Abstract
When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.
Highlights
Snow avalanches, debris flows, pyroclastic flows and submarine landslides often occur on inclines covered by a static layer of erodible granular material
This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate
Frictional hysteresis (Daerr & Douady 1999; Daerr 2001; Aranson & Tsimring 2002; Pouliquen & Forterre 2002; Edwards et al 2017, 2019) results in a range of thicknesses over which both static and moving layers of material can coexist in a granular flow
Summary
Debris flows, pyroclastic flows and submarine landslides often occur on inclines covered by a static layer of erodible granular material. Daerr & Douady (1999) and Daerr (2001) termed the retrogressive failures ‘uphill-propagating avalanches’ These were investigated theoretically by Bouchaud & Cates (1998) using the Bouchaud, Cates, Ravi Prakash and Edwards (BCRE) model (Bouchaud et al 1994) in which a granular material is separated into two phases; rolling and static. The uphill-propagating front was a travelling wave solution with the onset occurring through a discontinuity in the speed of the retrogressive waves, i.e. the wave speed immediately jumped up from zero (for no upslope propagation) to a finite non-zero value Their theoretical prediction for the existence of uphill waves agreed with the experimental data of Daerr & Douady (1999) and Daerr (2001), using one fitting parameter. Retrogressive failures are not just of fundamental scientific interest, and provide an important test case for the friction law proposed by Edwards et al (2017, 2019)
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