Differential model for 2D turbulence

We present a phenomenological model for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order differential equation. This equation respects the scaling properties of the original Navier-Stokes equations and it has both the $-5/3$ inverse-cascade and the $-3$ direct-cascade spectra. In addition, our model has Raleigh-Jeans thermodynamic distributions, as exact steady state solutions. We use the model to derive a relation between the direct-cascade and the inverse-cascade Kolmogorov constants which is in a good qualitative agreement with the laboratory and numerical experiments. We discuss a steady state solution where both the enstrophy and the energy cascades are present simultaneously and we discuss it in context of the Nastrom-Gage spectrum observed in atmospheric turbulence. We also consider the effect of the bottom friction onto the cascade solutions, and show that it leads to an additional decrease and finite-wavenumber cutoffs of the respective cascade spectra which agrees with existing experimental and numerical results.