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 Alfvenic 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 $zeta$.
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