Surface forced internal waves and vortices in uniformly stratified and rotating fluids

Abstract

The motion of an initially quiescent, incompressible, stratified and/or rotating uid of semi-infinite extent due to surface forcing is considered. The stratification parameter N and the Coriolis parameter f are constant but arbitrary and all possible combinations are considered, including N = 0 (rotating homogeneous fluid), f = 0 (non-rotating stratified fluid) and the special case N = f. The forcing is suction or pumping at an upper rigid surface and the response consists of geostrophic flows and inertial-internal waves. The response to impulsive point forcings (Green's functions) is contrasted with the response to finite-sized circularly symmetric impulsive forcings. Early-time and large-time behaviour are studied in detail. At early times transient internal waves change the vortices that are created by pumping/suction at the surface. The asymptotically remaining vortices are determined, a simple expression for what fraction of the initial energy is converted into internal waves is derived, as well as wave energy fluxes and the dependence of the flux direction on the value of N/f. The internal wave field is to leading order in time a distinct pulse, and rules for the arrival time of the pulse, its amplitude, its motion along a ray of constant frequency and decay with time, are given for the far field. A simple formula for the total wave energy distribution as a function of frequency is derived for when all waves have propagated away from the forcing.