  We investigate two-dimensional turbulence in finite-temperature trapped
Bose-Einstein condensates within damped Gross-Pitaevskii theory. Turbulence is
produced via circular motion of a Gaussian potential barrier stirring the
condensate. We systematically explore a range of stirring parameters and
identify three regimes, characterized by the injection of distinct quantum
vortex structures into the condensate: (A) periodic vortex dipole injection,
(B) irregular injection of a mixture of vortex dipoles and co-rotating vortex
clusters, and (C) continuous injection of oblique solitons that decay into
vortex dipoles. Spectral analysis of the kinetic energy associated with
vortices reveals that regime (B) can intermittently exhibit a Kolmogorov
$k^{-5/3}$ power law over almost a decade of length or wavenumber ($k$) scales.
The kinetic energy spectrum of regime (C) exhibits a clear $k^{-3/2}$ power law
associated with an inertial range for weak-wave turbulence, and a $k^{-7/2}$
power law for high wavenumbers. We thus identify distinct regimes of forcing
for generating either two-dimensional quantum turbulence or classical weak-wave
turbulence that may be realizable experimentally.
