A new density Variance-Mach number relation for subsonic and supersonic isothermal turbulence

The probability density function of the gas density in subsonic and supersonic, isothermal, driven turbulence is analyzed using a systematic set of hydrodynamical grid simulations with resolutions of up to 1024³ cells. We perform a series of numerical experiments with root-mean-square (rms) Mach number M ranging from the nearly incompressible, subsonic (M = 0.1) to the highly compressible, supersonic (M = 15) regime. We study the influence of two extreme cases for the driving mechanism by applying a purely solenoidal (divergence-free) and a purely compressive (curl-free) forcing field to drive the turbulence. We find that our measurements fit the linear relation between the rms Mach number and the standard deviation (std. dev.) of the density distribution in a wide range of Mach numbers, where the proportionality constant depends on the type of forcing. In addition, we propose a new linear relation between the std. dev. of the density distribution σ_ρ and that of the velocity in compressible modes, i.e., the compressible component of the rms Mach number, M_comp. In this relation the influence of the forcing is significantly reduced, suggesting a linear relation between σ_ρ and M_comp, independent of the forcing, and ranging from the subsonic to the supersonic regime.