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The generation of internal tides over steep continental slopes

A numerical procedure is presented for the calculation of internal tides generated by the interaction of surface tide with bottom topography which is tangent to the direction of internal tidal energy propagation at some depth. This procedure, together with that of Baines (1973), permits the calculation of internal tides generated by (virtually) arbitrary topography with horizontal scale greater than 1 km, and a wide range of realistic density stratifications. The procedure is applied to continental slopes with simple linear and quartercircle profiles, and constant stratification. For these cases, the largest internal tidal velocities and energy densities occur in regions around characteristics emanating from the tangential corner point; on the shallow shelf side the energy flux is a maximum in this region, but on the deep side it is a minimum and is distributed more evenly with depth. The total energy flux is greater than the maximum for flat-bump topography of comparable height by a factor of order 2-3. It increases nearly exponentially with height but is less sensitive to shape provided the slope is greater than critical, and is greater on the deep than on the shallow side by a factor of order 10. Calculations for more realistic density stratifications yield similar results. The procedure is also applied to a real continental slope for which observations have been made by Wunsch & Hendry (1972), with stratification representing summer and winter conditions. The velocity fields and associated energy fluxes differ significantly from those of simple geometries, and are also sensitive to the seasonal density changes in the upper 50 m. It is suggested that internal tidal generation will give rise to two mixing processes, one associated with the boundary layer near the tangent point and the other with shear instability in the velocity profile. Instability of the theoretical profiles according to the Richardson number criterion may be readily achieved in oceanic conditions. The reflexion of an internal wave from a concave corner is discussed in an appendix, where it is shown that no singularities occur unless the radius of curvature is very large.