A Turbulent-Entropic Instability and the Fragmentation of Star-Forming
  Clouds
  The kinetic energy of supersonic turbulence within interstellar clouds is
subject to cooling by dissipation in shocks and subsequent line radiation. The
clouds are therefore susceptible to a condensation process controlled by the
specific entropy. In a form analogous to the thermodynamic entropy, the entropy
for supersonic turbulence is proportional to the log of the product of the mean
turbulent velocity and the size scale. We derive a dispersion relation for the
growth of entropic instabilities in a spherical self-gravitating cloud and find
that there is a critical maximum dissipation time scale, about equal to the
crossing time, that allows for fragmentation and subsequent star formation.
However, the time scale for the loss of turbulent energy may be shorter or
longer, for example with rapid thermal cooling or the injection of mechanical
energy. Differences in the time scale for energy loss in different star-forming
regions may result in differences in the outcome, for example, in the initial
mass function.
