be stressed here is that *the proof of the inequality involves both the linearity of the dissipation terms and the twin conservation laws (of enstrophy and energy), and is not inherent solely to the twin conservation laws.* With a nonlinear dissipation term, that could result from neglected three-dimensional effects, the flux inequality could be violated within the inertial range. Since the transition wavenumber $k_t$ is expected to be approximately equal to the wavenumber where the flux inequality becomes an equation, an inertial range violation of the flux inequality would give $k_t \ll k_d$.

As was shown by @article:Tung:2007, in the quasi-geostrophic two-layer model, the relationship between $D_E (k)$ and $D_G (k)$ becomes quite complicated, so it may be possible to violate the flux inequality, thus resulting in a significant separation between $k_t$ and $k_d$. If that occurs, we can expect $k^{-5/3}$ scaling in the gap created between $k_t$ and $k_d$. In @article:Tung:2007, we have identified asymmetric dissipation as the only mechanism that can break the flux inequality. By asymmetric dissipation we mean that the dissipation operators acting on the top and bottom velocity equations are different: indeed, in the two-layer quasi-geostrophic model there is an Ekman dissipation term acting at 