Is molecular cloud turbulence driven by external supernova explosions?
  We present high-resolution ($\sim$ 0.1 pc), hydrodynamical and
magnetohydrodynamical simulations to investigate whether the observed level of
molecular cloud (MC) turbulence can be generated and maintained by external
supernova (SN) explosions. The MCs are formed self-consistently within their
large-scale galactic environment following the non-equilibrium formation of
H$_2$ and CO including (self-) shielding and important heating and cooling
processes. The MCs inherit their initial level of turbulence from the diffuse
ISM, where turbulence is injected by SN explosions. However, by systematically
exploring the effect of individual SNe going off outside the clouds, we show
that at later stages the importance of SN driven turbulence is decreased
significantly. This holds for different MC masses as well as MCs with and
without magnetic fields. The SN impact also decreases rapidly with larger
distances. Nearby SNe ($d$ $\sim$ 25 pc) boost the turbulent velocity
dispersions of the MC by up to 70 per cent (up to a few km s$^{-1}$). For $d$
$>$ 50 pc, however, their impact decreases fast with increasing $d$ and is
almost negligible. For all probed distances the gain in velocity dispersion
decays rapidly within a few 100 kyr. This is significantly shorter than the
average timescale for a MC to be hit by a nearby SN under solar neighbourhood
conditions ($\sim$ 2 Myr). Hence, at these conditions SNe are not able to
sustain the observed level of MC turbulence. However, in environments with high
gas surface densities and SN rates like the Central Molecular Zone, observed
elevated MC dispersions could be triggered by external SNe.
