  Molecular clouds are to a great extent influenced by turbulent motions in the
gas. Numerical and observational studies indicate that the star formation rate
and efficiency crucially depend on the mixture of solenoidal and compressive
modes in the turbulent acceleration field, which can be quantified by the
turbulent driving parameter b. For purely solenoidal (divergence-free) driving
previous studies showed that b=1/3 and for entirely compressive (curl-free)
driving b=1. In this study, we determine the evolution of the turbulent driving
parameter b in magnetohydrodynamical simulations of molecular cloud formation
and evolution. The clouds form due to the convergence of two flows of warm
neutral gas. We explore different scenarios by varying the magnitude of the
initial turbulent perturbations in the flows. We show that the driving mode of
the turbulence within the cloud strongly fluctuates with time and exhibits no
clear correlation with typical cloud properties, such as the cloud mass and the
(Alfven) Mach number. We specifically find that $b$ strongly varies from b=0.3
to b=0.8 on timescales t<5 Myr, where the timescale and range of variation can
change from cloud to cloud. This rapid change of b from solenoidal to
compressive driving is primarily associated with global contraction of the
cloud and subsequent onset of star formation. We conclude that the effective
turbulence driving parameter should be treated as a free parameter that can
vary from solenoidal to compressive in both time and space.
