Long Wave Run-Up on Random Beaches

The estimation of the maximum wave run-up height is a problem of practical importance. Most of the analytical and numerical studies are limited to a constant slope plain shore and to the classical nonlinear shallow water equations. However, in nature the shore is characterized by some roughness. In order to take into account the effects of the bottom rugosity, various ad hoc friction terms are usually used. In this Letter, we study the effect of the roughness of the bottom on the maximum run-up height. A stochastic model is proposed to describe the bottom irregularity, and its effect is quantified by using Monte Carlo simulations. For the discretization of the nonlinear shallow water equations, we employ modern finite volume schemes. Moreover, the results of the random bottom model are compared with the more conventional approaches.

Figure 1

A sample realization of random bottoms for $\sigma =5\times {10}^{-2}$ and various values of the regularity parameter $r=1$, 2, 4, 5, and 8 starting correspondingly from the top.

Figure 2

The shoreline motion in the case of a smooth shore (the dashed line) and a particular realization of the random bottom with $\sigma ={10}^{-2}$, $r=1$ (solid line).

Figure 3

Probability density distribution for $\sigma ={10}^{-3}$ [left image, $\mathbb{E}(R_{\max })=0.78088$; 5% and 95% quantiles are equal to 0.777 34 and 0.784 16, respectively] and $\sigma ={10}^{-2}$ [right image, $\mathbb{E}(R_{\max })=0.42281$; 5% and 95% quantiles are equal to 0.378 57 and 0.462 61, respectively]. Note the difference in the vertical scales and in the horizontal extent of two distribution functions.

Figure 4

Maximum run-up value as a function of the perturbation characteristic magnitude $\sigma$ for the irregular case $r=1$ (dashed line) and the regularized noise $r=6$ (solid line). For comparison, the red dash-dotted line represents the maximum run-up value for the smooth bottom case.

Figure 5

The maximum run-up value as a function of the regularity parameter $r$ for several values of the perturbation magnitude $\sigma$ (increasing from the top).

Figure 6

Comparison of the run-up reduction effect for various ad hoc friction terms and the random bottom perturbation model in the nonregularized case $r=1$. The horizontal axis represents the friction coefficient $c_f$ for deterministic computations and $\sigma$ for the random roughness model (solid line).