figure 1.  Figure 3c is a shaded relief and contour map of the solution to equation (3) obtained by our discrete algorithm method. The shaded relief image has been constructed with the same vertical exaggeration (30x) as Figure 3b to provide a direct, side-by-side comparison. The similarity of the location and shape of the contours indicate that the overall solution is in good agreement with the observed topography. The location of the major divide, offset from center by approximately 100 km, is also reproduced in the model indicating that the asymmetry of the ice sheet is primarily the result of bed topographic control. One major difference between the observed and modeled topography, however, is the steepness of divides and the more angular appearance of the modeled topography. This angularity can be traced to the poor fit between the power function and the data of Figure 1a. In the lower left corner of Figure 1b, representing areas of low slope, the data fall far below the power function indicating that the threshold basal shear stress near divides is significantly lower than the values predicted by the power function. As a result, the model solution overestimates the ice surface slopes near divides by as much as a factor of 2. However, this does not introduce significant errors in the elevations of these regions because the slopes in either case are small. 4. Application of the Model to Mars 4.1. Methods  The model’s ability to accurately reproduce ice sheets such as the one in Greenland is the basis for further testing of the model on Mars. To do this, we selected 10 partially ice covered impact craters in which ice has overtopped the crater rim. Craters were specifically chosen in which relatively accurate predictions of the bed topography underneath the ice could be made and in which the ice margins could be clearly determined. All of the craters are located near or on the edges of the SPLD which are composed of ice and dust [e.g., Mellon, 1996; Durham et al., 1999; Nye, 2000; Nye et al., 2000]. The locations of each crater are indicated with black boxes in Figure 4.  As in the Greenland ice sheet example, reconstructions of ice thickness and morphology with the thresholdsliding model require three inputs: aDEM of bed topography, a ‘‘mask’’ grid defining the ice margin, and a function defining the basal shear stress. The bed topographies for the model could be determined due to the radial symmetry typical of impact craters. Craters were specifically chosen in which ice could clearly be seen to cover part but not all of the crater floor.
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Unless there has been significant subglacial erosion, the symmetry of the exposed crater form can be used to predict the morphology of the crater floor obscured beneath the ice. The software RiverTools [RIVIX, 2005] was used to extract multiple profiles through each crater from digital elevationmodels (DEMs) of the south polar cap and surrounding SPLDbased on altimetry data acquired by theMars Global Surveyor Mars Orbiter Laser Altimeter (MOLA) Instrument (data available at http://www.pds_geosciences.wustl.edu/ missions/data). Each profile started at the center of the crater and was extended out past the crater rim along surfaces not obscured by ice. A custom computer program was then used to average the extracted profiles and create 3-D bed topography assuming radial symmetry.  The margins of the ice were identified from the DEM based on changes in elevation and differences in surface texture. Profiles were extracted along the floors of the craters to determine the locations of sudden increases in elevation indicating the edge of the ice. Ice margins were also identified where the texture changed from the rough crater floor to the relatively smooth and layered appearance of the ice. These observations were later confirmed with higher resolution MOC and THEMIS images. The ice margins were converted into a binary ‘‘mask’’ grid defining the grid points allowed to accumulate ice as was done when applying the model to the Greenland ice sheet.  For the basal shear stress, we investigated the effect of both constant basal shear stress and the spatial variability of basal shear stress, t(S), on model simulations for each crater. Gravity was changed to match the gravity on Mars and ice density was given a value of 0.92 g/cm3 based on the assumption that the ice in the SPLD is primarily H20 ice [e.g., Pathare et al., 2005; Koutnik et al., 2005] and has the same physical properties as H20 ice on earth [Greve et al., 2003]. Forward modeling was then used to construct families of ice lobes corresponding to a series of basal shear stress conditions.  Once completed, the elevation of the ice in each pixel of the model reconstructions was subtracted from the elevation of the ice in the corresponding pixel of the original DEM to determine an average ice elevation difference per pixel. This was done for the ice-covered areas within the crater only and excluded the surrounding ice-covered terrain where bed topography was not well constrained. This method enabled us to quantitatively determine which values of basal shear stress most accurately reconstructed the ice thickness and to establish the uniqueness of the results. Figure 5 shows plots for two impact craters of the average ice elevation difference per pixel (in meters) for a series of basal shear stresses. As can be seen in Figure 5, similar ice thicknesses are produced for a basal shear stress range of roughly ±0.1 bar. Model results were also analyzed visually to determine which values of basal shear stress best reconstructed the original ice morphology. 4.2. Determining the Bed Topography for Craters More Than Half Covered With Ice  In determining the bed topographies for our model reconstructions, we encountered two different scenarios: crater floors that were less than half covered by ice and crater floors that were half or more than half covered with ice. For craters less than half covered with ice, the symmetry of the observed crater form could easily be used to estimate the morphology of the crater floor buried beneath the ice lobe as was already described. However, about half of the craters in our study were more than half covered with ice. In these cases, topographic information could be obtained from the rim to the base of the crater but information regarding any features potentially existing in the center of the crater floor was obscured by ice. To resolve this, it was necessary to investigate the characteristic dimensions of impact structures in the south polar region of Mars.  Impact craters can be divided into different categories with different profiles and features. Two main categories are simple craters and complex craters. Simple craters have relatively smooth, bowl-shaped cross sections with upraised rims while complex craters are characterized by one or more central mountains (central peak) surrounded by a relatively flat floor inside of a terraced rim. As complex craters increase in size, other features, such as peak rings, may appear [e.g., Baldwin, 1949; Pike, 1988; Melosh, 1989]. Whether or not a simple or complex crater is formed by an impactor depends on factors such as the size of the impactor and the gravity of the planet it is impacting [e.g., Barlow and Bradley, 1990]. For Mars, the transition from a simple to a complex crater typically occurs at diameters of roughly 6 to 7 km [e.g., Garvin et al., 2003]. In this study we look at impact craters on Mars within the diameter range of _10–90 km and are therefore only concerned with complex crater morphologies that may or may not contain a central peak. The dimensions of central peaks, such as the basal diameter of the central peak (Dcp), and height of the central peak above the observed depth of the crater (hcp), typically form in predictable ratios to the diameter (D) of the crater as measured from its rim crests (Figure 6a). The accepted relationships for these features on Mars as a whole are (see the full text in pdf) where all distances are in kilometers [Garvin et al., 2003]. These ratios, however, may change depending on factors such as the velocity of the impactor, the composition of the impacted surface, or the presence of subsurface volatiles [e.g., Barlow and Bradley, 1990; Melosh, 1989, Cordell et al., 1974; Cintala et al., 1976; Hale and Head, 1981].  In order to make accurate predictions as to the bed topography under the ice in the central part of the craters, central peaks were measured and compared for multiple complex craters in the area surrounding the SPLD. The locations of these craters are indicated with white boxes in Figure 4 and the results are shown in Figures 6b and 6c where D is plotted against Dcp and hcp, respectively.
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Least squares power function fits are marked by the solid lines in the plots (Figures 6b and 6c) and indicate the following relationships: (see the full text in pdf) where all distances are in kilometers. The difference between the Garvin et al.  relationship and our measured relationship of Dcp to D is insignificant. Therefore, for determining the diameter of potential central peaks beneath the ice, we used the Garvin et al.  equation (4). Our measured relationship of hcp to D indicates that in comparison to nonpolar craters on Mars, craters in the area surrounding the SPLD have central peaks that are lower in elevation for smaller craters (D < _15 km) and higher in elevation for larger craters. The reasons for this are beyond the scope of this paper but may include the presence of subsurface volatiles, viscous creep relaxation, and/or infilling or erosional processes in the polar regions [e.g., Garvin et al., 2000; Thomas et al., 1992; Pathare et al., 2005; Schaller et al., 2005]. Therefore, to determine the height of potential central peaks beneath the ice, equation (7) was used.  To simulate a crater with a flat floor (no central peak), the lowest elevation point on each profile was extended across the length of the crater floor. To simulate a crater floor with a central peak, the lowest elevation point on each profile was again extended and equations (4) and (7) were used to create a central cone structure of the appropriate dimensions simulating the approximate shape and elevation of a central peak. A custom computer program was then used to separately average both sets of profiles and create two different 3-D bed topographies for each crater; one representing a flat crater floor and one representing a central peak at the center of the crater floor (Figures 7a and 7b). 4.3. Results  The majority of the model results indicated a constant basal shear stress range of 0.5–0.7 bar (craters A through E). Several outliers also indicated constant basal shear stress ranges of 1.1–1.4 bars (craters F and G) and 1.9–2.4 bars (craters H, I, and J). With the exception of craters H, I, and J, the use of constant basal shear stress, t, with the model produced better matching ice morphologies than did a slope-dependent shear stress, t(S). Therefore we will refer to our results in terms of constant basal shear stress unless otherwise indicated. Results are summarized in Table 1 and Figures 8 and 9 show shaded relief images for each crater and our most accurate model simulation followed by the best matching profiles for comparison of ice thickness and morphology. The location and direction of each profile is indicated by the black arrows in each crater. Elevation data is in relation to the zero datum elevation on Mars and has an offset of _8000 m.  The ice thickness in the majority of the craters (craters A–E) was best reconstructed using a constant basal shear stress of _0.6 bar (Figure 8). In general, the ice thickness, particularly near the rims of the craters, and the ice geometry are well reconstructed by the model. For crater D, even the small bump at the ice margin has been simulated. In a few instances however, (craters A and E), the ice surface slope near the margins tapers either too gradually or too steeply in the model simulations.  A threshold basal shear stress of _1.25 bars was used to model craters F and G (Figure 9a). For both of these craters, the ice thickness, particularly near the crater rim, matches very well. A constant threshold basal shear stress of _2.15 bars was used for the reconstructions of ice morphology for craters H, I, and J (Figure 9b). In the profiles it can be seen that the ice thickness was generally well simulated. However, the actual ice surface slopes are relatively flat with steep slopes at the margin creating a blunt edge while, in the simulations, the ice tapers more gradually and consistently to the ice margin. The reasons for the ice surface slopes in craters H, I, and J are unclear. When comparing the dimensions of these three craters to those of the other craters in this study, their flat and blunt ice surface slopes were not found to be related to crater diameter, depth, or the slopes of the crater rims. However, the ice morphology was best simulated when spatial variations in basal shear stress were taken into account. This was done by using the shear stress relationships t = 9S0.55, t = 7S0.55, and t = 11S0.55 for craters H, I, and J, respectively (Figure 9b). In comparison to the model simulations using a constant basal shear stress, the use of a slope-dependent shear stress maintained the same average ice thickness but created slightly more blunt edges near the ice margins.  On the basis of the model results, the crater rims in the bed topography under the ice are too pronounced. Most likely, the actual crater rims under the ice have been altered or eroded more extensively than the parts of the rims that can be observed directly. Since we did not have a method for accurately estimating the subglacial erosion, the exposed parts of the rim were averaged for each crater and used as the entire rim in the bed topography. The profiles in Figures 8 and 9 were therefore drawn starting just inside the rims and extended toward the center of the crater roughly perpendicular to the ice margin to show only the areas of ice coverage for which a well constrained bed topography was used. Quantitatively, the presence or absence of a central peak beneath the ice did not change the value of basal shear stress inferred from the simulation that best reconstructed the ice thickness. However, in a few instances (craters C, E, and F), the presence of a central peak in the bed topography produced the best matching morphology.  In summary, the threshold-sliding model was successful at creating accurate reconstructions for both the Martian examples and the Greenland ice sheet. Overall, a constant shear stress proved more accurate than a slopedependent shear stress for the Martian simulations.
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However, this may be due to the smaller spatial scales of these reconstructions (i.e., tens, not hundreds of kilometers). Data from the Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS) on the Mars Express orbiter should provide information about the bed topography beneath both Martian polar caps. Once this information becomes available, the threshold-sliding model can be calibrated with the larger spatial scale of the Martian polar caps to determine whether t or t(S) is more appropriate when using the model to reconstruct the thickness and morphology of large expanses of ice on Mars. 5. Discussion  The basal shear stresses inferred from the model reconstructions represent the basal shear stress values at which a threshold is reached. Any additional stress will cause some type of deformation or movement of the ice. The threshold-sliding model is able to account for basal sliding of the ice with no internal deformation, internal deformation of the ice with no basal sliding, or a combination of both, but it does not give any information about which of these processes is taking place. Although our results showed a range of values, the majority of the reconstructions in this study indicated an average threshold basal shear stress of _0.6 bar or almost 1/3 that found for the Greenland ice sheet (_1.41 bars) on Earth. The causes behind the overall lower shear stresses associated with ice on Mars are unclear and factors that influence the yield strength of the ice and/or the coefficient of friction at the base of the ice should be investigated.  One possible explanation for the low basal shear stresses on Mars is a reduced yield strength of the ice. On Earth, it has been found that the crystal sizes of ice vary directly with temperature and indirectly with strain rates. Potentially, lower temperatures, such as those found on Mars, can cause ice to form smaller crystal sizes which in turn could cause a lowering of the yield strength and thus an increase in the internal deformation rate [Hooke, 2005]. In addition, the orientation of ice crystals has been found to have a significant effect on shear strain rates [Hooke, 2005; Li et al., 1996] and impurities, such as dust, salt, clathrates, or a proportion of carbon dioxide ice mixed in with the water ice, can cause softening or stiffening of the ice thus influencing the yield strength [e.g., Greve and Mahajan, 2005, Greve et al., 2004].  Other possible explanations for the low basal shear stresses found in our study could involve a lower coefficient of friction at the base of the ice which would lead to an increase in basal sliding. For example, producing meltwater at the base of a glacier increases the water pressure and results in a reduction in resistive drag at the bed and a decrease in the viscosity of the ice [Hooke, 2005]. Causes of basal melting could include climate changes, obliquity changes, frictional melting due to cap movement, or melting due to pressure from the overlying ice cap [Fishbaugh and Head, 2002; Greve et al., 2004]. Basal melting could also result from modification of the melting point due to impurities in the ice or warmer temperatures at the base of the ice resulting from reduced heat conductivity [Greve and Mahajan, 2005; Greve et al., 2004; Fishbaugh and Head, 2002] or a temporary heat source under the ice such as a volcanic eruption or a tectonothermal event [Greve et al., 2004; Fishbaugh and Head, 2002]. However, in simulations conducted by thermodynamic models, basal temperatures were consistently found to be far below pressure melting [Greve et al., 2004]. Although melting may not occur, basal temperatures could still be a principal parameter affecting ice rheology and the rate of ice flow as simulations have shown that higher temperatures, although still below pressure melting, can produce higher rates of flow [Paterson, 1994; Greve et al., 2004].  The majority of the model results indicated a constant basal shear stress of _0.6 bar which is consistent with the calculated values of _0.5 bar for areas near the divides in the Greenland ice sheet (Figures 1a and 8). Several simulations also indicated basal shear stress values of _1.25 and _2.15 bars which are similar to values found closer to the margins of the Greenland ice sheet (Figures 1a and 9). Potentially, these three values could be reflecting different basal conditions on Mars that are comparable to the different basal conditions existing between the divides and the margins of the Greenland ice sheet. When investigating the bed topography for each of the craters, no significant differences in the ratio of crater depth to crater diameter or surface slope could be found. Zwally and Saba  calculated basal shear stresses for the Martian north polar cap from observed surface slope and gravitational forcing and also found three distributions of shear stress, although these values were lower than those found in this study. Even though the average threshold basal shear stress values for the Earth and Mars examples in this study differ by _1/3, the total range in values found for Mars, 0.5–2.4 bars, is consistent with the range of values calculated for the Greenland ice sheet, 0.5–3 bars (Figure 1a), and all but three of the Martian simulations inferred values that fall within the range of 0.5 to 1.5 bars reported by Nye [1952a, 1952b]. When reconstructing former ice sheets and glaciers on Mars (using modern conditions), families of reconstructions should be investigated for this full range of threshold basal shear stresses. When reconstructing large expanses of ice, the spatial variability in basal shear stresses, t(S), as well as constant basal shear stresses, t, should be investigated.  A possible source of error in our results could be caused by the depths used for the crater bed topography. The averaged bed topography used as input for the model was based on current crater depths. Since we do not know exactly when the craters became covered with ice, possible alterations to the crater depth and morphology since the time of ice coverage have not been taken into account. Most likely these alterations would be small and would not significantly influence our results as was the case with the addition of a central peak to the bed topography of craters more than half covered with ice.  Another limitation of the model in reconstructing past ice sheets on Mars is that the results of this study infer rheological conditions of modern ice which may be different from past ice.
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However, these experiments provide a starting point for model parameter estimation. For future work, knowledge of the temperature dependence of glacial flow parameters on Earth will allow us to increase or decrease the inferred modern values corresponding to a range of climatic conditions. 6. Summary  Using the threshold-sliding model, we reconstructed the thickness and morphology of the Greenland ice sheet and ice draping the rims and partially filling impact craters on the margins of the layered deposits in the south polar region of Mars. An equation for shear stress as a function of surface slope, t = 15S0.55, was used when reconstructing the thickness of the Greenland ice sheet while a constant basal shear stress of _0.6 bar was used for the majority of the Mars examples. The ice thicknesses of several outliers in the Martian examples were also reconstructed using constant shear stress values of _1.25 and _2.15 bars. When reconstructing former ice sheets and glaciers on Mars (using modern conditions), families of reconstructions should be investigated for this full range of constant threshold basal shear stresses. When reconstructing large expanses of ice, the spatial variability in basal shear stresses should also be investigated. The reasons for the lower values of basal shear stress found for the Mars examples are unclear but could involve higher rates of internal deformation and/or increased rates of basal sliding.
Frosted granular flow: A new hypothesis for mass wasting in martian gullies Chris H. Hugenholtz Abstract Recent gully deposits on Mars have been attributed to both wet and dry mass wasting processes. In this paper frosted granular flow (FGF) is presented as a new hypothesis for recent mass wasting activity in martian gullies. FGF is a rare type of granular flow observed on a talus slope in the Province of Québec, Canada [Hétu, B., van Steijn, H., Vandelac, P., 1994. Géogr. Phys. Quat. 48, 3–22]. Frost reduces dynamic inter-particle friction, enabling flows to mobilize onto relatively low slope gradients (25–30°) compared to those involving dry granular flow of the same material (35–41°). Resulting erosional and depositional features include straight to sinuous channels, levees and digitate to branching arrangements of terminal deposits. Similar features are commonly found in association with geologically-young gully systems on Mars. Based on terrestrial observations of FGF processes the minimum criteria required for their occurrence on Mars include: (i) readily mobilized, unconsolidated sediment at the surface; (ii) an upper slope gradient at or near the angle of repose; (iii) frost accumulation at the surface; and (iv) triggering by rock fall. All four conditions appear to be met in many areas on present-day Mars though triggering mechanisms may vary. Compared to terrestrial FGFs, which are lubricated by thin liquid films at inter-particle contacts, those occurring on Mars are more likely lubricated by vaporization of CO2 and small amounts of H2O frost that becomes incorporated in the translating mass. Some recent mass wasting activity in martian gullies, therefore, could be interpreted as the product of FGF. Keywords: Mars; Mars, surface; Earth; Geological processes 1. Introduction A number of hypotheses have been presented regarding the development of hillside gullies on Mars ([Malin and Edgett, 2000], [Gaidos, 2001], [Musselwhite et al., 2001], [Costard et al., 2002], [Hoffman, 2002], [Christensen, 2003], [Treiman, 2003], [Shinbrot et al., 2004], [Heldmann et al., 2005] and [Pelletier et al., 2008] among others). From these contributions three broad categories of mass wasting processes are recognized: liquid flows, vapor-assisted flows and dry flows. A major challenge continues to be the link between the process(es), the composition of the mobilized material and the resultant landforms.
Geomorphological features of gullies (alcoves, channels and debris aprons) provide the most convincing support for hypotheses predicated on liquid flows, but atmospheric pressure and temperature in most areas on present-day Mars suggest extremely limited potential for the stability of H2O or CO2 in liquid phase at the surface where gullies are located ([Ingersoll, 1970], [Haberle et al., 2001], [Mellon and Phillips, 2001] and [Stewart and Nimmo, 2002]). Interpretations of contemporary liquid-supported mass flows on the martian surface are therefore problematic.
Until recently (Malin et al., 2006), no convincing evidence had been presented to indicate that gullies on Mars are active under the constraints of the current atmospheric conditions. Costard et al. (2002) showed that favorable conditions for liquid water stability and surface runoff were possible several hundred thousand years ago when Mars' obliquity was higher ([Touma and Wisdom, 1993] and [Laskar and Robutel, 1993]). Therefore, in the context of geomorphic timescales investigated on Earth (Schumm and Lichty, 1965) gullies could be relict features and the possibility they formed in the ancient past by liquid flows of sediment–water mixtures is reasonable.
From sequential images taken by the Mars Orbital Camera (MOC), Malin et al. (2006) recently documented evidence of contemporary mass wasting activity in two southern mid-latitude gullies. From an exhaustive review of thousands of images over several hundred sites, Malin et al. (2006) distinguished two new bright gully deposits along southeast- and northeast-facing walls of craters in the Terra Sirenum and Centauri Montes regions, respectively. Geomorphic characteristics of the deposits suggest emplacement by sediment–liquid (H2O) mixtures, but the physical mechanism responsible for supporting liquid water at the surface in these regions remains to be fully developed (Malin et al., 2006).
Pelletier et al. (2008) determined that one of the recent bright gully deposits described by Malin et al. (2006) is consistent with a dry granular flow emplacement mechanism. Comparable runout distance and deposit morphology (fingering) were reproduced by combining a photogrammetrically-derived digital elevation model (DEM) with a two-dimensional numerical model of a dry granular flow. The model is predicated on several assumptions, the most important of which are the location and size of the detachment region (100 m2) and the triggering mechanism (rock fall). Surface expression indicating recent mass wasting in the supposed detachment and erosion zones, however, is not obvious in HiRise image PSP_001846_1415 (cf. [McEwen et al., 2007] and [Pelletier et al., 2008]). This contrasts significantly with the well-preserved distal deposit. Additionally, the high resolution DEM developed by Pelletier et al. (2008) shows small-scale details of gullies 0.3 to 1.5 m deep, but signs of erosion or rock fall are not evident in the supposed detachment region. Thus, if the detachment region is incorrectly characterized there might not be enough momentum in a dry granular flow to produce the same runout distance and terminal deposit morphology.
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Because much of our interpretation of gullies on Mars is conditioned by terrestrial analogs, exploring all aspects of terrestrial mass wasting processes is extremely important. Recent terrestrial analog research has focused mainly on water-carved gullies in polar regions and icy debris fans in Alaska ([Hartmann et al., 2003] and [Marchant and Head, 2007]; Kochel and Trop, 2008). However, one analog process not yet investigated in this context is frosted granular flow (FGF). Originally observed and described by Hétu and Vandelac (1989) and Hétu et al. (1994), FGF is a rare type of terrestrial granular flow in which the translation of debris is facilitated by thin frost coatings on clasts.
This paper describes some of the process-form characteristics of FGFs that are relevant to recent and contemporary gully processes on Mars. The scope of this paper is by no means synoptic as the development of gullies and mass wasting deposits on Mars is likely polygenetic, reflecting a continuum of processes and conditions operating over both cyclic and graded timescales (Schumm and Lichty, 1965). However, FGF is proposed as a viable hypothesis for recent mass wasting features in martian gullies, particularly given the constraints of Mars' current atmospheric conditions, which limit the potential for liquid H2O or CO2 at the surface. The FGF hypothesis is also relevant to the interpretation of recent gully deposits as dry granular flows where the detachment region might be small or located lower on the slope (cf. Malin et al., 2006, HiRise image PSP_003596_1435). In the absence of water or sufficient momentum in a translating mass flow, vapor from sublimating frost might provide the necessary fluid conditions to mobilize debris onto lower slope gradients.
2. Frosted granular flow (FGF)
FGFs were first described by Hétu and Vandelac (1989) and Hétu et al. (1994) in the Gaspé region of the Province of Québec, Canada (Fig. 1). In their paper published in the French language, Hétu et al. (1994) referred to the process as “frost-coated clast flow.” Because the process is a form of granular flow, however, the term “frosted granular flow” is preferred and used throughout the remainder of this paper.
FGFs occur on maritime talus slopes in the St. Pierre valley (Fig. 1B), which lies close to the Gulf of the Saint Lawrence River. The talus is comprised of weathered Ordovician shale and greywacke. Steep (50–70°) cliffs bordering the upper section of the talus slopes are dissected by numerous chutes, which serve as transfer paths for weathered material supplying the talus surface below. Chutes are smaller versions of the alcoves commonly associated with gully systems on Mars (cf. Malin and Edgett, 2000). The profile form of the talus slope described by Hétu et al. (1994) is concave-up, with more than 80% of the slope lying below the angle of repose. The lower 40% of the talus slope is covered by Boreal forest. Weathered material accumulates on the upper section where local slope gradient approximates the angle of repose (39–41°). Below the accumulation zone the main portion of the talus surface is characterized by a network of channels. The channels show progressive transformations from straight to sinuous where the local slope gradients drop below 31° (Hétu et al., 1994). The lower slope section (<27°) is comprised of a broad digitate to branching arrangement of terminal deposits. The grain size of sediment sampled from the channels, levees and terminal deposits ranges from very fine sand (0.007 cm) to individual clasts up to 20 cm (b-axis). The average grain size of all three morphological features is 3 cm (b-axis), indicating that FGF processes mainly involve gravel-sized sediment.
FGFs have been observed in the fall and spring when the air temperature oscillates around the freezing point (0 °C) and the talus surface is generally snow-free. Three meteorological conditions have been associated with FGF activity (Hétu et al., 1994, p. 17): (i) freezing of surface moisture, which produces relatively thin ice coatings on clasts; (ii) freezing rain accumulating across the talus surface producing a thick, near-continuous (up to 3 mm) ice coating; and (iii) frost accumulating on the talus surface. The latter triggered widespread FGF activity in the winter of 1989–1990 according to Hétu et al. (1994). During this period, FGFs occurred across 40% of the slope studied by Hétu et al. (1994), or some 65,000 m2. Thus, they are not an isolated phenomenon and have played a formative role in the evolution of the talus slope.
FGFs initiate on the upper portion of snow-free talus slopes and mobilize for distances up to 500 m. The triggering mechanism is localized rock fall impact on the talus surface. Once mobilized, FGFs develop into small flows 1–3 m wide, moving at velocities of 3–6 m s−1. At sites with repeated FGF activity the flows often follow pre-existing channels or diverge and create new ones when stagnant lobes or obstacles are encountered. As the flows translate downslope, small clasts and sand-sized sediment are filtered to the base by mechanical sieving, while large clasts concentrate at the surface and margins. Consequently, sediment that is deposited on the channel floor is much finer than sediment comprising the levees and terminal lobes. FGFs can occur as discrete, continuous flows involving a single bore of sediment or as a series of discontinuous bores that mobilize down the same channel (Fig. 2). In the middle section of the talus slope some flows stagnate upon encountering vegetation or an older terminal lobe. Eventually, with enough debris accumulating on and behind the stagnant front, the local shear strength is exceeded and the flow resumes. Along the lower section FGFs are diverted around stagnant terminal lobes, producing a broad digitate to branching pattern of terminal deposits (Fig. 2). Levees are between 0.2 to 0.75 m tall, suggesting that FGFs involve a relatively thin mobilized debris layer. The role of frost is to reduce dynamic inter-particle friction through lubrication and by smoothing clast surfaces. These two factors enable FGFs to mobilize onto relatively low slope gradients (25–30°) compared to those involving dry granular flow of the same material (35–41°).
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3. FGF and recent gully activity on Mars
Recent activity in southern mid-latitude gully systems on Mars indicate that conditions are suitable for contemporary mass wasting processes ([Malin et al., 2006] and [Costard et al., 2007]). From repeat MOC images of thousands of gullies at hundreds of sites, Malin et al. (2006) observed two new bright features that formed between 1999–2005 and 2001–2005, respectively (Fig. 3). Both features exhibit digitate to branching terminal deposits, and show signs of diverting around large boulders encountered in the path of the flows. Malin et al. (2006) suggested that these characteristics are evidence of thin, slow-moving, liquid-charged debris flows. In keeping with the early hypothesis of Malin and Edgett (2000), Malin et al. (2006) proposed that the source of the liquid is groundwater. Additionally, Costard et al. (2007) recently identified signs of contemporary gully activity from a HiRise image taken over a crater in the Terra Sirenum region (Fig. 4). The image revealed dark, elongate streaks over seasonal frost on steep slopes in gully alcoves, suggesting contemporary mass wasting activity. The full extent of the mass wasting features noted by Costard et al. (2007) is unknown because of significant shadowing in the gully alcove and main channel. McEwen et al. (2007) noted that other craters in the southern mid-latitudes also show signs of recent gully activity (Fig. 5). Although the specific time period in which these deposits were emplaced is unknown, they appear to be recent because they exhibit the same bright appearance and digitate to branching morphology as those described by Malin et al. (2006). Collectively, these observations indicate that a small handful of martian gullies are presently, or were very recently, active. It remains to be determined whether these mass wasting features correspond to primary (formative) or secondary (modifying) processes in the development of martian gully systems.
Morphological characteristics of the terrestrial FGFs reported by Hétu et al. (1994) closely resemble mass wasting features observed in martian gully systems (Fig. 3 and Fig. 5), suggesting that the processes may be related. In particular, terrestrial FGFs exhibit well-defined straight to sinuous channels, levees and digitate to branching arrangements of terminal deposits. These features are frequently cited as key morphological components of martian gullies ([Malin and Edgett, 2000] and [Malin and Edgett, 2001]; [Costard et al., 2002], [Hartmann et al., 2003] and [Arfstrom and Hartmann, 2005]). While levees have not been well documented in past reports, several examples within gully channels and on debris aprons are shown in Fig. 6. Levees indicate viscous flow and typically develop from frictional drag and lateral shearing of lobate mass flow surges (Sharp, 1942). Like the examples in Fig. 6, terrestrial FGFs also tend to produce levees (Hétu et al., 1994). The most striking morphological similarity between terrestrial FGFs and recent gully deposits on Mars, however, is the arrangement of terminal deposits. Fig. 2 shows multiple branches of terminal deposits emanating from individual, FGF-generated channels. A very similar pattern is noted at the terminus of recently active martian gullies shown in Fig. 3 and Fig. 5. Although liquid-supported flows are also known to produce digitate terminal deposits (Hooke, 1967; [Blair and McPherson, 1992] and [Blair and McPherson, 1998]; Hugenholtz et al., 2007; Kochel and Trop, 2008 among others), terrestrial FGF deposits demonstrate that similar features can be produced in the absence pf large quantities of water-charged debris.
Based on the research by Hétu et al. (1994), the minimum environmental conditions required for the development of a FGF can be summarized by the following criteria: (i) readily mobilized, unconsolidated sediment at the surface; (ii) a slope gradient in the triggering zone at or near the angle of repose; (iii) frost accumulation at the surface; and (iv) triggering by rock fall impact.
Although the grain size and composition of unconsolidated sediment in recently active martian gullies is not well known, occurrences of aeolian bedforms, surface boulders and boulder tracks (Fig. 7) indicate that unconsolidated sediment, comprising a range of grain sizes, is locally present at the surface. Bedrock outcrops in the alcoves of recently active gullies also confirm the presence of steep slopes ([Malin and Edgett, 2000] and [Malin and Edgett, 2001]; Treiman, 2003). Malin et al. (2006) indicate that slope gradients in the vicinity of the two new terminal deposits is between 20° and 30°, which is similar to the gradients of the FGF terminal deposits (25–30°) reported by Hétu et al. (1994). Furthermore, lateral slopes of many gully channels are relatively steep and show signs of sloughing and failure (Fig. 8). Seasonal frost accumulation up to several centimeters thick (CO2 and small amounts of H2O) is also prevalent on near pole-facing slopes in the areas of recent gully activity ([Schorghofer and Edgett, 2006] and [Costard et al., 2007]). Diurnal frost is likely to occur on all slope orientations in these areas (cf. Schorghofer and Edgett, 2006). Finally, the identification of a large number of well-preserved boulder tracks at two sites with recent gully deposits on Mars suggests the possibility of present-day rock fall activity (Fig. 7). Collectively, these environmental factors indicate that the minimum criteria for a FGF are met on present-day Mars.
4. FGF on Mars
Terrestrial FGF typically occurs when air, saturated with water vapor, comes into contact with a cold surface prompting solid deposition of the water vapor as frost. As a FGF mobilizes, thin films of liquid water generated by frictional heating at inter-particle contacts lubricate the flow, enabling mobilization onto relatively low slope gradients. On present-day Mars, however, the situation is different. In areas with recent gully activity, cold temperatures and low atmospheric pressure are unlikely to permit the development of thin liquid films at inter-particle contacts during a FGF. In fact, recent gully deposits appear to be relatively thin (Malin et al., 2006), suggesting limited potential for friction-induced melting of frost within a mobilized mass flow.
Frost in areas with recent mass wasting activity on Mars is also very different from the terrestrial analog. On Earth a maritime, humid climate favors frequent episodes of frost accumulation, but this type of climatic setting is not required for frost to accumulate on hyper-arid Mars as long as the near-surface is cold and in contact with a vapor-saturated atmosphere.
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Thus, while the atmospheric conditions are markedly different, the process of surface frost accumulation is similar. Furthermore, surface temperatures on near pole-facing slopes in the southern mid-latitude regions of Mars suggest that the frost is likely to consist of condensed CO2 with minor amounts of H2O (Schorghofer and Edgett, 2006). A CO2 flow model of mass wasting and gully formation in martian polar regions was presented by Hoffman (2002). The model proposes that pole-facing, frost-covered slopes thaw from the bottom upwards, leading to sublimation and a near-surface buildup of CO2 vapor. Thawing is initiated by absorption of incoming solar radiation. The overall process is similar to the model proposed for the development CO2 jets and dark spots (“spiders”) in the polar regions of Mars (Kieffer et al., 2006). Hoffman (2002) suggests that the vapor triggers near-surface instability, eventually leading to an avalanche that rushes down a gully, progressively growing by incorporating additional CO2 frost and sediment encountered along the path of the mass flow. Thus, during mobilization additional CO2 vapor is generated by frictional heating and churning of warm sediment, lubricating the flow and producing fluid-like behavior. Laboratory experiments with mixtures of silt and solid CO2 have produced similar behavior owing to rapid sublimation and lubrication by inter-granular vapor (Hsu, 1975). Although Hoffman's (2002) CO2 flow model was developed for polar martian gullies, many of the principles could apply to the development of a FGF in the mid-latitude regions. In particular, frosted sediment incorporated in the translating mass would produce additional vapor along inter-particle contacts, thereby lubricating the flow and enabling fluid-like behavior. An important difference with Hoffman's model, however, is whether the mobilized mass flow travels on a cushion of CO2 vapor (Hoffman, 2002, p. 322). Even under ideal conditions in the southern mid-latitudes, sublimation within thin frost covers is unlikely to generate significant volumes of trapper CO2 vapor to cushion the base of a mass flow. Instead, it is likely that warming by solar radiation produces small amounts of CO2 vapor within the frost and near-surface sediment. During FGF mobilization the trapped vapor reduces inter-particle friction much like the thin liquid films that play a role in terrestrial FGFs. Thus, while the fluid may differ in FGFs produced on Earth and Mars, the overall effect of lubricating the translating mass is similar. Most of the CO2 vapor within a translating FGF probably diffuses rapidly into the martian atmosphere unless the detached mass is initially thick. In this regard, the proportion of fine-grained sediment is likely to play an important role in FGF mobility on Mars. Laboratory experiments on gas-fluidized granular flows show that fines-rich flows can have negligible internal friction due to high pore vapor pressure, which reduces inter-particle frictional contacts and increases mobility (Roche et al., 2005). This enables the mass flows to behave as inviscid fluid gravity currents for most of their emplacement. Under the same initial conditions (mass, velocity, slope, etc.), therefore, martian FGFs comprising fine-grained sediment are likely to have greater mobility than those involving coarse-grained sediment. Rock fall impact is one of several possible triggering mechanisms for a FGF on Mars. Though not all recent bright mass wasting deposits in martian gullies occur in areas with demonstrable signs of recent rock fall activity, many of them are located near boulder tracks, suggesting this process is active in many areas (Fig. 7). Other possible triggers include vapor-induced instability (Hoffman, 2002), avalanching of CO2 frost (Ishii and Sasaki, 2004), and point-source defrosting (Costard et al., 2007). The trigger proposed by Ishii and Sasaki (2004) is further supported by HiRise image PSP_007338_2640, which shows possible avalanching associated with defrosting CO2. All of these triggers can occur within and below the gully alcoves, or along steep channel slopes. Small scarps along the walls of gully channels suggest that many mass flows have initiated within the channels, well below the alcoves (Fig. 8). This might explain why some of the recent bright deposits are visible only along the terminus zones of gullies (Fig. 3 and Fig. 5). 5. Conclusions The FGF hypothesis is appealing for several reasons. First, large volumes of liquid water runoff at the surface are not required to mobilize sediment in gullies. Thus, while hypotheses predicated on liquid flows are hindered by atmospheric conditions, FGF is not dependent on the availability of liquid water at the surface or in the shallow subsurface. Second, the FGF hypothesis agrees well with the orientation and concentration of gullies on near pole-facing, mid-latitude slopes. These slopes favor relatively thick (several centimeters) seasonal frost accumulation at the surface (Schorghofer and Edgett, 2006). More importantly, however, the FGF hypothesis does not require that gullies occur only on near pole-facing, mid-latitude slopes since the process can occur on any steep slope where frost accumulates on unconsolidated sediment. Third, the morphological characteristics of terrestrial FGFs closely resemble those commonly associated with martian gully systems (i.e., straight to sinuous channels, levees and digitate to branching arrangements of terminal deposits). Collectively, these characteristics indicate that recent mass wasting features on Mars can be explained by the FGF hypothesis. This new hypothesis does not propose that all recent mass wasting features observed in martian gully systems are the product of FGF activity or, for that matter, any one particular mass wasting process (cf. Pelletier et al., 2008). Indeed, many gullies display morphological features that could represent liquid-supported or dry granular flow ([Malin and Edgett, 2000], [Treiman, 2003] and [Pelletier et al., 2008]). As on Earth, a wide variety of processes are likely to contribute to the development of gully systems on Mars. Therefore, the FGF hypothesis is proposed as one of several mass wasting processes that play a role in the formation and subsequent modification of martian gully systems. The hypothesis is particularly appealing in the context of recent and contemporary mass wasting processes because it does not require mobilization of sediment by a liquid. Furthermore, in the absence of sufficient momentum, sublimating frost and CO2 vapor might provide the necessary inter-granular lubricant to mobilize debris onto lower slope gradients.
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New kinematic models for Pacific-North America motion from 3 Ma to present, I: Evidence for steady motion and biases in the UVEL-1A model Charles DeMets
Abstract. We use velocities derived from 2-4.5 years of continuous GPS observations at 21 sites on the Pacific and North American plates along with a subset of the NUVEL-1A data to examine the steadiness of Pacific-North America motion since 3.16 Ma, the transfer of Baja California to the Pacific plate, and the magnitude of biases in the NUVEL-1A estimate of Pacific-North America motion. We find that Pacific-North America motion has remained steady since 3.16 Ma, but at rates significantly faster than predicted by NUVEL-1A. In the vicinity of Baja California, our GPS-derived model and recent seafloor spreading rates in the southern Gulf of California both indicate that the NUVEL-1A model underestimates Pacific-North America rates by 4+_2m m yr- •. Steady Pacific-North America motion since 3.16 Myr and increasing seafloor spreading rates since3.58 Myr in the Gulf of California imply that Pacific-North America motion was partitioned between seafloor spreading in the Gulf of California and decelerating slip a long faults in or offshore from the Baja peninsula.
Introduction The degree to which plates can change their motions over geologically brief intervals is a key unanswered question in plate kinematics, one with important implications for the forces that drive plate motion. Ongoing geodetic measurements of plate velocities and refinements in models of several-million-year-average global plate motions [e.g. DeMets et al., 1990, 1994] will enable future tests for geologically-rececnt changes in the velocities of most of the tectonic plates. Here, we revisit questions about recent changes in motion between the Pacific and North American plates, where geodetic measurements are more mature and more widespread and thus enable stronger tests for recent plate motion changes. DeMets [ 1995] presents evidence that seafloor spreading rates across the Gulf of California, which separates Baja California from the North American plate, have accelerated by -15% since 3.58 Myr and now significantly exceed the Pacific-North America rate predicted by the 3.16 Myr-average NUVEL-1A model [DeMets et al., 1994]. This can be interpreted in at least two ways. If the Baja peninsula has moved with the Pacific plate since 3 Myr, the observed spreading acceleration directly records a post-3Ma acceleration of Pacific-North America motion. DeMets  instead proposes that Pacific-North America (hereafter abbreviated PA-NA) motion since 3.16 Myr has been steady, that the NUVEL-1A model significantly underestimates 3.16 Myr-average PA-NA motion, and that slip between these two plates in the vicinity of Baja California has been partitioned between accelerating seafloor spreading in the Gulf of California and decelerating slip along faults west of Baja California. In support of this interpretation, DeMets  demonstrates that a revised model for PA-NA motion, one derived from a subset of the NUVEL-1A plate kinematic data that excludes demonstrably biased data from the Gulf of California and other plate boundaries, predicts 3.16 Myr-averag PA-NA motion that agrees with in uncertainties with the observed rate of seafloor spreading in the Gulf of California since 0.78 Ma. This implies that PA-NA motion has remained constant since at least 3.16 Ma. The seemingly incompatible kinematic evidence for steady PA-NA motion and accelerating seafloor spreading rates between the Baja peninsula and North American plate is reconciled by assuming that the Baja peninsula moved relative to the Pacific plate prior to 780,000 years ago. Here, we combine new GPS-derived velocities at 21 sites from the Pacific and North American plates with a subset of the NUVEL-1A plate motion data to undertake an independent test for steady PA-NA motions ince 3.16 Myr. The models we derive for instantaneous (geodetic) PA-NA motion and motion since 3.16 Ma (geologic) predict velocities in the Gulf of California and a long the Baja peninsula that differ insignificantly by, no more than 1.5 mm yr- I and 2ø. In a forthcoming paper (Dixon et al., “New kinematic models for Pacific-North America motion from 3 Ma to present, II: Tectonic implications for Baja and Alta California”, to be submitted to GRL, 1999), we use GPS velocities for sites from Baja and Alta California and our improved estimates for PA-NA motion to describe the present tectonics o f Baja California and Alta California and directly estimate the status of Baja California's transfer to the Pacific plate. GPS data analysis and site velocities To solve for Pacific and North American plate angular velocities, we use data from permanent GPS stations located on the Pacific (5) and North American (16) plates (Fig. 1 and Table 1). All stations have 2.0 or more years of data, current through early September, 1998. Data were analyzed at the University of Miami using GIPSY analysis software from the Jet Propulsion Laboratory [Zumbergee t al., 1997], high-precision on-fiducial satellite orbits and clocks, and procedures described by Dixon et al. . Station velocities are given in Table 1 and are defined in ITRF96 [Sillard et al., 1998]. We selected sites far from deforming zones within North America to minimize the effect of any fault-induced elastics train or distributed deformation on our site velocities. We also incorporate a new model for white noise and time correlated noise in GPS coordinate time series for globally distributed sites [ Mao et al., 1998], which builds on earlier work by Zhang et al. . We solve for angular velocities that best fit GPS station velocities, seafloor spreading rates, and plate slip directions using techniques described by DeMets et al.  and Ward . The data are inverted to solve for one or more angular velocities that simultaneously minimize the weighted, least-square missfit to data from one or more plates and satisfy the condition of plate circuit closure wherever it applies. The best-fitting angular velocities and model uncertainties are given in Table 2. The misfits of the North American plate angular velocity to the 16 North American site velocities range from 0.0- 2.6 mm yr- • (Table 1) and average1 .0 mm yr- •. The misfits of the Pacific plate angular velocity to the five Pacific site velocities range from 0.1-4.0 mm yr- • (Table 1) and average2 .1 mm yr- •. Values of reduced chis-quare for the North American and Pacific angular velocities are 0.42 and 1.26, respectively, indicating that the station velocity uncertainties may be modestly overstated and slightly underestimated, respectively. Given the small number of data, we conclude that the site velocity and thus model uncertainties are approximately correct. The locations of the Pacific and North American sites with respect to the best-fitting poles of rotation (Fig. 1) strongly constrain the rotation poles and angular rotation rates for both plates. Geodetic rates for both plates show the expected sinusoidal increase with angular distance from the pole and the radial rate components scatter symmetrically about their expected value of zero.
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To assess the robustness of our solution, we also derived North American and Pacific plate angular velocities from GPS $6- station velocities from the Jet Propulsion Laboratory (Mike Heftin, pers. commun., 1998). The predictions four best-fitting North American angular velocity differ from those of the angular velocity we derived from the JPL data by only 0 -1.3m my -r• . The site distributions – and observation time –spans at the North American sites are thus sufficient to give robust solutions for the North American plate angular velocity regardless of small differences in analytical techniques or station selection. Velocity differences at the five Pacific plate sites are n- 49 somewhat larger, from 1-3 mm yr- , likely reflecting differences in our analytic techniques and the time spans of the data used to solve for individual station velocities. None of our conclusions change significantly if we use 46 velocities from JPL instead of our own. A rigorous test for changes in Pacific-North 0 i 2 3 4 America motion since 3 Ma Tim(eM a) To test for significant changes in PA-NA motion since 3.16M a, we used a variant of the statistical test for plate circuit closur [Geordon et al., 1987] to compare motion over geodetic intervals (several years) and geologic itintervals (several million years). Short-term Pacific-North America motion is determined from the 21 GPS velocities from Pacific and North American plate sites (Table 1). Pacific-North America motion since 3.16 Myr is constrained by a modified subset of the NUVEL-1A data that excludes all 77 PA-NA data and thus estimates PA-NA motion solely from global plate circuit closures. If motion has remained steady since 3.16 Myr, then a model that fits both sets of data simultaneously should differ insignificantly from a model that fits the geodetic and geologic data separately. A significant difference in the fits of the two models would indicate that geodetic and geologic plate velocities cannot be combined without violating global plate circuit closure requirements. We use only 906 of the 1122 NUVEL-I(A) data to solve for the PA-NA closure-fitting angular velocity. The data that we use and reasons for excluding the remaining data are described by DeMets [ 1995] and are not repeated here. In addition to these changes, we also adjusted data uncertainties such that reduced chi-square for individual data types have their expected values of 1.0. Simultaneously fitting the 42 (2'21) geodetic velocity components and the 906 plate kinematic data described by DeMets  gives a weighted least-square misfit Z2 of 810.2. Fitting both sets of data separately and summing their individual misfits gives a combined Z2 of 807.4. Three additional parameters are adjusted in the latter model. Applying the F-ratio test for plate circuit non-closure gives F = 1.1. This is lower than the value for F, 2.6, that would indicate that the fits of the two models differ significantly at the 95% confidence level. A model in which the two sets of data are fit simultaneously thus fits the data nearly as well as a model in which they are fit separately. Pacific-North America motion estimated over the past few years and the past few million years is thus identical within uncertainties. Repeating the above test usinga ll the NUVEL-1A data except for the 77 PA-NA data gives a different result. Models that fit geodetic and geologic data simultaneously and separately differ significantly at the 98.5% confidence level. The NUVEL-1A data are thus not consistent with steady PA-NA motion since 3.16M yr; however, this is almost certainly an artefact of systematic biases in some of the NUVEL-1A data. Figure 2 reinforces these results. The velocities of the five Pacific plate sites relative to the North American plate are fit nearly as well by the angular velocity we derived from the modified N UVEL-1A datas by the angular velocity that best fits the GPS velocities. In contrast, t he NUVEL-1A PA-NA angular velocity fits the data more poorly, presumably due to systematic biases in some of the NUVEL-1A data [ DeMets1, 995]. Changes to geologic kinematic data that reflect advances since 1990 in our understanding of global plate kinematics are presently underway (C. DeMets and R . Gordon,1 999].
Our determinations for PA-NA motion for the present and3 .16 Ma (Figs.2 and3 ) both give rates of 51-53m m yr- • in the southern Gulf of California, each with standard errors of +2 mm yr- •. These rates are consistent with the 51.4+2m m yr- average rate of seafloor spreading in the southern Gulf of California since 0.78 Ma [DeMets, 1995] and with the predictions of PA-NA angular velocities (Fig. 3) that were derived from VLBI data [Gordon, 1995] and GPS data that were available as of several years ago [Argus and Heftin, 1995]. All of these observations suggest that Pacific-North America motion has been steady at-52+2 (95% limits) mm yr- since at least 3.16 Myr. Newly available PA-NA rotations for longer intervals predict that displacement rates in the southern Gulf of California have been 5 2-57 mm yr-1 since -8 Ma, in good accord with the results reported here (J. Stock, pers. commun., 1999). Most kinematic evidence ( Fig. 3) thus indicates that PA-NA motion in Baja and Alta California is -4+2 mm yr- • faster than predicted by NUVEL-1A. Although the PA-NA rate we predict significantly exceeds that derived by Larsone t al. , we view the results reported here as a direct outcome of improvements in GPS site distribution, longer GPS time series, improvements in analytical techniques, and geodetic reference frames not available to Larson et al. . The evidence for accelerating seafloor spreading between the Baja peninsula and North American plate (Fig. 3) and steady PA-NA motions ince3 .16M yr can be reconciled if we assume that the Baja peninsula moved relative to the Pacific plate prior to-1 Myr. Faults west of the peninsula that display evidence for Holocene slip (e.g. the Tosco-Abreojos and San Isidro fault zones described by Spencer and Normark [ 1979] and Legg et al. ) presumably accommodated some or all of this motion. Assuming that steady PA-NA motions ince3 .16 Myr has been partitioned between faults in the Gulf of California and faults west of the Baja peninsula, increasing seafloor spreading rates in the Gulf of California imply a corresponding decrease in slip along faults to the west. The present slip rates along these faults are best addressed by comparing the velocities of coastal sites to the predictions of the Pacific plate angular velocity derived here. We address this in our forthcoming paper.
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Global Plate Motion Frames: Toward a Unified Model Trond H. Torsvik
 Plate tectonics constitutes our primary framework for understanding how the Earth works over geological timescales. High-resolution mapping of relative plate motions based on marine geophysical data has followed the discovery of geomagnetic reversals, mid-ocean ridges, transform faults, and seafloor spreading, cementing the plate tectonic paradigm. However, so-called ‘‘absolute plate motions,’’ describing how the fragments of the outer shell of the Earth have moved relative to a reference system such as the Earth’s mantle, are still poorly understood. Accurate absolute plate motion models are essential surface boundary conditions for mantle convection models as well as for understanding past ocean circulation and climate as continent-ocean distributions change with time. A fundamental problem with deciphering absolute plate motions is that the Earth’s rotation axis and the averaged magnetic dipole axis are not necessarily fixed to the mantle reference system. Absolute plate motion models based on volcanic hot spot tracks are largely confined to the last 130 Ma and ideally would require knowledge about the motions within the convecting mantle. In contrast, models based on paleomagnetic data reflect plate motion relative to the magnetic dipole axis for most of Earth’s history but cannot provide paleolongitudes because of the axial symmetry of the Earth’s magnetic dipole field. We analyze four different reference frames (paleomagnetic, African fixed hot spot, African moving hot spot, and global moving hot spot), discuss their uncertainties, and develop a unifying approach for connecting a hot spot track system and a paleomagnetic absolute plate reference system into a ‘‘hybrid’’ model for the time period from the assembly of Pangea (!320 Ma) to the present. For the last 100 Ma we use a moving hot spot reference frame that takes mantle convection into account, and we connect this to a pre– 100 Ma global paleomagnetic frame adjusted 5! in longitude to smooth the reference frame transition. Using plate driving force arguments and the mapping of reconstructed large igneous provinces to core–mantle boundary topography, we argue that continental paleolongitudes can be constrained with reasonable confidence. Citation: Torsvik, T. H., R. D. Mu¨ller, R. Van der Voo, B. Steinberger, and C. Gaina (2008), Global plate motion frames: Toward a unified model, Rev. Geophys., 46, RG3004, doi:10.1029/2007RG000227. 1. INTRODUCTION  Plates form the outer shell of the Earth, and their past movements may be traced using geological data. Plate tectonics is a paradigm that attempts to describe the complex dynamic evolution of the Earth in terms of rigid lithospheric plates. A simplified form of the theory invokes the Earth’s heat engine to drive plate motions: mantle material heated by isotopic decay rises at spreading ridges where plates diverge and cool during seafloor spreading. The mantle is cooled by subduction of old, cold lithosphere and is then isotopically heated to rise once again ad infinitum. The theory of plate tectonics has proved successful both theoretically and practically, providing a scientific framework for diverse geological disciplines. It is now an important challenge to integrate plate tectonics into mantle dynamics in order to allow a full dynamic treatment of Earth motion and deformation on all scales. Much progress has been made in understanding the dynamics of mantle convection, plate tectonics, and plumes, but a fully integrated model incorporating both plate motions and mantle dynamics has yet to be realized. Even though links between mantle activity and plate tectonics are becoming more evident, notably through subsurface tomographic images and advancements in mineral physics, there is still no generally accepted mechanism that consistently explains plate tectonics in the framework of mantle convection.  The development of a unifying geodynamic model requires the establishment of a global plate motion model. However, the relative motion between tectonic plates must first be determined from fracture zones and ocean floor magnetic anomalies, the oldest of which are only Jurassic in age (!175 Ma) (section 2). Only then can plates be restored to their paleopositions on the globe using paleomagnetic data (section 3), ‘‘absolute’’ plate rotations from hot spot tracks (if one considers hot spots fixed), or, alternatively, using tracks of hot spots that move because of plume advection in the mantle (sections 4 and 5). (Italicized terms are defined in the glossary, after the main text.) However, the hot spot track method cannot be used prior to !130 Ma, which is the age at the end of the oldest known track in the South Atlantic. That leaves paleomagnetism, with its known limitation that it cannot determine motions in longitude, as the only quantitative way of positioning objects on the globe during older times.  Multiple paleomagnetic and hot spot–mantle reference frames have been published and compared over the past decades, but many were constructed without appropriate consideration of results based on different data sets and methods. In this paper we combine interdisciplinary knowhow in developing paleomagnetic and hot spot reference frames, and most importantly, we compare reference frames (section 6) that are generated with the same timescales and plate circuit closure. Ultimately, we combine hot spot and paleomagnetic frames in order to develop a hybrid global reference frame for plate motions back to the time when Pangea assembled (section 7). We illustrate how this hybrid frame can be used to explore links between surface phenomena and deep mantle heterogeneities and briefly discuss the possible causes of kinks, cusps, and longer-duration small circle segments in the global apparent polar wander (APW) path for the Pangea supercontinent (section 8). 2. RELATIVE PLATE MOTIONS AND PLATE MOTION CHAINS  The relative motions between tectonic plates can be determined from marine geophysical data by the matching of fracture zones and magnetic anomalies of the same age, corresponding to patterns of paleoridge and paleotransform segments at a given reconstruction time. Usually, the geophysical data quality varies substantially, and identification errors can occur; therefore, the quality of the computed rotations needs to be assessed against the quality of input data. Since Bullard et al.  published the first set of computer-generated reconstructions and defined the uncertainties attached to the inferred rotations, several other methods have been proposed to account for uncertainties in plate rotations [Hellinger, 1981; Stock and Molnar, 1983].
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Many of the rotations included in our study were calculated using Hellinger’s  criteria for goodness of fit, associated with uncertainties based on the statistical approach developed by Chang  (see Table 1 for references to quantitative reconstructions). This method requires that isochrons (i.e., magnetic and fracture zone data of the same age) are divided into great circle segments (Figure 1a). Even though fracture zones are expected to follow small circles in plate tectonic theory, Hellinger  chose to fit both paleo-mid-ocean ridge and fracture zone segments to great circles because this greatly simplifies the least squares fitting routine. The length of fracture segments used in this approach is so short that the difference between a small circle versus a great circle segment is negligible in this context. The sum of squares of the weighted distances of fixed data points (from one plate) and rotated data points (from the other plate) to the great circle segments is minimized in order to derive the rotation parameters and their uncertainties [Hellinger, 1981]. The uncertainty in a rotation is described by a covariance matrix, which depends on plate boundary geometry, the number of data points, and data uncertainties [Chang et al., 1992]. This method allows one to combine independently calculated rotations and their uncertainties and to compute the resulting rotation with an uncertainty region that reflects the errors in the input rotations.  The Hellinger  criteria for goodness of fit have been used mainly for deriving best fit rotations from conjugate magnetic anomalies and fracture zone data. For matching boundary between continental and oceanic crust segments a visual fit is usually preferred because the geometry of a continent-ocean boundary (COB) can be very sinuous and difficult to break into great circle segments, as required by Hellinger’s  methodology. Therefore, predrift rotations mostly do not have uncertainties attached to them. However, plate circuits can be used to derive the amount of prebreakup displacement (and uncertainties). As an example we used the rotations between North America and Greenland and between North America and Eurasia to determine the relative motion and its uncertainties between Greenland and Eurasia before breakup (Figure 1b). According to our kinematic model the position of the COBs should be found within an area that is 45 to 77 km wide (from south to north); the uncertainty of reconstructed points is given by the stage pole uncertainty ellipse calculated for stage pole 31 to 25 (67 to 55 Ma). A rotated Eurasian COB at 55 and 57 Ma (white lines in Figure 1b) fits the end limits of the oldest uncertainty ellipse. Because the ellipse shows the uncertainty of the location of the Eurasian COB at 55.9 Ma, this might indicate that the time of breakup occurred between 55 and 57 Ma.  Most Euler rotations include insignificant predrift extension prior to initiation of seafloor spreading; as a result the majority of Pangea reconstructions essentially use Jurassic Euler rotations with minor post-Permian intraplate deformation. The paleomagnetic coverage from two adjacent plates is usually not precise enough to determine this deformation, but in a few rare cases it has been possible to construct predrift relative motion models by fitting portions of APW paths (Figure 2). In this example, late Paleozoic APW segments from North America and Europe match each other well in the Bullard et al.  reconstruction for the interval from 310 to 240 Ma. However, no single fit accommodates the early Mesozoic APW segments for North America and Europe. The fit for 240–210 Ma in Figure 2b was obtained with a gradually changing set of reconstructions using interpolated Euler poles (Table 1). Employing published Jurassic stage poles [e.g., Royer et al., 1992] for pre-Jurassic times results in APW paths that are markedly divergent (Figure 2a); this must clearly be in error given the fact that Laurentia had already collided with Baltica- Avalonia in the middle-to-late Silurian [Torsvik et al., 1996] and remained attached to it during Pangea times.  On the basis of the data listed in Table 1 we have calculated Euler rotations relative to a fixed southern Africa; interpolated rotations (5 Ma intervals) are listed in Table 2. As an example we include a reconstruction for 200 Ma that also shows our plate motion chains with respect to a fixed southern Africa (Figure 3a). We have also calculated relative velocities of a few selected plates in our analyses. As examples we show in Figure 3b calculated plate velocities between North America and NWAfrica (histogram), Europe (squares), and Greenland (circles). Our model (Table 1) includes predrift extension of 1.75 cm/a between Europe and North America during most of the Triassic and the Early Jurassic. The bulk of predrift extension occurs in the Cretaceous (red squares in Figure 3b) followed by an increase in relative velocities (3.5 cm/a) between 50 and 60 Ma, which coincides with the initial opening (rift to drift) of the northeast Atlantic (!55 Ma). Note that North America–NWAfrica predrift extension from 220 to 180 Ma (orange part of the histogram, 0.7 cm/a) was recorded by the formation of complex rift systems (e.g., Newark, Connecticut, and Fundy basins) and was contemporaneous with the Central Atlantic Magmatic Province (!200 Ma) that affected vast areas in North America, NW Africa, SW Europe, and South America. 3. GLOBAL PALEOMAGNETIC FRAME 3.1. Plate of Choice Anchoring the Global Reference Framework  Paleomagnetic reconstructions derived from paleopoles or APW path segments constrain the paleolatitude and the angular orientation of a continent, but its paleolongitude remains unconstrained. However, this degree of freedom can be minimized by selecting an appropriate reference plate; in other words, if one can determine which plate (or continent) has moved longitudinally the least since the time represented by a reconstruction, then that plate should be used as the reference plate [Burke and Torsvik, 2004]. Africa was surrounded on nearly all sides by midocean ridges after the breakup of Pangea: hence, the ridge push forces should roughly cancel (see also section 9). 3.2. Paleomagnetic Data Selection  Paleomagnetic data were compiled from original sources and graded according to Van der Voo’s classifica- tion system [Van der Voo, 1988, 1993].
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In brief, this classification system includes seven reliability criteria: (1) well-determined age and the assumption that the magnetization age equals the actual rock age, (2) sufficient number of samples and adequate statistics, (3) proper demagnetization techniques and documentation, (4) field tests to constrain the age of the magnetization, (5) structural control and tectonic coherence with the involved craton or block, (6) presence of reversals, and (7) no resemblance to paleopoles of younger age. For example, a quality factor Q $ 3 (7 is best) means that at least three of these quality criteria are satisfied. Some criteria are obviously more important than others when constructing APW paths, and no paleomagnetic poles that knowingly fail criterion 1 are included in our analysis.  The data compilation for Laurussia (North America, Greenland, and stable Europe) and Gondwana follows Van der Voo , Torsvik et al. [2001b], Si and Van der Voo , Torsvik and Van der Voo , and Van der Voo and Torsvik  with the additional inclusion of a few new data entries and revised ages for certain poles (notably from Europe [Van der Voo and Torsvik, 2004]). Only poles with Q $ 3 are included, and several paleopoles have been updated with new isotopic age information whenever available. We did not include late Paleozoic–early Mesozoic paleomagnetic data from Siberia because we are as yet uncertain whether Siberia was fully and tightly joined with Laurussia and the rest of Pangea at the dawn of the Mesozoic [Torsvik and Andersen, 2002; Van der Voo and Torsvik, 2004; Cocks and Torsvik, 2007]. Inclusion of Siberian Trap poles (!251 Ma) [Bowring et al., 1998] based on the assumption of coherence between Siberia and the rest of Pangea, however, would not critically affect our global analysis.  Our paleopole compilation is listed in Table 3. Each paleopole was rotated to southern African coordinates (Figure 3a), using the parameters of Tables 1 and 2 while interpolating to the same age as the paleopole. The global compilation, comprising 419 paleomagnetic poles with Late Carboniferous and younger ages, is shown as south poles in Figure 4. The scatter of poles can be considerable; note that the Late Permian–Early Triassic poles from Gondwana generally show more easterly pole longitudes than do poles rotated from Laurentia (Figure 4b). Possible explanations for this are discussed in section 3.4. 3.3. APW Paths  APW paths are expected to average out random noise and to determine basic patterns of APW. The two most common methods for generating such paths are the running mean (moving window) and the spherical spline method. In the running mean method, paleomagnetic poles from a continent are assigned absolute ages, a time window is selected, and then all paleomagnetic poles with ages falling within the time window are averaged (Figure 5a). Using Fisher  statistics, 95% confidence ellipses (known as A95) can then be calculated for each mean pole.  A spherical spline on the surface of a sphere can be fitted to paleomagnetic poles [Jupp and Kent, 1987] and weighted according to the precision of the paleopole entries. The precision of the path [Silverman and Waters, 1984] can be estimated when angular errors are used for weighting. However, the real uncertainty surrounding individual paleopole positions is a combination of angular errors, age uncertainties (set at zero for this paper), and uncertainties surrounding the geomagnetic field recording of complexities such as the averaging of secular variation. Torsvik et al. , for example, developed a routine to give weight to the data according to their Q factor (section 3.2) so that the APW path is firmly anchored to the most reliable data.  Our global APW path (in southern African coordinates, Figure 5a and Table 4) extends back to 320 Ma when the Pangea supercontinent was initially being assembled. A global APW path was first constructed using the running mean method since this is the simplest method and can easily be reproduced by others. We used a window length of 20 Ma and employed 10 Ma increments, causing only moderate smoothing. Increased window length leads to a higher degree of smoothing.  We compare our path of Figure 5a to the APW paths of Besse and Courtillot  and Schettino and Scotese  and calculate the great circle distance (shortest distance on a sphere) between mean poles of the same age (Figure 5b). The mean great circle distance for the entire 200 Ma interval is 3.9! ± 3.3! with respect to Besse and Courtillot , with a peak of 11.1! at 200 Ma (Figure 5b). The mean great circle distance is 4.2! ± 3.3! for our comparison with Schettino and Scotese , with a peak of 12.8! at 150 Ma. Most of the differences are likely to be statistically insignificant, but the open loop between 110 and 170 Ma observed in our running mean APW path (Figure 5a) appears as a hairpin with a sharp cusp in the other two paths (Figures 5c and 5d). The reasons for this can be found in a combination of different data selection, degree of smoothing, and different Euler rotation parameters.  With global data sets that have considerable spread and ‘‘unclear’’ age progressions, spherical splines (e.g., weighted purely by angular errors) produce sinuous APW paths unless they are severely smoothed. Conversely, a spherical spline weighted by the Q factor [Torsvik et al., 1996] will produce APW paths that are anchored to the most reliable data, and this procedure ‘‘reproduces’’ the Late Jurassic–Early Cretaceous loop using moderate-tohigh smoothing parameters (red line in Figure 5a). However, a shortcoming of this procedure is that the assignment of weights to Q factors does not produce output angular uncertainties along the path that have a physical meaning. We therefore opted to first calculate a running mean path (10 Ma window, with increments of 10 Ma, so that there is no window overlap in order to avoid presmoothing) and found that this leads to a better age progression. Only then we applied the spherical spline algorithm to this running mean path weighted by the mean angular errors (A95). This results in angular output uncertainties that have physical meaning.
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The outcome of this procedure is shown in Figure 5c as the red path with yellow cones of confidence every 10 Ma. Using a moderate-to-high smoothing parameter, we generated a path similar to that of Schettino and Scotese . However, smoothing can lead to removal of real, short-duration features in the path. In order to explore this issue we attempted to ‘‘reproduce’’ the more smoothed Schettino and Scotese  APW path simply by increasing the window length with the running mean method. We found that our APW path can mimic their path if we use a window length of at least 50 Ma (red line with blue cones of confidence ovals in Figure 5d).  We calculated the APW for 10 Ma bins since the Carboniferous from the running mean path of Figure 5a. This magnitude of APW is shown in Figure 5e as the maroon histogram pattern, whereas the APW calculated from the smoothed spherical spline path of Figure 5c is shown as the black transparent histogram pattern in Figure 5e. The latter removes practically all temporal variation in APW, and notably, it eliminates the 110–100 Ma peak (!13 cm/a), which is so visible in the running mean-based path. A detailed examination of paleopoles between 90 and 120 Ma (as used to calculate the running mean path and shown in Figure 6a) clearly illustrates that this peak is real (even when smoothed because of the 10 Ma moving window overlap) because there is indeed a ‘‘systematic’’ progression of the Cretaceous poles for this interval. Hence the spherical spline path has smoothed away this important short-time APW feature. 3.4. Plate Motion Chains and the Pangea Enigma  Different Euler rotations for relative plate motions can produce significant differences between global APW paths. Most of our Euler rotations (Tables 1 and 2) differ from those of Besse and Courtillot , where finite (Euler) rotations were largely recomputed from the Mu¨ ller et al.  model described in section 4. Conversely, many Euler rotations of Schettino and Scotese  were calculated by inversion of the digital ocean seafloor grid of Mu¨ ller et al. . Different choices will naturally affect the resultant global APW path. Moreover, we have extended our APW path back to 320 Ma, which introduces some additional enigmas.  Paleomagnetic poles are calculated under the routine assumption that the time-averaged geomagnetic field is that of a geocentric axial dipole. However, Van der Voo and Torsvik  have suggested that nondipole field contributions may have persisted through significant periods of Earth history. Most Earth scientists agree that before the onset of breakup, the Jurassic ‘‘Pangea A’’ reconstruction is the correct one, in which NW Africa is located adjacent to the Atlantic margin of North America (Figure 3a). For Permian times, however, the paleomagnetic poles do not agree with the Pangea A fit of Figure 3a, as can be seen by examining the discrepancy between the poles from Gondwana and those for Laurussia in Figures 4b and 6b. To reconcile the paleomagnetic misfit, one has the option of (1) modifying the Pangea A reconstruction or (2) dismissing much of the database as contaminated by later magnetic overprints or (3) arguing that the mean pole positions are imprecise because of rock magnetic recording problems, such as those caused by sedimentary inclination shallowing [e.g., Kodama, 1997; Rochette and Vandamme, 2001; Torsvik and Van der Voo, 2002; Kent and Tauxe, 2005]. A fourth solution is to hypothesize significant octupole field contributions, in which case the Pangea misfit is caused by inferring a too far northerly position of Gondwana from the geocentric axial dipole hypothesis, as well as a too far south position for Laurussia [Van der Voo and Torsvik, 2001, 2004; Torsvik and Van der Voo, 2002]. Latitudinal errors caused by 5–10% octupole field contributions are comparable to inclination shallowing effects in sediments. All four of these explanations have been presented in previous publications [see Van der Voo, 1993; Muttoni et al., 2003; Irving, 2004; Van der Voo and Torsvik, 2004, and references therein]. The largest misfit is seen at 250 Ma (Figure 6b), likely because the data are indeed not very reliable for this interval, as argued by Muttoni et al. .  In general, the number of poles from Laurussia is larger (and, on average, of higher data quality) than similarly aged poles from Gondwana, and they will therefore bias the global path. Since the incorporation of nondipole contributions is controversial, we opted in this paper to only analyze geocentric axial dipole–based APW paths. However, we should mention that the debate about the various Pangea reconstructions has not subsided [see Irving, 2004] and that different Pangea fits labeled A2 and B (or less commonly C or D) remain favored even in recent literature. Since nondipole field contributions remain an open possibility, we refer to Table 4 (see column labeled SAFR) for a comparison of our geocentric axial dipole–based APW path with one that incorporates time-dependent nondipole fields (0–17.5% octupole contributions) [Torsvik and Van der Voo, 2002]. On average, however, this alternative path differs by only 2.0! ± 1.3! (great circle distance between poles of same mean age). 3.5. Paleomagnetic Euler Pole Analysis  Movements of continents, APW paths (in the absence of true polar wander), tracks of hot spots (if fixed to the mantle), and ocean fracture zones all should represent segments of small circles if the Euler pole is kept constant at the same location. This is portrayed for a sphere with only two plates in Figure 7: Plate F is fixed, while Plate M moves and rotates around a low-latitude Euler pole in the Southern Hemisphere. Paleomagnetic poles determined from young rocks on Plate M should plot near the Earth’s spin axis, whereas paleopoles derived from older rocks are situated on an APW path that represents a small circle around the Euler pole. If the moving plate (Plate M) is underlain by a hot spot, a chain of volcanic seamounts should also describe a small circle centered on the same Euler pole [cf. Gordon et al., 1984; Butler, 1992].  It is reasonable to assume that continents may drift around the pivotal axes of such unchanging Euler poles for, say, some tens of millions of years, and one can therefore attempt to construct APW paths by fitting small circles to paleomagnetic pole sequences; abrupt changes in the balance of forces driving and resisting plate motions should then be reflected in the APW paths as hairpins or cusps [Irving and Park, 1972; Gordon et al., 1984].
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Such analyses, however, must be undertaken in the reference frame of the plate in question, i.e., the plate (continent) on which the changing forces are acting. The analysis, referred to as the paleomagnetic Euler pole method, has commonly been performed on individual paleopoles from a single plate; however, below we test global running mean poles for small circle segments.  In southern African coordinates (Figure 5a) we find seven small circle segments (300–250, 250–220, 220–190, 190–170, 170–130, 120–50, and 50–0 Ma) with root mean square (RMS) differences less than 1! (0.68! > RMS > 0.28!) for the last 300 Ma. Segment length varies between 30 and 70 Ma in duration, and small circle intersections coincide with segment boundaries except for those at 120–130 Ma and 50 Ma.  Gordon et al.  originally applied the concept of Euler pole rotations [McKenzie and Parker, 1967] to the Mesozoic APW path of Laurentia, and we therefore show a detailed example of this method applied to North America. Figure 8a shows our global paleomagnetic running mean path in North American coordinates (Table 4). The three segments from 300 to 190 Ma are found to be similar to those of southern Africa and all other plates we have analyzed. This is to be expected because the continents moved together as parts of Pangea, with only minor predrift extension invoked prior to 190 Ma (Figure 3b). The Jurassic section, however, differs substantially; the abrupt 170 Ma cusp seen for southern Africa (Figure 5a) is not present. Instead, there is a change in trajectory at 150 Ma where an intersection occurs between the 190–150 and the 150– 130 Ma segments, but there is no trace of a distinct cusp. Similarly for southern Africa, we can fit small circles to the segments of 120–50 Ma (RMS = 0.81!) and 50–0 Ma (RMS = 0.23!), but the magnitude of Laurentian APW between 120 and 50 Ma (0.13!/Ma) is much less than that of southern Africa (0.45!/Ma). Our analysis is very different from any earlier paleomagnetic Euler pole analysis because we use global running mean poles and not just North American poles [see Beck and Housen, 2003], which show lower pole latitudes during the Middle-Late Jurassic than those predicted from a global model. We illustrate this in Figure 8a where the red dashed line represents a running mean APW path, calculated exclusively from North American poles (Table 3). In this purely North American running mean path the global 250 Ma cusp has vanished (no longer influenced by Gondwana poles, see Figure 6b); the 220 Ma knickpoint is deteriorating (the segments being replaced by a single small circle from 270 to 190 Ma, RMS = 0.58!), whereas the 190, 120, and 50 Ma cusps are recognized, just as they were in the global running mean path. However, the Middle-Late Jurassic section differs markedly, with a knickpoint near 160 Ma, located at lower latitude. This is relevant to the debate about the reliability of the Laurentian Jurassic APW segment [e.g., Ekstrand and Butler, 1989; Witte and Kent, 1990; Van Fossen and Kent, 1990; Van der Voo, 1992; Courtillot et al., 1994].  On the basis of North American data alone and different pole selection criteria, Beck and Housen  identified three small circle tracks (245–200, 200–160, and 160–125 Ma) and their two intersections (or cusps, both are pronounced), named J1 and J2 (following Gordon et al. ). Using this nomenclature but on the basis of our global running mean path data, J1 (!190 Ma in our global analysis) occurs shortly before seafloor spreading began in the central Atlantic (rift to drift and the first real breakup phase of Pangea) and !10 Ma after eruption of the Central Atlantic Magmatic Province. This J1 cusp at 190 Ma is associated with the second biggest burst in APW (Figure 8c), linked to a peak in angular rotation that initiated a prolonged period of clockwise rotation of !50 Ma duration. Conversely, J2 (!150 Ma) coincides with local high seafloor spreading rates in the central Atlantic (Figure 3b) and occurs shortly before a northward velocity increase that coincides with the highest amount of APW for North America during the entire Mesozoic (Figures 8c and 8d). The Cretaceous is marked by decreasing APW, and the Cretaceous cusp at !125 Ma (labeled C in Figures 8c–8e and Figure 3b) marks the beginning of the so-called North American Cretaceous stillstand that lasted until !70 Ma as judged from our global running mean path (Figures 8b and 8c). This apparent stillstand ended when seafloor spreading (rift to drift) occurred in the Labrador Sea between North America and Greenland at!67Ma (see Table 1 and Figure 3b). The C cusp is associated with the onset of the highest seafloor spreading peak in the central Atlantic (Figure 3b). In an ‘‘absolute’’ sense we argue that the North American Cretaceous stillstand is only apparent and that it was caused by a dominating component of paleo-east-to-west drift (section 7). Finally, a Tertiary cusp (labeled T at 50 Ma in Figures 3b and 8c–8e) at around 50 Ma is linked to seafloor spreading in the North Atlantic and the accompanying westward drift of the North American plate. The older cusps (denoted T1 and T2 in Figures 8c–8e and 3b) will be discussed in section 8.2. 4. African Fixed Hot Spot Frame  Wilson  first suggested that linear chains of seamounts and volcanoes, which display age progression, are caused by focused spots of melting in the mantle, termed ‘‘hot spots.’’ Morgan  proposed that hot spots may be caused by mantle plumes upwelling from the lower mantle, which in his model remain fixed relative to each other over geologically long periods of time (‘‘fixed hot spot hypothesis’’). During the past 40 years a multitude of marine geophysical data and isotope ages of seamounts from volcanic hot spot tracks have been collected, allowing reconstructions of plate motions relative to hot spots since the Cretaceous.  Early attempts to reconstruct all major tectonic plates with respect to one collective set of hot spots (presumed fixed relative to each other) led to the realization that models constructed for Pacific plate motions over hot spots could not be reconciled with the motions over the hot spots in the African–Indian Ocean domain for the last 80 Ma [Duncan, 1981; Morgan, 1981]. Subsequently, evidence has accumulated that hot spots underlying the Pacific cannot have remained fixed relative to the Atlantic-Indian hot spots [Molnar and Stock, 1987; Tarduno and Gee, 1995; Tarduno and Cottrell, 1997; DiVenere and Kent, 1999; Torsvik et al., 2002; Tarduno et al., 2003].
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However, hot spots within the Atlantic-Indian domain appear to have moved much less dramatically relative to each other. On the basis of this inference, Mu¨ ller et al.  used an interactive technique to derive a ‘‘best fit’’ model in a qualitative sense for motions of the major plates in the Atlantic-Indian domain relative to hot spot tracks with a clear age progression.  Even though this model is widely used, it has some well-recognized shortcomings: the Late Tertiary portion of this model was not well constrained by radiometric ages because of the lack of published age dates for the post– 30 Ma portion of most hot spot tracks. For reconstruction times predating 80 Ma, the only available hot spot tracks with a reasonably well known age progression in the Atlantic- Indian oceans are those of the New England seamount chain (linked to the Great Meteor hot spot) and the Walvis Ridge/ Rio Grande Rise (linked to the Tristan hot spot, Figure 9), both in the Atlantic Ocean. Therefore, the absolute motion of the Indian, Australian, and Antarctic plates relative to the mantle has to be computed solely based on plate motion chains for these times. However, when the Mu¨ ller et al.  model was constructed, pre–80 Ma relative plate motions in the Indian Ocean were poorly known because of a lack of data in crucial areas, in particular offshore Antarctica in the Enderby Basin and south of the Kerguelen Plateau. This has recently improved as a sequence of Mesozoic magnetic anomalies was mapped and modeled in the Enderby Basin, starting at about 130 Ma [Gaina et al., 2003, 2007]. The Mu¨ ller et al.  model was based on the assumption of post–120 Ma breakup between India and Madagascar, placing India too far north from 130 to about 90 Ma. Disagreements between the hot spot and paleomagnetic reference frames have been documented for India [Mu¨ ller et al., 1994; Torsvik et al., 1998] and for Australia [Idnurm, 1985], suggesting either an incorrect relative plate motion model, relative motion between hot spots through time, or true polar wander or any combination of these three issues. As an example, paleopoles for India from the Rajmahal Traps (Table 3) result in a paleolatitude of the traps at their time of formation (!116 Ma) at 47!S, whereas the Mu¨ ller et al.  model places them at about 40!S.  A comparison of mid-Cretaceous (122–80 Ma) paleolatitudes of North America and Africa from paleomagnetic data with those from hot spot tracks [Van Fossen and Kent, 1992] provided evidence for an 11!–13! discrepancy, suggesting that Atlantic hot spots likely did not remain fixed relative to the Earth’s spin axis before 80 Ma. With our more recent analyses using a different time interval and compared to the Mu¨ ller et al.  model, we find that the hot spots moved southward as much as 18! between 100 and 130 Ma [Torsvik et al., 2002]. Others argue that this apparent southward movement was caused by true polar wander (TPW) [see Pre´vot et al., 2000; Camps et al., 2002; Tarduno and Smirnov, 2002].  The Mu¨ller et al.  plate motion model results in a relatively sharp turn in plate motion directions (e.g., of Australia and Antarctica) relative to the mantle at about 80 Ma, which originates from the bend between the New England seamount chain and the Corner seamounts at roughly 80 Ma in the central Atlantic. An equivalent bend in fracture zones is not found in either the Pacific or Indian oceans. Bends in hot spot tracks, which are not seen in relative motions of the plates involved, are likely because of time-varying velocities of local mantle (hot spot) motion relative to the mean mantle, as has been suggested as having caused the bend in the Hawaiian-Emperor chain [Norton, 1995; Tarduno and Cottrell, 1997]. It follows that the central Atlantic bend may be due to a slowdown in southward motion of the mantle underlying at least the Atlantic Ocean and its bordering continents at around 80 Ma and that fixed hot spot models therefore need to be replaced by models that take into account the motion of hot spots in a convecting mantle.  The plate motion model used by Mu¨ller et al.  was based on the timescales of Berggren et al.  and Kent and Gradstein . In order to compare Mu¨ ller et al.  with more recent models for the purpose of this paper, we first need to translate it to more recent timescales [Cande and Kent, 1995; Gradstein et al., 1994]. This presents some difficulty, as Mu¨ ller et al.’s  model is based on a combination of relative and radiometric age dates, because samples from hot spot tracks are radiometrically dated, whereas the ages of plate reconstructions used in the model depended upon the ages of magnetic anomalies in the ocean basins, which, in turn, were dated with an older magnetic reversal timescale. For this paper we adjusted the ages given by Mu¨ller et al.  to the Cande and Kent  timescale for times back to chron 34 (83 Ma according to Cande and Kent  and 83.5 Ma according to Gradstein et al. ). Earlier ages were translated to the Gradstein et al.  timescale. We assigned the age of 83.5 Ma to chron 34 because the Gradstein et al.  timescale takes both Mesozoic and Cenozoic ages into account, thereby providing better constraints on chron 34 compared to the Cande and Kent  timescale, which does not take into account ages older than chron 34. The difference between Mu¨ller et al.  and our revised model, however, is small, and the largest differences (at !40 and !80 Ma) are less than 1! of arc when modeling the Tristan track (Figure 9). Exact input ages and Euler poles for the revised model are listed in Table 5, while interpolated Euler poles are shown in Figure 10 (African fixed hot spot) and listed in Table 6 (in the African fixed hot spot column). The pre–80 Ma portion of the fixed hot spot model, however, is notably weak, and our revised model still places India too far north at !116 Ma. Therefore, we will argue in section 5 that the African fixed hot spot frame should be replaced by a moving hot spot frame.
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5. MOVING HOT SPOT FRAMES 5.1. Introduction  Testing models for motions between individual hot spots, or between regional groups of them, requires mantle convection models constrained by surface boundary conditions based on known plate motions. Steinberger and O’Connell [1997, 1998] pioneered the modeling of hot spot motions and TPW based on mantle flow models. Their technique to take differential motion of individual hot spots into account essentially uses a two-step approach: In the first step, large-scale mantle flow is computed based on mantle density heterogeneities derived from seismic tomography and viscosity structure as well as known plate motions. These flow computations yield certain predictions, e.g., geoid, and matching these with observations serves the goal of making the flow models as realistic as possible. Since the geoid is sensitive to relative viscosity variations with depth and not absolute viscosity values, flow is better constrained in direction than magnitude. The flow models are also used to advect density heterogeneities and in this manner extended backward in time. Since the rotation axis will remain aligned with the axis of maximum nonhydrostatic moment of inertia, which, in turn, is inferred from the predicted geoid, a prediction of TPW is a by-product of this first step. In a second step the motion of plume conduits embedded in large-scale flow, and thus the motion of hot spots, the points where the plume conduits reach the lithosphere, is computed. The conduit is assumed to be initially vertical, and motion of each conduit element is a superposition of advection and buoyant rising. Predicted hot spot motion is often similar to mantle flow at some depth, typically the upper part of the lower mantle. As for the flow models, directions of hot spot motion can be inferred with greater confidence than the amount of hot spot motion. In fact, observational limits on the speed of relative hot spot motion can help to constrain the speed of mantle flow. Steinberger  extended this type of modeling to a larger number of hot spots and mantle flow schemes. This approach is most reliable for the Tertiary and has provided estimates for TPW and the motion of individual hot spots relative to each other for the last 68 Ma.  When these models are extended back to the Cretaceous, however, pure backward advection of mantle density anomalies becomes increasingly unreliable [Conrad and Gurnis, 2003], as more of the past mantle temperature anomalies may have diffused away and therefore cannot be reconstructed by extrapolation. Other methods, such as variational data assimilation and adjoint methods [e.g., Bunge et al., 2003; Ismail-Zadeh et al., 2006], will be needed to reconstruct mantle density and flow further back in time. Nevertheless, hot spot motion can still be computed prior to 68 Ma but with additional uncertainty. Test runs for models from 120 Ma to the present, either including or excluding the advection of mantle density anomalies, have shown that there are many similarities between the two types of models. This indicates that meaningful predictions for relative hot spot motion can be made based on a simple mantle convection model, constrained by time-dependent plate motions and mantle density heterogeneities. This strategy was used by O’Neill et al.  to model the motion of plumes relative to each other in a convecting mantle in the context of an ‘‘interactive inversion’’ strategy. O’Neill et al.  explored the large parameter space inherent in these models to search for those mantle convec- tion models that provide the best fit to observed hot spot tracks and their age progression, while minimizing the disagreements between model-based hot spot paleolatitudes and paleolatitudes from paleomagnetic data. Furthermore, O’Neill et al.  adopted the Hellinger  criteria of fit for deriving best fit absolute plate rotations based on track geometries, radiometric ages, and the moving locations of plumes in a convecting mantle to derive covariance matrices for absolute rotations of plates in the Indo-Atlantic domain for the last 120 Ma. While the original Hellinger  method uses fracture zones and isochrons as two orthogonal data sets to determine relative plate motions, in the O’Neill et al.  method, fracture zones are replaced by hot spot tracks, and ‘‘isochrons’’ are constructed based on age data along hot spot tracks. While fracture zones are the flow lines of relative plate motion, the geometry of hot spot tracks, corrected for computed hot spot motion, marks the flow lines of absolute plate motion. And while in the Hellinger  method isochrons on two plates are matched, O’Neill et al.  match instead their constructed isochrons with the modeled hot spot location at the respective times. In practical terms, application of the method is not straightforward; for example, age data along hot spot tracks are often sparse and may not always represent passing of the plate over the hot spot. Posterosional volcanism may still occur millions of years afterward; furthermore, both age data and track geometry may be influenced by plume-ridge interaction, i.e., flow of material from a plume to a nearby spreading ridge. These and other limitations and difficulties are discussed further by O’Neill et al. .  While the two-step approach was motivated by computational limitations, a fully dynamic computation of hot spot motion in large-scale flow is now feasible where not only the influence of the mantle on the plume is taken into account but also the effect of the plume on its surrounding medium [Tan et al., 2006]. Also, large-scale mantle flow models are becoming more realistic because of inclusion of temperature-dependent and strain rate–dependent viscosities [e.g., Becker, 2006; Cˇ adek and Fleitout, 2003] and further constraints, in particular related to seismic anisotropy [e.g., Becker et al., 2006; Behn et al., 2004]. 5.2. African Moving Hot Spot Frame  In the method of O’Neill et al.  each reconstruction is statistically independent from reconstructions at younger or older ages. This implies that smoothness is not imposed on the set of absolute rotations derived for a given plate. Therefore, a bias in the data used for a given reconstruction may result in a best fit rotation that is implausible in terms of plate kinematics. For instance, a consecutive set of rotation poles may lie approximately on a small circle, with one rotation being situated far off this small circle path, indicating that it is an outlier if there are no independent data supporting a plate kinematic event, such as a change in plate motion direction, or rate, at this time.
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The set of 12 Euler poles initially derived by O’Neill et al.  (OMS2005), from 10 to 120 Ma in 10 Ma intervals (black circles in Figure 10a), includes four rotations that are geologically or kinematically implausible, i.e., for 20, 70, 80, and 110 Ma (red circles in Figure 10a, OMS2005 rejected). These rotations are removed from this model and have been replaced by interpolated rotations (gray open circles in Figure 10a, OMS2005 interpolated, see Table 6), whereas the 130 Ma rotation pole has been extrapolated. The resulting Euler pole path can be approximated by three small circle segments from 0 to 40, 40–90, and 90–120 Ma; however, considering how large the errors are, none of these bends in the Euler pole path are significant. The largest difference between the O’Neill et al.  and our revised African fixed hot spot Euler pole paths is in the 0–60 Ma portions of the paths (Figure 10a, green versus black symbols). However, different age progressions in the Euler angles (Table 6) lead to very different mean plate velocities for southern Africa during the Cretaceous. The African fixed hot spot Euler rotations result in increasing plate velocities in the Late Cretaceous (peaking at !80 Ma), whereas O’Neill et al.  predicts decreasing velocities and shows a minimum where the African fixed hot spot frame predicts the highest velocities. The running mean paleomagnetic frame (thin dashed line in Figure 10b) shows gross similarities with O’Neill et al.  (solid black line) for the interval !110–50 Ma, but the two Tertiary peaks in the running mean path are not seen in any of the other reference frames (Figure 10b). 5.3. Global Moving Hot Spot Frame  The construction of the global reference frame of Steinberger et al.  was motivated by the long known fact that when a plate motion chain through West Antarctica is used (Figure 3a, model 1), the fixed hot spot African reference frame does not agree with a fixed hot spot Pacific reference frame. If African absolute plate motions are chosen such that hot spot tracks in the African hemisphere are fitted to observations, while hot spot tracks in the Pacific hemisphere are predicted through the plate motion chain, then the Pacific track predictions do not agree with the observations there: in particular, the predicted Hawaiian track is somewhat south of the observed track between Hawaii and the Hawaiian-Emperor bend at !50 Ma [Sharp and Clague, 2006]. Not only does the predicted track not have as pronounced a bend, it is also substantially farther southwest than the Emperor Chain for times prior to the bend. Wessel et al.  and Whittaker et al.  recognized that the recent redating of bend initiation to !50 Ma correlates the bend with major tectonic events from around the Pacific, such as South Pacific triple-junction reorganization at chrons 22–21 (49.7–47.9 Ma), Farallon- Pacific fracture zone bends at chrons 24–21 (53.3–47.9 Ma), and the direction change and proposed halt of Pacific-Kula plate spreading at chrons 24–20/19 (53.3–43.8/41.5 Ma) as well as a major reorganization of relative plate motions between Australia and Antarctica. Nevertheless, some regard the bend to be partly or fully caused by hot spot motion [e.g., Molnar and Stock, 1987; Norton, 1995], whereas Whittaker et al.  argued subduction of the Pacific- Izanagi spreading ridge and subsequent Marianas/Tonga- Kermadec subduction initiation may have been the ultimate causes of these events. If a Pacific absolute plate motion model is chosen such that hot spot tracks in the Pacific hemisphere are fitted and hot spot tracks in the African hemisphere are predicted through the plate motion chain, the predicted tracks do not agree with the observed tracks either.  Steinberger et al.  found that their geodynamical model typically predicts a southward motion of the Hawaiian hot spot with velocities up to a few centimeters per year, contrasting with slower motion for other hot spots. Thus, their model of hot spot motion, in combination with the plate motion chain that connects Africa and the Pacific via East Antarctica and Marie Byrd Land (West Antarctica), allowed a fit of hot spot tracks globally for times after the age of the Hawaiian-Emperor bend. In this model, no motion occurs between East and West Antarctica prior to 43.8 Ma (model 1, Figure 3a). For times prior to 43.8 Ma an east-west misfit between predicted and observed Hawaiian hot spot track remains. Consequently, Steinberger et al.  explored the use of an alternative plate motion chain that connects Africa and the Pacific via East Antarctica, Australia, and the Lord Howe Rise for times 46.3 Ma and older (model 2, Figure 3a). Between the Lord Howe Rise and the Pacific the 46.3 Ma rotation was adopted from Sutherland , and no motion was assumed prior to 46.3 Ma. For times 43.8 Ma and younger the plate motion chain through East Antarctica and Marie Byrd Land is maintained, and rotations are interpolated between 43.8 and 46.3 Ma. The two plate motion chains differ: for the southwest Pacific plate motion chain, model 2 predicts intra-Antarctic motion prior to 43.8 Ma, with extension in the Ross Sea area, whereas model 1 does not involve movements between East and West Antarctica before 43.8 Ma. With the plate motion chain of model 2, Steinberger et al.  were able to achieve an acceptable fit to hot spot tracks globally back to about 65 Ma. Prior to that a misfit between predicted and observed Hawaiian track remains, which can only be eliminated if further motions between the Lord Howe Rise and the Pacific are introduced.  With this plate motion chain an ‘‘absolute’’ African plate motion is determined such that the fit to the Hawaiian, Louisville, Reunion, and Tristan hot spot tracks is optimized in a least squares sense: locations and ages of dated samples, as compiled from various sources, are included in this optimization. For the Reunion track, samples from both the Indian and African plates were used, with sample locations of the Reunion track on the Indian plate rotated using India-Africa finite (Euler) rotations for their respective ages. For the Tristan track (Figure 11), only samples from the African plate were used. For the Hawaiian and Louisville tracks (Figures 11 and 12), sample locations were rotated using Pacific-Africa finite (Euler) rotation parameters for their respective ages.
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This means that, in essence, the tracks are computed as if all four hot spots remained located underneath the African plate all the time. For these four tracks rotated onto the African plate, the best fitting African plate motion is determined with a least squares method as described by Steinberger . Parameters used in the optimization include African plate rotation rates and latitude-longitude pairs of stage rotation poles for three time intervals (0–43.8, 43.8–61.2, and 61.2–83.5 Ma). For each time interval a constant rotation rate is assumed. This method requires that appropriate uncertainties in both space and time are assigned to each data point. Uniform spatial uncertainties of 50 km have been assigned to each data point, whereas in terms of temporal uncertainty, published age errors have been used for the Hawaii and Louisville tracks. For the Tristan and Reunion tracks a uniform temporal uncertainty of 0.5 Ma is assigned. Present hot spot locations are entered assuming a 50 km spatial uncertainty but zero time uncertainty. Each track is given equal weight in the optimization. With these choices for the uncertainties the resulting best fitting African plate motion gives a better visual agreement between predicted and observed tracks than when using published age errors.  The African plate motion in the ‘‘global moving hot spot’’ framework (Table 7), as determined here, differs from that of Steinberger et al.  because relative plate motions are slightly different. Furthermore, optimized African plate rotations are determined here for the three time intervals mentioned above. These interval boundaries are chosen to occur at those times when relative plate rotations also change. The direction change at !60–62 Ma corresponds to a change in the age progression along the Louisville chain [Koppers et al., 2004]. The ‘‘fixed hot spot’’ tracks in Figure 12 show that without further smoothing the inferred Pacific plate motions exhibit short-term fluctuations that bear no apparent relation to actual plate motions, because they do not correspond to features in the observed hot spot tracks and would, at any rate, be difficult to explain dynamically. We therefore introduce a few ‘‘ad hoc’’ changes in order to obtain a smoother, and hence likely more realistic, Pacific plate motion history. First, we assign the total rotation between Lord Howe Rise and Pacific plate from Sutherland  (latitude of 49.8!, longitude of “1.6!, and angle of 49.0!) to 51.7 Ma instead of 46.3 Ma. At 43.8 Ma the rotation (latitude of 50.1!, longitude of “2.7!, and angle of 47.7!) is inferred through the South Pacific plate motion chain, and for the interval between 43.8 and 51.7 Ma it is interpolated. We note that there is no evidence for deformation between Lord Howe Rise and the Pacific as early as 51.7 Ma; however, the required deformation is quite small and presumably within error bounds (R. Sutherland, personal communication, 2005) because the two rotations above are quite similar. This change simplifies Pacific plate motion around the time of the Hawaiian- Emperor bend and removes a kink (shown in green in Figure 12) in the Hawaiian track. Inferred Pacific plate motion around the time of the bend (43.8–51.7 Ma) is quite slow (!2.5–3 cm/a). (See Figure 11 (bottom); note that this diagram shows mean plate speeds; the actual point speed around the bend is as low as 1.7 cm/a between 50 and 45 Ma.) Second, from among the Lord Howe Rise versus Australian rotations for magnetic anomalies 24o (o indicates older end of the anomaly, 53.3 Ma) through 33y (y indicates younger end of the anomaly, 73.6 Ma) we only use anomaly 27o (61.2 Ma) and 30y (65.5 Ma). These rotation poles are at intermediate locations and for times when the spreading direction appears to have changed from rotating clockwise to counterclockwise and back to clockwise. This change leads to a considerably smoother and more realistic Hawaiian track in Figure 12 (violet versus red or white versus orange). Short-term changes in Tasman Sea spreading may hence be related to deformation within continental crust bounding the Tasman Sea (especially in New Zealand) and not to short-term changes of Australian or Pacific plate motion. Third, we exclude the southern Africa versus East Antarctica rotation at the time of anomaly 28 (63.1 Ma), which was published by Bernard et al. . This rotation is quite different from the rotations before and after and also differs from that of Royer et al. . At the location of Hawaii the omission of this 63.1 Ma rotation does not change Pacific plate motion by much, but it removes a kink (white versus violet track in Figure 12) in the Louisville track. Figure 11 shows the ‘‘moving hot spot’’ Tristan and Hawaiian tracks for the smoothed model (white tracks in Figure 12). Back to about 75 Ma, the difference between predicted and observed Hawaii-Emperor hot spot tracks is now less than about 300 km (Figure 11) and thus is probably less than uncertainties arising from the oceanic part of the plate motion chain [e.g., Cande et al., 1995]. In the model without smoothing, larger misfits occur before !65 Ma. We also note that with southward motion of the Hawaiian hot spot between about 75 and 50 Ma of a few degrees more than modeled here, the entire Emperor chain could be fit in terms of geometry, ages, and paleolatitudes [Tarduno et al., 2003]. However, a more detailed analysis and justification of the plate motion chain modifications will be required before fully endorsing this smoothed model.  Steinberger et al.  were concerned with the past 83.5 Ma, and they made no attempt to determine a reference frame that considered hot spot motion before chron 34, primarily because the Hawaii-Emperor track does not extend further back in time but also because the uncertainties are much larger for older times. Here, we extend the reference frame to times before 83.5 Ma, using rotation rates relative to hot spots that are assumed fixed. The extension is done separately for the Pacific [Duncan and Clague, 1985] and Africa (section 4). The predicted plate velocity for southern Africa in this frame is quite similar to that in the African moving hot spot framework [O’Neill et al., 2005] until !40 Ma; for 60–83.5 Ma it is similar to the velocity in the African fixed hot spot frame, and prior to 83.5 Ma it is by construction identical (Figure 10b).
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6. COMPARISON OF RECONSTRUCTION FRAMES  The largest differences between the revised African fixed hot spot, the African moving hot spot, and the global moving hot spot reference frames are seen in the Tertiary portions of the Euler pole ‘‘paths’’ (Figure 10). However, error ellipses calculated from the O’Neill et al.  model demonstrate that the majority of the computed Euler poles are not statistically different at the 95% confidence level.  In order to compare hot spot and paleomagnetic frames the most common approach is to rotate the mean paleomagnetic poles using the hot spot reconstruction parameters. This is commonly referred to as plotting paleomagnetic poles in a hot spot frame [see Torsvik et al., 2002]. In the absence of errors in the rotation parameters, TPW, or other complexities all paleomagnetic poles should plot at 90!N in the hot spot frame. Here, we use a novel approach and compare uncertainty ellipses computed by Hellinger’s method (centered at 90!N, with blue ovals in Figure 13) for the African moving hot spot frame, with errors of the mean paleomagnetic poles (A95, pink circles in Figure 13). Although there are marked differences between the rotated global mean paleomagnetic poles and the rotation axis for the Early Cretaceous (>100 Ma), it is evident that for Tertiary times (with only one exception at !50 Ma) one cannot argue for systematic and statistically significant differences; the average great circle distance is only 3.7! ± 1.6! (Figure 13b). At 100 Ma, great circle distance is still only 3!, whereas the two data sets are significantly different at 120 Ma (there are no Hellinger error ellipses for 110 and 130 Ma).  Compared with the African fixed hot spot frame, the African moving hot spot frame is an important improvement because it significantly reduces the difference between the global mean paleomagnetic poles and the rotation axis for the Early Cretaceous (compare red and black lines in Figure 13b, showing a reduction of the great circle distance at 120 Ma from !17! to !10.5!), although this remains a statistically significant difference at the 95% confidence level. We note, though, that there are substantial uncertainties in modeled mantle flow and hot spot motion this far back in time, and these uncertainties are not included in the assessment of significance. This is important and has implications for some recent and passionate debates concerning Cretaceous TPW [Pre´vot et al., 2000; Tarduno and Smirnov, 2001, 2002; Camps et al., 2002; Torsvik et al., 2002].  Given plate rotations and boundaries, it is also possible to compute mean lithospheric rotations in different reference frames. For example, combining our global moving hot spot reference frame with NUVEL [DeMets et al., 1990] plate boundaries, we find for the past 5 Ma an average of 0.165!/Ma around an axis 40!S, 38!E. Such a net rotation of the lithosphere relative to the deeper mantle can result from lateral viscosity variations [Ricard et al., 1991; O’Connell et al., 1991]. We regard this net rotation mainly to be a consequence of the large size of the Pacific plate. The Pacific plate is subducting in the west and north and has ridges in the east and, as an entirely oceanic plate, is presumably underlain by a low-viscosity asthenosphere. This qualitatively explains its direction and relatively fast speed of motion (Figure 12). Because of its large size the Pacific plate dominates the mean lithospheric rotation. Hence the axis of mean lithospheric rotation that we find is similar to the axis of Pacific plate rotation (Table 8). More quantitatively, Becker  finds predicted mean lithospheric rotation for a number of geodynamic models to be similar to our result both in direction and magnitude. Since its rotation pole is in the Southern Hemisphere, this net rotation contains a ‘‘westward drift’’ component. As this westward drift is mainly caused by motion of the Pacific plate, it does not imply westward motion of other plates, such as the African plate. Other authors [e.g., Doglioni et al., 2005] have proposed different reference frames and obtain much higher values of westward drift. 7. TOWARD A HYBRID REFERENCE FRAME  Hot spot frames are arguably not very robust prior to 100 Ma, but at 100 Ma the African moving hot spot frame accommodates the global mean paleomagnetic pole location pretty well, so that the position of Africa in Figure 14 is very similar in the moving hot spot frames and in the global paleomagnetic frame; only the fixed hot spot frame produces a different position of Africa. Since it is permissible in paleomagnetic reconstructions to adjust the longitude, we correct for a 5! longitude offset observed at 100 Ma between the African moving hot spot (blue in Figure 14a) and the global paleomagnetic reference frames. In this way we produce the first global hybrid reference frame where we combine the paleomagnetic frame for 320–110 Ma with the moving hot spot frame for #100 Ma (Table 6).  Predicted mean plate speeds for southern Africa based on this hybrid model average to 3.5 ± 1.5 cm/a (Figure 15b), compatible with ‘‘normal’’ plate tectonic speeds, but there is a velocity spike when the two reference frames are merged (6.7 cm/a at 105 ± 5 Ma). This mid- Cretaceous velocity peak is not an artifact of frame change since the peak is seen independently in the paleomagnetic frame (Figure 10b) and predicted by maximum APW at this time (section 3 and Figure 5e). A change in velocity but with different magnitude (!6–13 cm/a) is found for all continental plates at this time. For Africa this event is related to a strong counterclockwise rotation (Figure 15c) with a rotation pole near the equator (Figure 16) that could be interpreted as TPW at 100–110 Ma. This issue has been addressed by Steinberger and Torsvik  who further developed the reference frames of this paper by considering TPW.  One other significant issue emerges from our consideration of the paleomagnetic data rotated into the hybrid moving hot spot framework: the well-known Cretaceous stillstand in Laurentia’s APW path (Figure 8b). We argue that this pattern represents pure east-to-west drift of the continent. Note that the North American plate motion history is very different in the hybrid model from the fixed hot spot model.
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The latter predicts northward drift during the Cretaceous, with a major bend at 80 Ma [see also Torsvik et al., 2001a]. In contrast, the hybrid moving hot spot model predicts a strong component of westward drift. The sharp change in ‘‘absolute’’ motion at !50 Ma for both North America and Europe (Figure 17) is linked to the opening of the NE Atlantic. 8. HYBRID FRAME: EXAMPLES 8.1. Linking Plate Motions to the Deep Earth  An ‘‘absolute’’ plate motion reference frame is essential in order to explore potential links between plate tectonics and processes operating in the deep Earth. Volcanism unrelated to plate boundaries or rifts has been widely attributed to mantle plumes from the deep mantle. Torsvik et al.  explored the spatial relation between large igneous provinces and the deep mantle by comparing large igneous provinces (LIPs) at eruption time with shear wave anomalies near the core-mantle boundary. Testing all reference frames developed in this study, Torsvik et al.  showed that practically all LIPs erupted at the Earth’s surface for the past 200 Ma lay over the margins of the African or the Pacific large low-shear-velocity provinces [Garnero et al., 2007] at the core-mantle boundary. In Figure 18 we reconstruct 17 LIPs in the Indo-Atlantic realm using our new hybrid frame and extending the analysis to the last 300 Ma. In situ locations of approximated LIP centers (small annotated circles with gray or light blue background colors) are scattered and show no obvious link to the African large low-velocity provinces (LLSVPs). Conversely, correcting for plate motion since eruption time (large annotated circles with white or blue background colors) demonstrates a strong spatial link between LIP surface eruption locations and the deep mantle; that is, all LIPs in the Indo-Atlantic realm (except the North Atlantic Igneous Province, GI in Figure 18) project radially downward onto or close to the margin of the African LLSVP.  Eruption locations vertically above the edge of one or other of the Earth’s two LLSVPs at the core-mantle boundary characterize nearly all the LIPs erupted since 300 Ma [Burke and Torsvik, 2004; Torsvik et al., 2006, 2008], and for that reason it can be argued that LIPs are derived from deep mantle plumes. The hybrid plate reference frame used to reconstruct LIPs in Figure 18 is based on the African moving hot spot (0–100 Ma) and the paleomagnetic (>100 Ma) frame. The paleomagnetic frame is based on keeping the African plate fixed in longitude before 100 Ma but adjusted 5! in longitude. Despite the ‘‘zero’’ Africa assumption (sections 3.1 and 9), paleomagnetically reconstructed LIPs show a close correspondence with today’s deep mantle shear wave tomography. However, longitude is strictly not known, and all LIPs older than 100 Ma can therefore theoretically be adjusted in an E–Wsense and with different magnitudes since LIPs have different ages. 8.2. Pangea  An important growth phase occurred for Pangea during the Late Carboniferous when Laurussia, Gondwana, and intervening terranes collided. Although some continental elements were probably still adjusting their positions along the Pangea perimeter, Pangea had essentially accomplished its ‘‘all-Earth’’ mission by Early Permian time. However, the China blocks were still only loosely connected with Pangea within the Paleo-Tethys Ocean [Torsvik, 2003; Torsvik and Cocks, 2004]. Our 250 Ma Pangea reconstruction is shown overlaying the present-day shear wave velocity anomalies near the core-mantle boundary in Figure 19, assuming that lower mantle heterogeneities have remained stationary for hundreds of millions of years [Burke and Torsvik, 2004; Torsvik et al., 2006] (see section 8.1). Our global hybrid model predicts that the bulk of Pangea was centered above the present-day African low-velocity region and that the peri-Pangea subduction rim was essentially located above high-velocity zones, feeding the subduction graveyards in the deepest mantle [Richards and Engebretson, 1992].  Africa was at the heart of Pangea, and two clear 90! kinks (near cusps at 220 and 190 Ma) in the southern African APW path (Figure 15a) may relate to changes in the balance of plate motion forces that are representative of Pangea breakup. A third cusp-like feature (at !250 Ma, see Figure 15a (labeled T1 in Figure 15b)) should be considered with some care since there is a major discordance between Laurussian and Gondwana poles of this age (Figure 6b); moreover, increased smoothing treatments of the APW path diminish the impact of this feature (Figures 5c and 5d).  Considerable counterclockwise rotation of Pangea and terrane displacements in the Tethys occur during the Permo-Triassic. An additional, unresolved question is whether Siberia was fully joined to Pangea before the eruption of the Siberian Traps (!251 Ma). The Neo-Tethys probably began opening at !265 Ma [Stampfli and Borel, 2002] and was well developed as a young oceanic basin by 250 Ma (black area in Figure 19), while Paleo-Tethyan oceanic crust was being subducted beneath Eurasia. All other subduction zones inferred for the Permo-Triassic have Panthalassa’s oceanic lithosphere plunging down under Pangea’s perimeter. In other words, no subducting slabs are known to have been attached to Pangea’s continental lithosphere with outward directed plunges.  It is possible and natural to link the 220 and 190 APW kinks (Figure 15a) to the destruction of much of the Paleo-Tethys and the transition from rift to drift in the central Atlantic (Pangea breakup). Paleo-Tethys had essentially vanished by the Early Norian (!220 Ma) [Stampfli and Borel, 2002] as a result of the collisions of many peri- Gondwana terranes, collectively called Cimmeria [S¸engo¨r and Natal’in, 1996], with Eurasia. The Early Jurassic witnessed the assembly of the Asian part of Pangea but simultaneously saw the breakup of Pangea in the central Atlantic; both of these events reflect changes in plate motion forces that are likely candidates to explain the 190 Ma cusp.  Cause and effect can be enigmatic, and APW tracks, separated by cusps or kinks, can also be the result of TPW, best represented by a rotation around an equatorial axis close to a supercontinental center of mass [Steinberger and Torsvik, 2008]. For the late Paleozoic–early Mesozoic during which Pangea was a supercontinental entity drifting as a whole with respect to the rotation axis, TPW has been speculatively proposed by Marcano et al. .
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The white arrows in Figure 19 illustrate the generalized velocity field that remained in effect for much of the counterclockwise rotation episode of about 0.4!/Ma that lasted until the latest Triassic. In order to establish TPW with any certainty one needs the velocity field for oceans as well as continents, and in the Permo-Triassic this is obviously not possible because the plate configurations inside the Panthalassa ocean clearly remain unknown. However, Marcano et al.  argued that if the Panthalassa hemisphere participated in the same rotation shown by the Pangea hemisphere, no evidence for significant convergence between Panthalassa and Pangea at the latter’s leading edge should exist. Examining the regional geology along the Siberian-Baltic-Laurentian Arctic margins, they found that evidence for convergence in the 295–205 Ma interval was rather scant and therefore concluded that slow TPW (! 4 cm/a) for this interval could not be ruled out. 9. CONCLUSIONS AND FUTURE CHALLENGES  On the basis of revised plate motion chains we have recomputed and compared four different plate reference frames for Africa (paleomagnetic, African fixed hot spot, African moving hot spot, and global moving hot spot). We find that the African moving hot spot frame compares most favorably with the global paleomagnetic frame; considering the uncertainties in both reference frames, they are essentially identical for the last 100 Ma. For older times the moving hot spot frame is uncertain because simple backward advection is increasingly inappropriate for reconstructing past mantle density anomalies. Given the magnitude of the error limits in both mantle and paleomagnetic reference frames, it is premature to conclude true polar wander fromtheir differences except for the Early Cretaceous (130–110 Ma). With respect to the African moving hot spot frame the mean paleomagnetic poles for 110–130 Ma show a discrepancy of approximately 10!. Mantle models are arguably not very robust before 100 Ma, but 10! is a considerable reduction when compared to the use of the African fixed hot spot frame where the discrepancy is !18!. The fixed hot spot frame should no longer be used.  The African moving hot spot frame is modeled back to 130 Ma. The ‘‘global’’ moving hot spot frame (incorporating the Pacific realm) is valid only back to 83.5 Ma; prior to this time it is extended by using rotation relative to assumed fixed hot spots. Our smoothed global moving hot spot frame produces a more realistic Pacific plate motion history, in which the difference between predicted and observed Hawaiian hot spot tracks is probably less than uncertainties arising from the plate motion chain for the last 75 Ma. Since the African moving hot spot and global paleomagnetic frames are exclusively based on data from the Indo-Atlantic realm, we decided to merge these two frames in our hybrid model.  We have constructed the first hybrid ‘‘absolute’’ reference frame model for the last 320 Ma: we use the African moving hot spot frame for the last 100 Ma and then the global paleomagnetic frame adjusted 5! in longitude to smooth the frame transition. All hot spot–based ‘‘absolute’’ plate motion models (Figure 14) result in minimal longitudinal motion of Africa (compared to most other plates), thus confirming the lack of significant longitudinal motion inferred from consideration of plate driving forces (section 3.1). It is less certain whether the ‘‘zero-longitudinal motion’’ approximation for Africa, corrected for the longitudinal motion of Africa during the past 100 Ma (Figure 14), holds before Pangea broke up. On the one hand, Forsyth and Uyeda  showed that plates move faster when there are subducted slabs attached and that the larger the continental area on a plate is, the slower it tends to move.With little or no subducting slabs attached (section 8.2) and a very large continental area, Pangea is expected to have moved slowly. On the other hand, average N–S velocities derived from our global paleomagnetic path are as high as 3.6 ± 2.1 cm/a (320–180 Ma) for a central Pangea location in NW Africa. Nevertheless, in the absence of arguments for better reference points, we regard zero longitudinal average motion of Pangea as the best possible assumption.  The hybrid model is a step forward to quantifying plate kinematics and the time-dependent plate velocity field and is essential for providing surface boundary constraints for mantle convection models and testing the relationship between surface magmatism and deep Earth processes, with the caveat that motions based on the paleomagnetic frame may also contain contributions due to TPW [Steinberger and Torsvik, 2008]. Both the fact that plumes only seem to arise from the edges of the large low-shear-velocity provinces (LLSVPs) and that these LLSVPs appear not to have moved by much over the past 300 Ma (section 8.1) provide a challenge to be explained in future mantle dynamic models. At face value the mantle flow models used to compute advection of plumes would also predict changes of LLSVPs with time. Just as the observational limits of relative hot spot motion provide a constraint on the speed of mantle flow and hence on mantle viscosity, the observational limits on LLSVP motion and deformation will also allow us to gain information regarding their rheology.  Our hybrid reference system is yet to be tested using forward mantle convection models, which ‘‘stir’’ the mantle by prescribing the global plate subduction history as a surface boundary condition. An accurate absolute plate motion model in a mantle reference system would result in an accumulation of subducted slab material in the mantle through geological time that is in agreement with global seismic tomography images. An absolute reference frame in agreement with subducted slabs may be termed a ‘‘subduction absolute reference frame.’’ Such a reference system, based on using subducted slab locations from seismic tomography as an additional plate kinematic model constraint, could be the next step toward developing a unified plate motion/mantle convection model through time. A unified geodynamic/plate tectonic modeling approach would enable the mapping of mantle-driven surface topography (dynamic topography) in space and time and ultimately provide fundamental insights into the driving forces of plate tectonics.