Internal tide generation at the continental shelf modeled using a modal decomposition: Two-dimensional results

Stratified flow over topography is studied, with oceanic applications in mind. A model is developed for a fluid with arbitrary vertical stratification and a free surface, flowing over three-dimensional topography of arbitrary size and steepness, with background rotation, in the linear hydrostatic regime. The model uses an expansion of the flow fields in terms of a set of basis functions, which efficiently capture the vertical dependence of the flow. The horizontal structure may then be found by solving a set of coupled partial differential equations in two horizontal directions and time, subject to simple boundary conditions. In some cases, these equations may be solved analytically, but, in general, simple numerical procedures are required. Using this formulation, the internal tide generated by a time-periodic barotropic tidal flow over a continental shelf and slope is calculated in various idealized configurations. The topography and fluid motion are taken to be independent of one coordinate direction and the fluid to be either two-layer or uniformly stratified. For the two-layer case, expressions for the shoreward and oceanward energy fluxes associated with the internal tide are derived. For the uniformly stratified case, it is studied numerically how the accuracy of the solutions depends upon the number of basis functions used, and it is shown that good solutions and energy flux estimates can often be obtained with only a few basis functions. In both cases, the results show that the position of the coastline, through its effect on the form of the barotropic tide, significantly influences the strength of the internal tide generation.