Vergleich dreier Differenzen-Verfahren zur numerischen Integration der ,,primitiven Gleichungen“ für ein barotropes divergentes Modell der Atmosphäre

Abstract

Integrations of the hydrostatic system of equations for the divergent barotropic model of an atmosphere confined within latitudinal walls are performed with three finitedifference schemes of second-order accuracy. Scheme I is the so-called leap-frog scheme, commonly used in numerical forecasting, scheme II is based on a method proposed by Lax & Wendroff and N. A. Phillips. Scheme III was discussed by one of the authors. Applying for all schemes the same grid system with respect to the field- and timecoordinates and allowing for no smoothing or friction it is possible to compare objectively the behaviour of the difference equations. Initially a geostrophically balanced jet stream field consisting of three zonal wave numbers is prescribed; the flow is barotropically unstable.

The comparison is made with integrals of (?) eddy kinetic energy, including it's spectral parts, (b) zonal kinetic energy, (c) kinetic energy, and (d) total energy. The latter is a conservative property for the differential system. These integrals yield a good insight into the appearance of barotropic instability. This problem will be treated in a subsequent paper. All three methods give practically equal results within the first week of forecasting time; then scheme I begins to show numerical instability followed by scheme III about one week later. Scheme II seems to be completely stable, shows, however, as expected, a slight tendency of damping relatively short waves. Scheme III requires one field less to be stored in the computer than schemes I and II.