Spectral reduction for two-dimensional turbulence

A new method for predicting the statistical properties of fluid turbulence, called Spectral Reduction, is described. Collections of Fourier wavenumbers are represented by certain nonuniformly spaced sample modes that interact via enhanced coupling coefficients. The approximation reduces to the exact Navier-Stokes equation as a control parameter, the number of fundamental wavenumbers associated with each sample mode, tends to unity. Even at large values of this parameter, the time-averaged predictions of the theory can be shown to recover the statistics of the exact dynamics to high accuracy. Preliminary results from the numerical implementation of Spectral Reduction for two-dimensional homogeneous turbulence axe very encouraging; for example, the method is used to illustrate a recent modification to Kraichnan's logarithmically corrected two-dimensional enstrophy cascade. Higher-order statistics, such as the kurtosis, are also examined: for weak forcing, it is found that the probability distribution function is highly non-Gaussian