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 _ { \ln \rho / \bar { \rho } } ^ { 2 } = \ln \left ( 1 + b ^ { 2 } \mathcal { M } ^ { 2 } \right ) , 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 / \bar { \rho } } \approx b \mathcal { 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 \approx 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 .