Statistical effect of interactions between resolvable and unresolvable scales of two-dimensional motion

This paper deals with a problem related to that of “subgrid-scale parameterization” of the statistical effects of unresolvable small-scale motions on the evolution of motions on resolvable scales. The approach is based on a second-moment closure in which the variables are an ensemble-averaged vorticity field, the mean transport of vorticity by small-scale deviations from the mean motion, the variance of small-scale deviations from the mean vorticity and their characteristic scale. This closure, which applies specifically to two-dimensional inviscid flow, differs from those based on the equations for the Reynolds stresses, in that it does not involve correlations between pressure and velocity fluctuations. It is found that, in the early stages of evolution, the mean transport of vorticity by small-scale motions is highly dependent on the local scale of the mean vorticity field relative to the characteristic scale of deviations from that mean. In particular, the statistical effect of small-scale motions is counter-gradient diffusive transport of mean vorticity if the local scale of the mean vorticity field is large, but down-gradient if the scale of the mean vorticity field is only slightly larger than the scale of deviations from that mean. Although the interpretation of our results for long times is not as simple as that described above, it is at least clear that mean vorticity is not universally transported down-gradient by the subgrid-scale motions.