Finite Difference Approximation for the Barotropic Instability Problem

Abstract

The validitiy of finite difference approximations in solving the dynamic instability problem of non-divergent barotropic currents is examined numerically.
We first study the instability of a sine-curve symmetric zonal current confined between two rigid walls. The relation between the solutions of the finite difference equations and those of the original differential equation is discussed. Then the accuracy of the finite difference method in describing the instability is examined by varying the number of subdivisions. For a sufficiently accurate description of the instability, a large number of subdivisions, at least 20, are required for this velocity profile.
We then study the instability of a symmetric and an antisymmertic zonal currents extending to infinity. The effect of rigid boundaries placed at a finite distance from the central shearing wind belt on the critical wavelengths is examined by varying the position of the boundaries. For both velocity profiles, it is shown that the exact boundary conditions at infinity can be replaced by rigid boundary conditions at a distance equal to the half width of the shearing wind belt.