Multicomponent turbulence, the spherical limit, and non-Kolmogorov spectra

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Abstract
A set of models for homogeneous, isotropic turbulence is considered in which the Navier-Stokes equations for incompressible fluid flow are generalized to a set of N coupled equations in N velocity fields. It is argued that in order to be useful these models must embody a new group of symmetries, and a general formalism is laid out for their construction. The work is motivated by similar techniques that have had extraordinary success in improving the theoretical understanding of equilibrium phase transitions in condensed matter systems. We consider two classes of models: a simpler class (model I), which does not contain an exact Galilean symmetry, and a more complicated, extended class (model II), which does. The key result is that these models simplify when N is large. The so-called spherical limit N→∞ can be solved exactly, yielding closed sets of nonlinear integral equations for the response and correlation functions. For model I, these equations, known as Kraichnan’s direct interaction approximation equations, are solved fully in the scale-invariant turbulent regime. For model II, these equations are more complicated and their full solution is left for future work. Implications of these results for real turbulence (N=1) are discussed. In particular, it is argued that previously applied renormalization group techniques, based on an expansion in the exponent y that characterizes the driving spectrum, are incorrect and that the Kolmogorov exponent ζ has a nontrivial dependence on N, with ζ(N→∞)=3/2 for both sets of models. This value is close to the experimental result ζ≃5/3, which must therefore result from higher-order corrections in powers of 1/N. Prospects for calculating these corrections are briefly discussed: though daunting, such calculations might provide a controlled perturbation expansion for the Kolmogorov, and other, exponents. Our techniques may also be applied to other nonequilibrium dynamical problems, such as the Kardar-Parisi-Zhang equation for interface growth, and perhaps to turbulence in nonlinear wave systems.