An exact relation for density fluctuations in compressible turbulence

We derive an exact relation for density fluctuations in statistically stationary compressible turbulence. In wealdy compressible turbulence, the relation identifies two contributions, corresponding to the acoustic mode and the pseudo-sound mode, respectively, to the density power spectrum, providing a unifying picture for the origin of density fluctuations in turbulent flows with Mach number less than or similar to 1. Using numerical simulations of driven turbulence, we verified the validity of the exact relations, and examined the contributions of the acoustic and pseudo-sound terms as a function of the Mach number. For simulations of supersonic turbulence, the exact relations provide a tool to quantify the artificial reduction of the density variance by numerical viscosity. The artificial suppression of density fluctuations increases with increasing Mach number, due to the necessity of applying larger numerical diffusion to stabilize stronger shocks. The exact relation also helps to theoretically establish the relation between the density variance and the density-weighted Mach number, /(rho) over bar (2) = b(2)M(rho)(2), in supersonic turbulence, which is of crucial importance for the modelling of star formation. Combining with simulations of supersonic turbulence with solenoidal driving and removing the artificial suppression of density fluctuations by numerical viscosity, our exact relation gives an estimate of b similar or equal to 0.4 for the b parameter.