The field theoretic renormalization group (RG) and the operator product expansion are applied to the model of a transverse (divergence-free) vector quantity, passively advected by the "synthetic" turbulent flow with a finite (and not small) correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity; it contains as special cases the kinematic dynamo model, linearized Navier-Stokes (NS) equation, the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic NS equation. The statistics of the advecting velocity field is Gaussian, with the energy spectrum E(k) ∝ k 1−ε and the dispersion law ω ∝ k −2+η , k being the momentum (wave number). The inertial-range behavior of the model is described by seven regimes (or universality classes) 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 one to construct a regular perturbation expansion in ε and η; the actual calculation is performed to the first order (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. The relevance of these results for the real developed turbulence described by the stochastic NS equation is discussed.
The effect of a random velocity field on the kinetics of the single-species annihilation reaction A+A--> is analyzed near two dimensions with the aid of the perturbative renormalization group. The previously found asymptotic behavior induced by density fluctuations only in the diffusion-limited reaction is shown to be unstable to any velocity fluctuations (including thermal fluctuations near equilibrium) in spatial dimensions d=d(c)=2. Four different stable long-time asymptotic regimes induced by the combined effect of velocity and density fluctuations are identified and the corresponding decay rates calculated in the leading order.
Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function ∝ δ(t − t ′ )k 4−d−y and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier-Stokes equation with a random Gaussian force with the correlation function ∝ δ(t − t ′ )k 4−d−y ′ . Depending on the relations between the exponents y and y ′ and the space dimensionality d, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in y, y ′ and ε = 4 − d).
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