SHOCK-GENERATED VORTICITY IN THE INTERSTELLAR MEDIUM AND THE ORIGIN OF THE STELLAR INITIAL MASS FUNCTION
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The main observational evidence for turbulence in the interstellar medium
(ISM) and molecular clouds is the power-law energy spectrum for velocity
fluctuations, E(k) \propto k^{\alpha}. The Kolmogorov scaling exponent,
\alpha=-5/3, is typical. At the same time, the observed probability
distribution function (PDF) of gas densities in both the ISM as well as in
molecular clouds is a log-normal distribution, which is similar to the initial
mass function (IMF) that describes the distribution of stellar masses. We
examine the density and velocity structure of interstellar gas traversed by
curved shock waves in the kinematic limit. We demonstrate mathematically that
just a few passages of curved shock waves generically produces a log-normal
density PDF. This explains the ubiquity of the log-normal PDF in many different
numerical simulations. We also show that subsequent interaction with a
spherical blast wave generates a power-law density distribution at high
densities, qualitatively similar to the Salpeter power-law for the IMF.
Finally, we show that a focused shock produces a downstream flow with energy
spectrum exponent \alpha=-2. Subsequent shock passages reduce this slope,
achieving \alpha \approx -5/3 after a few passages. Subsequent dissipation of
energy piled up at the small scales will act to maintain the spectrum very near
to the Kolomogorov value. Therefore, fully-developed turbulence may not be
required to explain the observed energy spectrum and density PDF. We argue that
the self-similar spherical blast wave arising from expanding HII regions or
stellar winds from massive stars may ultimately be responsible for creating the
high mass, power-law, tail in the IMF.