Dynamics of granular avalanches caused by local perturbations

Surface flow of granular material is investigated within a continuum approach in two dimensions. The dynamics is described by the nonlinear coupling between a mobile layer and an erodible bed of static grains. Following previous studies, we use mass and momentum conservation to derive St. Venant–like equations for the evolution of the thickness $R$ of the mobile layer and the profile $Z$ of the bed. This approach allows the rheology in the flowing layer to be specified independently, and we consider in detail the two following models: a constant plug flow and a linear velocity profile. We study and compare these models for nonstationary avalanches triggered by a localized amount of mobile grains on a bed of initially constant slope. We solve analytically the nonlinear dynamical equations by the method of characteristics. This enables us to investigate the temporal evolution of the avalanche size, amplitude, and shape as a function of model parameters and initial conditions. In particular, we can compute their large time behavior as well as the condition for the formation of shocks.

Figure 1

Rolling grains are locally deposited on an erodible sand bed which has initially a uniform slope $\mu +m$, where $\mu$ is the tangent of the angle of repose.

Figure 2

Flowing layer thickness $R(t,x)$ (gray) and bed profile $Z(t,x)$ (crosses). The grains flow from the right to the left. The dashed box is the control volume for which mass and momentum conservation laws are written.

Figure 3

Bold curve: plot of the Lambert’s function $W\left(x\right)$. Thin curves: small and large $x$ expansions $W\left(x\right)\sim x-x^2$ and $W\left(x\right)\sim \ln x-\ln \ln x$, respectively.

Figure 4

Model $\mathcal{P}$: Boundaries between different sectors in the $t\text{−}x$ plane.

Figure 5

Model $\mathcal{P}$: Characteristic curves with constant $\alpha$ for a perturbation $R_0\left(x\right)$ of Eq. (9) with $r_0=0.1$, $\delta =5.0$, and slopes(a) $m=-0.05$, (b) $m=0.05<1∕\delta$, (c) $m=0.15<1∕\delta$, and (d) $m=0.25>1∕\delta$. The shaded area indicates the region with shocks. The dashed curves mark the boundaries between the different sectors, cf. Fig. 4.

Figure 6

Model $\mathcal{P}$ with $b=0$: Fixed time profiles $Z(t,x)-mx$ and $R(t,x)+Z(t,x)-mx$ so that the gap between two corresponding curves represents the thickness of the mobile phase. Plot (a) is for $R_0\left(x\right)$ of Eq. (9) and (b) for a Gaussian $R_0\left(x\right)$, both for $m=-0.05$. The profiles are for times $t=0$, 5, 10, 20, and 50. Here and in the following plots of profiles the plotting intensity decreases with increasing time.

Figure 7

Analog of Fig. 6 but for $m=0.05$ and times $t=0$, 5, 20, 40, and 60.

Figure 8

Analog of Fig. 6 but for $m=0.15$ and times $t=0$, 10, 20, 40, and 60 for (b) only. In plot (a) a shock occurs at the uphill end of the avalanche (shock time $t_s=41.71$). Note that for the Gaussian $R_0\left(x\right)$, plot (b), shocks are generated only for larger values of $m$, see text.

Figure 9

Model $\mathcal{P}$ with finite $b=0.5$: Profiles for a Gaussian $R_0\left(x\right)$ and (a) $m=-0.05$ (for $t=0$, 5, 10, 20, and 45), (b) $m=0.05$ (for $t=0$, 10, 20, 30, and 45). The dashed curves represent the corresponding profiles for $b=0$ of Sec. 3A and are shown for comparison.

Figure 10

Model $\mathcal{P}$ at the angle of repose $(m=0)$ for finite horizontal stress, $b=0.5$: Profiles for a Gaussian $R_0\left(x\right)$ at times $t=0$, 15, 30, and 60. The dashed curves correspond to the absence of horizontal stress, $b=0$, where the perturbation $R_0$ propagates without changing its shape.

Figure 11

Model $\mathcal{L}$ with $b=0$: Profiles for a Gaussian profile $R_0\left(x\right)$ with $r_0=0.7$, $\delta =5.0$, and slope (a) $m=-0.1$ (b) $m=0.1$, both for times $t=0$, 3, and 7. The shock time is $t_s=\sqrt{e∕2}\delta ∕r_0=8.32$. For negative $m$ the perturbation has stopped at the time $t=-r_0∕m=7.0$, i.e., the profile shown for the latter time is the final bed profile. Note that for positive $m$ the thickness of the moving layer shows an overall linear increase proportional to $mt$ since for a Gaussian profile, $R_0\left(x\right)$ is in fact small but finite everywhere. For clarity, we indicated explicitly the thickness $R$ of the mobile layer at $t=3$.

Figure 12

Model $\mathcal{L}$ with finite $b=0.5$ for the same parameters as in Fig. 11. The plots are for times (a) $t=0$, 3, and 6, and (b) $t=0$, 3, and 7. For comparison, the corresponding profiles for $b=0$ are shown as dashed curves. Notice that the peak appears rather sharp due to the relative elongation of the vertical axis. Inset: For qualitative comparison, experimental results as given by Fig. 23 of Ref. 16 for the surface velocity $v$ and the local height $h$ of the mobile layer which corresponds to $R$ in our notation. A linear relation between $v$ and $R$ is observed, and the shape of $R$ resembles that of our analytical result. (The direction of flow is inverted in the experiment compared to our model.)