Large-amplitude topographic waves in 2D stratified flow

Abstract

Our fundamental understanding of steady, stratified flow over two-dimensional (2D) topography rests on the pioneering works of G. Lyra and R. Long. Within linear theory, Lyra established the far-field radiation conditions that determine the downstream pattern of buoyancy waves. Soon after, Long discovered that the steady, nonlinear streamfunction for special cases of stratified, 2D flow could satisfy the same equations as linear theory, subject to an exact topographic boundary condition. Fourier methods are currently used to compute solutions to Long's theory for arbitrary topography in the near-hydrostatic or small-amplitude topographic parameter regimes. It is not generally appreciated however, that these methods encounter difficulties for flows that are both strongly non-hydrostatic and beyond linear amplitudes. By recasting Long's theory into a linear integral equation, this difficulty is shown to be a computational barrier associated with an ill-conditioning of the Fourier method. The problem is overcome through the development of a boundary integral computation which relies on some lesser known solutions from Lyra's original analysis. This method is well-conditioned for strongly non-hydrostatic flows, and is used to extend the exploration of critical overturning flows over Gaussian and bell-shaped ridges. These results indicate that the critical value of the non-dimensional height () asymptotes to a finite value with increasing non-hydrostatic parameter ().