  We study the impact of compressibility on two-dimensional turbulent flows,
such as those modeling astrophysical disks. We demonstrate that the direction
of cascade undergoes continuous transition as the Mach number Ma increases,
from inverse at zero Ma, to direct at infinite Ma. Thus, at Ma of order one
comparable amounts of energy flow from the scale of the pumping to large and
small scales, in accord with previous data. For supersonic turbulence with
large Ma, the cascade is direct, as in three dimensions, which results in
multifractal density field. For compressible flows of conducting fluids,
imposing external magnetic field allows to manipulate the flow producing
possibly large changes even at small Mach number. Thus Zeldovich antidynamo
theorem, by which at zero Ma the magnetic field is zero in the steady state,
must be used with caution. Real flows have finite Ma and, however small it is,
for large magnetic flux through the disk, the magnetic field changes the flow
appreciably, or rearranges it completely. For large Ma, relevant for
astrophysical disks, the magnetic field energy is enhanced by a large factor as
compared to estimates based on the mean field. We propose to use
two-dimensional Burgers turbulence, whose three-dimensional counterpart is used
for studies of the large-scale structure of the Universe, as a model for
supersonic thin accretion disks.
