Anomalous scaling in anisotropic turbulence

We present a short review of the work conducted by our group on the subject of anomalous scaling in anisotropic turbulence. The basic idea that unifie s all the applications discussed here is that the equations of motion for c orrelation functions are always linear and invariant to rotations, and ther efore the solutions foliate into sectors of the symmetry group of all rotat ions (SO(3)). We have considered models of passive scalar and passive vecto r advections by a rapidly changing turbulent velocity field (Kraichnan-type models) for which we find a discrete spectrum of universal anomalous expon ents, with a different exponent characterizing the scaling behavior in ever y sector. Generically the correlation functions and structure functions app ear as sums over all these contributions, with nonuniversal amplitudes whic h are determined by the anisotropic boundary conditions. In addition we con sidered Navier-Stokes turbulence by analyzing simulations and experiments, and reached some interesting conclusions regarding the scaling exponents in the anisotropic sectors. The theory presented here clarifies questions lik e the restoration of local isotropy upon decreasing scales. We explain when the local isotropy is fully restored and when the lingering effects of the anisotropic forcing appear for arbitrarily small scales. (C) 2000 Elsevier Science B.V. All rights reserved.