Adjustment of unbalanced vortices in uniformly rotating and stratified fluids

The motion of an initially quiescent, incompressible, stratified and/or rotating fluid 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) and f=0 (non-rotating stratified fluid). 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 finite-sized circularly symmetric impulsive forcings is considered. Initial conditions are unbalanced geostrophic vortices which adjust through radiation of internal waves. 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. Remarkably, the amplitude of the vortices amplifies by a factor N/f after the adjustment is completed, which can put them well into the inertially (centrifugally) unstable range if anticyclonic and N >> f.