Structure of the three-point correlation function of a passive scalar in the presence of a mean gradient

The three-point correlation function of a passive scalar advected by a random, incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological Hopf equations. Numerical solutions are provided in the case where the velocity is Gaussian, white in time, and with a power law in space, [[v((r) over right arrow)-v((0) over right arrow)](2)]similar to r(2-epsilon), and in the model introduced by Shraiman and Siggia [C. R. Acad. Sci. 321, 279 (1995)]. Anomalous scaling exponents are found in both models. The numerics agrees with all the available analytic, perturbative results. In addition, the angular dependence of the correlation function is explicitly determined. In the Batchelor limit of random advection by a smooth velocity field, the exponent is found to remain very close to 1, as found experimentally. In this limit, the three-point correlation function is found to be very well represented, away from collinearity, by an explicit integral representation.