Analysis of the Nonlinear Shallow Water Equations Over Nonplanar Topography

Abstract

The role played by the beach bottom profile on coastal inundation phenomena is analyzed here by means of approximate analytical solutions of the nonlinear shallow water equations (NSWEs) over uneven bottoms. These are obtained by only using the assumptions of small waves at the seaward boundary and small topographic forcing. Our work, built on the Carrier and Greenspan [1] hodographic transformation and on the solution of the boundary value problem (BVP) for the NSWEs proposed by Antuono and Brocchini [2], focuses on the propagation of nonlinear non-breaking waves over quasi-planar beaches. Since the terms associated with the perturbed bottom only appear in the second-order perturbed solutions, the breaking conditions for the planar-beach bathymetry also predict well the breaking occurring on the nonplanar beaches analyzed here. The most important results, concerning the shoreline position and the near-shoreline velocity, are given for both pulse-like and periodic input waves propagating over two types of nonplanar bathymetries. The solution proposed here is a fundamental benchmark for any numerical and theoretical analyzes concerned with estimates of wave run-up on beaches of complex shape.

Shallow water dynamics on linear shear flows and plane beaches

“…Also, three-dimensional effects were included in recent work [16], where a general approach was put forward to study the problem on a bay of arbitrary cross-section. The work laid down in [3] makes use of analysis techniques to estimate the Jacobian function associated to an arbitrary bottom profile, and thus proves that at least in theory, that the restriction to planar or convex beaches is not necessary. One problematic issue with the approach of Carrier and Greenspan is that it is difficult to treat the boundary-value problem.…”

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

“…In this context, Antuono et al [36] studied an analytical solution of the shallow water ow over uneven topography, and this is for analysis of wave run-up on a beach. e current shallow water model is applied to simulate a dam break over an initially dry oodplain with three humps, recommended by Kawahara and Umetsu [37] for this challenging problem that involves complex ow hydrodynamics, bottom topography, and wetting and drying.…”