Multiscale asymptotics and analysis for atmospheric flow models with moisture

Latent heat conversions due to phase changes of water in the atmosphere do to a great extent influence the energy balance. Of particular interest here are hot towers, which are large cumulonimbus clouds that live on small horizontal scales having a diameter less than one kilometer. It is common belief by now that these hot towers are to a great extent responsible for the vertical transport of heat into the upper troposphere within the intertropical convergence zone and it is therefore extremely important to gain a good understanding of their life cycles. One main goal of this project is thus the extension of the multiscale model of Klein and Majda by introducing layers in the vertical direction accounting for different orders of magnitude of the saturation mixing ratio as the key quantity in the moisture source terms and by simultaneously improving the scaling assumptions. Moreover we will use this refined modelling approach to analyse the modulation of gravity waves by hot towers in more detail. It was discovered that moisture reduces the vertical energy transport by internal gravity waves, and a profound theoretical basis might bear a great potential for improving global circulation models. Whereas the first part of the project is concerned with formal asymptotics for moist atmospheric flow models, it is a second aim to also proceed in developing further the rigorous analysis. A first step in this direction will be the investigation of the global well-posedness of the moisture balances that are used as a basis for the above asymptotic expansions. We plan to extend the recent results of Coti Zelati et al. for the coupled dynamics of temperature with a single balance equation for humidity based on very elaborate techniques of partial differential equations to physically more involved moisture dynamics describing the processes of water vapor, cloud water and rain water. Succeeding in this problem, we expect that we can also couple the moisture dynamics to the primitive equations, which result from the viscous compressible governing equations using only one simiplification, namely the assumption of hydrostatic balance. The primitive equations have been demonstrated to serve in general as a very good approximation and therefore build the basis for weather forecast and climate models. A rigorous analysis for this set of equations is therefore of particular interest not only from a mathematical point of view.

Since atmospheric flows are extremely complex, model reductions by scale analysis have a long history in meteorology. There is a huge variety of scales in atmospheric dynamics, such as e.g. microscale airflow over more or less rough surfaces, or clouds having typical diameters of a few kilometers, whereas weather fronts are associated to the larger so called synoptic scale well known from weather charts. For each choice of scale made, depending on what type of phenomena should be described, a different model is obtained from the dominant force balances. A huge challenge thereby constitutes the modelling of precipitation, which still causes one of the largest uncertainties in weather forecast and climate models. Of particular interest are hot towers, which are large cloud columns with small horizontal extent. They reveal enormous updrafts in their cores and come along with high precipitation yields. Moreover these deep convective clouds constitute the building blocks of larger scale convective storms, such as squall lines with thunderclouds forming along a line, or on the next larger scale the hurricanes. For studying such complex cloud phenomena involving the interaction of processes occuring on different length and time scales the key technique is multiple scales asymptotics. In collaboration with R. Klein we in this project in comparison to existing studies not only incorporated moisture into the asymptotical framework via balance equations for water vapor, cloud water and rain, but also refined the thermodynamics by taking into account the different gas constants and heat capacities for dry air and water components. For deep convective clouds these thermodynamic details, often neglected in the literature, were demonstrated to be essential by leading to different force balances. Moreover this setting provides the basis for studying further the case of organised convection as occuring in the core of storm systems. Systematic averaging procedures in the multiscale asymptotics derivations allow to quantify the modulation of the larger scale flow by the moisture processes in the cloud regions. Such theoretical studies could in particular serve as a basis for the further development of parameterisations of moisture for forecast models, which have grid spacing larger than the typical cloud processes. Another aim in this project was to rigorously analyse such moisture dynamics models used as a basis for weather forecast models by proving not only the existence of solutions, but also their uniqueness. Such theoretical studies, as carried out in collaboration with R. Klein, J. Li and E. Titi, provide important validity arguments for the models being used in applications. The findings in this project pave the way for many further investigations on cloud dynamics, such as the derivation of multiscale models for storm systems like squall lines or hurricanes and the analysis thereof.