A stability analysis of a family of baroclinic semi-Lagrangian forecast models

Abstract

There are three sets of physical modes, namely the usual gravity and slow ("Rossby') modes, corresponding to the three solutions of a third-order (in time) normal-mode differential equation. For one-, two-, and three-term extrapolation of quantities, there are also zero, one, and two computational modes, respectively, since the normal-mode difference equation is then of higher order than third. The following conclusions hold equally well for both the Bates et al. and McDonald and Haugen model formulations, which although different in detail behave very similarly. The slow mode is stable and slightly damped (by interpolation) for all schemes, and the gravity modes are unconditionally stable in the absence of extrapolated terms. When the extrapolated terms are included, however, the gravity modes become unstable in the absence of damping mechanisms. Introducing both divergence damping and a time decentering of the scheme (with a judicious choice of coefficients) stabilizes these modes. -from Authors