We investigate the properties of highly compressible turbulence, the compressibility arising from a small effective polytropic exponent γe due to cooling. In the limit of small γe, the density jump at shocks is shown to be on the order of eM2, much larger than the M2 jump associated with high Mach number flows in the isothermal regime. In the absence of self-gravity, the density structures that arise 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. The transition from wavelike to object-like behavior requires a change in the relative phase of the density and velocity fields analogous to that in the development of an instability. 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, such as the Coriolis force at large scales and the Lorentz force at small scales in the interstellar medium, or on the presence of local sources of heating. In the presence of self-gravity, we suggest that turbulence can produce bound structures for γe 2(1 - n–1), where n is the typical dimensionality of the turbulent compressions. We support this result by means of numerical simulations in which, for sufficiently small γe, 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. Finally, we discuss models of the interstellar medium at the kiloparsec scale including rotation, which restores a high-γe behavior.
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