  Turbulence in the quantum (superfluid) regime, similarly to its classical
counterpart, continues to attract a great deal of scientific inquiry, due to
the yet high number of unresolved problems. While turbulent states can be
routinely created in degenerate atomic gases, there is no generic scheme to
produce turbulence in fluids of light. Under paraxial propagation, light in
bulk nonlinear media behaves as a two-dimensional superfluid, described by a
nonlinear Schr\"{o}dinger equation formally equivalent to the Gross-Pitaevskii
model of a weakly interacting Bose gas, where photon-photon interactions are
mediated by a third order (Kerr) nonlinearity. Here, we develop the theory
describing the onset of a kinetic instability when two paraxial optical fluids
with different streaming velocities interact via the optical nonlinearity. From
numerical simulations of the nonlinear Schr\"{o}dinger equation, we further
characterize the onset of the instability and describe its saturation in the
form of vortex nucleation and excitation of turbulence. The experimental
observation of such effects is also discussed. The class of instabilities
described here thus provide a natural route towards the investigation of
quantum (superfluid) turbulence, structure formation and out-of-equilibrium
dynamics in superfluids of light.
