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

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 regime epsilon = alpha/h much less than 1, mu = h/l much less than 1 (where h is the channel depth and alpha,l are the peak amplitude and the width of the obstacle respectively), the parameter A = epsilon/mu(p) (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 = 0(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 mu --> 0) lee-wave amplitude. Comparison with numerical results indicates that the asymptotic theory often remains reasonably accurate even for moderately small values of mu and epsilon 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 much less than 1) is rather limited.