Symmetries and the approach to statistical equilibrium in isotropic turbulence

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Abstract
The relaxation in time of an arbitrary isotropic turbulent state to a state of statistical equilibrium is identified as a transition to a state which is invariant under a symmetry group. We deduce the allowed self-similar forms and time-decay laws for equilibrium states by applying Lie-group methods (a) to a family of scaling symmetries, for the limit of high Reynolds number, as well as (b) to a unique scaling symmetry, for nonzero viscosity or nonzero hyperviscosity. This explains why a diverse collection of turbulencemodels, going back half a century, arrived at the same time-decay laws, either through derivations embedded in the mechanics of a particular model, or through numerical computation. Because the models treat the same dynamical variables having the same physical dimensions, they are subject to the same scaling invariances and hence to the same time-decay laws, independent of the eccentricities of their different formulations. We show in turn, by physical argument, by an explicitly solvable analytical model, and by numerical computation in more sophisticated models, that the physical mechanism which drives (this is distinct from the mathematical circumstance which allows) the relaxation to equilibrium is the cascade of turbulence energy toward higher wave numbers, with the rate of cascade approaching zero in the low wave-number limit and approaching infinity in the high wave-number limit. Only the low-wave-number properties of the initial state can influence the equilibrium state. This supplies the physical basis, beyond simple dimensional analysis, for quantitative estimates of relaxation times. These relaxation times are estimated to be as large as hundreds or more times the initial dominant-eddy cycle times, and are determined by the large-eddy cycle times. This mode of analysis, applied to a viscousturbulent system in a wind tunnel with typical initial laboratory parameters, shows that the time necessary to reach the final stage of decay is astronomically large, and would require a wind tunnel of astronomical length.

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