Topographic control of stratified flows: Blocking, upstream jets, isolating layers and downslope flows

We consider stratified flows over isolated obstacles in the limit of strong stratification and tall obstacles for which the total kinetic energy of the flow is insufficient to draw all of the fluid up and over the obstacle crest and so the fluid below some height is arrested or blocked. Optimal solutions to the nonlinear, hydrostatic, Boussinesq equations are developed for steady, density-stratified, topographically controlled flows characterized by blocking and upstream influence. These flows are jet-like upstream and are asymmetric and thinning as they accelerate over the crest. A stagnant, uniform-density isolating layer, surrounded by a bifurcated uppermost streamline, separates the accelerated flow from an uncoupled flow above. The flows are optimal in the sense that, for a given stratification, the solutions maximize the topographic rise above the blocking level required for hydraulic control while minimizing the total energy of the flow. Hydraulic control is defined mathematically by the asymmetry of the accelerated flow as it passes the crest. A subsequent analysis of the Taylor-Goldstein equation shows that these sheared, non-uniformly stratified flows are indeed subcritical upstream, critical at the crest, and supercritical in the downslope flow with respect to gravest-mode, long internal waves. The solutions are not just relevant to atmospheric flows over mountains but also arrested wedge flows, selective withdrawal, stratified towing experiments and oceanic flows over topography.