A class of phenomenological Hopf equations describing mixing of a passive scalar by random flow close to the Batchelor limit (i.e., advection by random strain and vorticity) is analyzed. In the Batchelor limit multipoint correlators of the scalar are constructed explicitly by exploiting the SL $(N,R)$ symmetry of the Hopf operator. Hopf equations close to this “integrable” limit are solved via singular perturbation theory based on matched asymptotic expansions. The solution for the three-point correlator exhibits anomalous scaling indicating persistence of the small scale anisotropy for the scalar. In addition to the exponent, the full configuration dependence of the correlator is obtained.

• Received 6 August 1997

DOI:https://doi.org/10.1103/PhysRevE.57.2965