WAVE INTERACTIONS IN STRATIFIED SHEAR FLOWS

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Resonant interactions between triads of internal gravity waves propagating in a stratified shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If wn, κ˜n (n = 1,2,3) are the local frequencies and wavenumbers respectively then the resonance conditions are that w1 + w2 + w3 = 0, κ˜1 + κ˜2 + κ˜3 = 0. If the medium is only weakly inhomogeneous, to leading order the resonance conditions are satisfied globally, and the equations governing the wave amplitudes are well known and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. Near each such resonance surface the wave amplitudes An (n=1,2,3) satisfy equations of theβr∂Ar∂τ=Ap*Aq* exp (-12iSτ2)where {r, p, q} is a cycle permutation of {1,2,3}, τ is a co-ordinate normal to the resonance surface, and βr is a real interaction coefficient. Numerical and analytical solutions of the interaction equations are obtained which demonstrate that the interactions are either explosive, when all modes generally grow as the resonance surface is approached, or the interaction is contained, when the modes exchange action during the interaction, depending on whether or not the coefficients βr (r = 1,2,3) all have the same sign, or not. The explosive case is either real or apparent depending on whether the approach to the resonance surface can be linked to a well-posed initial condition, or not. For the weak background shear flows considered here all the interactions are either only apparently explosive, or contained. The results are applied to the hierarchy of wave interactions which can occur near a critical layer, with the aim of determining to what extent a critical layer can reflect, or transmit, wave energy.