Last updated: 2008
The Runge-Kutta (RK) method is implemented in COSMO as the basic time integration scheme and will be used for high resolution applications with COSMO and possibly replace the Klemp Wilhelmson (1978) scheme later on. It offers a substantial gain in accuracy at no additional costs. The method is quite new and differs from the RK scheme used in WRF, for example by having less conservation properties and a smaller approximation order in the vertical.
Further work is required to investigate the advantages (and possible problems to be solved) thoroughly and to investigate further developments, such as third (or even higher) order in the vertical and conservation properties. Within the WRF group possibilities of increasing the efficiency of RK are seen. Such developments should be followed. With an increased order a better interface to physics as well as the observation of other approximation conditions should become more essential. Interactions with fine scale orography should be investigated more thoroughly.
The aim of this project is to create an improved and properly checked version of RK in the timescale of 3 years. The work is divided into 14 tasks:
Up to now, COSMO verifications with 7 km resolution show a higher positive pressure bias for the RK core than for the Leapfrog core, whereas other variables show comparable behavior. This could give a hint of problems with the lateral boundary conditions. Other possible measurements could be the improvement of the lower boundary formulation for the pressure or a reformulation of the fast waves solver (these could also candidates to solve problems in task 6).
An extensive verification should accompany the other tasks.
A tool for inspection of the conservation properties of COSMO shall be developed with the possibility to set an arbitrarily chosen cubus (in the transformed grid, i.e. terrain-following; perhaps also in the Cartesian grid) into the model. Then integrals over the inner volume and integrals of surface fluxes shall be calculated.
A Courant-number independent advection algorithm for the moisture densities (therefore formulated as a conservation law) shall be developed analogous to the proposal of Skamarock (2005). The Bott (1989)-schemes could be taken as a basic advection scheme.
The aim is to determine the temporal and spatial order of convergence of the RK scheme in combination with advection schemes of higher order. Tests are done for linear mountain flows (analytic solution available), non-linear mountain flows (analytic solution only in special cases), non-linear mountain flow with precipitation.
This task depends also from task 8.
The reason for the occurrence of unrealistic "cold pools" in Alpine valleys has to be detected. In general the behavior of the dynamical core with steep slopes should be investigated.
Filtering of orography can to a certain extend avoid the "cold pool" -problem addressed in task 6. Beyond this it should be clarified what is the effective resolution of the model and what are the consequences for the choice of the filter scale for orography and also for other topographical features.
In meso-gamma models with the new feature of resolving convection a better representation of vertical processes seems to be important. The up to now used implicit centered difference scheme of 2 order could not longer be adequate for this task. Moreover there is a discrepancy between the horizontal and vertical order of the advection schemes (5th order to 2nd order).
Problems with reduced precipitation in meso-gamma simulations could be due to a non-adequate coupling between the physical processes and the dynamical core. This has especially to be obeyed for a relatively large time step of 30 sec. for 2.8 km resolution in relation of the development of convective cells and the micro physical changing rates. (Task 4 of the QPF-priority project will support this.)
Alternatively to the Wicker-Skamarock type of splitting, which is closely related to the Runge-Kutta method, another approach is possible: It consists of treating the full wave information including vertical advection of perturbation pressure and temperature within the fast-waves part and using the Runge-Kutta method only for horizontal advection as proposed in Gassmann (2005).
The program code is already available and first testing is under way by Almut Gassmann (Uni Bonn) and Gerd Vogel (DWD, Potsdam). Comparison with the other Runge-Kutta-methods over a wider range of cases or test suites is intended to highlight the advantages and disadvantages of different dynamical core formulations.
A first step is the implementation of mass conservation (rho' as prognostic variable; or, rho'-Theta'-dynamics). This is a very ambitious task, because a higher order discretization of a conservative scheme is unknown up to now.
Perhaps in combination with compact discretization.
Cases occurred, where the up to now used 'quasi-2D' divergence filtering lead to unstable results. But a complete abandoning of the divergence filtering (as proposed by A. Gassmann for her dynamical core) also leads to several instabilities. This was also shown by stability analyses of the RK-core by M. Baldauf. P. Prohl (DWD) could demonstrate, that the Bryan-Fritsch test case of a rising warm bubble is unstable with 'quasi-2D' divergence damping but becomes stable only with a full 3D (isotropic) version (realized with a preliminary explicit formulation). For operational use an implicit version of 3D divergence damping is necessary.
(this task was defined as a subtask in Task 10; but Task 10 concentrates more and more to the A. Gassmann dynamical core)
Bug of DFI has been repaired in the Leapfrog-version of the COSMO-model. This has still to be done in RK.