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Low-order Boussinesq models based on $\unicode[STIX]{x1D70E}$ coordinate series expansions

Published online by Cambridge University Press:  01 June 2020

James T. Kirby Open the ORCID record for James T. Kirby [Opens in a new window]
James T. Kirby*
Affiliation:
Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware, NewarkDE 19716, USA
*
†Email address for correspondence: kirby@udel.edu
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Abstract

We derive weakly dispersive Boussinesq equations using a $\unicode[STIX]{x1D70E}$ coordinate for the vertical direction, employing a series expansion in powers of $\unicode[STIX]{x1D70E}$. We restrict attention initially to the case of constant still-water depth $h$ in order to simplify subsequent analysis, and consider equations based on expansions about the bottom elevation $\unicode[STIX]{x1D70E}=0$, and then about a reference elevation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}}$ in order to improve linear dispersion properties. We use a perturbation analysis, suggested recently by Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38), to show that the resulting models are not subject to the trough instability studied there. A similar analysis is performed to develop a model for interfacial waves in a two-layer fluid, with comparable results. We argue, by extension, that a necessary condition for eliminating trough instabilities is that the model’s nonlinear dispersive terms should not contain still-water depth $h$ and surface displacement $\unicode[STIX]{x1D702}$ separately.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Internal waves Instability: Nonlinear instability
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , R3
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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