Finite-amplitude effects on steady lee-wave patterns in subcritical stratified flow over topography

Abstract

The flow of a continuously stratified fluid over a smooth bottom bump in a channel of finite depth is considered. In the weakly nonlinear-weakly dispersive régime ε = a/h [Lt ] 1, μ = h/l [Lt ] 1 (where h is the channel depth and a, l are the peak amplitude and the width of the obstacle respectively), the parameter A = ε/μp (where p < 0 depends on the obstacle shape) controls the effect of nonlinearity on the steady lee wavetrain that forms downstream of the obstacle for subcritical flow speeds. For A = O(1), when nonlinear and dispersive effects are equally important, the interaction of the long-wave disturbance over the obstacle with the lee wave is fully nonlinear, and techniques of asymptotics ‘beyond all orders’ are used to determine the (exponentially small as μ → 0) lee-wave amplitude. Comparison with numerical results indicates that the asymptotic theory often remains reasonably accurate even for moderately small values of μ and ε, in which case the (formally exponentially small) lee-wave amplitude is greatly enhanced by nonlinearity and can be quite substantial. Moreover, these findings reveal that the range of validity of the classical linear lee-wave theory (A [Lt ] 1) is rather limited.