Renormalization-group analysis of turbulence.

The direct interaction approximation (DIA), due to Kraichnan [1], was the first field-theoretical approach to the theory of turbulence. Formulated in terms of the Dyson equation, the DIA is characterized as the lowest-order approximation that includes nonlinear corrections to the propagator for the mode v(k,a>). It was shown [1] that, in the inertial range, the DIA gives the energy spectrum is E(k) oc &~/. This result contradicts both experimental data and the Kolmogorov theory of turbulence which gives E(k) oc fc/, perhaps with small corrections due to intermittency. The source of this discrepancy between the DIA and the Kolmogorov theory has long been understood [2]. The DIA does not distinguish between dynamic and kinematic interactions between eddies of widely separated length-scales. Small eddies are convected by large eddies in a purely kinematic way, which should not lead to energy redistribution between scales. The spurious effect of large-scale convection on small scales has been removed from the DIA by use of a Lagrangian description of the flow. This Lagrangian History Direct Interaction Approximation (LHDIA) [3] leads to the Kolmogorov |-energy spectrum with the Kolmogorov constant CK = 1.77 [see (11) below], which is in reasonable agreement with experiment [4]. However, application of the LHDIA to the problem of turbulent diffusion of a passive scalar does not lead to quantitive agreement with experimental data: The turbulent Prandtl number P t calculated from the LHDIA [4] is roughly 0.14, much smaller than the experimentally observed P t « 0.7-0.9. In 1977 Forster, Nelson, and Stephen [5] used dynamic renormalization group (RNG) methods, originally developed for the description of the dynamics of critical phenomena [6] to derive velocity correlations generated by the NavierStokes equation with a random force term. The ideas expressed in [5] have been used by others in the context of hydrodynamic turbulence [7-10]. The problem is formulated as follows: Consider the d-dimensional space-time Fouriertransformed Navier-Stokes equation for incompressible flow