Low-order Boussinesq models based on coordinate series expansions

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.