Distinguished limits, multiple scales asymptotics, and numerics for atmospheric flows

Abstract

Atmospheric flow theory relies to a large extent on scale analyses and perturbation methods. These analyses explore the smallness of various non-dimensional characteristic numbers, such as the Froude-, Mach-, and Rossby-numbers and various length and time scale ratios. Notably the Mach number, being the ratio of a characteristic flow velocity and a typical sound speed, is independent of the considered length and time scales, and it is almost everywhere small within the atmosphere. This observation suggests the following approach towards deriving a hierarchy of simplified asymptotic model equations for atmospheric flows: 1. Use the Mach number, M, as the basic asymptotic expansion parameter. 2. Consider flows on increasingly larger scales, beginning with fluid motion on scales of a few meters, and ending with synoptic scale or even global atmospheric dynamics. 3. As the considered length scale increases, the above-mentioned scale dependent characteristic numbers will decrease. Whenever one of these numbers, say K, becomes small and justifies an estimate K=Ma as M -> 0, a new distinguished limit can be introduced and the associated flow regime can be explored through low Mach number asymptotics. This presentation will first show that this procedure allows one to recover many, if not most, of the well-established simplified model equations for atmospheric flow dynamics in a unified fashion. By introducing multiple scales expansions in space and/or time, we reveal additional interactions between phenomena acting on different length and time scales. Implications of these results for the construction of efficient and accurate numerical integration schemes for atmospheric flow dynamics will be summarized in the end.