A new instability for Boussinesq-type equations

Abstract

A wide class of problems for free-surface gravity waves fall into a weakly dispersive regime, in which wavelength is large compared to water depth, and wave phase speed differs by a small amount from the speed of shallow-water waves. The resulting problem is treated naturally using Taylor series expansions of dependent variables in the vertical coordinate, leading to a class of models that are collectively referred to here as Boussinesq-type models. Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38) have recently shown that certain members of this broad class of models are subject to a high-wavenumber instability, which can grow rapidly when the elevation of the wave trough is sufficiently depressed below the mean water surface. This newly revealed instability may provide an explanation for the modelling community's frequent observations of noisy behaviour in Boussinesq-type model calculations.