other hand, the two-layer model is indeed realistic at the small synoptic-scale wavenumbers, where the forcing takes place, so using it to explain the rates of energy and potential enstrophy injection at the forcing range is a fair argument.

As we have explained above, in the Tung-Orlando theory, the location of the transition wavenumber $k_t$ is directly determined by the relative magnitude of the downscale potential enstrophy flux $\gn$ over the downscale energy flux $\gee$. A different mechanism underlies an SQG model that was recently proposed by @article:Smith:2006. In their model, there is only one cascade, whose scaling exponent changes with wavenumber $k$, because the self-similar scaling of the model’s nonlinear term changes with increasing $k$. As a result, in the Tulloch-Smith SQG model the transition wavenumber $k_t$ is strictly constrained to coincide with $k_R$, because the transition in the scaling of the nonlinear term of the model occurs at $k_R$. This dynamics of the Tulloch-Smith SQG model is analogous to that of the LANS $\ga$-model of three-dimensional Navier-Stokes turbulence [@article:Wingate:2005], in which, once again there is a single downscale energy cascade with $k^{-5/3}$ scaling in the energy spectrum, with a transition to a steeper $k^{-3}$ slope at higher wavenumbers 