  We analyse the formation and the dynamics of quantum turbulence in a
two-dimensional Bose-Einstein condensate with a Josephson junction barrier
modelled using the Gross-Pitaevskii equation. We show that a sufficiently high
initial superfluid density imbalance leads to randomisation of the dynamics and
generation of turbulence, namely, the formation of a quasi-1D dispersive shock
consisting of a train of grey solitons that eventually breakup into chains of
distinct quantised vortices of alternating vorticity followed by random
turbulent flow. The Josephson junction barrier allows us to create two
turbulent regimes: acoustic turbulence on one side and vortex turbulence on the
other. Throughout the dynamics, a key mechanism for mixing these two regimes is
the transmission of vortex dipoles through the barrier: we analyse this
scattering process in terms of the barrier parameters, sound emission and
vortex annihilation. Finally, we discuss how the vortex turbulence evolves for
long times, presenting the optimal configurations for the density imbalance and
barrier height in order to create the desired turbulent regimes which last as
long as possible.
