Differential models for 2D turbulence

We present two phenomenological models for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order and a second-order differential equations respectively. Both equations respect the scaling properties of the original Navier-Stokes equations and it has both the -5/3 inverse-cascade and t -3 direct-cascade spectra. In addition, the fourth order equation has Raleigh-Jeans thermodynamic distributions, as exact steady state solutions. We use the fourth-order 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 obtain 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.