5:15 PM - 7:15 PM
[ACG47-P06] Coastal trapped wave modes revisited
Keywords:coastal trapped wave modes, numerical calculation, eigenvalue-dependent boundary value problem, dispersion relation
A low-frequency (ω << f) "coastal trapped wave" (CTW) is an extension to "shelf wave" with stratification or a coastal Kelvin wave with bottom slope. When both stratification N(z) and the bottom slope h(x) are involved, the x-z structure of the wave is not separable, and one needs to solve a 2D eigenvalue problem that involves N(z) and h(x).
The well-known Fortran program developed by Brink and Chapman (1987) uses the sigma coordinates (σ = z/h(x)) and solves a discretized version of the equations and boundary conditions for one eigenpair at a time by a search method starting from an initial guess of the eigenvalue (characteristic wave speed c_n). Today our PCs are powerful enough to directly solve the entire matrix eigenvalue problem to obtain all the eigenpairs at once (Tanaka 2023). Orthogonality of the eigenvectors, however, doesn't exactly hold and some eigenvectors are unphysical with imaginary eigenvalues.
In the present study, we develop a simpler numerical method that uses the z coordinates, with which we can prove that all eigenvectors are orthogonal and that the eigenvalues are real and have the right sign. We discuss mathematical properties of the original continuous eigenvalue problem and the discretized version that lead to these results.
We also show a preliminary comparison of our numerical solutions to Rhines's (1970, Geophys Fluid Dyn) dispersion relation, which as far as we know hasn't been done yet.
The well-known Fortran program developed by Brink and Chapman (1987) uses the sigma coordinates (σ = z/h(x)) and solves a discretized version of the equations and boundary conditions for one eigenpair at a time by a search method starting from an initial guess of the eigenvalue (characteristic wave speed c_n). Today our PCs are powerful enough to directly solve the entire matrix eigenvalue problem to obtain all the eigenpairs at once (Tanaka 2023). Orthogonality of the eigenvectors, however, doesn't exactly hold and some eigenvectors are unphysical with imaginary eigenvalues.
In the present study, we develop a simpler numerical method that uses the z coordinates, with which we can prove that all eigenvectors are orthogonal and that the eigenvalues are real and have the right sign. We discuss mathematical properties of the original continuous eigenvalue problem and the discretized version that lead to these results.
We also show a preliminary comparison of our numerical solutions to Rhines's (1970, Geophys Fluid Dyn) dispersion relation, which as far as we know hasn't been done yet.