Renormalization group and operator product expansion in turbulence: Shell models

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Abstract
A general role of the renormalization group (RG) in the theory of fully developed turbulence is proposed, with the simple case of the shell models as an illustrative example. A Wilson-type RG is defined, i.e., a transformation in a space of shell-dynamics ‘‘subgrid models’’ with fixed uv cutoff, for a class of theories with fixed mean dissipation and strength of quadratic nonlinearity. It is explained that, if a zero-viscosity limit exists, then its ‘‘subgrid’’ dynamics below the cutoff is necessarily (near) a fixed point of the RG transformation. Conversely, any RG fixed-point subgrid model is associated to a zero-viscosity limit. By means of an ‘‘asymptotic completeness’’ assumption for the fixed point, a high shell-number expansion is established, analogous to the operator product expansion (OPE) of field theory. This expansion predicts characteristic ‘‘multifractal scaling’’ for shell variable moments and also relations between inertial and dissipation range scaling exponents. Furthermore, under the plausible assumption of an ‘‘additive OPE,’’ a predicted scaling form for two-point moment correlations is established. The results of this paper are nonperturbative but only of a qualitative character, based upon precise assumptions about the fixed-point theory. However, we also discuss the possibility of an implementation of RG by numerical methods (Monte Carlo, decimation, etc.) or perturbation expansion to test the assumptions and to make a quantitative evaluation of the scaling exponents. The relation of RG to naive ‘‘cascade ansatz’’ is also discussed.