  We study scaling relations of compressible isothermal strongly magnetized
turbulence using numerical simulations with resolution 512$^3$. 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. $\boldsymbol{u} \equiv \rho^{1/3} \boldsymbol{v}$, proposed in
Kritsuk et al. (2007) obeys the Kolmogorov scaling, i.e. ${\cal E}_{u}(k)\sim
k^{-5/3}$ for the high Mach number turbulence. Similarly, we find that the
exponents of the third order structure functions for $\boldsymbol{u}$ stay
equal to unity for all Mach numbers studied. The scaling of higher order
correlations obeys the She-L\'{e}v\^{e}que (1994) scalings corresponding to the
two-dimensional dissipative structures, and this result does not change with
the Mach number either. In contrast to velocity $\boldsymbol{v}$ which exhibits
different scaling parallel and perpendicular to the local magnetic field, the
scaling of $\boldsymbol{u}$ 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.
