The stationary condition (Hopf equation) for the $(n+1\mathrm{})$-point correlation function of a passive scalar advected by turbulent flow is argued to have an approximate $\mathrm{SL}\left(n,R\right)$ symmetry which provides a starting point for our phenomenological theory in which less symmetric terms are treated perturbatively. The large scale anisotropy is found to be a relevant field, in contradiction with Kolmogorov phenomenology, but in agreement with the large scalar skewness observed in shear flows. Exponents are not universal, yet quantitative predictions for experiments to test the $\mathrm{SL}\left(n,R\right)$ symmetry can be formulated in terms of the correlation functions.

• Received 20 February 1996

DOI:https://doi.org/10.1103/PhysRevLett.77.2463