On the nonlinear reflexion of a gravity wave at a critical level. Part 1

Abstract

In this paper we examine the nonlinear interaction of a forced internal gravity wave in a stratified fluid with its critical level. The representative Richardson number J is taken to be large and the undisturbed state consists of a hyperbolic-tangent velocity profile and an almost constant density gradient. It is assumed that at large values of a non-dimensional time t the flow outside the critical layer is steady, consisting of the mean shear together with a disturbance periodic in x that corresponds to the single harmonic of the incident wave of small amplitude ε. The requirements of a match across the critical layer lead to a reflected wave and a transmitted wave both of whose amplitudes are O(εe^−νπ) when 1 [Lt ] t [Lt ] ε^−2/3, where ν = (J − ¼)^½. For ν [Gt ] 1 the layer therefore acts as a wave absorber, and the purpose of this investigation is to ascertain whether this property persists on an even longer time scale. At times t = O(ε^−2/3) the layer has thickness O(ε^2/3) and the first few terms of an expansion in powers of ε^2/3t show that higher harmonics are forced on the outer flow, and the reflexion and transmission coefficients develop with time. The leading-order correction to these coefficients is calculated explicitly; that to the transmission coefficient is again exponentially small in ν though that to the reflexion coefficient is O(ν⁻¹). The reflexion coefficient is therefore increasing and the critical layer begins to restore wave energy to the outer flow. Owing to the complexity of the calculation higher-order corrections are not obtained here, but the results presented are in agreement with predictions of earlier workers that the layer acts as an absorber and a reflector but not as a transmitter.