A Rational Approach to Cumulus Parameterization

Traditionally, cumulus parameterizations in large-scale atmospheric models are simplified treatments based on plausibility arguments and followed by extensive tuning to optimize overall model behavior. The results of this approach are not always satisfactory, and have stimulated the development of so-called super-parameterizations, in which a cumulus ensemble model run in each grid box takes the place of the cumulus parameterization. Though some of the early results of this approach are encouraging, it seems unlikely that computing power will increase sufficiently in the near-term to make this approach generally feasible at spatial resolutions needed to resolve most tropical phenomena.

Assuming that cumulus ensemble models can themselves be verified against observation to sufficient accuracy, an alternative approach is to use such models to improve cumulus parameterizations. However, in order to do so, the cumulus models must be run in a realistic context, so that the full range of behaviors possible in the large-scale model is exercised. Sobel and Bretherton (2000), Derbyshire et al. (2004), and Raymond and Zeng (2000, 2005) have developed such a context, called the weak temperature gradient (WTG) approximation by Sobel and Bretherton. In this context the effects of buoyancy redistribution by gravity waves in the tropics are mimicked by relaxing the average virtual temperature profile of the model to some reference tropical profile. This relaxation may be thought of as being due to the adiabatic cooling from some vertical velocity, called the WTG vertical velocity. Vertical advection and (optionally) lateral entrainment and detrainment from the surroundings follow from this vertical motion, and are used to modify the average humidity field as well. After determining how the cumulus ensemble model responds in this context to known forcing factors for deep convection, such as surface heat fluxes and environmental humidity profiles, the cumulus model is replaced by a cumulus parameterization and adjustments are made to the parameterization to make it mimic the cumulus ensemble model.

In the present paper I use this technique to test a "toy" cumulus parameterization previously used in an equatorial beta plane model to simulate the Madden-Julian oscillation. In particular, the parameterization is tuned to reproduce the dependence of the cumulus ensemble model's equilibrium mean precipitation rate on surface wind speed over an ocean with fixed sea surface temperature. This dependence has a strong effect on the development of the Madden-Julian oscillation (MJO) in the model as well as smaller scale disturbances such as easterly waves, and the tuning of the cumulus parameterization using this technique results in more realistic simulations. The wide variability in predicted MJO behavior across models may be the result of (among other things) variability in this relationship.

This is just one example of how WTG plus cumulus ensemble models may be used to improve cumulus parameterizations. Systematic application of this technique has the potential to move the subject of cumulus parameterization out of the realm of "black magic".

REFERENCES

Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix, J.-M. Piriou, J.-L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130, 3055-3079.

Raymond, D. J., and X. Zeng, 2000: Instability and large-scale circulations in a two-column model of the tropical troposphere. Quart. J. Roy. Meteor. Soc., 126, 3117-3135.

Raymond, D. J., and X. Zeng, 2005: Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation. Quart. J. Roy. Meteor. Soc., 131, 1301-1320.

Sobel, A. H., and C. S. Bretherton, 2000: Modeling tropical precipitation in a single column. J. Climate, 13, 4378-4392.