The Interaction of Model Dynamics and Numerical Errors in a Nondivergent Global Barotropic Model

Abstract

The nondivergent barotropic vorticity equation on a sphere supplies the framework for a series Of experiments designed to study the suitability of numerical methods commonly used in the modeling and prediction of the atmosphere. Specifically, making use of idealized initial conditions and a scale-dependent spatial filter we investigate the capabilities and limitations of a typical second-order finite-difference approximation in numerical time integrations. From comparisons with analytic solutions, it is found that for dynamically stable flows numerical methods such as second-order finite-difference approximations, together with a Shapiro-type filter, are adequate in yielding approximate solutions to the modeling differential equations. For dynamically unstable flows, numerical errors are amplified as part of the dynamics of the unstable system. The use of finite-difference approximations may yield solutions which bear no resemblance whatsoever to the true solution of the differential equations in spite of the maintenance of computational stability. It is postulated that interactions among long-wave computational modes and physical modes in a numerical model may prove to be another major obstacle in numerical prediction of an unstable flow.