Stably stratified rotating flow through a group of obstacles

Abstract

Abstract A linear calculation is presented of uniform stably stratified flow (velocity U 0, Brunt-Vaisala frequency N) through a region of resistance on a rigid plane with vertical scale Hand horizontal scale Dwhen the Froude number F = U 0/HNis much less than unity, and when the velocity perturbation is small compared with U 0. It is shown that that largest vertical perturbation occurs to the streamlines within a “summit layer”, of thickness O(U 0/N), at the top of the region and that this perturbation is transmitted upwards and downwards by internal waves. This calculation is a model of the flow through groups of hills (which is illustrated with some small laboratory experiments) and it helps explain some unsolved problems about strongly stratified flow over a single hill. The first-order correction, below the summit layer, to flow through and around groups of hills can be larger than the O(F2) term given by Drazin (1961). On extending the theory, it is found that the effect of a rotation with angular velocity Ω (and the consequent Coriolis acceleration) on the flow is a singular perturbation when F « l; the far-field perturbation decays slowly on the length scale NH/2Ω(≫ H), under the action of inertial waves; the lateral deflections of the streamlines can be large even if ΩD/U« 1, depending on the vertical distribution of the resistance. This theory implies that topography may have an effect on the flow over a much wider scale than is commonly supposed or modelled, and, more generally, the theory provides a mechanism for showing how disturbances in stratified flows can induce motions on much larger scales—i.e. an “up-scale” energy transfer mechanism. This result is implicit in the important paper of Merkine (1975). Key words: Froude numberinertial waveshills