Tidal conversion at a submarine ridge

The radiative flux of internal wave energy (the tidal conversion) powered by the oscillating flow of a uniformly stratified fluid over a two-dimensional submarine ridge is computed using an integral-equation method. The problem is characterized by two nondimensional parameters, A and B. The first parameter, A, is the ridge half-width scaled by mu h, where h is the uniform depth of the ocean far from the ridge and mu is the inverse slope of internal tidal rays (horizontal run over vertical rise). The second parameter, B, is the ridge height scaled by h. Two topographic profiles are considered: a triangular or tent-shaped ridge and a polynomial ridge with continuous topographic slope. For both profiles, complete coverage of the (A, B) parameter space is obtained by reducing the problem to an integral equation, which is then discretized and solved numerically. It is shown that in the supercritical regime (ray slopes steeper than topographic slopes) the radiated power increases monotonically with B and decreases monotonically with A. In the subcritical regime the radiated power has a complicated and nonmonotonic dependence on these parameters. As A -> 0 recent results are recovered for the tidal conversion produced by a knife-edge barrier. It is shown analytically that the A -> 0 limit is regular: if A << 1 the reduction in tidal conversion below that at A = 0 is proportional to A(2). Further, the knife-edge model is shown to be indicative of both conversion rates and the structure of the radiated wave field over a broad region of the supercritical parameter space. As A increases the topographic slopes become gentler, and at a certain value of A the ridge becomes critical; that is, there is a single point on the flanks at which the topographic slope is equal to the slope of an internal tidal beam. The conversion decreases continuously as A increases through this transition. Visualization of the disturbed buoyancy field shows prominent singular lines (tidal beams). In the case of a triangular ridge these beams originate at the crest of the triangle. In the case of a supercritical polynomial ridge, the beams originate at the shallowest point on the flank at which the topographic slope equals the ray slope.