Development of Grain Avalanches

We present a new theoretical approach aimed to describe the dynamical properties of avalanches. Starting from simple microscopic considerations, we propose an analytical derivation accounting for the avalanche growth as a function of time. The solution resembles very closely recent experimental observations. We show in particular that the two observed regimes of growth proceed from a singularity in the rear front velocity as a function of the slope.

Figure 1

The cone of equation $x^2+(y-2\overline{v}t+\frac{5}{2}\overline{v}\tau {)}^2=(\frac{1}{2}\overline{v}\tau {)}^2$ is suited to represent schematically the growth of avalanches (regime I, $\theta >{\theta }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}$). After a time $\tau$ has elapsed since the triggering, the rear front has moved upwards, according to the equation $x^2+(y+\frac{1}{2}\overline{v}\tau {)}^2=(\frac{1}{2}\overline{v}\tau {)}^2$. Next, each destabilized grain falls along the steepest slope over a length $2\overline{v}(t-\tau )$ until time $t$. The projection of the cone onto the plane $\tau =0$ corresponds to the avalanche expansion at time $t$.

Figure 2

Avalanche boundary at time $t$ (regime I, $\theta >{\theta }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}$) according to the present modeling. The head front tip moves with velocity $2\overline{v}$, while the rear front moves upwards with velocity $\overline{v}\cos \alpha$. The order of magnitude of $\overline{v}$ is given by $\sqrt{gd}$. We superimposed for comparison a real photograph taken by the authors of Ref. 11.

Figure 3

Left: Total area overrun by the avalanche from its triggering to time $t$ in regime II ($\theta <{\theta }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}$) according to the present modeling. We superimposed for comparison a real photograph taken by the authors of Ref. 11. The value of $\overline{v}$ is taken to be $5.5\mathrm{c}\mathrm{m}/\mathrm{s}$, which compares well with $\sqrt{gd}=4.9\mathrm{c}\mathrm{m}/\mathrm{s}$, and the value of the ratio $v_{\perp }/\overline{v}$ is approximately 0.13. Right: Superposition of the avalanche boundaries for three different times. The dashed line is the envelope of the successive avalanche boundaries.