From granular collapses to shallow water waves: A predictive model for tsunami generation

In this article, we present a predictive model for the amplitude of impulse waves generated by the collapse of a granular column into a water layer. The model, which combines the spreading dynamics of the grains and the wave hydrodynamics in shallow water, is successfully compared to a large dataset of laboratory experiments, and captures the influence of the initial parameters while giving an accurate prediction. Furthermore, the role played on the wave generation by two key dimensionless numbers, i.e., the global Froude number and the relative volume of the immersed deposit, is rationalized. These results provide a simplified, yet comprehensive, physical description of the generation of tsunami waves engendered by large-scale subaerial landslides, rockfalls, or cliff collapses in a shallow water.

Figure 1

(a) 3D view of the experimental setup, showing the initial granular column of height $H_g$ and width $L_0$, the initial fluid depth $h_0$, the channel width $W$, and the maximum wave amplitude $A_m$. (b) 2D view of the final granular deposit of height $H_{\infty }$, runout length $L_{\infty }$, immersed volume $\mathrm{\Delta }{V}_{\infty }$, and front position ${\ell }_{\infty }$ at $z=h_0$. (c), (d) Impulse waves generated by the collapse of a column with $H_g=39$ cm and $L_0=10$ cm: (c) bore wave for $h_0=3$ cm and (d) solitary wave for $h_0=10$ cm (movies are available in Supplemental Material [48]).

Figure 2

Evolution of the maximum wave amplitude $A_m$ with the fluid depth $h_0$, for three different initial granular columns of same width $L_0$ but different height $H_g$. The horizontal dashed lines indicate the saturation of $A_m$ at large $h_0$. Full symbols refer to shallow water waves, while empty symbols correspond to deep water waves. The shaded area corresponds to deep water waves, where $A_m\lesssim 0.45h_0$, and the black dashed line (- -) represents the observed limit between bore and solitary waves ($A_m\simeq 0.96h_0$).

Figure 3

(a) Relative final height $H_{\infty }/L_0$ and (b) runout distance $\mathrm{\Delta }{L}_{\infty }/L_0$ reached by the final deposit as a function of the aspect ratio of the initial column $a=H_0/L_0$. (—) Power laws $H_{\infty }/L_0=\alpha a^n$ (with $\alpha =1$ and $n=1$ when $a\lesssim 0.93$, and $\alpha =0.95$ and $n=1/3$ when $a\gtrsim 0.93$), and $\mathrm{\Delta }{L}_{\infty }/L_0=\beta a^m$ (with $\beta =1.65$ and $m=1$ when $a\lesssim 3$, and $\beta =2.38$ and $m=2/3$ when $a\gtrsim 3$). ($\circ$) Experiments of bore and solitary waves which are not significantly affected by the presence of the solid step. ($\times$) Bore and solitary waves with a noticeable influence of the solid step, and nonlinear transition waves.

Figure 4

(a) Relative maximum amplitude $A_m/h_0$ of the wave as a function of the relative final extension of the granular front $\mathrm{\Delta }{\ell }_{\infty }/h_0$, with (—) $A_m/h_0=0.50\sqrt{\mathrm{\Delta }{\ell }_{\infty }/h_0}$. (b) Comparison between the measurement of $\mathrm{\Delta }{\ell }_{\infty }$ and the model prediction $\mathrm{\Delta }{\ell }_{\infty ,\mathrm{th}}$. (—) $\mathrm{\Delta }{\ell }_{\infty }=0.82\mathrm{\Delta }{\ell }_{\infty ,\mathrm{th}}$. (c) $A_m$ as a function of $\sqrt{\mathrm{\Delta }{\ell }_{\infty ,\mathrm{th}}{h}_0}$, with (—) Eq. (4). Symbols designate ($▹$) bore and ($\square$) solitary waves. The shaded area in (a) corresponds to the deep water waves region where $A_m\lesssim 0.45h_0$.

Figure 5

(a), (b) Maximum wave amplitude $A_m$, given by Eq. (5), as a function of the water depth $h_0$, for (a) different initial height $H_0$ with the same aspect ratio $a=3$ and (b) different $a$ with $H_0=40$ cm. (c), (d) $A_m/h_0$ as a function of (c) ${\mathrm{Fr}}_0$ as given by Eq. (6) and (d) $\mathrm{\Delta }{V}_{\infty ,\mathrm{th}}/\left(W{h_0}^2\right)$ as given by Eq. (7), for $H_0=40$ cm and different $a$. Shaded areas are the same as in Fig. 2.

Figure 6

(a) $A_m$ as a function of $h_0$, for the experiments of Fig. 2. (b) $A_m/h_0$ as a function of the global Froude number ${\mathrm{Fr}}_0$. (c) Comparison between the experimental and predicted immersed volumes, with (—) $\mathrm{\Delta }{V}_{\infty }=0.65\mathrm{\Delta }{V}_{\infty ,\mathrm{th}}$. (d) $A_m/h_0$ as a function of the relative volume $\mathrm{\Delta }{V}_{\infty }/\left(W{h_0}^2\right)$ of the immersed deposit, with (- -) the empirical law $A_m/h_0=0.25{[\mathrm{\Delta }{V}_{\infty }/\left(W{h_0}^2\right)]}^{0.8}$ from [43]. The colored solid lines in (a), (b), and (d) correspond to the model predictions for the same initial granular columns as in the experiments, and the shaded areas are the same as in Fig. 2.

Figure 7

Critical Froude number ${\mathrm{Fr}}_0^c=\sqrt{H_0/h_0^c}$ at which the maximum wave amplitude is reached for the collapse of a granular column, as a function of the initial aspect ratio $a$. The line corresponds to Eqs. (A1) for the three domains separated by the two vertical dotted lines at $a=0.93$ and $a=3$.