Evolution of high Reynolds number two-dimensional turbulence

Kraichnan's (1967) predictions concerning a simultaneous direct enstrophy cascade and inverse energy cascade for high Reynolds number two-dimensional turbulence are tested numerically using a variant of the eddy-damped quasi-normal approximation. For the initial-value problem, an analytic study using this theory shows that, in the zero-viscosity limit, energy and enstrophy are conserved for arbitrarily long times, contrary to the three-dimensional case, where the energy is conserved for only a finite time, after which it is dissipated. Non-local effects in the enstrophy inertial range, which are difficult to treat by conventional numerical schemes (Leith 1971; Leith & Kraichnan 1972), are shown to be representable by an additional diffusion term in the spectral equation. The resulting equation, including non-local effects, is integrated numerically. When enstrophy and energy are continuously injected at a fixed wavenumber, it is shown numerically that a quasi-steady regime is obtained where enstrophy cascades to large wavenumbers across a k−3 inertial range with zero energy transfer while energy flows indefinitely to small wavenumbers across a $k^{-\frac{5}{3}}$ inertial range with zero enstrophy transfer.