  We investigate an inequality constraining the energy and potential enstrophy
flux spectra in two-layer and multi-layer quasi-geostrophic models. Its
physical significance is that it can diagnose whether any given multi-layer
model that allows co-existing downscale cascades of energy and potential
enstrophy can allow the downscale energy flux to become large enough to yield a
mixed energy spectrum where the dominant $k^{-3}$ scaling is overtaken by a
subdominant $k^{-5/3}$ contribution beyond a transition wavenumber $k_t$
situated in the inertial range. The validity of the flux inequality implies
that this scaling transition cannot occur within the inertial range, whereas a
violation of the flux inequality beyond some wavenumber $k_t$ implies the
existence of a scaling transition near that wavenumber. This flux inequality
holds unconditionally in two-dimensional Navier-Stokes turbulence, however, it
is far from obvious that it continues to hold in multi-layer quasi-geostrophic
models, because the dissipation rate spectra for energy and potential enstrophy
no longer relate in a trivial way, as in two-dimensional Navier-Stokes. We
derive the general form of the energy and potential enstrophy dissipation rate
spectra for a generalized symmetrically coupled multi-layer model. From this
result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic
model, where the dissipation terms for each layer consist of the same
Fourier-diagonal linear operator applied on the streamfunction field of only
the same layer, the flux inequality continues to hold. It follows that a
necessary condition to violate the flux inequality is the use of asymmetric
dissipation where different operators are used on different layers. etc.
