Low-order Boussinesq models based on σ coordinate series expansions

We derive weakly dispersive Boussinesq equations using a coordinate for the vertical direction, employing a series expansion in powers of . We restrict attention initially to the case of constant still-water depth in order to simplify subsequent analysis, and consider equations based on expansions about the bottom elevation , and then about a reference elevation 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 and surface displacement separately.