Probes of turbulent driving mechanisms in molecular clouds from
  fluctuations in synchrotron intensity
  Previous studies have shown that star formation depends on the driving of
molecular cloud turbulence, and differences in the driving can produce an order
of magnitude difference in the star formation rate. The turbulent driving is
characterised by the parameter $\zeta$, with $\zeta=0$ for compressive,
curl-free driving (e.g. accretion or supernova explosions), and $\zeta=1$ for
solenoidal, divergence-free driving (e.g. Galactic shear). Here we develop a
new method to measure $\zeta$ from observations of synchrotron emission from
molecular clouds. We calculate statistics of mock synchrotron intensity images
produced from magnetohydrodynamic simulations of molecular clouds, in which the
driving was controlled to produce different values of $\zeta$. We find that the
mean and standard deviation of the log-normalised synchrotron intensity are
sensitive to $\zeta$, for values of $\zeta$ between $0$ (curl-free driving) and
$0.5$ (naturally-mixed driving). We quantify the dependence of zeta on the
direction of the magnetic field relative to the line of sight. We provide
best-fit formulae for $\zeta$ in terms of the log-normalised mean and standard
deviation of synchrotron intensity, with which $\zeta$ can be determined for
molecular clouds that have similar Alfv\'enic Mach number to our simulations.
These formulae are independent of the sonic Mach number. Signal-to-noise ratios
larger than $5$, and angular resolutions smaller than $5\%$ of the cloud
diameter, are required to apply these formulae. Although there are no firm
detections of synchrotron emission from molecular clouds, by combining Green
Bank Telescope and Very Large Array observations it should be possible to
detect synchrotron emission from molecular clouds, thereby constraining the
value of $\zeta$.
