Mass invariant in a compressible turbulent medium

Predicting the measurable statistical properties of density fluctuations in a supersonic compressible turbulent flow is a major challenge in physics. In 1951, Chandrasekhar derived an invariant under the assumption of the statistical homogeneity and isotropy of the turbulent density field and stationarity of the background density. Recently, Jaupart and Chabrier [Astrophys. J. Lett. 922, L36 (2021)] extended this invariant to nonisotropic flows in a time-evolving background and showed that it has the dimension of a mass. This invariant Minv is defined by Minv = E(ρ)Var( ρ E(ρ) )(lρ c )3, where ρ is the density field and lρ c is the correlation length. In this article, we perform numerical simulations of homogeneous and isotropic compressible turbulence to test the validity of this invariant in a medium subject to isotropic decaying turbulence. We study several input configurations, namely different Mach numbers, injection lengths of turbulence, and equations of state. We confirm that Minv remains constant during the decaying phase of turbulence. Furthermore, we develop a theoretical model of the density field statistics which predicts without any free parameter the evolution of the correlation length with the variance of the log-density field beyond the assumption of the Gaussian field for the log density. Noting that Minv is independent of the Mach number, we show that this invariant can be used to relate the non-Gaussian evolution of the log-density probability distribution function to its variance with no free parameters.