Effect of stair-step and piecewise linear topography on internal wave propagation in a geophysical flow model

The paper describes the effect of a stair-step representation of bottom topography, common in most numerical models, on the propagation of internal waves in a geophysical flow. An analytical solution was used to predict the effect of stair-step and piecewise linear representations of topography on the reflection and transmission of internal waves. The Estuary Lake and Coastal Model (ELCOM) was used to simulate the transmission of a train of small-amplitude internal gravity waves propagated along a variable depth rectangular channel containing a linearly stratified fluid. The effect of the different obstacle approximations was examined in terms of the interference of the resulting wave number content, and hence the associated reflection coefficients, which could be directly compared to the analytical theory. It was shown that for Gaussian obstacles of varying lengths and heights, the full bottom cells and partial bottom cells solutions from ELCOM agree with the stair-step and piecewise linear solutions from the theory, respectively. It was shown that when Gaussian obstacles with weak slopes are represented by stair steps, an aliasing that is similar to that in discrete Fourier transform can occur, resulting in reflection coefficients that are much higher than the exact solution. On the other hand, piecewise linear approximations produce significantly attenuated aliasing. An immersed boundary technique that enforces a no-flux condition at the actual sloping bottom, similar to that of the analytical solution, is found not to produce better numerical results.