Tsunami Squares Approach to Landslide-Generated Waves: Application to Gongjiafang Landslide, Three Gorges Reservoir, China

Abstract

We have developed a new method, named “Tsunami Squares”, for modeling of landslides and landslide-generated waves. The approach has the advantages of the previous “Tsunami Ball” method, for example, separate, special treatment for dry and wet cells is not needed, but obviates the use of millions of individual particles. Simulations now can be expanded to spatial scales not previously possible. The new method accelerates and transports “squares” of material that are fractured into new squares in such a way as to conserve volume and linear momentum. The simulation first generates landslide motion as constrained by direct observation. It then computes induced water waves, given assumptions about energy and momentum transfer. We demonstrated and validated the Tsunami Squares method by modeling the 2008 Three Gorges Reservoir Gongjiafang landslide and river tsunami. The landslide’s progressive failure, the wave generated, and its subsequent propagation and run-up are well reproduced. On a laptop computer Tsunami Square simulations flexibly handle a wide variety of waves and flows, and are excellent techniques for risk estimation.

Appendix 1: Inclusion of wave dispersion

Long wave theory assumes that the depth-averaged horizontal acceleration of a water column is proportional to the gradient of the fluid surface as stated in Eq. (21):

$$\frac{{\partial {\mathbf{v}}({\mathbf{r}},t)}}{\partial t} = - {\mathbf{v}}({\mathbf{r}},t) \bullet \nabla_{h} {\mathbf{v}}({\mathbf{r}},t) - g\nabla_{h} \zeta ({\mathbf{r}},t)\quad$$

(21)

According to linear dispersive wave theory (Ward and Day 2010), the depth-averaged horizontal acceleration of a water column is proportional to the gradient of the fluid surface smoothed over a dimension comparable with the water depth. Linear dispersion can be accommodated in tsunami squares simply by replacing Eq. (21) by Eq. (22)

$$\frac{{\partial {\mathbf{v}}({\mathbf{r}},t)}}{\partial t} = - {\mathbf{v}}({\mathbf{r}},t) \bullet \nabla_{h} {\mathbf{v}}({\mathbf{r}},t) - g\nabla_{h} \zeta_{\text{smooth}} ({\mathbf{r}},t)\quad$$

(22)

where

$$\begin{gathered} \zeta_{\text{smooth}} ({\mathbf{r}},t) = \zeta ({\mathbf{r}},t) \times S({\mathbf{r}}) \hfill \\ \quad = \int {\zeta ({\mathbf{r^{\prime}}},t)S({\mathbf{r}} - {\mathbf{r^{\prime}}}){\mathbf{d}}} {\mathbf{r^{\prime}}}\quad \hfill \\ \end{gathered}$$

(23)

and

$$S({\mathbf{r}}) = \text{Re} \int\limits_{k} {\frac{{e^{{{\text{ik}} \cdot r}} \,\,\tanh ({\text{kh}})}}{{4\pi^{2} \,\,\,\,\,\,\,\,\,\,\,{\text{kh}}}}} {\text{d}}{\mathbf{k}}$$

(24)

This indicates that short wave (kh ≫ 1) contributions to the surface gradient impart less depth-averaged acceleration to the water column than do longer wave (kh ≪ 1) contributions. As a result, short waves fall behind long waves, as dictated by linear dispersive theory. For the applications in this paper, water wave dispersion is not important.