The improvement of the stability of the non-hydrostatic version of ALADIN

Pierre Bénard

A complete study to evaluate (and try to correct) the weaknesses of the non-hydrostatic (NH) version of ALADIN has started in spring 2000 and is still under progress. This study involves several actors and topics, and many results were obtained. Hence we present her a short synthesis on the main steps accomplished in this field for those who did not follow the whole story.

Beginning Spring 2000:

The starting point was the evaluation of the current state of the NH model. In typical NWP conditions, the model was found stable in Eulerian, but unstable in Leap-Frog (3-TL) SL when used with long time-steps as allowed by the SL technique. Moreover the model was unconditionally unstable when used with two time-level (2-TL) SL scheme (K. Yessad).

April 2000:

The essence of the instability was successfully reproduced in progressively more and more academic (hence simple) frameworks. The observed behaviour clearly pointed out a lack of robustness of the time-marching scheme for the free fast-propagating waves, when the complete nonlinear model (M) significantly departed from the linear model (L*), for which the evolution is treated implicitly (T. Szabo, P. Bénard).

Two main sources of discrepancy between M and L* can be distinguished :

• (i) the reference state (X*) chosen to linearize the model is too "far" from the real state (X) of the atmosphere;
• (ii) the linearized model is too "far" from the complete model, i.e. even if X is close to X*, M and L* do not behave similarly.

Summer 2000:

The source (ii) was attacked first. Three major discrepancies were listed :

• the form of the hydrostatic pressure-gradient term,
• the form of the geopotential-gradient term,
• the form of the non-hydrostatic pressure-gradient term.

The third one was shown to be detrimental to the robustness, and it was decided to remove it. This requires to abandon the angular momentum conservation (AMC) property for the non-hydrostatic part of the pressure force, a drawback which was evaluated as minor in comparison to the involved lack of robustness (P. Bénard). This correction allowed to remove some vertical smoothing of the wind at the bottom boundary which was necessary up to now to maintain the stability of the model (C. Smith). The second discrepancy was also removed from the code, even if apparently not directly related to the robustness (R. Bubnova). The first discrepancy is active only under some conditions, It involves the AMC property for the hydrostatic part of the flow, and is less important for robustness. Hence it is not modified for time-being.

At the end of the summer 2000, the model was definitively able to run stably with long time-steps, provided X was chosen close to X*.

Autumn 2000:

The source (i) was then attacked. It was first proven that the problems linked to the pressure discrepancy between X and X* could (provisionally at least) be evacuated through the choice of a pure "s" coordinate instead of the current hybrid one (P. Bénard, J. Vivoda). Hence only the thermal discrepancies between X and X* remained to be considered.

A first strategy to obtain a stable scheme was defined. The idea was to damp the instability of the model by using an iterative scheme which converges toward a fully implicit scheme. This strategy is likely to be operative since fully implicit schemes are normally stable. However, the main concern of this method is the efficiency, since the convergence may be slow, or may require small time-steps to be ensured.

It was shown that under the typical conditions of NWP, this strategy could be applied with an acceptable efficiency. For this, the X* must be chosen carefully, thus allowing a reasonably small number of iterations to bound the growth-rate inside a fixed acceptable interval. With this strategy, the robustness of the model was shown to be possible with an overcost factor of 2-3 compared to the present model (J. Vivoda, P. Bénard).

January 2001:

Parallel to these studies on iterative schemes, two major events occurred at the beginning of 2001:

• It was found, then shown, that the model could be stabilized by iterating only a very small part of the nonlinear model (namely two terms), and this with only one iteration. This demonstrated that the "fully implicit" strategy defined during the autumn was not necessarily the only one possible and that it was possible to obtain stable schemes with very little iterated content. However, the reason why this new scheme was stable was unclear (R. Bubnova, P. Bénard).
• The fact that the choice of the vertical coordinate and prognostic variables had a large impact on the robustness became more and more clear during this study. For example, the transformation of the model into a pure (more stable) s-coordinate model requires not only the change of the coordinate by itself, but also the change of both hydrostatic and non-hydrostatic pressure variables (J. Vivoda, P. Bénard).

Due to the new lighting brought by these two points, a class of stable models could be defined through a change of the set of prognostic variables, while keeping the usual time-marching schemes (semi-implicit, or one-iteration alias "predictor/corrector" scheme). The stability of the reformulated model was established for isothermal and realistic profiles, but still in an academic context. These changes require very little code modification, and the global data flow is unaffected (P. Bénard).

The robustness of these formulations remain now to be demonstrated for progressively more and more realistic (and potentially more unstable) frameworks up to the real NWP context, maybe requiring some more adaptations or restrictions.

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