Numerical simulation of supercritical trapped internal waves over topography

Abstract

We consider the steady flow of a stratified fluid over topography in a fluid of finite vertical extent, as typified by experimental flumes with a rigid lid or the ocean under the rigid lid approximation. We do not specify a functional form of the upstream stratification or background current and derive a general version of the Dubreil–Jacotin–Long equation appropriate for the problem. This elliptic equation is strongly nonlinear and we develop an efficient, pseudospectral, iterative method for its numerical solution. The method allows us to compute laminar, trapped waves with amplitudes more than 50% of the depth of the fluid. We find that when either a background shear current is present or the topography is narrow enough, multiple steady states are possible and we confirm this finding by using integrations of the full time-dependent Euler equations. We discuss instances of waves with closed streamlines, finding that the presence of shear allows for waves with vortex cores that persist for long times in time-dependent simulations and match well with solutions of the steady theory. In contrast, streamline overturning in the absence of upstream shear only occurs for flows that are stratified near the surface and in this instance, time-dependent simulations yield unsteady cores that do not match steady results very well.