Numerical simulations of the flow of a continuously-stratified inertial effects

A high-resolution spectral numerical scheme is developed to solve the two-dimensional equations of motion for the ow of a density stratified, incompressible and inviscid fluid. It incorporates the inertial terms ne- glected in the Boussinesq approximation. Thus it aims, inter alia , to extend the numerical simulations of Rottman et. al. [12] and Aigner et. al. [1].To test its validity, the code is used for two applications. One is the resonant flow over isolated bottom topography in a channel of finite depth, which has been studied extensively in the Boussinesq approximation. The inclusion of inertial effects, that is the influence of the stratification on the acceleration terms, discarded in the Boussinesq approximation, allows the comparison of the solution to the unsteady governing equations with the fully nonlinear, but weakly dispersive resonant theory of Grimshaw and Yi [7]. The focus is on topography of small to moderate amplitude and slope, and for conditions such that the flow is close to linear resonance for either of the first two internal wave modes. We also determine the ver- tical position where wave-breaking occurs. The other application is the propagation of large-amplitude internal solitary waves with vortex cores, again in a channel of finite depth. We aim to verify the existence and permanence of these types of waves derived by Derzho and Grimshaw [5]. Furthermore the time-dependent solution provides measurements of the structure of the vortex core and maximum adverse velocity at the top boundary.