Acoustic emissions of nearly steady and uniform granular flows: a proxy for flow dynamics and velocity fluctuations
Vincent Bachelet, Anne Mangeney, Renaud Toussaint, Julien DeRosny, Maxime Farin, Clément Hibert
mmanuscript submitted to
JGR-Earth Surface
Acoustic emissions of nearly steady and uniformgranular flows: a proxy for flow dynamics and velocityfluctuations
Bachelet, V. , Mangeney, A. , , Toussaint, R. , , DeRosny, J. , Farin, M. , andHibert, C. Institut de Physique du Globe de Paris, Universit´e Paris-Diderot, Sorbonne Paris Cit´e, CNRS (UMR7154), 75005 Paris, France ANGE team, Inria, Lab. J.-L. Lions, CNRS, 75005, Paris, France Institut de Physique du Globe de Strasbourg, Universit´e de Strasbourg/EOST, CNRS, 67000,Strasbourg, France PoreLab, Njord Centre, Department of Physics, University of Oslo, Oslo, Norway Institut Langevin, ESPCI Paris, CNRS, PSL Research University, 75005, Paris, France
Key Points: • We analyze the high-frequency emissions and particle agitation of quasi steadygranular flows on constant slopes. • Scaling laws between granular temperature, average velocity, shear rate andinertial number are derived. • A simple physical model for the acoustic emissions and acoustic efficiency ofsteady flows is developed and tested.
Corresponding author: Toussaint, R., [email protected] –1– a r X i v : . [ c ond - m a t . s o f t ] J a n anuscript submitted to JGR-Earth Surface
Abstract
The seismic waves emitted during granular flows are generated by different sources:high frequencies by inter-particle shocks and low frequencies by global motion and largescale deformation. To unravel these different mechanisms, an experimental study hasbeen performed on the seismic waves emitted by dry quasi steady granular flows. Theemitted seismic waves were recorded using shock accelerometers and the flow dynamicswere captured with a fast camera. The mechanical characteristics of the particle shockswere analyzed, along with the duration between shocks and the correlations in theparticle motion. The high-frequency seismic waves (1-50 kHz) were found to originatefrom particle shocks and waves trapped in the flowing layer. The low-frequency waves(20-60 Hz) were generated by the oscillations of the particles along their trajectories,i.e. from cycles of dilation/compression during the shear. The profiles of granulartemperature (i.e. the square of particle velocity fluctuations) and average velocitywere measured and related to the average properties of the flow as well as to the slopeangle and flow thickness. These profiles were then used in a simple steady granularflow model to predict the radiated seismic energy and the energetic efficiency, i.e.the fraction of the flow potential energy converted to seismic energy. Scaling lawsrelating the seismic power, the shear strain rate and the inertial number were derived.In particular, the emitted seismic power is proportional to the granular temperature,which is also related to the mean flow velocity.
Gravitational flows such as landslides, debris avalanches and rockfalls representone of the major natural hazards threatening life and property in mountainous, vol-canic, seismic and coastal areas, with large events possibly displacing several hundredthousand people. They play a key role in erosion processes on the Earth’s surface.Gravitational instabilities are also closely related to volcanic, seismic and climatic ac-tivity and thus represent potential precursors or proxies for changes in these activitieswith time, as shown for example for the Piton de la Fournaise volcano, La R´eunion Is-land [Durand et al., 2018, Hibert et al., 2014, 2017a] or for the Soufri`ere Hills volcano,Montserrat Island [Calder et al., 2005].Research involving the dynamic analysis of gravitational mass flows is advancingrapidly. One of its ultimate goals is to produce tools for detection of natural insta-bilities and for prediction of velocity, dynamic pressure and runout extent of rapidlandslides. However, the theoretical description and physical understanding of theseprocesses in a natural environment are still open and extremely challenging problems[see Delannay et al. [2017] for a review]. In particular, the origin of the high mobil-ity of large landslides is still unexplained with different hypotheses proposed in theliterature (acoustic fluidization, flash heating, etc.) [Lucas et al., 2014]. The lack offield measurements relevant to the dynamics of natural landslides prevents us fromfully understanding the processes involved and from predicting landslide dynamicsand deposition. Indeed, these events are generally unpredictable, but have a stronglydestructive power. Furthermore, data on the deposits are not always available due tosubsequent flows, erosion processes and site inaccessibility.In this context, the analysis of the seismic signal generated by natural instabilitiesprovides a unique way to detect and characterize these events and to discriminatebetween the physical processes involved. When flowing down the slope, landslidesgenerate seismic waves in a wide frequency range that are recorded by local, regional orglobal seismic networks, depending on the event size [Allstadt et al., 2018, Okal, 1990].As a result, the recorded seismic signal, with frequencies ranging from about 0.006Hz to 30 Hz, carries key information on the landslide dynamics to distances far fromthe source. However, inferring information from the seismic signal to characterize the –2–anuscript submitted to
JGR-Earth Surface
Mt Lituya
SKAG (140 km)
Mt Dall
BYR (115 km)
Sheemahant glacier
BBB (152 km)
TimeTime (a)(c)
Time A m p li t ude [ - m / s ] M o m en t u m [ N s ] A m p li t ude [ - m / s ] M o m en t u m [ N s ] A m p li t ude [ - m / s ] M o m en t u m [ N s ] A m p li t ude [ - m / s ] M o m en t u m [ N s ] Time
Lamplugh glacier
BESE (121 km) (b)(d)
Figure 1.
Seismic signal envelope (gray), smoothed envelope (red) and inverted momentum(blue) from the inversion method proposed by Ekstr¨om & Stark [2013] for landslides on a) MtDall, b) Mt Lituya, c) the Sheemahant glacier and d) the Lamplugh glacier. landslide source suffers from uncertainties related to the respective effects of the meanflow dynamics, grain-scale processes, topography, mass involved and wave propagationon the recorded signal. It is commonly speculated that grain impacts on the substrategenerate high frequencies ( > < T < -120 s) by recovering the force that thelandslide applies to the ground from signal deconvolution, e.g. Allstadt [2013], Ekstr¨om& Stark [2013], Hibert et al. [2017b], Kanamori & Given [1982], La Rocca et al. [2004],Lin et al. [2010], Moretti et al. [2012], Yamada et al. [2013], Zhao et al. [2015]. The timehistory of this force is essentially related to the acceleration/deceleration of the flowalong the topography. Comparing this force with the force simulated with landslidemodels makes it possible to recover landslide characteristics and dynamics such asits volume and timing, the friction coefficients involved, the role of erosion processes,bedrock nature (rock or ice) and topography [Favreau et al., 2010, Moretti et al., 2015,2012, Schneider et al., 2010, Yamada et al., 2018, 2016].The high-frequency signal is much more difficult to interpret, in particular due tothe strong effect of topography and Earth heterogeneity along the path of the seismicwaves from the source to the receiver. For these reasons, mainly empirical relationshipshave been proposed between the high-frequency signal and landslide characteristics[Dammeier et al., 2011, Deparis et al., 2008, Norris, 1994]. This high-frequency signalis however more generally recorded because of the lower price of short period seismome-ters and because small landslides (with volumes < ) only generate frequencieslarger than about 1 Hz. Recent studies show correlations between the high-frequencysignal (energy, envelope, etc.) and the mean properties of the flow (potential energylost, force, velocity, momentum, etc.) estimated using landslide models [Hibert et al.,2014, 2011, Levy et al., 2015] or from inversion of low-frequency seismic data [Hibertet al., 2017b]. In particular, Hibert et al. [2017b] observed that the flow momen-tum is generally proportional to the high-frequency envelope of the signal. However,sometimes, in particular during the deceleration phases, a high-frequency signal canbe observed even if the force inverted from the seismic signal, which is proportionalto the landslide acceleration, is almost zero, leading to an apparent zero-velocity (seegray area in Fig. 1). However zero acceleration could correspond to constant velocityflows which would generate seismic waves, possibly due to grain agitation. Huang –3–anuscript submitted to JGR-Earth Surface et al. [2007] compared the high-frequency seismic signals generated by rock impactsand debris flows (grain/fluid mixtures) and concluded that one of the main sources ofground vibration caused by debris flows was the interaction of rocks or boulders withthe channel bed. However, the complexity of natural landslides and the difficulty toobtain accurate measurements of their dynamics makes it nearly impossible to quantifythe link between grain scale physical processes such as velocity fluctuations and thegenerated seismic signal. More generally, measurements of particle agitation, calledgranular temperature in the kinetic theory of granular flows, and its link with meanflow properties in dense flows, are still open questions, closely related to the rheologyof granular materials [see e.g. Andreotti et al. [2013], Delannay et al. [2017] for reviewpapers].We propose to address this issue here by developing laboratory experiments torecord and quantify seismic (i.e. acoustic) waves generated by almost steady and uni-form granular flows. Only a few experiments on granular flows and generated acousticwaves have been conducted. These experiments however make it possible to test phys-ical interpretations of the characteristics of the seismic signal generated by naturallandslides and to quantify the energy partitioning between the flow and the seismicemissions. Furthermore, such experiments provide a unique way to check models ofgranular flows and seismic wave generation in a simple configuration, before tacklingnatural applications. On a 8-meter long channel, Huang et al. [2004] investigated theacoustic waves generated by the friction and impacts of rocks of about 100 g to 1 kgon a granular bed filled with water and slurry and by a mixture of gravel and wa-ter/slurry. They recorded similar frequencies for individual rock motion and debrisflows, as observed in the field by Huang et al. [2007]. Their measurements also showedthat the amplitude of the acoustic signal increases with the gravel size. However, thecomplexity of the materials involved and the lack of measurements at the grain scalemade it difficult to capture the origin of the generated signal and to quantify the linkbetween the acoustic measurements and the flow properties.A series of experiments of granular impacts on various beds showed that Hertztheory quantitatively explains the generated acoustic signal on smooth beds [Farin etal., 2015]. These experiments also showed that power laws issued from this theory makeit possible to relate empirically the acoustic energy to the properties of the impactor(mass, velocity) on smooth, rough and erodible beds [Bachelet et al., 2018, Farinet al., 2016, 2015]. More specifically, the frequency/energy of the acoustic signal isshown to decrease/increase with the mass and velocity of the impactor, respectively, asobserved for debris flows [Okuda et al., 1980] or for single block rockfalls [Hibert et al.,2017c]. Finding these quantitative relationships between acoustic and flow propertieswas only possible because of the joint accurate measurements and calculations of thegrain motion and absolute value of the radiated energy using coupled optical andacoustic methods. In this way, Farin et al. [2018, 2019] showed that the seismic powervaries in the same manner as the flow velocity in granular collapses on inclined planes.In particular, after the first acceleration/deceleration phase of the mass, the seismicpower increases with increasing slope in the same way as the downslope velocity andthe agitation of the particle at the flow front. Measurements of grain-scale fluctuationswere however not performed in these 3D experiments.In a quite different setting involving granular materials sheared in a torsionalrheometer, Taylor & Brodsky [2017] found that the square of the acceleration measuredwith their accelerometers divided by the number of particles was proportional to I × d ,where d is the particle diameter and I the so-called inertial number, defined as theratio between the time scale related to shear and the time scale related to particlerearrangement under confining pressure. However, Taylor & Brodsky [2017] neithercalculated absolute values of the acoustic energy nor measured the characteristics ofthe flow such as velocity fluctuations, mean velocity profiles, etc. –4–anuscript submitted to JGR-Earth Surface
Table 1.
Parameters of the quasi-steady and quasi-uniform flows obtained in our 9 experiments(referred by the index 1-9): slope angle of the channel θ , thickness of the flow h , downslope ve-locity of the surface particles V xs , average downslope velocity (cid:104) V x (cid:105) , average shear rate (cid:104) ˙ γ (cid:105) andaverage inertial number < I > . Note that here √ gd (cid:39) .
14 m/s and (cid:112) d/g (cid:39) .
014 s.
Index θ [ ◦ ] ( ± . h/d ( ± . V xs / √ gd ( ± . (cid:104) V x (cid:105) / √ gd ( ± . (cid:112) d/g (cid:104) ˙ γ (cid:105) ( ± . (cid:104) I (cid:105) ( ± . . . .
15 0 .
65 0 .
12 0 . . . .
05 0 .
55 0 .
10 0 . . . .
35 0 .
80 0 .
12 0 . . . .
50 0 .
75 0 .
15 0 . . . .
85 0 .
90 0 .
16 0 . . . .
95 1 .
00 0 .
17 0 . . . .
02 0 .
50 0 .
11 0 . . . .
95 0 .
90 0 .
18 0 . . . .
45 1 .
10 0 .
21 0 . The experimental set-up consists of a 1.5 m long chute made of poly(methylmethacrylate) (PMMA), inclined at an angle θ to the horizontal, with rigid side walls10 cm apart. Granular flows are initiated by opening a gate that releases glass particlesof diameter d = 2 mm and density ρ = 2500 kg m − , initially stored in a tank (Fig.2). The resulting flow thickness is related but not equal to the height of the gatethat varies between h g = 4 . h g = 8 . h g and the slope angle of the channel θ that varies between θ = 16 . ◦ and θ = 18 . ◦ . In this range of inclination angles, almoststeady and uniform flows can be observed at about 70 cm from the gate as discussedbelow. The characteristics of these flows are summarized in Table 1. At this position,a Photron SA5 ® fast camera (5000 frames per second) records the flow during 2 swith a field of view of around 50 mm by 50 mm. Simultaneously, two accelerometers(bandwidth 10 Hz-54 kHz) record the radiated acoustic waves. These accelerometersare glued, using the same phenyl salicylate as for the particles of the rough surface,on the back of a 10 cm × . –5–anuscript submitted to JGR-Earth Surface c m c m c m Acousticallyisolatedplate θ slope angle F l o w A cc e l e r o m e t e r s Particletank Fastcamera xy c m . c m Gate opening c m Gluedparticles seal h g Figure 2.
Set-up composed of a narrow inclined channel in which granular flows are createdby opening the gate of the upstream tank that contain glass particles. The same particles areglued to the bottom plate to obtain a rough surface. The flow properties are measured by a fastcamera and the generated acoustic waves by accelerometers fixed on the channel bottom.–6–anuscript submitted to
JGR-Earth Surface
Our objective is to obtain deep quantitative insights into the mean propertiesof the flow and into its fluctuations and heterogeneity, in order to further interpretthe generated acoustic signal in terms of grain scale and mean flow dynamics. Beforeanalysis of these measurements in section 4, let us detail below the optical and acousticmethods used here to measure flow and acoustic characteristics, respectively. To illus-trate the methods, we focus in this section on the two ’extreme’ cases: experiment 1 at θ = 16 . ◦ with flow thickness h = 3 . V xs = 0 .
30 m s − or ex-periment 2, and experiment 9 at θ = 18 . ◦ and with h = 3 . V xs = 0 .
48 m s − (Table 1). The flows in all our experiments almost reach a steady and uniform regime:their heights vary by only one particle diameter in space and time (see Fig. A1in the Appendix). However, the average heights decrease by around half a particlediameter in the flow direction between x = 0 and x = 25 d (Fig. A1c in the Appendix),corresponding to an angle of 1 ◦ . As a result, our flows are not fully steady as discussedin section 4.1. They would have stopped if the channel had been long enough. We measured particle velocities V = ( V x , V y ) by Correlation Image Velocimetry(CIV) and Particle Tracking Velocimetry (PTV). CIV divides the images from themovie into boxes and calculates the average displacement into each box by correlationof the graymap between successive images (Fig. 3a). The size of the boxes is a crucialparameter. Boxes too large miss individual particles whereas boxes too narrow do notallow good correlations. Similarly to Gollin et al. [2015a], the size of the boxes waschosen to be equal to 1 .
14 particles. The overlap between boxes is 75%. We used thecode developed by Thielicke & Stamhuis [2014].On the other hand, PTV detects and follows the particle positions, making itpossible to record their trajectories (Fig. 3b). The particles are semi-transparent andcause complex reflection effects. Consequently, a compromise must be made betweenthe number and relevance of detections. PTV shows that particles are essentiallyorganized into layers that do not really mix during the flow. Mean velocities (cid:104) V (cid:105) =( (cid:104) V x (cid:105) , (cid:104) V y (cid:105) ) are therefore calculated by averaging the measurements within each layer(over 1 particle diameter in the y -direction), the borders of which are clearly visible onthe PTV images (Fig. 3b). As done for calculating the mean thickness, the averagingis performed over about 16 particles in space in the downslope direction and over thewhole experiment duration (2 s).Velocity fluctuations δV are computed over the same intervals (2 s, 16 particlesin the x -direction and 1 particle in the y -direction) by taking the standard deviationof the norm of the velocities: δV = (cid:113) δV x + δV y (1)where δV i = (cid:10) ( V i − (cid:104) V i (cid:105) ) (cid:11) the variance of the velocity along the i -direction, with i = x, y . For granular systems, the measurement of velocity fluctuations may lead to scaledependency effects due to gradients developing in the flow (see e.g. Artoni & Richard[2015a]). Indeed, the thickness w of the layers within which the velocity fluctuationsare calculated affects the estimates. Following Glasser & Goldhirsch [2001], we showedthat the size dependency starts for w > d (see Fig. B1 of Appendix B). In thefollowing, we will consider velocity fluctuations calculated with a window size w = d .Note that when velocity fluctuations are calculated over a smaller time window (e.g. w = 0 . d ), the layering of the flow clearly appears and resembles what was observed –7–anuscript submitted to JGR-Earth Surface by Weinhart et al. [2013] (Fig. B1, Appendix B). Note that velocity fluctuations ofabout 0 . √ gd are measured near the bottom where the mean velocity is zero. Thisgives an order of magnitude of the error in the measurement of velocity fluctuations( ∼ .
01 m s − ).The profiles of the mean velocity in the downslope (cid:104) V x (cid:105) and normal (cid:104) V y (cid:105) direc-tions obtained using CIV and PTV only differ by maximum 10 % as illustrated inFig. 3c. In contrast, velocity fluctuations may differ by up to a factor of two betweenthe two methods. This discrepancy has also been observed by Gollin et al. [2015b]and Gollin et al. [2017] and seems to be due to the average nature of CIV, which istherefore less relevant to measure fluctuations. As a result, PTV measurements will beused in the following, as in Pouliquen [2004], except for mapping of the spatio-temporaldistribution of velocity fluctuations (Fig. C1). The set-up can only measure the surface packing fraction φ D at the lateral walls(Fig. 3de). The complex light reflections makes it impossible to extract the volumepacking fraction φ D from φ D as proposed by Sarno et al. [2016]. Nevertheless, as istypically observed, we measure an almost constant packing fraction within the flow anda decrease when approaching the free surface (Fig. 3e). Due to the strong uncertaintyin our measurements, the change of φ D when increasing the slope angle (i.e. whenthe inertial number changes) is hard to capture, even though a decrease of φ D withincreasing inertial number is visible near the surface, in agreement with the literature[Midi, 2004]. Calculation of the volume fraction shows the layering of the granularflows observed for example in Artoni & Richard [2015a] and Weinhart et al. [2013]. During the flow, vertical oscillations of the particles can be observed relatedto compression/dilatation effects occurring when one layer passes over another (seeMovies 1 and 2 in supplementary material). The frequency of these oscillations canbe captured in the PTV measurements of the trajectory of the particles located at thesurface (Fig. 4). Indeed, several oscillations can be observed before loosing the particletracking owing to the relatively high velocity of these particles. On the contrary, for theparticles located deeper in the flow, the oscillations generally occur when the particletracking has already been lost. The oscillation frequency is calculated by filtering theparticle trajectory with two filters and taking the median of 1 /T , where T (cid:39) .
02 s isthe time separating successive maxima and minima of the trajectory (Fig. 4). Moreprecisely, the first filter is a normalized median filter adapted from Westerweel &Scarano [2005] for vectors applied on each trajectory component (neighborhood radiusof 5 successive positions, noise threshold level of 0 .
10 and threshold of 1), and thesecond filter is a second order zero-phase low pass filter (cut-off frequency of 50 Hz).The median filter has been chosen to suppress random fluctuations.
The elastic waves generated by the granular flows and by their interactions withthe bottom are recorded by two accelerometers glued to the isolated plate (Fig. 5a).It is assumed here that the accelerometers mainly record the vibrations generated bythe section of granular flow over the plate. Isolation of the plate from the rest of theflume was verified by comparing the signals recorded by accelerometers glued to thesetwo elements. –8–anuscript submitted to
JGR-Earth Surface (a)
CIVPTV
20 x/d5 10 15 20 250 24810121416 y / d Figure 3. (a) Example (experiment 2) of the velocity field calculated by CIV (red arrows)and (b) superposition of particle trajectories obtained with PTV during 2 s. The organization ofthe flow into a superposition of layers is clearly visible. In (b), red lines indicate the separationbetween layers. (c) Example (experiment 2) of mean downslope (cid:104) V x (cid:105) and normal (cid:104) V y (cid:105) velocityprofiles as a function of the position above the bottom y . The associated velocity fluctuations arerepresented by the horizontal error bars. Vertical error bars correspond to the thickness of thelayer within which the velocity has been averaged. Comparison between the measurements madeby CIV (blue line) and PTV (red line). (d) and (e) Surface packing fraction of the particles incontact with the lateral wall: (d) manual picking of the particles of flow 1 ( θ = 16 . ◦ , h/d = 17 . θ = 18 . ◦ , h/d = 16 .
5) are plotted in the full red line.–9–anuscript submitted to
JGR-Earth Surface t [s] y / d trajectorymaximaminima Figure 4.
Example (experiment 2) of vertical particle oscillations captured by PTV for a par-ticle located at the surface of the flow: The smoothed trajectory shows the average period of theoscillations T (cid:39) .
02 s.
The radiated elastic power over duration ∆ t is Π el = W el / ∆ t , where W el isthe radiated elastic energy. The acoustically isolated plate is small compared to thecharacteristic viscoelastic attenuation length of energy in PMMA. As a result, thewaves are reflected many times at the boundaries of the plate leading to a diffuseelastic field, i.e. homogeneously distributed over the plate and equipartitioned. Theelastic energy can then be computed using the diffuse field theory proposed by Farinet al. [2016]: W el = M γ v g × (cid:90) ∆ t v z ( t )d t (2)where M ≈
80 g is the mass of the isolated piece of plate, γ ≈ − its averageviscoelastic attenuation and v g ≈ − the average group velocity of the radiatedacoustic waves ( A Lamb waves). The value of γ is obtained by measuring the responseof the plate at various distances with a source and a vibrometer and the value of v g bycalculating the dispersion relation of the A Lamb modes of the plate following Royer& Dieulesaint [2000]. A large time window ∆ t = 0 . el , with error bars representing the standard deviation. The fastfluctuations will be characterized in the next section. An example of radiated elasticpower computation is presented in Fig. 5a. The spectrogram of the acoustic signal shows a mean frequency of around 20 −
30 kHz, corresponding to the average height of the dark bands in Fig. 5f,g, that slightlyincreases when the slope angle increases. The mean frequency is calculated as follows: f mean = (cid:82) ∞ | ˜ A z ( f ) | f d f (cid:82) ∞ | ˜ A z ( f ) | d f (3)within time windows of 2 ms (Fig. 5b,c), where ˜ A z is the Fourier transform of theacceleration a z ( t ) of the vertical vibration. Amplitude spectra are not studied beyond –10–anuscript submitted to JGR-Earth Surface
54 kHz because of the limit of the accelerometer responses (signal polluted by theaccelerometer resonances).Vertical stripes can be identified on the spectrograms (Fig. 5f,g). The distancebetween these stripes decreases as the slope angle increases. This corresponds to a so-called modulation amplitude frequency of around 25 −
50 Hz, about 500 to 1000 timessmaller than the mean frequency defined in Eq. (3). To calculate the modulationfrequency, we first extract the envelope of the signal (absolute value of the signal) andapply a low pass filter (cut-off frequency empirically fixed at 75 Hz). Then, the meanoscillation frequency is determined by fitting a Gaussian in the Fourier space (Fig.5d,e).
Our objective here is to capture the relationship between mean flow propertiesand fluctuations that are expected to play a role in acoustic emissions. Note thatthe flow measurements are made at the side walls. It is well known in regard of thebulk that the wall boundaries significantly affect the mean flow quantities and theirfluctuations, as will be discussed below (see e.g. Artoni & Richard [2015b], Fern´andez-Nieto et al. [2018], Jop et al. [2005, 2007], Mandal & Khakhar [2017], Taberlet et al.[2003]).
The nearly uniform and steady flows confined in a narrow channel inclined atslope angles between 16 . ◦ and 18 . ◦ obtained here are similar to those observed byHanes & Walton [2000] in similar settings. In these flows, the downslope velocity V x ( y )is maximum at the free surface, decreasing down to zero near the bottom (Fig. 6).Such convex velocity profiles are observed in flows confined in narrow channels (seee.g. Ancey [2001], Courrech du Pont et al. [2003], Jop et al. [2005, 2007], Mandal& Khakhar [2017], Midi [2004], Taberlet et al. [2003]) and differ from Bagnold-likevelocity profiles obtained for steady and uniform flows in wide channels (see Midi[2004] or Fig. 4 of Fern´andez-Nieto et al. [2018]). These profiles have a shape that canbe approximately fitted by the velocity profiles assumed in Josserand et al. [2004] todescribe heap flows: 1 − V x ( y (cid:48) ) V x ( y (cid:48) = 0) = (cid:32) − e − y (cid:48) /Y φ M φ m − e − y (cid:48) /Y (cid:33) / (4)where y (cid:48) = y h − y and y h is the height of the flow surface, Y a fitting parameter, φ M = 0 .
65 and φ m = 0 . h , where the horizontal velocity is non-zero atthe base. Second order polynomials ( V x / √ gd = a ∗ ( y/d ) + b ∗ ( y/d )) give even betterresults, especially near the bottom. We will therefore use these polynomial fits tocalculate the shear strain rate ˙ γ = ∂V x /∂y .The shear strain rate ˙ γ decreases from the surface down to the bottom (Fig. 7b).Granular flows are characterized by the inertial number I = ˙ γd/ (cid:112) P/ρ s , where P isthe pressure taken here as hydrostatic ( P = ρ s φg cos( θ )( h − y )): I ( y ) = ˙ γ ( y ) d (cid:112) φg cos( θ )( h − y ) (5)The packing fraction is approximated by φ = 0 . –11–anuscript submitted to JGR-Earth Surface a z [ m s - ] a z [ m s - ] (a)(b) (c)(d) (e) signalgaussian fi t f mean e n v ( a z ) [ m s - ] | F T { e n v ( a z )} | [ m / s ] (f)(g) Figure 5.
Acoustic signal of the flow number 2: (a) acceleration of the vibration (blue) andassociated elastic power (red), (b) enlargement of the acoustic signal and (c) associated spectra,(d) envelope (red) of the acoustic signal (blue) and (e) associated spectra of the envelope. (f) and(g) Spectrogram of the signal of (f) experiment 1 ( θ = 16 . ◦ , h = 3 . V xs = 0 .
30 m s − )and (g) experiment 9 ( θ = 18 . ◦ , h = 3 . V xs = 0 .
48 m s − ). The dark colors show themean frequency (a few tens of kHz) with its mean value represented by a light blue horizontalline, whereas the so-called modulation frequency, 1000 times smaller, is related to the distancebetween the vertical stripes (a few tens of Hz).–12–anuscript submitted to JGR-Earth Surface y / d y / d y / d / V x / V x L/d=3.94R =0.98a*=7.39e-3b*=-1.62e-2 L/d=5.27R =0.97L/d=4.27R =0.98L/d=4.00R =0.99L/d=3.03R =0.99L/d=3.61R =0.99 a*=5.05e-3b*=1.58e-2a*=1.03e-2b*=1.42e-2a*=9.13e-3b*=-2.04e-2 a*=9.82e-3b*=9.42e-3a*=1.29e-2b*=-1.57e-20 0.1 0.2 0.305101520 0 0.2 0.4051015 0 0.2 0.4051015 (b)(e)(h) L/d=4.06R =0.98a*=6.44e-3b*=-1.10e-2L/d=4.18R =0.98a*=9.36e-3b*=4.03e-3L/d=3.77R =0.99a*=1.16e-2b*=-9.81e-4 / V x Figure 6.
Velocity profiles of all the experiments with letters (a) to (i) referring to flows1 to 9, corresponding to the angles (a-c) θ = 16 . ◦ , (d-f) θ = 17 . ◦ and (g-i) θ = 18 . ◦ ,each associated with increasing flow thicknesses (see Table 1 for details). Two theoretical pro-files have been fitted: the ones given by Eq. (4) in dashed lines and a 2nd order polynomial( V x / √ gd = a ∗ ( y/d ) + b ∗ ( y/d )) in full lines. For all polynomial fits, R ≥ . JGR-Earth Surface y = 0.28 x + 0.08R = 0.98 y = 1.67 x + 0.05 R = 0.3 y = 2.03 x + 0.09R = 0.9 y = x R = 0.64 flow 1flow 2 flow 3flow 4 flow 5flow 6 flow 7flow 8 flow 9 Figure 7. (a) Normalized fluctuating speed δV / √ gd , (b) Normalized shear rate (cid:112) d/g ˙ γ and(c) Inertial number I , computed using the second order polynomials that provide the best fit, asa function of flow depth y/d , for all of the experiments (colors). (d) to (f) Normalized fluctuatingspeed δV / √ gd as a function of (d) the flow normalized average speed | V | / √ gd , (e) the normal-ized shear rate (cid:112) d/g ˙ γ and (f) inertial number I . In panels (d) to (f), dashed lines show fits ofthe data with linear laws. In panel (f), the dash-dotted line shows a power-law (square root) fitof the data. –14–anuscript submitted to JGR-Earth Surface
The high-frequency acoustic signal generated by granular flows is expected to bemainly due to particle collisions, even though friction may play a role that will notbe considered here [Huang et al., 2007, Michlmayr et al., 2013]. Such collisions occurwhen neighboring particles have different velocities, in particular related to velocityfluctuations. Velocity fluctuations, or their squared values called granular temperature[Goldhirsch, 2008], T = (cid:10) δV (cid:11) (6)where (cid:104) . (cid:105) means the average over volume and time ranges, are known to be significant ingranular flows. Granular temperature is however generally not accounted for explicitlyin the rheology of dense granular flows, except in the extended kinetic theory [e.g. Berzi,2014, Gollin et al., 2017]. Indeed, the relationship between velocity fluctuations and theinertial number or other mean flow quantities has not yet been thoroughly investigatedin dense granular flows. Indeed, they are difficult to measure experimentally, and evenmore in the field [Berzi & Jenkins, 2011, Hill & Tan, 2014]. The acoustic power, mucheasier to measure, may provide a unique tool to obtain quantitative measurements ofgranular temperature as will be investigated below.Fig. 7a shows that velocity fluctuations decrease from the surface to the bottomfor all experiments and increase with slope angle. Using discrete element modeling,Hanes & Walton [2000] showed that the granular temperature profile is very differentat the side wall than within the core of the flow: the granular temperature at thesurface is the same at the side walls and across the flow but it increases with depth inthe middle of the flow, while it decreases at the side walls, as observed here.Even though δV looks regular when averaged over volume and time, Figs. C1(a)and (b) in the Appendix and Movies 3 and 4 in the supplementary material illustratethe existence of transient vortices of velocity fluctuations in our experiments, as ob-served by Kharel & Rognon [2017]. The size and intensity of these transient vorticesseem to be related to the flow regime, leading to strong variations of velocity fluctua-tions (in space and time) when approaching jamming, and thus possibly contributingto generate acoustic emissions in these regions. The correlation length of these velocityfluctuations is around 1 grain diameter in the y -direction and can reach up to 8 d inthe x -direction, decreasing with increasing slope (see Fig. C2 in Appendix C). Granular temperature is expected to scale with the square of the shear strainrate so that δV ∝ ˙ γ [see e.g. Andreotti et al., 2013, Pouliquen, 2004]. Such a linearrelationship between δV and ˙ γ seems indeed to be satisfied (Fig. 7e), in very goodagreement with what was found at the surface of granular flows by Pouliquen [2004]or in other configurations [Midi, 2004]. If we try to fit the data by a power law, weget a power equal to 2 with high R . A higher R is found when trying to relatethe velocity fluctuations to the mean downslope velocity (cid:104) V x (cid:105) (Fig. 7d). The slightlyhigher R obtained when relating velocity fluctuations to the mean velocity comparedto the strain rate may result from errors related to the calculation of the gradient ofthe measured velocity profile. The power law between velocity fluctuations and theinertial number is less clear, with a smaller R (Fig. 7f). This is also possibly due tothe errors in the calculation of I . As a result, velocity fluctuations averaged in timeand along one layer of grains scale very well with shear rate and with mean velocityand to a lesser extent with the inertial number: δV ∝ < V x > ∝ ˙ γ ∝ I . . (7) –15–anuscript submitted to JGR-Earth Surface
Our objective is to quantitatively relate the characteristics of both the seismicsignal and the flow to (i) get physical insights into the sources of acoustic emission and(ii) propose empirical scaling laws that can be used to recover flow properties fromthe recorded acoustic waves. As the range of configurations (slope angle, thickness)investigated here is not very large, it is hard to discriminate between power laws orlinear trends. We will therefore systematically test these two types of empirical fitsand quantify the associated R . Let us first discuss the order of magnitude of the expected frequencies in themeasured acoustic signal generated by the granular flows and their potential causes,based on our setup and on the observation of flow dynamics described in the previoussections.The main frequency of the signal is expected to be caused by particle collisionsbetween two spheres of diameter d at relative velocity δV and to approximately scalewith the inverse of the contact time calculated by the Hertz contact theory [Farin etal., 2015]. For impacts on smooth plates, this leads to the following expression for themain frequency f Hertz = a (cid:48) d − δV / , (8)where a (cid:48) ≈ . (cid:18) E ∗ πρ (cid:19) / ≈ . S . U . ) (9)and E ∗ is the effective Young modulus such that 1 /E ∗ = (cid:0) − ν s (cid:1) /E s + (cid:0) − ν p (cid:1) /E p . ν s = 0 . ν p = 0 . E s = 74 GPa and E p = 4 . ρ = 2500 kg / m is the bulk density of the particles.This leads here to 30 kHz < f Hertz <
48 kHz for 0 . × √ gd < δV < √ gd , with √ gd = 0 .
14 m s − . However, the frequency of an impact on a rough bed is less thanon a smooth bed as shown in Fig. 9 of Farin et al. [2018]. In their case, involvingsteel particles, the mean frequency for impacts on rough beds was about 2/3 to halfof the mean frequency over a smooth bed. If we assume similar behavior in our case,we could expect impact frequencies of about 15 kHz < f Hertz <
30 kHz. Note thatattenuation within the granular media, which is higher for high frequencies, may alsodecrease the measured mean frequency.In contrast, the vertical oscillations of the particles due to the motion of onelayer over another (see section 3.1.3, Fig. 5fg) are a possible cause of the so-calledmodulation frequencies 33 Hz < f mod <
52 Hz, about 500 times smaller than f mean ,shown in Fig. 8. This modulation frequency is of the order of magnitude of δV /d ,corresponding to a typical frequency between collision events.On the other hand, frequencies around f h (cid:39) − h = 3 cm, if we assume an acoustic wave velocity in granular flows of 100-200 m s − (see e.g. Hostler [2004], Hostler & Brennen [2005], Mouraille & Luding [2008]). Notethat the velocity of acoustic signals in granular material varies strongly depending onthe confining pressure, packing fraction, material involved, etc. Liu & Nagel [1993]found values varying from about 60 to 280 m s − depending upon the kind of velocitymeasured, van den Wildenberg et al. [2013] between 80 m s − and 150 m s − andBonneau et al. [2008] between 40 m s − and 80 m s − . –16–anuscript submitted to JGR-Earth Surface
Observations show that the flow thickness oscillates slightly with time (see Fig.A1 in the Appendix), possibly due to compression/dilatation waves in the media or tothe complex heterogeneity of the flow (see section 4.2 and Fig. C1 in the Appendix).The typical period of these oscillations is 1 s, possibly generating waves at frequencies f flow (cid:39) Hz [Gimbert et al., 2014]. Turbulent vortices formclose to the flowing static interface due to the shear stress applied by the flow onthe bed. The formed vortices enlarge by coalescence until they reach the thicknessof the flow, then break up into smaller vortices, transferring flow energy towards thesmaller scales [Kolmogorov, 1941]. The highest frequencies generated by the vorticesare related to the minimum vortex size, i.e. the Kolmogorov microscale, which may notbe reachable in a granular flow because the minimum vortex scale is in theory at leasttwo particle diameters 2 d . Therefore, in granular flows we expect lower frequenciesgenerated by vortices than those that can be observed in a liquid flow. The typicalsize of the observed vortices in our granular flows is about 5-8 d (cid:39) − . − . Therefore, these granularvortices may generate waves at frequencies f v (cid:39) −
100 Hz.Finally, the resonance of the 10 cm × . f p (cid:39)
15 kHz and f p (cid:39)
10 kHz if we assume a wave velocity in theplate of 1000 m s − . Let us now analyze the frequency content of the measured signaland compare it to these expected frequencies. At high frequencies ( f > . < f mean <
27 kHz. This clearly corre-sponds to the order of magnitude of the frequencies 15 kHz < f
Hertz <
30 kHz dueto collisions in the Hertz contact theory (frequency range highlighted in light grayin Fig. 8(a) and 8(i)). The values of f mean are closer to the maximum expectedfrequencies, corresponding to higher velocity fluctuations, and thus to the particleslocated near the free surface. Fig. 9 shows that the mean frequency f mean increaseswith the amplitude of the mean velocity fluctuations and mean inertial number. Themean values are obtained by averaging the quantities over the flow depth. Fittingthis increase with affine or power law ( f mean ∝ δV . ) relationships give approxi-mately the same R (cid:39) .
58 (see details in Fig. 9). The power exponent 0 .
14 is notfar from the theoretical value predicted by the Hertz theory, i.e. 1 / . a (cid:48) /d (cid:39) × is not so far from the coefficient of the powerlaw fit 3 × ∗ √ gd . (cid:39) . × , supporting the interpretation that these meanfrequencies come from the Hertzian contact between particles of relative velocity δV .There is a better collapse of the experimental data when f mean is represented as afunction of I , i.e. f mean ∝ I or f mean ∝ I . , or as a function of shear strain rate,i.e. f mean ∝ ˙ γ or f mean ∝ ˙ γ . , leading to R (cid:39) .
82 and 0 .
83, respectively. The fitof f mean as a function of ˙ γ gives R (cid:39) .
79. Even though no clear peaks appear inthe high-frequency spectra, some peaks are observed at frequencies 3 < f <
10 kHzfor almost all the flows, which may correspond to waves trapped within the flowinggranular layer 3 < f h < f p (cid:39)
10 kHz and f p (cid:39)
15 kHz) as illustrated inlight pink in Fig. 8(b), (d), (e), (g), and (h). –17–anuscript submitted to
JGR-Earth Surface (a) (b) (c)(d) (e) (f)(g) (h) (i) f Hertz f Hertz flow 1 flow 2 flow 3flow 4 flow 5 flow 6flow 7 flow 8 flow 9
Figure 8.
High-frequency ( f > θ = 16 . ◦ , (d-f) θ = 17 . ◦ and(g-i) θ = 18 . ◦ , each associated with increasing flow thicknesses (see Table 1 for details). Theorange lines correspond to the mean frequencies. Light gray areas in Fig. (a) and (i) correspondto the range of expected Hertz’s frequencies f Hertz , light pink areas to the frequency range asso-ciated with plate resonance f p and f p and light green areas to the frequency range associatedwith waves trapped in the granular layer f h . –18–anuscript submitted to JGR-Earth Surface (a) (b) y = 9.2x + 22.4R = 0.57y = 29.7 x R =0.58 y = 37x + 23R = 0.82y = 34.1 x R = 0.8330 f mod [Hz] f b e a d s y [ H z ] θ = 16.5° θ = 17.2° θ = 18.1°h/d = 17.6 (flow 1)h/d = 17.95 (flow 2) h/d = 19.95 (flow 3) h/d =15.35 (flow 4)h/d = 14.55 (flow 7)h/d =15.15 (flow 8) h/d = 16.65 (flow 6)h/d = 16.6 (flow 5)h/d = 16.5 (flow 9) (c) (d)
35 40 45 50
Figure 9.
Mean frequency f mean as a function of (a) normalized average fluctuating speed < δV > / √ gd and (b) the average inertial number < I > . (c) Vertical particles oscillationfrequency f yparticles as a function of the frequency of the acoustic amplitude modulation. (d)Acoustic modulation frequency f mod as a function of the average inertial number < I > .–19–anuscript submitted to JGR-Earth Surface (a) (b) (c)(d) (e) (f)(g) (h) (i) f f ff f ff f f f mod f v f v f flow low 1 low 2 low 3low 4 low 5 low 6low 7 low 8 low 9 f mod Figure 10.
Low-frequency ( f <
100 Hz) spectral amplitude measured for all flows. Letters(a) to (i) refer to flow numbers 1 to 9 corresponding to angles (a-c) θ = 16 . ◦ , (d-f) θ = 17 . ◦ and (g-i) θ = 18 . ◦ , each associated with increasing flow thicknesses (see Table 1 for details). Theorange curves correspond to the Gaussian fits (see Fig. 5e). Light gray areas in Fig. (a) and (i)correspond to the frequency range associated with particle oscillations f mod , light pink zones onall the figures correspond to the frequency range of flow oscillations f flow and light green zonesto frequency range of vortices f v . In the low-frequency range, Fig. 10 shows clear frequency peaks between 28 Hzand 50 Hz. These frequencies of the acoustic amplitude modulation are clearly in therange of the frequencies f mod associated with the vertical oscillation of the particlesat the surface of the flow (Fig. 9c). Indeed, despite high error bars, they are bothbetween approximately 30 and 60 Hz (frequency range of f mod is highlighted in lightgray in Figs. 10(a) and 10(i)). The acoustic amplitude modulation frequency increasesas a function of the inertial number (Fig. 9d). Almost all the flows exhibit an increaseof spectral amplitude at frequencies between 1 Hz to 3 Hz. This may correspond tothe frequencies of flow oscillations f flow (cid:39) f v (cid:39) −
100 Hz. –20–anuscript submitted to
JGR-Earth Surface
We investigate here the relationship between the acoustic power and the proper-ties of the flow averaged over the granular depth. Figs. 11(a) and (b) show that theacoustic power increases with the velocity fluctuations and the inertial number. Ourdata are compatible with affine or power law relationships. The best power laws thatfit the data are Π el ∝ δV . ∝ I . . (10)As velocity fluctuations are related to mean velocity by (Fig. 7d) δV = 0 . < V x > +0 . , (11)the seismic power also scales as Π el ∝ < V x > . .In the field, the seismic power calculated from the signal measured at seismic sta-tions can be related to the mean flow velocity deduced from inversion of low-frequencyseismic data [Allstadt, 2013, Hibert et al., 2017b]. Field experiments consisting in therelease of single blocks of different masses have also shown a correlation between thevelocity of the block before impact v and the seismic energy E s released at the source[Hibert et al., 2017c]. From this dataset, we looked for the exponents α and β givingthe best fitting regression line between E s and m α × v β , where m is the mass of theblock and v the velocity before impact. When considering the modulus of the velocity,we found that the seismic energy scales as E s ∝ | v | . (Figure 12a). When consideringonly the modulus of vertical component of the velocity before impact v z , the seismicenergy scales as E s ∝ | v z | . (Figure 12b). These exponent values are not so far fromthose of our laboratory measurements, Eq. (10), even though they were obtained forsingle blocks and not for granular flows. Note that similar scaling laws linking theseismic wave characteristics to the dynamic properties have been found for granularflows and single blocks for natural events (e.g. [Hibert et al., 2017b, 2017c, Schneideret al., 2010]). We propose a simple model making it possible to recover the radiated elasticpower from the velocity fluctuations (i.e. square root of the granular temperature) ofthe particles, based on the understanding of the seismic source gained above. We as-sume that (i) the elastic waves are generated during binary collisions between particlesof adjacent layers at speeds corresponding to the particles fluctuation velocities, (ii)collisions are described by the Hertz contact law and the radiated elastic energy is thework done by the impact force during the contact [Farin et al., 2015, Johnson, 1987],and (iii) the acoustic waves propagate from the layer where they are generated downto the bottom of the channel, with attenuation γ g .The attenuation in granular material varies strongly depending on the confiningpressure, packing fraction, etc. Different values are reported in the literature varyingbetween 15 −
150 m − : e.g. Voronina & Horoshenkov [2004] found γ g = 100 m − and Hostler & Brennen [2005] found values between 25 m − and 50 m − . The totalanalytical elastic power is obtained by summing up the contributions of all layers:Π Hertzel = n (cid:88) i =1 N i W iel,Hertz e − γ g y i (12)where W iel,Hertz is the elastic energy radiated during the impact of a particle of thelayer i , y i the height of the center of the layer i , e − γ g y i the exponential decay of thewave energy with depth, N i the number of impacts per time unit in the layer i and n the number of layers. –21–anuscript submitted to JGR-Earth Surface (a) (b) y = 4.2 10 -5 x - 1.0 10 -5 R = 0.78y = 1.0 10 -4 x R = 0.62 (c) (d) θ = 16.5°θ = 17.2°θ = 18.1° h/d = 17.6 (flow 1)h/d = 17.95 (flow 2) h/d = 19.95 (flow 3) h/d =15.35 (flow 4)h/d = 14.55 (flow 7)h/d =15.15 (flow 8)h/d = 16.65 (flow 6)h/d = 16.6 (flow 5)h/d = 16.5 (flow 9)y = 1.4 10 -4 x - 5.4 10 -5 R = 0.83y = 1.3 10 -3 x R = 0.8y = xR = 0.78y = 1.5 10 -3 xR = 0.79 (e) / Figure 11.
Radiated elastic power Π el as a function of (a) normalized average velocity fluc-tuations < δV > / √ gd and (b) average inertial number < I > . (c) Experimental Π el versusanalytical elastic power Π Hertzel for granular attenuation γ g = 100 m − . Comparison with the line y = x in red. (d) Model efficiency Π Hertzel / Π el calculated from the best linear fit between Π Hertzel and Π el for a given attenuation as a function of the attenuation coefficient γ g . The vertical blackdashed line highlights the case of γ g = 100 m − , the value for which the model gives about thesame result as the measurements, i.e. Π Hertzel / Π el (cid:39)
1. (e) Comparison between the measuredradiated elastic power Π el and analytical kinetic power Π k .–22–anuscript submitted to JGR-Earth Surface m |v| -4 -3 -2 -1 E s m |v z | -4 -3 -2 -1 E s R = 0.65 R = 0.67 a b Figure 12. a) Energy of the seismic signal generated at each individual block impact as afunction of the block mass m α and the modulus of the velocity before impact v β , with the valuesof α and β inferred to get the best fit by linear regression between those quantities; b) Similarderivation as a), considering only the vertical component of the velocity v z . The elastic energy radiated during an impact is computed from Hertz contacttheory [Farin et al., 2015] for impacts on plates W iel,Hertz = a (cid:18) d (cid:19) ( δV ( y i )) / (13)with δV ( y i ), the velocity fluctuation in the layer i , and a , a prefactor involving theelastic parameters of the particles and the PMMA plate [Bachelet et al., 2018] a ≈ . (cid:112) Bρ p h p (cid:18) E b − ν b ) ρ b (cid:19) / ≈ . × (I . S . U . ) (14)The number of impacts per time unit in layer i is given by: N i = φlLπ ( d ) f i (15)with the first term corresponding to the number of particles above the plate isolatedacoustically and f i , the number of impacts per particle and per time unit. Impactsare assumed to occur when a particle overrides another particle of the layer below attheir relative downslope velocity f i = V x ( y i ) − V x ( y i − ) d = ˙ γ ( y i ) (16)Combining expressions (12), (13), (15) and (16) leads to the final expression of theanalytical radiated elastic powerΠ Hertzel = a φlL π d (cid:88) i ˙ γ ( y i ) δV ( y i ) / e − γ g y i (17)Using Eq. (16), the acoustic power is expected to scale asΠ el ∝ δV / (cid:39) . ∝ ˙ γ . ∝ I . , (18) –23–anuscript submitted to JGR-Earth Surface as our optical observations showed that δV ∝ ( ˙ γd ) ∝ I . . This is in very goodagreement with the scaling observed in Fig. 11a that suggests Π el ∝ δV . ∝ ˙ γ . ∝ I . even though, as said above, the narrow range of our experiments makes it verydifficult to discriminate between the affine and the power law relationship.To compare our observations with those of Taylor & Brodsky [2017], we have tomultiply what they called acoustic energy E a by ω , where ω is the pulsation becausetheir definition of energy is a term proportional to the square of the acceleration,rather than the square of velocity. As a result, to determine their energy, we have tomultiply our scaling by δV / owing to Eq. (8). This would lead to E a ∝ δV / (cid:39) . and possibly E a ∝ I , if we assume the same relationship between the mean quantitiesin their experiments. Their observations however suggest that E a ∝ I . This differencemay be due to the fact that their setting is very different from ours, to the uncertaintyof their calculation of I (i.e. they calculate ˙ γ by dividing the imposed velocity of theshear cell by 5 d for all experiments) or to the limitations of our simple model.The key parameter in the calculation of Π Hertzel is the attenuation factor. If wetake γ g = 100 m − , we obtain a very good agreement with the measured acousticpower (Fig. 11c). However, the value of Π Hertzel is very sensitive to γ g as shown inFig. 11d. For example if γ g = 50 m − , Π Hertzel (cid:39) . el . Figs. 13(a) and (b) showthat with γ g = 100 m − , the main contribution to the acoustic power comes from thegrains near the surface, while with γ g = 300 m − , it comes from the grains located inthe middle of the granular layer where velocities and velocity fluctuations are small.Because the measured mean frequencies are closer to those associated with the highestvelocity fluctuations within the flow (see section 5.1.2), a larger contribution of thegrains near the surface seems more probable. This suggests a granular attenuationcloser to 100 m − . Precise attenuation measurements will be a crucial step to furthervalidate our simple model and will be performed in the future.Another key issue is the difference between the fluctuations measured near theside walls and those within the flow as shown in the discrete element simulations ofHanes & Walton [2000] and discussed in section 4.2. To assess the change in theacoustic power calculation if measurements were performed in the flow center, wecalculate Π Hertzel by taking the same value δV xs of the fluctuating velocity δV at thefree surface and by assuming that δV increases (instead of decreases) linearly downto the bottom to reach δV ( d ) = 1 . δV xs to mimic the simulations of Hanes & Walton[2000] (their Fig. 15). This assumption corresponds toΠ Hertzel = (cid:88) i φlLπ ( d/ f i a ( d/ (1 . δV xs (1 − y i /h ) + δV xs y i /h ) / e − γ g y i (19)Assuming that the collision frequency is f i = δV i /d = (1 . δV xs (1 − y i /h )+ δV xs y i /h ) /d further leads to Π Hertzel = a φlLhd π δV / xs (cid:90) (1 . − . z ) / e − γ g hz dz (20)Note that when we make this assumption on the δV ( y ) profile, the main contributionto the acoustic power comes from slightly below the middle of the granular layer,regardless of the attenuation coefficient γ g = 100 m − or γ g = 300 m − (Figs. 13(c)and (d)). Finally, in order to quantify the part of the kinetic energy of the flow convertedinto elastic energy, we assume that the conversion coefficient from kinetic to elasticenergy, i.e. the energy ratio or acoustic efficiency, W el /E k = ξ generated by eachimpact is the same. We then replace the term W iel,Hertz in expression (12) by ξ E ik , –24–anuscript submitted to JGR-Earth Surface γ = 100 m -1 γ = 100 m -1 γ = 300 m -1 γ = 300 m -1 Side SideMiddle Middle
Figure 13. (a),(b) Analytical acoustic power Π el ( y i ) per layer (i.e. as a function of depth y/d ) computed using the fluctuating speed δV measured along the side of the flow in experiment1, for (a) γ g = 100 m − and (b) γ g = 300 m − . (c),(d) Analytical acoustic power Π el ( y i ) perlayer computed assuming a linear granular temperature profile increasing with depth, as might beobserved in the middle of the flow, for (c) γ g = 100 m − and (d) γ g = 300 m − . In each panel,attenuation exp( − γ g y ) is also represented. –25–anuscript submitted to JGR-Earth Surface where E ik is the average kinetic energy of particles of layer number i :Π Hertzel = ξ Π k = ξ (cid:88) i N i E ik e − γ g y i (21)Fig. 11e shows that the measured acoustic power varies as an affine function of Π k and that the energy ratio is ξ = 0 . × − . For individual particle impacts, Bacheletet al. [2018] showed that the energy ratio ξ decreases from about 10 − for impacts ona rough bed to 10 − for impacts on an erodible bed of a thickness equal to 10 particlediameters. As the highly energetic impacts are mostly at the surface here, at leastnear the side walls, we may consider that the particle impacts a bed of 15 d . If we usethe empirical relationship proposed by Bachelet et al. [2018], ξ = 0 . e − . e ∗ , where e ∗ is the number of layers of grains of the erodible bed, we find in our case, for e ∗ = 15,that ξ = 1 . × − , in agreement with our results from granular flows. This supportsthe simple model proposed here. The energy ratio of 1 . × − is very similar to whatis observed in the field for rockfalls. As an example, values of ξ (cid:39) − − − werefound for rockfalls on La R´eunion Island [Hibert et al., 2011], on Montserrat Island[Levy et al., 2015] and in the French Alps [Deparis et al., 2008]. As seismic waves generated by landslides are continuously recorded by seismicnetworks, detailed analysis of these signals provides a new way to collect data on thedynamics and rheology of natural flows. This is however only possible if quantitativerelationships between the flow properties and the acoustic signal characteristics areestablished.In the experiments reported here, we provide new quantitative insights into theorigin of the acoustic signals generated by almost steady and uniform granular flows.By measuring precisely and synchronously the flow and generated waves with opticaland acoustic sensors respectively, we have identified the essential physical sources ofthe waves. We have shown that the high-frequency signal (tens of kHz for mm-sizeglass particles) corresponding to the mean frequency of the signal is essentially relatedto particle collisions. This mean frequency is shown to be roughly proportional tothe inertial number I . The measured acoustic power is well reproduced quantitativelywith a simple model of particle impacts using Hertz theory and involving the relativeparticle velocity corresponding to flow velocity fluctuations. As our experiments showthat velocity fluctuations roughly correlate with the mean flow velocity, our resultssuggest that mean flow velocity and velocity fluctuations could be determined frommeasurements of the high-frequency seismic signal. The conversion coefficient fromkinetic to elastic energy, i.e. the energy ratio or acoustic efficiency, is around 10 − inthe experiments. This values is in rough agreement with field measurements wherevalues of 10 − − − have been found for the ratio between seismic energy and potentialenergy lost [e.g. Hibert et al., 2011, Levy et al., 2005], given that kinetic energy istypically one order of magnitude smaller than potential energy lost (see Figures 6(a),(b) of Farin et al. [2018]).More precisely, our results suggest that the emitted seismic power is proportionalto the granular temperature (square of velocity fluctuations). Beyond the interpreta-tion of the generated acoustic signal in terms of granular flow properties, the measure-ment of velocity fluctuations and their link with mean properties may help improveour understanding of the behavior of natural flows near boundaries. Indeed, [Artoni &Richard, 2015b] suggested that velocity fluctuations are a key ingredient to be includedin models describing dense granular flows in the vicinity of an interface and appear inscaling laws reproducing the effective friction at lateral walls. More specifically, forcefluctuations related to velocity fluctuations may trigger slip events even if the systemis globally below the slip threshold [Artoni & Richard, 2015b]. Furthermore, velocity –26–anuscript submitted to JGR-Earth Surface fluctuations, i.e. granular temperature, is a key parameter of the kinetic theory. Itsmeasurement in dense granular flows will help constrain attempts to extend this theoryto dense granular flows [Berzi, 2014].Finally, a thousand times lower frequency (tens of Hz) is also identified in theacoustic signal and is shown to correspond to the displacement of particles over oneanother, related to the relative motion of the grain layers. This seems to result fromthe quasi monodisperity of the particles involved in these experiments. Further stud-ies should investigate the role of particle size and shape on the generated acousticsignals and extend the range of bed slopes (i.e. velocities) so as to be able to betterdiscriminate scaling laws between the flow and acoustic signal quantities. –27–anuscript submitted to
JGR-Earth Surface
Notation a (cid:48) Coefficient depending on elastic parameters (I.S.U.) (see Eq. 9) a z Vibration acceleration of the plate (m s − ) B Bending stiffness (J) ˜ A z Amplitude spectrum of the vibration acceleration (m s − /Hz) E s , E p , E ∗ Young’s moduli (Pa) E a Acoustic energy (m s − ) [from Taylor & Brodsky, 2017] E k Kinetic energy (J) f Frequency of the vibration signal (Hz) f i Number of impacts per particle and per time unit (s − ) f mean Mean frequency (Hz) (see Eq. (3)) f Hertz
Theoretical mean frequency predicted by the Hertz impact model (Hz) (seeEq. (8)) f mod , f h , f flow , f v Characteristic frequencies generated by the granular flow (Hz)(see section 5.1.1) g Gravitational acceleration (m s − ) d Diameter of the particles (m) h Flow thickness (m) h g Gate elevation (m) h p Plate thickness (m) I Inertial number (-) (see Eq. (5)) L , l Dimensions of the acoustically isolated plate (m) M Mass of the acoustically isolated plate (g) N i Number of impacts per unit time in particle layer i (s − ) n Number of particle layers (-) P Hydrostatic pressure (Pa) T Period of the signal (s) V x , V y Downslope and normal speeds of the particles (m s − ) < V x > , < V y > Speeds averaged within one layer (m s − ) V xs Downslope speed of the surface particles (m s − ) v g Group speed of A mode in PMMA v g ≈ − ) v z Vibration speed of the plate (m s − ) x , y Downslope and normal positions of the particles (m) w Thickness of one layer of particles in the flow (m) W el Radiated elastic energy (J) ˙ γ Shear rate (s − ) γ g Characteristic attenuation coefficient of acoustic energy in granular media (m − ) ∆ t Duration (s) δV x , δV y Downslope and normal fluctuating speeds of particles (m s − ) θ Slope angle ( ◦ ) ν p , ν s Poisson’s ratios (-) ξ Energy ratio, acoustic efficiency (-) Π el Radiated elastic power (J s − ) Π Hertzel analytical radiated elastic power (J s − ) Π k Kinetic power (J s − ) ρ , ρ p , ρ s Densities (kg m ) φ Packing fraction ( − ) ω Pulsation (s − ) –28–anuscript submitted to JGR-Earth Surface
We thank Xiaoping Jia, Sylvain Viroulet, Diego Berzi and Alexandre Valance forinsightful discussions on granular temperature and kinetic theory. We thank G¨oranEkstr¨om for sharing the inverted force history for the Mt Dall, Mt Lituya, SheemahantGlacier and Lamplughr Glacier landslides. We thank Alain Steyer for his great helpin mounting the setup. RT acknowledges the support of the INSU ALEAS and theFrance-Norway LIA D-FFRACT programs. The data acquired during the experimentsand the scripts to treat them are available on the repository [Bachelet et al., 2020].
Appendix A Heights of the Flows
The flow height is measured by tracking the particles at the free surface of theflow (Fig. A1a) (procedure similar to that used for particle tracking). Then, thespatial and temporal height obtained by repeating the procedure for all instants (Fig.A1b) is averaged over time (Fig. A1c) and space (Fig. A1d).
Appendix B Velocity Fluctuation Measurements: Window Effect
The estimate of total velocity fluctuations depends on the width w of the windowconsidered: δV ( y, t ) = 1 w (cid:90) y + w/ y − w/ ( V ( y (cid:48) , t ) − (cid:104) V (cid:105) ( y, t )) dy (cid:48) (B1)where (cid:104) V (cid:105) ( y, t ) is the average velocity in the center of the box. A first order expansion (cid:104) V (cid:105) ( y, t ) = (cid:104) V (cid:105) ( y (cid:48) , t ) − ˙ γ ( y )( y (cid:48) − y ) e x gives (the average vertical velocity equals zero): δV ( y, t ) = 1 w (cid:90) y + w/ y − w/ ( δ V ∗ ( y (cid:48) ) + ˙ γ ( y )( y (cid:48) − y ) e x ) dy (cid:48) (B2)with δ V ∗ ( y (cid:48) ) = V ( y (cid:48) , t ) − (cid:104) V (cid:105) ( y (cid:48) , t ). Developing the square leads to three terms I , I and I : I = δV ∗ ( y, t ) = 1 w (cid:90) y + w/ y − w/ δ V ∗ ( y (cid:48) ) dy (cid:48) (B3) I = 2 w (cid:90) y + w/ y − w/ ˙ γ ( y )( y (cid:48) − y ) δV x ( y (cid:48) ) dy (cid:48) (B4) I = 1 w (cid:90) y + w/ y − w/ ( ˙ γ ( y )( y (cid:48) − y )) dy (cid:48) = w ˙ γ ( y )12 (B5) I corresponds to the genuine velocity fluctuations averaged on the box. I can becomputed by a first order expansion of δV x ( y (cid:48) ): δV x ( y (cid:48) ) = δV x ( y ) + dδV x dy ( y )( y (cid:48) − y ) (B6)Thus: I = 2 w (cid:32) δV x ( y ) (cid:90) y + w/ y − w/ ( y (cid:48) − y ) dy (cid:48) + dδV x dy ( y ) (cid:90) y + w/ y − w/ ( y (cid:48) − y ) dy (cid:48) (cid:33) (B7)The first term equals zero, whereas the second can be neglected because of the secondorder.Finally, total velocity fluctuations estimate are given by the following expression: δV ( y, t ) = δV ∗ ( y, t ) + w ˙ γ ( y )12 (B8)The second term quantifies the error introduced by considering the average velocitytaken in y (the center of the box) instead of the value in y (cid:48) in formula (B1). Its –29–anuscript submitted to JGR-Earth Surface x/d t [ s ] x/d < h / d > t t [s] < h / d > x time average space average (b)(c) (d) x/d y / d (a) all instants Figure A1.
Heights of the flows: (a) example of flow interface detection (red line), (b) spaceand time height, thereafter averaged over (c) time or (d) space. Each color of panels (c) and (d)corresponds to a specific flow (see for example Fig. 11 for detailed legend).–30–anuscript submitted to
JGR-Earth Surface w = 0.2dw = 1dw = 2dw = 4dw = 6d
Figure B1.
Effect of the window size on the fluctuation velocity computation. expression is very similar to the one found by Weinhart et al. [2013] (Eq. (34)). Theonly difference comes from the choice of the averaging function, also called the coarse-graining function. We implicitly chose a gate equal to one in [ y − w/ , y + w/
2] and tozero elsewhere, whereas a more complex choice is usually selected for differentiability[Glasser & Goldhirsch, 2001, Weinhart et al., 2013].Thanks to expression (B8) and approximating δV ∗ by 2 . d ˙ γ , as suggested bythe linear fit in Fig. 7e, it is possible to deduce that the windows have an effect similarto that of δV ∗ when w = 5 d . For this reason, the window is negligible in our case (seeFig. B1) Appendix C Correlation Lengths within the Flow
To obtain quantitative measurements of the correlation length of velocity fluctu-ations we compute the downslope and vertical velocity correlations between two points M and M with coordinates ( x , y ) and ( x , y ): C V i ( M , M ) = (cid:80) t δV i ( M , t ) × δV i ( M , t ) (cid:112)(cid:80) t δV i ( M , t ) × (cid:112)(cid:80) t δV i ( M , t ) (C1)where i = x, y . Examples of downslope and vertical velocity correlations are presentedin Figs. C2(a) and (b) respectively. High correlations of the horizontal velocity overone particle thickness are clearly visible. To quantify this correlation, a correlationlength has been defined. It corresponds to the length at which the correlation reachesa given threshold. Unlike Pouliquen [2004] who chose a threshold of 0 .
05, we selecteda value of 0 . x -direction λ xx arehigher than one particle diameter. This suggests correlated motion of particles of thesame layer, supporting the layering observed in Fig. 3b. In agreement with Pouliquen[2004] and Staron [2008], correlation lengths decrease for increasing slope angles (Fig.C2c-e), as observed in Movies 3 and 4 (supplementary material). The correlationlengths collapse to zero under y/d = 5 because particle velocities are smaller thannoise. –31–anuscript submitted to JGR-Earth Surface
Note that for dry granular chute flows [Gardel et al., 2009] and for granular flowsin a fluid [Orpe & Kudrolli, 2007], significantly greater spatial correlations are observednear the boundaries, which may be the case here.
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