Gravitational contraction versus Supernova driving and the origin of the
  velocity dispersion-size relation in molecular clouds
  Molecular cloud observations show that clouds have non-thermal velocity
dispersions that scale with the cloud size as $\sigma\propto R^{1/2}$ at
constant surface density, and for varying surface density scale with both the
cloud`s size and surface density, $\sigma^2 \propto R \Sigma$. The energy
source driving these chaotic motions remains poorly understood. We describe the
velocity dispersions observed in a cloud population formed in a
kiloparsec-scale numerical simulation of a magnetized, supernova-driven,
self-gravitating, interstellar medium, including diffuse heating and radiative
cooling. We compare the relationships between velocity dispersion, size, and
surface density measured in the simulated cloud population to those found in
observations of Galactic molecular clouds. We find that external supernova
explosions can not drive turbulent motions of the observed magnitudes within
dense clouds. On the other hand, self-gravity also induces non-thermal motions
as gravitationally bound clouds begin to collapse in our model, and by doing so
their internal velocity dispersions recover the observed relations. Energy
conservation suggests that the observed behavior is consistent with the kinetic
energy being proportional to the gravitational energy. However, the clouds in
our model show no sign of reaching a stable equilibrium state at any time, even
for strongly magnetized clouds. We conclude that gravitationally bound
molecular clouds are always in a state of gravitational collapse and their
properties are a natural result of this chaotic collapse. In order to agree
with observed star formation efficiencies, this process must be terminated by
the early destruction of the clouds, presumably from internal stellar feedback.
