A new set of models for homogeneous, isotropic turbulence is considered in which theNavier-Stokes equations for incompressible fluid flow are generalized to a set of Ncoupled equations in N velocity fields. It is argued that in order to be useful thesemodels must embody a new group of symmetries, and a general formalism is laid outfor their construction. The work is motivated by similar techniques that have had extraordinarysuccess in improving the theoretical understanding of equilibrium phasetransitions in condensed matter systems. The key result is that these models simplifywhen N is large. The so-called spherical limit, N → ∞ can be solved exactly,yielding a closed pair of nonlinear integral equations for the response and correlationfunctions. These equations, known as Kraichnan's Direct Interaction Approximation(DIA) equations, are, for the first time, solved fully in the scale-invariant turbulentregime, and the implications of these solutions for real turbulence (N = 1) arediscussed. In particular, it is argued that previously applied renormalization grouptechniques, based on an expansion in the exponent, y, that characterizes the drivingspectrum, are incorrect, and that the Kolmogorov exponent ς has a nontrivial dependenceon N, with ς(N → ∞) = 3/2 This value is remarkably close to the experimentalresult, ς ≈ 5/3, which must therefore result from higher order corrections in powers of1/N. Prospects for calculating these corrections are briefly discussed: though daunting,such a calculations would, for the first time, provide a controlled perturbation expansionfor the Kolmogorov, and other, exponents. Our techniques may also be applied to other nonequilibrium dynamical problems, such as the KPZ equation for interfacegrowth, and perhaps to turbulence in nonlinear wave systems.
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