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High-order strongly nonlinear long wave approximation for variable bottom

Published online by Cambridge University Press:  06 February 2026

Wooyoung Choi Open the ORCID record for Wooyoung Choi [Opens in a new window]  and
Sunao Murashige Open the ORCID record for Sunao Murashige [Opens in a new window]
Wooyoung Choi
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
Sunao Murashige*
Affiliation:
Department of Mathematics and Informatics, Ibaraki University , Mito, Ibaraki 310-8512, Japan
*
Corresponding author: Sunao Murashige, sunao.murashige.sci@vc.ibaraki.ac.jp
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Abstract

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.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1028 , 10 February 2026 , A34
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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