Energy spectrum in compressible turbulence*

The general scaling of spectra in compressible turbulence is not well understood,
in particular the circumstances under which different theoretical proposals such as  
pseudo-sound and equipartition apply. The evidence supporting different scaling laws seems  
inconsistent across the literature, especially at high turbulent Mach numbers ($M_{t}$).
 From an asymptotic expansion based on two parameters characterizing compressibility (the traditional $M_{t}$ as a measure of scale separation between acoustic and turbulent processes,  and ${u^{'}_{d}}/{u^{'}_{s}}$ the ratio of dilatation to solenoidal rms velocity  
 as a measure of compressibility levels).   We identify the dominant processes in the governing equations and determine the behavior expected for the spectrum.  Using a very large DNS database with different forcing modes and a wide range of governing parameters  
(Taylor microscale Reynolds numbers, $Re_{\lambda}$ up to 170 and turbulent Mach numbers, $M_{t}$ up to 0.8) we support these  predictions and reconcile the seemingly contradictory results in the literature.  We will discuss the potential implications of these findings for compressible scalar mixing.