Low order models representing realizations of turbulence

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Abstract
The equations governing two-dimensional turbulence are written as an infinite system of ordinary differential equations, in which the dependent variables are the coefficients in the expansion of the vorticity field in a double Fourier series. The variables are sorted into sets which correspond to consecutive bands in the wavenumber spectrum; within each set it is supposed that the separate variables will exhibit statistically similar behaviour. A low order model is then constructed by retaining only a few variables within each set. Multiplicative factors are introduced into the equations to compensate for the reduced number of terms in the summations. Like the original equations, the low order equations conserve kinetic energy and enstrophy, apart from the effects of external forcing and viscous dissipation.A special case is presented in which the bands are half octaves and there is effectively only one dependent variable per set. Solutions of these equations are compared with conventional numerical simulations of turbulence, and agree reasonably well, although the nonlinear effects are somewhat underestimated.