Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter ε=η0/h0, where η0 is the characteristic amplitude of the wave, and h0 is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered.