Kelvin wave propagation along straight boundaries in C-grid finite-difference models
Discrete solutions for the propagation of coastally-trapped Kelvin waves are studied, using a second-order finite-difference staggered grid formulation that is widely used in geophysical fluid dynamics (the Arakawa C-grid). The fundamental problem of linear, inviscid wave propagation along a straight coastline is examined, in a fluid of constant depth with uniform background rotation, using the shallow-water equations which model either barotropic (surface) or baroclinic (internal) Kelvin waves. When the coast is aligned with the grid, it is shown analytically that the Kelvin wave speed and horizontal structure are recovered to second-order in grid spacing h. When the coast is aligned at 45 to the grid, with the coastline approximated as a staircase following the grid, it is shown analytically that the wave speed is only recovered to first-order in h, and that the horizontal structure of the wave is infected by a thin computational boundary layer at the coastline. It is shown numerically that such first-order convergence in h is attained for all other orientations of the grid and coastline, even when the two are almost aligned so that only occasional steps are present in the numerical coastline. Such first-order convergence, despite the second-order finite differences used in the ocean interior, could degrade the accuracy of numerical simulations of dynamical phenomena in which Kelvin waves play an important role. The degradation is shown to be particularly severe for a simple example of near-resonantly forced Kelvin waves in a channel, when the energy of the forced response can be incorrect by a factor of 2 or more, even with 25 grid points per wavelength.
wave boundary grid finite-difference convergence resolution