Asymptotic Models for Atmospheric Flows

Atmospheric flows feature length and time scales from 10−5 to 105 m and from microseconds to weeks and more. For scales above several kilometers and minutes, there is a natural scale separation induced by the atmosphere’s thermal stratification together with the influences of gravity and Earth’s rotation and the fact that atmospheric flow Mach numbers are typically small. A central aim of theoretical meteorology is to understand the associated scale-specific flow phenomena, such as internal gravity waves, baroclinic instabilities, Rossby waves, cloud formation and moist convection, (anti-) cyclonic weather patterns, hurricanes, and a variety of interacting waves in the tropics. Such understanding is greatly supported by analyses of reduced sets of model equations which capture just those fluid mechanical processes that are essential for the phenomenon in question while discarding higher-order effects. Such reduced models are typically proposed on the basis of combinations of physical arguments and mathematical derivations, and are not easily understood by the meteorologically untrained. This chapter demonstrates how many well-known reduced sets of model equations for specific, scale-dependent atmospheric flow phenomena may be derived in a unified and transparent fashion from the full compressible atmospheric flow equations using standard techniques of formal asymptotics. It also discusses an example for the limitations of this approach.Sections 3–5 of this chapter are a recompilation of the author’s more comprehensive article “Scale-dependent models for atmospheric flows”, Annual Reviews of Fluid Mechanics, 42 (2010).