Time integration

For each spectral mode n m in the final model equations can formally be written as


For n>0 the tendency vector Tnm can be expanded as




Here, T’nm represents all spectral tendencies owing to Coriolis force, nonlinearities, diffusion, gravity waves, and diabatic heating. The Matrix An describes the buoyancy oscillations of a horizontally uniform reference state, i.e. the internal gravity waves in a corresponding linearized model version without Coriolis force and without all so-called physical parameterizations. These gravity modes separate in spectral space and degenerate with respect to the zonal wave number m. An is zero up to the coupling between  (D1 n m,... ‚Dlev n m) and (T1 n m,... ‚Tlev n m , ps n m) The matrixes a1 and a2 depend on the reference state Tref(p) and on the distribution of model layers only (see Simmons and Burridge, 1981, appendix). In KMCM, we use the reference state shown in Fig. 2 (a). In ECHAM, an isothermal reference state is employed.

Following Hoskins and Simmons (1975), time stepping is performed using the semi- implicit leapfrog scheme. This scheme must be completed by a time filter (Asselin 1972) in order to damp the computational modes. This so-called semi-implicit method gains its efficiency from ‘stretching‘ the oscillatory terms. This is achieved by application of the implicit leapfrog scheme to the linear terms and the explicit scheme to the remainder. Dropping the wave number indices n m, introducing the time index i, and denoting the time step by ∆t, we get


Solving (A65) for yi+1 and introducing the time filter yields


Here, the unit matrix is abbreviated as E, and δ is a filter parameter which is usually set equal to 0.1. The leapfrog scheme requires two foregoing time steps to perform the next. In KMCM, the first time step is computed from the initial condition ŷ0 by performing one Eulerian time step followed by five semi-implicit time steps using ∆t/5.
Application of the explicit leapfrog scheme to diffusion tendencies can cause numerical instabilities that require short time steps in order to be controlled by the time filter. On the other hand, numerical stability of diffusion tendencies would be guaranteed by using an implicit leapfrog time step for these terms. KMCM compromises between numerical efficiency and a comprehensible source code by applying an Eulerian step for time integration of all tendencies owing to diffusion, dissipation, and the IGW parameterization. In other words, in (A66), T’i contains diffusion, dissipation, and IGW tendencies from time i-1. The particular time steps used in the T42/L24 and T29/L60 model versions are 15 and 7.30 minutes. Alternatively to semi-implicit time stepping, the model can be integrated by explicit standard schemes like the Runge-Kutta or the Bulirsch-Stoer method for instance. However, these schemes require shorter time steps and are less efficient.