Low Mach Number Modeling of Moist Atmospheric Flows
BackgroundIn general, climate and weather modeling is concerned with atmospheric phenomena at large scales, from regions spanning hundreds of kilometers to the entire globe. Complex, multi-physics phenomena encompass processes occurring within a broad spectrum of spatial and temporal scales. Even with today's supercomputers, subgrid parametrizations are still required to account for processes that cannot be resolved at the length scales in the simulation. We have developed a low Mach number model for moist atmospheric flows to serve as an efficient modeling tool for studying the small-scale processes underlying the construction of subgrid parametrizations.
Low Mach Number Approach
Small--scale atmospheric phenomena are typically characterized by relatively slow dynamics, that is, low Mach number flows for which the fast acoustic modes are physically irrelevant. Thus, numerical modeling of these flows does not typically require explicitly resolving fast--propagating sound waves.
Low Mach number equations evolve the system at the advective rather than the acoustic time scale, thus allowing more efficient numerical simulation than explicit solution of the fully compressible equations. While acoustic waves are eliminated, compressibility effects due to stratification, latent heat release, compositional changes and other thermal processes are retained.
Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint.
The low Mach number model described in  is valid for moist flows given by a mixture of dry air and water in both vapor and liquid states. The mathematical model can be solved by incorporating moist thermodynamics as described in  into the MAESTRO code, which was originally designed to simulate low Mach number stratified, reacting flows in astrophysical settings. Moist atmospheric flows are thus simulated using time steps set by the fluid motion and exploiting the parallel computing capabilities of MAESTRO to further enhance computational performance. Computational speed-ups by factors of 5 to 15 in total run time have been observed for the low Mach number model relative to the fully compressible model.
 Max Duarte, Ann S. Almgren, Kaushik Balakrishnan, John B. Bell, David M. Romps, "A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations," Monthly Weather Review, 142, pp. 4269--4283, 2014 [arxiv].
 Max Duarte, Ann S. Almgren, and John B. Bell, "A Low Mach Number Model for Moist Atmospheric Flows," Journal of the Atmospheric Sciences, 72(4), pp. 1605-1620, 2015 [arxiv].