Scaling Relations of Compressible MHD Turbulence

We study scaling relations of compressible strongly magnetized turbulence. We find a good correspondence of our results with the Fleck (1996) model of compressible hydrodynamic turbulence. In particular, we find that the density-weighted velocity, i.e. $u \equiv \rho^{1/3} v$, proposed in Kritsuk et al. (2007) obeys the Kolmogorov scaling, i.e. $E_{u}(k)\sim k^{-5/3}$ for the high Mach number turbulence. Similarly, we find that the exponents of the third order structure functions for $u$ stay equal to unity for the all the Mach numbers studied. The scaling of higher order correlations obeys the She-Leveque (1994) scalings corresponding to the two-dimensional dissipative structures, and this result does not change with the Mach number either. In contrast to $v$ which exhibits different scaling parallel and perpendicular to the local magnetic field, the scaling of $u$ is similar in both directions. In addition, we find that the peaks of density create a hierarchy in which both physical and column densities decrease with the scale in accordance to the Fleck (1996) predictions. This hierarchy can be related ubiquitous small ionized and neutral structures (SINS) in the interstellar gas. We believe that studies of statistics of the column density peaks can provide both consistency check for the turbulence velocity studies and insight into supersonic turbulence, when the velocity information is not available.