Gl. Eyink, LARGE-N LIMIT OF THE SPHERICAL MODEL OF TURBULENCE, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 49(5), 1994, pp. 3990-4002
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.