Passive advection of a vector field: Anisotropy, finite correlation time, exact solution, and logarithmic corrections to ordinary scaling

Abstract

In this work we study the generalization of the problem, considered in [Phys. Rev. E 91, 013002 (2015)], to the case of finite correlation time of the environment (velocity) field. The model describes a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow. Inertialrange asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and preassigned pair correlation function. Due to the presence of distinguished direction n, all the multiloop diagrams in this model are vanish, so that the results obtained are exact. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to the two nontrivial fixed points of the RG equations. Their stability depends on the relation between the exponents in the energy spectrum E ∝ k 1−ξ ⊥ and the dispersion law ω ∝ k 2−η ⊥ . In contrast to the well known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale L.

Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis neard= 4

“…In the RG+OPE approach they are connected with IR attractive fixed points of RG equations. To investigate them we will follow the logic of [19,20] and [30]: since the renormalization constants Z i and the RG functions of the reduced model (11) do not depend on the parameters, connected with the advection of passive field (10), it is possible to divide our task in two parts. The first part is devoted to investigation of the stochastic Navier-Stokes equation itself, i.e., we will deal only with the action functional (11).…”

“…In a number of papers the RG+OPE approach was applied to the case of passive advection by Kraichnan's ensemble (velocity field is a e-mail: n.antonov@spbu.ru b e-mail: n.gulitskiy@spbu.ru c e-mail: kontramot@mail.ru d e-mail: tomas.lucivjansky@uni-due.de taken isotropic, Gaussian, not correlated in time, having a power-like correlation function, and the fluid was assumed to be incompressible), see [7][8][9][10], and by numerous of its generalizations: largescale anisotropy, helicity, compressibility, finite correlation time, non-Gaussianity, and more general form of nonlinearity [6,[11][12][13][14][15][16]. Also this approach can be generalized to the case of non-Gaussian velocity field, governed by the stochastic Navier-Stokes equation -both for the scaling behaviour of the velocity field itself and for the advection of passive fields by this ensemble [17][18][19][20][21][22].…”

Eddy diffusivities of inertial particles in random Gaussian flows

“…Returning to the heavy particle case, in order to further simplify the expression for the eddy diffusivity, let us focus on a 2D carrier flow with a single wave-number k 0 . The correlation function we consider is [13]:…”

Anisotropic Turbulent Advection of a Passive Vector Field: Effects of the Finite Correlation Time

“…The analysis of the β-functions reveals four different IR attractive fixed points depending on the exponents ξ and η: if ξ < 0, η < 0, or if η > 0, η − ξ > 0, we have two gaussian fixed points with g * = 0 (regimes 1a and 2a, respectively; see Fig. 2); if ξ > 0, η < 0, or if η > 0, ξ − η > 0, we have two nontrivial regimes, called 1b and 2b, respectively; for detailes of the calculation see [16]. Thus we can conclude that the domains of IR stability in the vector model (5) coincide with the corresponding domains of IR stability in the scalar model considered in [7].…”