  Using observations from the first two years of the MOSFIRE Deep Evolution
Field (MOSDEF) survey, we study 13 AGN-driven outflows detected from a sample
of 67 X-ray, IR and/or optically-selected AGN at $z \sim 2$. The AGN have
bolometric luminosities of $\sim10^{44}-10^{46} ~\mathrm{erg~s^{-1}}$,
including both quasars and moderate-luminosity AGN. We detect blueshifted,
ionized gas outflows in the H$\beta$ , [OIII], H$\alpha$ ~and/or [NII] emission
lines of $19\%$ of the AGN, while only 1.8\% of the MOSDEF galaxies have
similarly-detected outflows. The outflow velocities span $\sim$300 to 1000 km
s$^{-1}$. Eight of the 13 outflows are spatially extended on similar scales as
the host galaxies, with spatial extents of 2.5 to 11.0 kpc. Outflows are
detected uniformly across the star-forming main sequence, showing little trend
with the host galaxy SFR. Line ratio diagnostics indicate that the outflowing
gas is photoionized by the AGN. We do not find evidence for positive AGN
feedback, in either our small MOSDEF sample or a much larger SDSS sample, using
the BPT diagram. Given that a galaxy with an AGN is ten times more likely to
have a detected outflow, the outflowing gas is photoionzed by the AGN, and
estimates of the mass and energy outflow rates indicate that stellar feedback
is insufficient to drive at least some of these outflows, they are very likely
to be AGN-driven. The outflows have mass-loading factors of the order of unity,
suggesting that they help regulate star formation in their host galaxies,
though they may be insufficient to fully quench it.
