$k$, because of the introduced distortion of the Navier-Stokes nonlinearity. In both cases, we are essentially dealing with a single homogeneous solution, associated with a single flux coefficient, which, to first approximation, we can consider bifractal.

Under the quasi-geostrophic models used by @article:Orlando:2003 and @article:Smith:2009, on the other hand, we are dealing with two independent homogeneous solutions each of which, to first approximation, can be considered monofractal. Because the two solutions are independent of each other, as far as the nonlinearity is concerned, it is possible for the transition wavenumber $k_t$, in principle, to have any arbitrary value, since its location is determined solely by the relative magnitude of the two homogeneous solutions against each other. From the viewpoint of this paper, the constraint $k_t \sim k_R$ is a weak indirect constraint that originates from the combined effect of anti-symmetric forcing and the large-scale Ekman dissipation term on the energy and potential enstrophy injection rates. This paper argues that the two-layer quasi-geostrophic model is consistent with placing $k_t$ near $k_R$.

The paper is organized as follows. In section 2 we review the two-layer quasi-geostrophic model and introduce various preliminaries. In section 3 we derive the potential enstrophy and energy forcing 