A Stability Analysis of a Family of Baroclinic Semi-Lagrangian Forecast Models

Abstract

The computational stability of a family of recently introduced semi-Lagrangian schemes for baroclinic models is analyzed to better explain their observed behavior and to provide additional theoretical justification. The linear stability analysis is a generalization of that presented in Bates et al. that includes the important impact of evaluating certain (nonlinear) terms using extrapolated quantities. 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, both with and without extrapolated terms, 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. The time decentering is the more efficient of these two damping mechanisms, and the values of the coefficients required for computational stability as determined from our analysis agree well with those determined empirically in the Bates et al. and McDonald and Haugen studies. Two-term extrapolation is to be preferred to both one- and three-term extrapolation, since the former is insufficiently accurate, whereas the latter requires unacceptably large damping coefficients.