LARGE-N LIMIT OF THE SPHERICAL MODEL OF TURBULENCE

We discuss a ''spherical model' of turbulence proposed recently by Mou and Weichman [Phys. Rev. Lett. 70, 1101 (1993)] and point out its clo se 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 --> + infinity of the sph erical 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 nonperturbati ve and is based on the Martin-Siggia-Rose functional formalism for ver tex reversion. Systematic corrections to the DIA equations for the sph erical model are developed in a 1/square-root N expansion for a ''self -consistent vertex.'' The coefficients of the expansion are given at e ach order as the solutions of linear, inhomogeneous functional equatio ns which represent an infinite resummation of terms in the expansion i n 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.