Dry Atmosphere Asymptotics

Abstract

Understanding and computing motions in the atmosphere is particularly challenging because of the multitude of physical processes and of the dierent 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 uid equations which govern atmospheric motion become singular. Because of this, numerical schemes for these equations exhibit a dramatic breakdown in eÆciency 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 ows and turns out to be the natural framework in whic...