Non-universal scaling for the direct cascade in 2D turbulent flow*

For 2D turbulent flows, Kraichnan-Leith-Batchelor’s (KLB) theory predicts an energy scaling $E(k) propto k^{-3}$ for the direct cascade. However, the presence of large-scale coherent structures leads to a power-law scaling with a different -- typically fractal -- exponent. The dominant physical mechanism underlying the direct cascade involves stretching and folding of vorticity filaments in the hyperbolic regions of the large-scale flow. We show that the deviations from the KLB predictions are due to the interaction between adjacent hyperbolic regions which tends to orient vorticity filaments with respect to the expanding and contracting directions of the large-scale flow. This orientational effect leads to the emergence of self-similar structure of small-scale vorticity characterized by a fractal scaling exponent.