Self-similarity and the direct (enstrophy) cascade in two-dimensional fluid turbulence

A widely used statistical theory of 2D turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha={-3}$, in the inertial range associated with the direct cascade. In the presence of large-scale coherent structures, a power-law scaling is observed, but the exponent often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory which describes the key physical mechanism behind the direct cascade and sheds new light on the relationship between the structure of the large-scale flow and the scaling of the small-scale structures in the inertial range. This theory also goes a step beyond KLB, to predict the upper and lower bounds of the inertial range as well as the energy scaling in the dissipation range.