Shock generated vorticity in the interstellar medium and the origin of the stellar initial mass function

Observations of the interstellar medium (ISM) and molecular clouds suggest these astrophysical flows are strongly turbulent. The main observational evidence for turbulence is the power-law energy spectrum for velocity fluctuations, $E(k)\propto k^\alpha$, with $\alpha\in [-1.5,-2.6]$. 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. In this paper 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 {\em downstream\/} flow with energy spectrum exponent $\alpha=-2$. Subsequent shock passages reduce this slope, achieving $\alpha\approx -5/3$ after a few passages. These results suggest that fully-developed turbulence may {\em not\/} be required to explain the observed energy spectrum and density PDF.