Lee waves in three-dimensional stratified flow

Abstract

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.