Near-critical reflection of internal waves

Using a matched asymptotic expansion we analyse the two-dimensional, near-c ritical reflection of a weakly nonlinear internal gravity wave from a slopi ng boundary in a uniformly stratified fluid. Taking a distinguished limit i n which the amplitude of the incident wave, the dissipation, and the depart ure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience he als the singularity which is presented by the solution of Phillips (1966) i n the precisely critical case. In the inviscid critical case, an explicit s olution of the initial value problem shows that the buoyancy perturbation a nd the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this sca le reduction, the stratification is 'overturned' and the Miles-Howard condi tion for stratified shear flow stability is violated. However, for all slop e angles, the 'overturning' occurs before the Miles-Howard stability condit ion is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualiz ations of the reflection process. In particular, the buoyancy field compute d from the analytic solution is in good agreement with visualizations repor ted by Thorpe & Haines (1987). One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansio n all of the apparently resonant nonlinear contributions cancel amongst the mselves). However, the reconstructed fields do contain nonlinearly driven s econd harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.