  We investigate the properties of highly compressible turbulence, the
compressibility arising from a small effective polytropic exponent $\gamma_e$
due to cooling. In the limit of small $\gamma_e$, the density jump at shocks is
shown to be of the order of $e^{M^2}$. Without self-gravity, the density
structures arising in the moderately compressible case consist mostly of
patches separated by shocks and behaving like waves, while in the highly
compressible case clearly defined long-lived object-like clouds emerge. When
the forcing in the momentum equation is purely compressible, the rotational
energy decays monotonically in time, indicating that the vortex-stretching term
is not efficient in transferring energy to rotational modes. This property may
be at the origin of the low amount of rotation found in interstellar clouds.
Vorticity production is found to rely heavily on the presence of additional
terms in the equations. In the presence of self-gravity, we suggest that
turbulence can produce bound structures for $\gamma_e < 2(1-1/n)$, where $n$ is
the typical dimensionality of the turbulent compressions. We support this
result by means of numerical simulations in which, for sufficiently small
$\gamma_e$, small-scale turbulent density fluctuations eventually collapse even
though the medium is globally stable. This result is preserved in the presence
of a magnetic field for supercritical mass-to-flux ratios. At larger polytropic
exponents, turbulence alone is not capable of producing bound structures, and
collapse can only occur when the medium is globally unstable. This mechanism is
a plausible candidate for the differentiation between primordial and
present-day stellar-cluster formation and for the low efficiency of star
formation.
