Internal tide generation at the continental shelf modelled using a modal decomposition: two-dimensional results

Stratified flow over topography is studied, with oceanic applications in mind. We develop a model for a fluid with arbitrary vertical stratification and a free surface, flowing over threedimensional 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, we calculate the internal tide generated by a time-periodic barotropic tidal flow over a continental shelf and slope, in various idealised configurations. We take the topography and fluid motion to be independent of one coordinate direction, and the fluid to be either two-layer or uniformly stratified. For the two-layer case, we derive expressions for the shoreward and oceanward energy fluxes associated with the internal tide. For the uniformly stratified case, we study numerically how the accuracy of the solutions depends upon the number of basis functions used, and show that good solutions and energy flux estimates can often be obtained with only a few basis functions. In both cases our results show that the position of the coastline, through its effect on the form of the barotropic tide, significantly effects the strength of the internal tide generation.