The adjustment of a rotating, stratified fluid subject to localized sources of mass

Abstract

AbstractThe time‐dependent response of a rotating, stratified fluid to localized, impulsive forcing is examined under the hydrostatic approximation and the assumption of two‐dimensionality. Analytic solutions of the linearized quasi‐Boussinesq equations are derived for quiescent and uniformly sheared flows when the axis of the forcing function lies in the direction of the mean wind shear. The mass forcing function, it is imagined, represents the effect of deep, penetrative convection at the cloud top.In the unsheared case, the phase lines of gravity waves emanating from the forcing region adopt a fanlike configuration symmetric about the vertical. The slopes of the phase lines become more horizontal as time progresses resulting in a vertical ‘fine structure’. As a consequence of this the value of the wave Richardson number may fall below 1/4 in the vicinity of the forcing function.When the basic‐state shear is chosen to be equal to the buoyancy frequency, the flow has neutral symmetric stability. In this case the response to impulsive heat and momentum forcing is biased towards modes with phase lines of orientation similar to that of the absolute momentum (or equivalently, potential temperature surfaces). A permanent, ageostrophic residual flow along these surfaces is revealed in the long‐time limit. In contrast, the response to mass forcing shows a far greater degree of symmetry about the vertical, with no residual ageostrophic flow.In general, for symmetrically stable flows, a geostrophic and hydrostatically balanced flow remains after the gravity‐wave energy has dispersed, consistent with the potential vorticity change caused by the mass forcing. However, the potential vorticity anomaly created is proportional in size to the basic state potential vorticity. For slantwise neutral flows therefore, mass forcing does not create a potential vorticity anomaly, and no balanced flow remains.