  Turbulence is the dominant source of collisional velocities for grains with a
wide range of sizes in protoplanetary disks. So far, only Kolmogorov turbulence
has been considered for calculating grain collisional velocities, despite the
evidence that turbulence in protoplanetary disks may be non-Kolmogorov. In this
work, we present calculations of grain collisional velocities for arbitrary
turbulence models characterized by power-law spectra and determined by three
dimensionless parameters: the slope of the kinetic energy spectrum, the slope
of the auto-correlation time, and the Reynolds number. The implications of our
results are illustrated by numerical simulations of the grain size evolution
for different turbulence models. We find that for the modeled cases of the
Iroshnikov-Kraichnan turbulence and the turbulence induced by the
magneto-rotational instabilities, collisional velocities of small grains are
much larger than those for the standard Kolmogorov turbulence. This leads to
faster grain coagulation in the outer regions of protoplanetary disks,
resulting in rapid increase of dust opacity in mm-wavelength and possibly
promoting planet formation in very young disks.
