The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form E(k) proportional to (1-epsilon), and the correlation time at the wave number k scales as k(-2+eta). It is shown that, depending on the values of the exponents epsilon and eta, the model in the inertial-convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior. The explicit asymptotic expressions for the structure functions and other correlation functions are obtained; the anomalous exponents, determined by the scaling dimensions of the scalar gradients, are calculated to the first order in epsilon and eta in any space dimension. For the first nontrivial regime the anomalous exponents are the same as in the rapid-change version of the model; for the second they are the same as in the model with time-independent (frozen) velocity field. In these regimes, the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator. For the last regime the exponents are nonuniversal (they can depend also on the amplitudes); however, the nonuniversality can reveal itself only in the second order of the RG expansion. A brief discussion of the passive advection in the non-Gaussian velocity field governed by the nonlinear stochastic Navier-Stokes equation is also given.

The field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by the Gaussian nonsolenoidal velocity field with finite correlation time, in the presence of largescale anisotropy. The energy spectrum of the velocity in the inertial range has the form E(k) ∝ k 1−ε , and the correlation time at the wavenumber k scales as k −2+η . It is shown that, depending on the values of the exponents ε and η, the model exhibits various types of inertial-range scaling regimes with nontrivial anomalous exponents. Explicit asymptotic expressions for the structure functions and other correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy) anomalous exponents, calculated to the first order in ε and η in any space dimension. These anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The anomalous exponents depend explicitly on the degree of compressibility.

The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. The velocity field is Gaussian, δ-correlated in time, and scales with a positive exponent ξ. Explicit inertial-range expressions for the magnetic correlation functions are obtained; they are represented by superpositions of power laws with non-universal amplitudes and universal (independent of the anisotropy and forcing) anomalous exponents. The complete set of anomalous exponents for the pair correlation function is found non-perturbatively, in any space dimension d, using the zeromode technique. For higher-order correlation functions, the anomalous exponents are calculated to O(ξ) using the renormalization group. The exponents exhibit a hierarchy related to the degree of anisotropy; the leading contributions to the even correlation functions are given by the exponents from the isotropic shell, in agreement with the idea of restored small-scale isotropy. Conversely, the small-scale anisotropy reveals itself in the odd correlation functions : the skewness factor is slowly decreasing going down to small scales and higher odd dimensionless ratios (hyperskewness etc.) dramatically increase, thus diverging in the r → 0 limit.

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