The response of a stratified rapidly rotating flow to a pulsating topography

Abstract

A theoretical study is made of the disturbance produced by an oscillating, shallow topographical feature in horizontal relative motion in a rapidly rotating, linearly stratified, unbounded fluid. For a sinusoidal surface oscillation, an explicit solution is obtained in terms of wavenumber spectra of the topography. The oscillating far-field behaviour is shown to consist of a large-scale, cyclonic component above the topography and a system of inertial waves behind the caustics, which spreads predominantly in the downstream direction. A significant property of the flow field is its dependence on a frequency threshold familiar from classical works on internal gravity waves in the absence of rotation, determined by the Brunt-Väisälä value. When the frequency is supercritical, a prominent circle of maximum disturbance appears in the far field, which provides the transition boundary between two distinct cyclonic structures and an upstream barrier to the propagating waves ahead of the obstacle. The circle has a radius depending on the relative magnitude of the pulsating frequency and the Brunt-Väisälä value, and is distinctly marked also by a phase jump in pressure and velocities. These features are substantiated by numerical examples of the full solution at a large but finite distance above the obstacle at supercritical frequencies. The circle of maximum disturbance signifies a preferential direction for energy propagation unaccounted for by group velocity. Its relation to the classical result of Görtler in the homogeneous case and that in the classical internal-gravity-wave theory are examined.