Numerical calculation of coastal trapped wave modes

A numerical method is developed to calculate coastal trapped wave (CTW) modes in the low-frequency limit ω ≪ f. The modes are solutions to a 2-dimensional eigenvalue problem. Useful properties like orthogonality are derived from the bilinear form associated with the operator of the eigenvalue problem. Our formulation uses the z coordinates and the CTW equation is discretized with a finite-difference method. We transform the set of the finite-difference equations in such a way that it has an analogous bilinear form and hence has an exact orthogonality condition and a few exact properties that the original differential equation has. By casting the set into a matrix form, we further prove that there are exactly as many modes as there are gridpoints in the vertical. We also prove another property that makes it trivial to filter out unphysical solutions. With our code, which have been made publicly available, all the numerical CTW modes are obtained at once for a given N(z) and bottom topography z = -h(x). By no means does our code supersedes Brin & Chapman's (1987), however, the latter being much more versatile.