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The density variance -- Mach number relation in supersonic, isothermal turbulence

We examine the relation between the density variance and the mean-square Mach number in supersonic, isothermal turbulence, assumed in several recent analytic models of the star formation process. From a series of calculations of supersonic, hydrodynamic turbulence driven using purely solenoidal Fourier modes, we find that the `standard' relationship between the variance in the log of density and the Mach number squared, i.e., sigma^2_(ln rho/rhobar)=ln (1+b^2 M^2), with b = 1/3 is a good fit to the numerical results in the supersonic regime up to at least Mach 20, similar to previous determinations at lower Mach numbers. While direct measurements of the variance in linear density are found to be severely underestimated by finite resolution effects, it is possible to infer the linear density variance via the assumption of log-normality in the Probability Distribution Function. The inferred relationship with Mach number, consistent with sigma_(rho/rhobar) ~ b M with b=1/3, is, however, significantly shallower than observational determinations of the relationship in the Taurus Molecular Cloud and IC5146 (both consistent with b~ 0.5), implying that additional physics such as gravity is important in these clouds and/or that turbulent driving in the ISM contains a significant compressive component. Magnetic fields are not found to change this picture significantly, in general reducing the measured variances and thus worsening the discrepancy with observations.