We show that the transition to a $k^{-1}$ spectrum in the enstrophy inertial range of generalized two-dimensional turbulence can be derived analytically using the eddy damped quasinormal Markovianized (EDQNM) closure. The governing equation for the generalized two-dimensional fluid system includes a nonlinear term with a real parameter $\alpha$. This parameter controls the relationship between the stream function and generalized vorticity and the nonlocality of the dynamics. An asymptotic analysis accounting for the overwhelming dominance of nonlocal triads allows the $k^{-1}$ spectrum to be derived based upon a scaling analysis. We thereby provide a detailed analytical explanation for the scaling transition that occurs in the enstrophy inertial range at $\alpha =2$ in terms of the spectral dynamics of the EDQNM closure, which extends and enhances the usual phenomenological explanations.

• Received 12 January 2016

DOI:https://doi.org/10.1103/PhysRevFluids.1.034403