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The Density Variance--Mach Number Relation in Supersonic Turbulence: I. Isothermal, magnetised gas

It is widely accepted that supersonic, magnetised turbulence plays a fundamental role for star formation in molecular clouds. It produces the initial dense gas seeds out of which new stars can form. However, the exact relation between gas compression, turbulent Mach number, and magnetic field strength is still poorly understood. Here, we introduce and test an analytical prediction for the relation between the density variance and the root-mean-square Mach number in supersonic, isothermal, magnetised turbulent flows. We approximate the density and velocity structure of the interstellar medium as a superposition of shock waves. We obtain the density contrast considering the momentum equation for a single magnetised shock and extrapolate this result to the entire cloud. Depending on the field geometry, we then make three different assumptions based on observational and theoretical constraints: B independent of density, B proportional to the root square of the density and B proportional to the density. We test the analytically derived density variance--Mach number relation with numerical simulations, and find that for B proportional to the root square of the density, the variance in the logarithmic density contrast, $σ_{\ln ρ/ρ_0}^2=\ln[1+b^2\mathscr{M}^2β_0/(β_0+1)]$, fits very well to simulated data with turbulent forcing parameter b=0.4, when the gas is super-Alfvénic. However, this result breaks down when the turbulence becomes trans-Alfvénic or sub-Alfvénic, because in this regime the turbulence becomes highly anisotropic. Our density variance--Mach number relations simplify to the purely hydrodynamic relation as the ratio of thermal to magnetic pressure $β_0$ approaches infinite.