The Velocity Statistics of Turbulent Clouds in the Presence of Gravity,
  Magnetic fields, Radiation, and Outflow Feedback
  The interaction of turbulence, magnetic fields, self-gravity, and stellar
feedback within molecular clouds is crucial for understanding star formation.
We study the effects of self-gravity and outflow feedback on the properties of
the turbulent velocity via the structure function over length scales from
$\sim$ 0.01 pc to 2 pc. We analyze a series of three-dimensional,
magnetohydrodynamical (MHD) simulations of star cluster formation. We find
outflow feedback can change the scaling of velocity fluctuations but still
roughly being in between Kolmogorov and Burgers turbulence. We observe that
self-gravity and protostellar outflows increase the velocity fluctuations over
all length scales. Outflows can amplify the velocity fluctuations by up to a
factor of $\sim$7 on scales $\sim$ 0.01 - 0.2 pc and drive turbulence up to a
scale of $\sim$ 1 pc. The amplified velocity fluctuations provide more support
against gravity and enhance fragmentation on small scales. The self-gravity's
effect is more significant on smaller dense clumps and it increases the
fraction of the compressive velocity component up to a scale of $\sim$ 0.2 pc.
However, outflow feedback drives both solenoidal and compressive modes, but it
induces a higher fraction of solenoidal modes relative to compressive modes.
Thus, with outflows, the dense core ends up with a slightly higher fraction of
solenoidal modes. We find that the compressible fraction is fairly constant
with about 1/3 on scales $\sim$ 0.1 - 0.2 pc. The combined effect of enhanced
velocity dispersion and reduced compressive fraction contributes to a reduction
in the star formation rate.
