Dispersive corrections to a modulation theory for stratified gravity waves

Abstract

Modulation theories, as used to describe the propagation of wavetrains, often possess a natural limitation in their tendency to finite-time breakdown via a wavenumber shock that occurs with the crossing of characteristics. For the geophysical example of linear wave propagation in an inviscid, density-stratified fluid, we demonstrate that the asymptotic corrections to modulation theory are sufficient for short-time regularization of the wavebreaking singularity. Computations show that the shock development is pre-empted by the emergence of spatial oscillations. In the asymptotic limit, these oscillations display a self-similar collapse of scale both in space and in time. Finally, an envelope analysis following a characteristic demonstrates that these new correction terms act locally as second-order wave dispersion.