Rotating, nonlinearly stratified flow past an isolated obstacle

Abstract

The effects of density profile nonlinearities upon topographic steering in rapidly rotating, stratified flows are considered. The vorticity equation describing the topographic interaction is derived for a stable density profile of arbitrary form, and numerical solutions, valid under the condition Ε≫E1/2, are obtained for shallow obstacles of parabolic shape, over a range of values of the (parabolic) density profile curvature, the bulk stratification, and the obstacle height, taken here to be O(Ε). (Here, Ε and E are the Rossby and Ekman numbers, respectively, of the flow.) The results (i) demonstrate that for linear and nonlinear density profiles there are vertical attenuations of the streamline deflections above the obstacle, and (ii) confirm the dependence of this attenuation upon the bulk stratification S in the fluid. It is found that near the top of the obstacle the streamline curvature increases as the value of S increases. Furthermore, it is shown that for a given value of S, the vertical extent of the topographically induced disturbance field is reduced as the degree of nonlinearity in the density profile is increased. The decay of vorticity with height above the obstacle is also shown to be very sensitive to the degree of nonlinearity, with nonlinear density gradients exhibiting much larger values of surface vorticity than linear gradient counterparts, under otherwise identical conditions. Good qualitative agreement is found with results of previous laboratory experiments by Davies and Rahm [Phys. Fluids 25, 1931 (1982)]. The consequences of neglecting Ekman suction effects in the analysis are discussed with regard to the boundary layer structures on the solid surfaces; inclusion of Ekman suction is shown to result in the occurrence of a wake in the nonlinear problem.