Turbulence with Pressure: Anomalous Scaling of a Passive Vector Field

Abstract

The field-theoretic renormalization group (RG) and the operator product expansion are applied to the model of a divergence-free vector quantity, passively advected by the ``synthetic'' turbulent flow with a finite correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity. The statistics of the advecting velocity field is Gaussian with powerlike energy spectrum and dispersion law. The inertial-range behavior of the model is described by seven regimes that correspond to nontrivial fixed points of the RG equations and exhibit anomalous scaling. The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field, which allows to construct a regular perturbation expansion. The actual calculation is performed in one-loop approximation, including the anisotropic sectors. Universality of the exponents, their (in)dependence on the forcing, effects of the large-scale anisotropy, compressibility and pressure are discussed. In particular, for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy: the more anisotropic is the contribution of a composite operator to a correlation function, the faster it decays in the inertial-range. Relevance of the results for the real developed turbulence described by the stochastic Navier-Stokes equation is discussed.