Inviscid critical and near-critical reflection of internal waves in the time domain

Abstract

A solution that describes the inviscid reflection of internal waves off a sloping bottom in time is derived under conditions of linearity and uniform stratification. The solution is well behaved even under critical conditions. In the region ky < Nt, where k is the along-slope wavenumber of the incoming wave, y is the slope-normal direction, N is the Brünt–Väisälä frequency and t is time, an approximation can be written in terms of Lommel's function of two variables. The analysis can easily be extended to the case of a beam of finite width. In the non-critical case, the streamfunction relaxes to the classical Phillips steady-state solution in the region y < cgt, where cg is the slope-normal component of the group velocity for waves at the forcing frequency. However, it is found that the region where the along-boundary component of the velocity relaxes to the Phillips solution is also bounded from below, leaving a region very close to the wall where the classical solution misses important elements of the reflectionprocess. This leads to interesting properties near the boundary, especially relatively to the formation of shear-driven unstable conditions.