A fully spectral framework for nonlinear water waves propagating over topography

Abstract

A problem of nonlinear water waves propagating over uneven bottom is considered. The proposed solution is based on a fully spectral Fourier-Galerkin method. Hence, higher-order terms appearing in free-surface and bottom boundary conditions are calculated entirely in a wave number space as convolution sums. In this way, the nonlinear terms may be efficiently determined using either a direct convolution method for smaller kernels or an FFT-based procedure for larger spectral domains. An implementation-ready form of a linear system of equations that bonds velocity potential coefficients at the surface and at the bottom is reported. In static topography conditions, the fast solution of the system is achieved due to precomputed factorization. The numerical model is applied to waves propagating over a bottom of various geometries, including abrupt topographies with upright slopes approximating transformation of waves over a shelf due to a considerable decrease in water depth. The accuracy of the solution is confirmed for a propagation of linear and nonlinear waves of permanent form including solitons, linear and nonlinear shoaling, reflection and transmission of linear waves at an underwater step, and landslide generated linear waves. An application of the method to a tsunami wave undergoing transformation over an abrupt bottom junction is presented along with the discussion on nonlinear wave processes strongly affecting the resulting transmitted tsunami profile.