Jozef VIVODA
The NH version of ALADIN is based on semi-implicit scheme, which is developed with both the Eulerian and semi-Lagrangian advection treatment. Two versions of leap-frog schemes (the semi-implicit Eulerian scheme and the three-time level semi-Lagrangian scheme (3TL SISL) ) and two-time level semi-implicit semi-Lagrangian scheme (2TL SISL) are implemented. The NH version of ALADIN is based on fully compressible equations and therefore sound waves are present in the solution. The semi-implicit scheme is supposed to also control sufficiently their evolution to allow the usage of the same length of time steps like in hydrostatic primitive equation (HPE) version of ALADIN. The length of time step should be restricted by maximum wind speed rather that by maximum phase speed, if semi-implicit scheme is applied. It was found in the past, that a traditional semi-implicit scheme does not control all the sources of sound waves. It was detected experimentally that semi-implicit treatment must be applied also on the nonlinear residuals of three-dimensional divergence (D3). Due to non-linear character of D3, the additional semi-implicit treatment was performed by simplified predictor corrector algorithm and iteration of Helmholtz solver. This appeared to be sufficient to integrate leap-frog schemes, but 2TL SISL scheme remained unstable, probably due to additional sources of instability. Idealized experiments (for example so-called SCANIA) showed recently that even three-time level schemes are not enough stable for their potential operational exploitation. The present work is concentrated on the stability problems in 2TL SLSI. The linear analysis of stability showed that the instability could be cured by a properly designed predictor-corrector scheme. The linear analysis of stability has been performed for the leap-frog schemes as well.
Linear stability analysis is based on Von Neumann method. The single wave component is analyzed to see growth rate of amplitude and dispersion of phase speed of that particular wave. Linear analysis of stability can be done for a linear part of the model only and due to this fact the equations system has to be linearized around generally chosen average state X' . In general, X' is different from the semi-implicit background state X* used for the semi-implicit linear system. There are two ways how to derive the linear model:
The second approach must be used for linear stability analysis. If an instability exists associated with purely nonlinear residuals, it is possible to study it experimentally only, since the linear stability analysis examines the residuals due to difference between X' and X*. The instability caused by these linear residuals is called SHB-type of instability. We assume that any nonlinear instability will be controlled by the chosen algorithm. The stability analysis of semi-implicit (SI) algorithm shows that:
model is unconditionally unstable and therefore PC algorithm has to be implemented
contrary to the HPE case, there is different behavior of thermal SHB instability. It is not enough for the stability that T* is chosen higher than a maximum temperature in the domain (usually 300K for 3TL and 350K for 2TL). The stability depends on magnitude of difference between T* and actual temperature T'. Therefore implementation of non-isothermal SI background should be considered.
in principle, when using a sigma coordinate, the instability associated with difference between p's and p*s should not exist. However, there is an additional dependency of linear residual on p*s in the system. The existing dependence causes instability in points where actual surface pressure p's is different from p*s . This is valid mainly for 2TL SISL scheme when instability is very strong and we believe that this is a source of instability that remains uncontrolled.
The thermal problem was studied with perfect pressure (p's equals to p's) and pressure problem was studied with perfect temperature (T* equals to T'). To solve the thermal problem, non-isothermal SI background was studied and it was found that:
the elimination of the linear system to one single equation for one variable is not possible. It is possible to eliminate the system until two equations for two variables. So instead of inversion of matrices with rank L we have to solve inversion of matrices with rank 2*L (here L is number of vertical levels) for each wave component. This only requires more memory but not performance, since it is done once (resp. two times) at the beginning of integration.
a new constrain (analogous to C1 in Bubnova,1995) was found. A constrain analogous to C2 does not need to be taken into account because C2 was related to the elimination for one variable which is not possible in this case.
There are new unknowns in discretized vertical operators fulfilling new C1, which cannot be expressed as simple relations of p*s and therefore approximations have to be done. Existing solution (NDLNPR=1) is a good approximation.
To solve the pressure problem, we have to redefine model equations. New prognostic variable independent of SI pressure has to be used instead of rescaled pressure departure. Following new variables were proposed:
Here p is true pressure and p is hydrostatic pressure. The stability analysis shows that using these two new variables solves pressure problem and only the thermal problem remains for sigma coordinate. We also analyzed the stability of PC algorithm. The traditional semi-implicit time step is used as a predictor. Corrector is constructed as an iteration of whole nonlinear model. It shows that:
one iteration is not enough to stabilize the model sufficiently with existing model formulation. There is a need to study PC algorithm with new model formulations.
It was found (by Pierre Benard) during the stability analysis, that existing discretization of pressure gradient force, supposed to conserve angular momentum, is also a source of instability, since it contains averaging of rescaled pressure departure gradient. It has no equivalent in linear SI model and therefore this part of discretization is not captured by SI control. There are two solutions to this problem:
to remove averaging in discretization of pressure gradient force (Benard proposal October 2000),
to implement the averaging operator into SI model (then elimination has to be rechecked).
New simplified 2d model ALADIN was written (only with Eulerian advection) to avoid complex data flow in ALADIN model. Dynamical core of the model is taken from ALADIN only the control routines are new. The aim of this development was to have available something easier for quick tests of ideas.
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