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,
ε. Of course, more familiar parameters, such as the Rossby and
Froude numbers may then be expressed in terms of ε 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).