The power spectrum of compressible turbulence: A story in symmetries

Abstract

This article presents a theoretical study of the scaling properties of the kinetic energy spectrum in compressible turbulence. From the fundamental symmetries and linear transformations of the microscopic action, we derive exact relations between the correlation functions and their generators. These relations put strong constraints on the possible scaling relations in the system as a function of scale. One of the main results of this study is that the action can be split between an incompressible part, which is the same as in the usual stochastic Navier-Stokes theory whatever the value of the Mach number, and a longitudinal part, whose behavior is to be compared to the three dimensional Burgers equation, which presents a much richer phase diagram as its usually discussed one dimensionalcounterpart.