On the nonlinear reflection of a gravity wave at a critical level. Part 2 - Brown, S. N.; Stewartson, K. (Journal of fluid mechanics, 1982)
In part 1 of this work (Brown & Stewartson 1980b) we examined the nonlinear interaction of a forced internal gravity wave in a stratified fluid with its critical level. Although the Richardson number J was taken to be large, the method described there was, in principle, applicable to all Richardson numbers and as such we did not take advantage of the asymptotic properties of the solution of the linearized equations. Here in part 2 we re-develop the linearized solution for a general basic shear and temperature profile when J [Gt ] 1 as the large-time limit of an initial-value problem for a wave incident from above the shear layer. On this time scale it is known that the reflection and transmission coefficients are 0(e−νπ), ν = (J −¼)½. It is shown that, when J [Gt ] 1, the solution in the neighbourhood of the critical layer consists only of algebraically decaying elements with a direction of propagation parallel to the layer (critical-level noise) below a certain level, but of critical-level noise and a wavelike term, corresponding to the imposed incident wave, above this level. On a longer time scale, specifically $t = 0(ε^{-\frac{2}{3}})$, where ε is the amplitude of the forced wave, the nonlinear terms are no longer negligible; the development of the reflection and transmission coefficients on this time scale is the subject of part 3 (Brown & Stewartson 1982).