large scales at the bottom layer but not at the top layer. Since the small-scale hyperdiffusion is not physically inherent in the quasi-geostrophic dynamics, there is no physical reason to prefer symmetric over asymmetric hyperdiffusion either. Unfortunately, there are still many open questions concerning the theory of the flux inequality. Consequently, the numerical results of @article:Orlando:2003 notwithstanding, there is still some uncertainty on whether the two-layer model can have a robust energy dissipation sink that can break the flux inequality in the inertial range.

It should be noted that when the same dissipation operator is used on both layers, it can be proved that the flux inequality is satisfied for all wavenumbers not in the forcing range. For that case, the prediction of @article:Charney:1971, that quasi-geostrophic turbulence will be isomorphic to two-dimensional turbulence, is expected to hold. This was confirmed in a numerical simulation by @article:Lindborg:2010, where the dissipation operator and the forcing term are both independent of the vertical coordinate. Recently, @article:Smith:2009 proposed a more sophisticated two-layer two-mode quasi-geostrophic model that has succeeded in reproducing the Nastrom-Gage spectrum. More importantly, using their quasi-geostrophic model, @article:Smith:2009 confirmed that it is possible for a downscale potential enstrophy cascade to coexist with 