  In star-forming clouds, high velocity flow gives rise to large fluctuations
of density. In this work we explore the correlation between velocity magnitude
(speed) and density. We develop an analytic formula for the joint probability
distribution (PDF) of density and speed, and discuss its properties. In order
to develop an accurate model for the joint PDF, we first develop improved
models of the marginalized distributions of density and speed. We confront our
results with a suite of 12 supersonic isothermal simulations with resolution of
$1024^3$ cells in which the turbulence is driven by 3 different forcing modes
(solenoidal, mixed and compressive) and 4 r.m.s. Mach numbers (1, 2, 4, 8). We
show, that for transsonic turbulence, density and speed are correlated to a
considerable degree and the simple assumption of independence fails to
accurately describe their statistics. In the supersonic regime, the
correlations tend to weaken with growing Mach number. Our new model of the
joint and marginalized PDFs are a factor of 3 better than uncorrelated, and
provides insight into this important process.

Astrophysics of Galaxies