Journal of Statistical Physics, Vol. 108, Nos. 3/4, August 2002. p. 591-645.
A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier-Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnan's eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnan's theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.