The density variance-Mach number relation in supersonic turbulence - I. Isothermal, magnetized gas

It is widely accepted that supersonic, magnetized 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 rms Mach number in supersonic, isothermal, magnetized 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 magnetized 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 p, B proportional to p1/2 and B proportional to p. We test the analytically derived density varianceMach number relation with numerical simulations, and find that for B proportional to p1/2, the variance in the logarithmic density contrast, , fits very well to simulated data with turbulent forcing parameter b= 0.4, when the gas is super-Alfvenic. However, this result breaks down when the turbulence becomes trans-Alfvenic or sub-Alfvenic, because in this regime the turbulence becomes highly anisotropic. Our density varianceMach number relations simplify to the purely hydrodynamic relation as the ratio of thermal to magnetic pressure beta 0 - lemniscate.