Coherent structures and the direct cascade in 2D turbulence

The classical theory of Kraichnan, Leith, and Batchelor predicts a universal power-law scaling for the direct (enstrophy) cascade in 2D turbulence. While power-law spectra are indeed observed in both experiments and simulations, the scaling exponent is found to be nonuniversal due to the presence of large-scale coherent structures. The direct cascade is dominated by the hyperbolic regions of the large-scale flow, where small-scale vorticity behaves like a passive scalar. For nearly-time-periodic large-scale flows, chaotic advection aligns vorticity filaments along the unstable manifolds of the saddle points. To investigate how the tangling of stable and unstable manifolds associated with different saddles leads to the emergence of a fractal structure and a fractal scaling exponent, we investigate a model problem which involves a passive scalar advected by a prescribed time-periodic flow that is qualitatively similar to large-scale flows found in DNS of 2D turbulence. This allows us to independently control the properties of the large-scale flow and investigate their impact on the scaling exponent.