A numerical solution to the non-linear problem of steady eastward barotropic flow past an island on the beta plane

Abstract

Uniform barotropic flow past an island on the beta plane is considered. The equations of motion contain terms for horizontal diffusion of momentum, the beta effect, and non-linear advection of momentum. A vorticity equation is formed and put into finite difference form utilizing centered differences in space and time. The DuFort-Frankel scheme is used for the lateral diffusion of vorticity terms, and the resulting system of finite difference equations are solved using the Gauss-Seidel method with successive over-relaxation. A condition that the island boundary be a streamline is imposed as well as the requirement that there be no tangential velocity component on the island boundary. Three cases are considered; the generation of von Karman wake, the generation of Rossby wake and the combination of the two. In the first case, the characteristics of the von Karman wake over a range of Reynolds numbers are found to agree with those available from theory and experimental results. In the second case, the Rossby wake is found to exist over a range of Island numbers from 0.4 to 3.5. Increasing the Island number increases the amplitude and decreases the wavelength of the stationary wave. A combination of both wakes is found for a low Island number (0.4) and a high Reynolds number (480). Other combinations yield a Rossby wake damped by frictional effects.