  In this paper we characterize the scaling of energy spectra, and the
interscale transfer of energy and enstrophy, for strongly, moderately and
weakly stably stratified two-dimensional (2D) turbulence under large-scale
random forcing. In the strongly stratified case, a large-scale vertically
sheared horizontal flow (VSHF) co-exists with small scale turbulence. The VSHF
consists of internal gravity waves and the turbulent flow has a kinetic energy
(KE) spectrum that follows an approximate $k^{-3}$ scaling with zero KE flux
and a robust positive enstrophy flux. The spectrum of the turbulent potential
energy (PE) also approximately follows a $k^{-3}$ power-law and its flux is
directed to small scales. For moderate stratification, there is no VSHF and the
KE of the turbulent flow exhibits Bolgiano-Obukhov scaling that transitions
from a shallow $k^{-11/5}$ form at large scales, to a steeper approximate
$k^{-3}$ scaling at small scales. The entire range of scales shows a strong
forward enstrophy flux, and interestingly, large (small) scales show an inverse
(forward) KE flux. The PE flux in this regime is directed to small scales, and
the PE spectrum is characterized by an approximate $k^{-1.64}$ scaling.
Finally, for weak stratification, KE is transferred upscale and its spectrum
closely follows a $k^{-2.5}$ scaling, while PE exhibits a forward transfer and
its spectrum shows an approximate $k^{-1.6}$ power-law. For all stratification
strengths, the total energy always flows from large to small scales and almost
all the spectral indices are well explained by accounting for the scale
dependent nature of the corresponding flux.
