  The probability density function (PDF) of the gas density in subsonic and
supersonic, isothermal, driven turbulence is analyzed with a systematic set of
hydrodynamical grid simulations with resolutions up to 1024^3 cells. We
performed a series of numerical experiments with root mean square (r.m.s.) Mach
number M ranging from the nearly incompressible, subsonic (M=0.1) to the highly
compressible, supersonic (M=15) regime. We study the influence of two extreme
cases for the driving mechanism by applying a purely solenoidal
(divergence-free) and a purely compressive (curl-free) forcing field to drive
the turbulence. We find that our measurements fit the linear relation between
the r.m.s. Mach number and the standard deviation of the density distribution
in a wide range of Mach numbers, where the proportionality constant depends on
the type of the forcing. In addition, we propose a new linear relation between
the standard deviation of the density distribution and the standard deviation
of the velocity in compressible modes, i.e. the compressible component of the
r.m.s. Mach number. In this relation the influence of the forcing is
significantly reduced, suggesting a linear relation between the standard
deviation of the density distribution and the standard deviation of the
velocity in compressible modes, independent of the forcing, ranging from the
subsonic to the supersonic regime.
