Alternative implementations of the semi-Lagrangian semi-implicit schemes in the ECMWF model



An important issue for the future development of global models is the degree of implicit solution required for optimum performance, and whether this will change the relative cost-effectiveness of spectral and finite-difference models. This is explored by using a predictor-corrector integration scheme in the European Centre for Medium-Range Weather Forecasts model, where both steps are integrated using the current semi-implicit procedure. This is equivalent to taking a first iteration towards a fully implicit scheme, using the existing semi-implicit method as a preconditioner. The results show that benefit can be obtained provided that both the nonlinear part of the gravity-wave terms and the semi-Lagrangian trajectory are recalculated. This suggests that more fully implicit schemes will be beneficial in numerical models. However, the cost of elliptic solvers will not dominate as significant recalculation of tendencies will be required at each iteration.

Another strategic issue is the robustness of semi-Lagrangian schemes at high resolution. These are the only advection schemes that preserve monotonicity that are available for spectral models. As resolution is increased, solutions of the equations always contain motions with the wind direction varying rapidly along the trajectory. These are typically not well treated at current operational resolutions, but limit the time step that can be used with semi-Lagrangian schemes. We illustrate the effect of selective damping of these motions, which may be desirable until much higher resolutions are reached.