High-order strongly nonlinear long wave approximation for variable bottom

Wooyoung Choi

doi:10.1017/jfm.2026.11135

2026-02-06

PRE

Wooyoung Choi

Sunao Murashige *

DOI：10.1017/jfm.2026.11135

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摘要

En 中文

This paper describes a high-order strongly nonlinear (SNL) model for long waves in the presence of a variable bottom; which is a generalisation of the model for a flat bottom (Choi 2022a; J. Fluid Mech. vol. 945; A15). This asymptotic model written in terms of the bottom velocity is obtained using systematic expansion with a single small parameter measuring the ratio of the water depth to the characteristic wavelength and is found linearly stable at any order of approximation. To test the high-order SNL model with a variable bottom; we solve numerically the first- and second-order models using a pseudo-spectral method to study the deformation or generation of long waves over a variable bottom. Specifically; we consider two examples: (i) the propagation of cnoidal waves over a fixed bottom topography; and (ii) the forced generation of solitary waves by a submerged topography moving steadily with a transcritical speed. The computed results are then compared with the fully nonlinear computation using a boundary integral method as well as the numerical solutions of the weakly nonlinear long wave model. It is found that the second-order SNL model for the bottom velocity is suitable for stable numerical computations and produces accurate solutions even for a relatively large-amplitude initial wave or submerged topography.

Keyword:

surface gravity waves

topographic effects