Supercritical rotating flow over topography

Abstract

The flow of a one-and-a-half layer Boussinesq fluid over an obstacle of nondimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a nondimensional parameter B (inverse Burger number). The supercritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is significantly greater than one is considered in the shallow water (small aspect ratio) limit. The linear drag exerted by the obstacle on the flow is shown to be M2/(FF2−1)×f(B/F2−1), where f is a function specific to each obstacle. Explicit expressions are given for several common obstacle shapes, and the results are checked against nonlinear flows simulated by a shock-capturing finite volume numerical scheme. For flows within the supercritical regime (F−1⪢M2/3) the linear drag result is found to remain accurate up to (at least) M≈0.7. The success of the linear drag theory can be explained because, in the supercritical regime, strong nonlinearities are displaced to the wake regions at the flanks of the obstacle. In the presence of weak rotation and for small obstacle height the development of the nonlinear wakes is governed by the Ostrovsky–Hunter (OH) equation. Across a wide range of parameter space the wake pattern is determined by a single parameter β=3M/BF2−1. Numerical solutions of the OH equation illustrate the dependence of flow patterns and wave breaking regions on β. Results are again verified by comparison with numerical solutions of the full nonlinear rotating shallow water equations.