Abstract
The turbulent passive advection under the environment (velocity) field with finite correlation time is studied. Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is investigated by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and prescribed pair correlation function. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to nontrivial fixed points of the RG equations and depend on the relation between the exponents in the energy energy spectrum e ∝ k ⊥ 1-ξ and the dispersion law ω ∝ k ⊥ 2-η . The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field itself. In contrast to the well-known isotropic Kraichnan model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: instead of power-like corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L . Due to the presence of the anisotropy in the model, all multiloop diagrams are equal to zero, thus this result is exact.
Highlights
Understanding the turbulent advection is a rich and challenging problem
This approximation corresponds to the passive field approximation: if we neglect the influence of advected field θ to the dynamics of the environment field v, the latter can be modeled by statistical ensembles with prescribed properties
One of the most convenient ways to study the anomalous scaling in various statistical models of the turbulent advection is to apply the field theoretic renormalization group (RG) and operator product expansion (OPE); see, e.g., the monographs [10, 11]
Summary
Understanding the turbulent advection is a rich and challenging problem. On the one hand, the violation of the classical Kolmogorov–Obukhov theory [1] is even more strongly pronounced for a advected field than for the velocity field itself; see, e.g., review [2]; on the other hand, the problem of passive advection appears to be easier tractable theoretically. We will investigate the “strongly anisotropic” model, which is obtained by introducing the velocity field v, oriented along a fixed direction n: u(t, x) = n × v(t, x⊥) This problem is closely related to the processes taken place in solar corona, e.g., with solar wind [4], and n is an “orientation of a large-scale flare” in this context. One of the most convenient ways to study the anomalous scaling in various statistical models of the turbulent advection is to apply the field theoretic renormalization group (RG) and operator product expansion (OPE); see, e.g., the monographs [10, 11] In this case the anomalous scaling is a consequence of the existence in the model of composite fields (“composite operators” in the quantumfield terminology) with negative scaling dimensions; see [12, 13] and references therein. The main result of the present paper is that the inertial-range behavior of vector fields advected by velocity ensemble with finite correlation time combines the features of both above models: as in the scalar case, there is a set of fixed points, governing the infrared (IR) behavior; as in the zero-time correlation model, the inertial-range behavior of vector fields has logarithmic corrections to ordinary scaling
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