Unified Scaling Theory for Local and Non-local Transfers in Generalized Two-dimensional Turbulence

The enstrophy inertial range of a family of two-dimensional turbulent flows, so-called -turbulence, is investigated theoretically and numerically. Introducing the large-scale correction into Kraichnan–Leith– Batchelor theory, we derive a unified form of the enstrophy spectrum for the local and non-local transfers in the enstrophy inertial range of -turbulence. An asymptotic scaling behavior of the derived enstrophy spectrum precisely explains the transition between the local and non-local transfers at 1⁄4 2 observed in the recent numerical studies by Pierrehumbert et al. [Chaos, Solitons & Fractals 4 (1994) 1111] and Schorghofer [Phys. Rev. E 61 (2000) 6572]. This behavior is comprehensively tested by performing direct numerical simulations of -turbulence. It is also numerically examined the validity of the phenomenological expression of the enstrophy transfer flux responsible for the derivation of the transition of scaling behavior. Furthermore, it is found that the physical space structure for the local transfer is dominated by the small scale vortical structure, while it for the non-local transfer is done by the smooth and thin striped structures caused by the random straining motions.