Large-Nlimit of the ‘‘spherical model’’ of turbulence

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Abstract
We discuss a ‘‘spherical model’’ of turbulence proposed recently by Mou and Weichman [Phys. Rev. Lett. 70, 1101 (1993)] and point out its close similarity to the original ‘‘random coupling model’’ of Kraichnan [J. Math. Phys. 2, 124 (1961)]. The validity of the direct-interaction-approximation (DIA) equations in the limit N→+∞ of the spherical model, already proposed by Mou and Weichman, is demonstrated by another method. The argument also gives an alternative derivation of DIA for the random-coupling model. Our proof is entirely nonperturbative and is based on the Martin-Siggia-Rose functional formalism for vertex reversion. Systematic corrections to the DIA equations for the spherical model are developed in a 1/ √N expansion for a ‘‘self-consistent vertex.’’ The coefficients of the expansion are given at each order as the solutions of linear, inhomogeneous functional equations which represent an infinite resummation of terms in the expansion in the bare vertex. We discuss the problem of anomalous scaling in the spherical model, with particular attention given to ‘‘spherical shell models’’ which may be studied numerically.