The turbulent velocity field in wave-number space is decomposed into two distinct fields. One is a purely chaotic field; while the other is a correction fteld, and carries all the phase information. Application of this decomposition to a thin shell of wave numbers in the dissipation range allows the elimination of modes in that shell, with the usual made-coupling problems being circumvented by the use of a conditional average. The (conditional) mean effect of the eliminated modes appears as an increment to the viscosity, with terms of order A. being neglected, where A, is a dimensionless measure of bandwidth thickness, such that O~A.~1. An iteration (with appropriate rescaling) to successively lower shells reaches a fixed point, corresponding to a renormalized turbulent viscosity. As previously reported [W.D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)],the spectrum of the purely chaotic field is found to take the Kolmogorov -5/3 power-law form, with a value for the Kolmogorov spectral constant of a = 1.6, independent of A, over the range of bandwidths for which the theory is valid.PACS number(s): 47.10.+ g, 47.25.Cg
A field is represented by its Fourier modes u(k, t) on the wave-number interval 0~k + k,". We seek to reduce the number of degrees of freedom needed to describe the system by eliminating those modes in the band of wave numbers k, k~k, ", the u+, while retaining their average effect on the remaining modes, the u . Because of mode-mode coupling, this requires, in principle, a conditional average over the u+, in a subensemble in which the u are held (approximately) constant. The conditional average can be related to the full ensemble average by means of the decomposition u+ =v++ 6+, where v+ is independent of u but has the same global mean properties as u+, while h, + represents the effect of mode coupling on the conditional average. Implementation of the formalism for any particular physical system requires an ansatz for the relationship between v+ and u+ such that the conditional average of h, + is small. This procedure is illustrated by its application to the Navier-Stokes equation, where v+ is taken to be the first-order expansion of u+ in a Taylor series about k = k,", thus permitting a renormalization group ca-lculation of the turbulent effective viscosity, as reported previously [Phys. Rev. Lett. 65, 3281 (1990)]. For systems like thisw, ith deterministic time evolution, the conditional average must be based on the weak, or imprecise, condition that u +4 is held constant, where 4 is small (but not zero) and has zero mean under the conditional average, in order to ensure chaotic u+. We show that our formalism, relating conditional and full ensemble averages, is independent of @ to second order in small quantities.PACS number(s): 05.20. -y, 05.45.+b, 47.25.Cg
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