Dry atmosphere asymptotics
N. Botta, R. Klein, A. Almgren (September 1999)
Understanding and computing motions in the atmosphere is particularly challenging because of the multitude of physical processes and of the different space and time scales involved. Nonetheless, atmospheric motion takes place in a very special regime: The Mach number M, the ratio of the wind speed to the speed of sound, varies from zero at rest to 0.3 in fast jet streams, hurricanes and tornadoes. Unfortunately, in the M -> 0 limit, the fully compressible fluid equations which govern atmospheric motion become singular. Because of this, numerical schemes for these equations exhibit a dramatic breakdown in efficiency and accuracy in the low Mach number regime.
We propose multiple scale low Mach number / low Froude number asymptotic analysis as a general framework for understanding the motion in the atmosphere on space scales ranging from a few meters to thousands of kilometers. The theory provides a consistent picture of slow atmospheric flows and turns out to be the natural framework in which popular approximations, traditionally obtained on the basis of simplifying assumptions or ad hoc scaling arguments, can be derived.
The analysis yields strong implications for discrete methods aiming at the numerical computation of atmosphere motions as, e.g., in numerical weather forecasting or climate modeling. There are two kinds of implications. On one hand one finds constraints on parameterizations of unresolved physical processes such as those for turbulent heat or for for velocity boundary conditions. These constraints apply to the the continuous as well as the discrete equations. On the other hand the asymptotic analysis suggests how to design efficient methods for computing low speed flows in the atmosphere. The theory provides guidelines for consistently "filtering" the equations in the various regimes of length and time scales and shows that the "dynamic range problem" can be overcome by introducing suitable multiple variables that mimic the asymptotic representation of the field variables.
