Multiple spatial scales in engineering and atmospheric low Mach number flows

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, epsilon. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of epsilon as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).