MODE-COUPLING THEORY, DYNAMIC SCALING, AND 2-DIMENSIONAL TURBULENCE

A self-consistent mode-coupling scheme, along with dynamic scaling ideas, is used to obtain a renormalized perturbation theory in the Eulerian framework from Wyld's perturbation theory of the forced Navier-Stokes equation. For the force-correlation behaving as k(-(d-4+y)), the Kolmogorov and Kraichnan-Batchelor scaling spectra of two-dimensional turbulence for the inverse energy cascade, E(k) = C epsilon(-2/3)k(-5/3), and the direct entropy cascade, E(k) = C'chi(-2)/(3)k(-3)(In k/k(0))(-1/3), are obtained for y = 4 and y = 6 respectively, including the logarithmic correction for the latter. Unlike the usual Eulerian formulations (e.g. the direct-interaction approximation), the theory is finite in the energy regime, while it becomes marginal in the enstrophy regime, leading to the logarithmic correction. Calculations yield C = 6.447 and C' = 1.923 at one-loop order, which are in exact agreement with those of field-theoretic renormalization group calculations [P. Olla, Phys. Rev. Lett. 67, 2465 (1991)]. However, a self-consistent treatment of the logarithmic scalings in E(lc) and the inverse response-time yields a different value: C' = 2.201. The theory is free of any external parameter; the choice of y(= 4 or 6) is dictated by the condition of conserved transfer of energy or enstrophy.