  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.
