  This work presents a rigorous framework based on coarse-graining to analyze
highly compressible turbulence. We show how the requirement that viscous
effects on the dynamics of large-scale momentum and kinetic energy be
negligible ---an inviscid criterion--- naturally supports a density weighted
coarse-graining of the velocity field. Such a coarse-graining method is already
known in the literature as Favre filtering; however its use has been primarily
motivated by appealing modeling properties rather than underlying physical
considerations. We also prove that kinetic energy injection can be localized to
the largest scales by proper stirring, and argue that stirring with an external
acceleration field rather than a body force would yield a longer inertial range
in simulations. We then discuss the special case of buoyancy-driven flows
subject to a spatially-uniform gravitational field. We conclude that a range of
scales can exist over which the mean kinetic energy budget is dominated by
inertial processes and is immune from contributions due to molecular viscosity
and external stirring.
