  We continue our investigation of an inequality constraining the energy and
potential enstrophy flux spectra in the two-layer quasi-geostrophic model. Its
physical significance is that it can diagnose whether any given model that
allows coexisting downscale cascades of energy and potential enstrophy can
reproduce the Nastrom-Gage spectrum, in terms of the total energy spectrum.
This inequality holds unconditionally for two-dimensional turbulence, however
it is far from obvious that it generalizes to multi-layer quasi-geostrophic
models. In previous work we considered the case of a two-layer
quasi-geostrophic model in which the dissipation terms for each layer are
dependent only on the streamfunction field of the corresponding layer. We now
generalize this configuration as follows: First, following a 1980 paper by
Salmon, we use an extrapolated Ekman term at the bottom layer which uses the
streamfunction field of both layers to approximate the streamfunction field at
the surface boundary layer. Second, for reasons explained in detail in the
paper itself, we use small-scale dissipation terms with different
hyperviscosity coefficients. Sufficient conditions for satisfying the flux
inequality are given under this more general dissipation configuration, and we
discuss the potential role of extrapolated Ekman dissipation and differential
small-scale dissipation in violating the flux inequality.
