  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.
