Spectral methods for coastal-trapped waves

This note describes efficient and accurate spectral numerical schemes to compute both propagating and evanescent free baroclinic coastal-trapped waves over general depth profiles for arbitrary vertical density profiles in horizontally semi-infinite domains. The general problem is recast into a linear eigenvalue problem for the along-shore wavenumber k, which can be solved directly, without initial guesses or searching, using any standard linear eigenvalue package to find real and complex eigenmodes simultaneously. An equivalent recasting gives the linearised eigenvalue problem for the frequency ω . A novel, nonlinear, boundary condition is derived that is particularly effective for modes whose offshore decay is weak, as in the long-wave limit. The resulting nonlinear eigenvalue problem is solved by a highly efficient Newton–Kantorovich algorithm.