The Free Kelvin Wave in Finite-Difference Numerical Models

Abstract

The effects of viscosity and finite- differencing on free Kelvin waves in numerical models (which employ the Arakawa B- or C-grid difference schemes) are investigated using the f-plane shallow-water equations with offshore finite-difference grids, (assuming alongshore geostrophy). Three nondimensional parameters arise: Δ [=(offshore grid spacing)/(Rossby radius)], ϵ characterizes the offshore lateral viscous effect and α the combined vertical and alongshore viscous effect. This study is more relevant to baroclinic Kelvin waves which tend to suffer poor offshore resolution because of their small Rossby radii. For inviscid models (ϵ = α = 0), as Δ increases (resolution worsens), the alongshore speed increases dramatically in the B-grid, but stays constant at the gravity wave speed in the C-grid. Models with damping only (α > 0, ϵ = 0) behave similarly. With lateral viscosity (ϵ > 0, α > 0), increasing ϵ decreases the speed in both the B- and C-grids—the drop in speed being less severe when the free-slip boundary condition is imposed instead of the no-slip one. As Δ increases, the speed declines in the B-grid, but in the C-grid, worsening resolution cancels the viscous slow-down, with speed rising to that when ϵ = 0. Our theory predicts the alongshore phase speed, the temporal decay rate and the offshore structure for B- and C-grid models of given viscosity and grid-spacing and of given boundary conditions (e.g., no-slip or free-slip). The predictions are checked against observations from two- and three-dimensional model—including the Bryan-Cox model—with good agreement.