The Column Density Variance in Turbulent Interstellar Media: A Fractal
  Model Approach
  Fractional Brownian motion (fBm) structures are used to investigate the
dependency of column density variance ({\sigma}_{\ln N}^2) in the turbulent
interstellar medium on the variance of three-dimensional density
({\sigma}_{\ln\rho}^2) and the power-law slope of the density power spectrum.
We provide quantitative expressions to infer the three-dimensional density
variance, which is not directly observable, from the observable column density
variance and spectral slope. We also investigate the relationship between the
column density variance and sonic Mach number (Ms) in the hydrodynamic (HD)
regime by assuming the spectral slope and density variance as functions of
sonic Mach number, as obtained from the HD turbulence simulations. They are
related by the expression {\sigma}_{\ln N}^2 = A{\sigma}_{\ln\rho}^2 =
Aln(1+b^2M^2), suggested by Burkhart & Lazarian for the magnetohydrodynamic
(MHD) case. The proportional constant A varies from ~ 0.2 to ~ 0.4 in the HD
regime as the turbulence forcing parameter b increases from 1/3 (purely
solenoidal forcing) to 1 (purely compressive forcing). It is also discussed
that the parameter A is lowered in the presence of a magnetic field.
