Stability of Kolmogorov scaling in anisotropically forced turbulence

Numerical investigation of scaling regimes in a model of an anisotropically advected vector field

“…In what follows, we shall present only the first part of the RG analysis, namely, we shall analyze the influence of the small-scale anisotropy on the infrared (IR) stability of the possible scaling regimes of the model. It will be seen that complexity of this task is also very close to the corresponding problem in the stochastic Navier-Stokes equation [15].…”

“…In action (7) the terms with new parameters χ 10 , χ 20 , and χ 30 are related to the presence of small-scale anisotropy and they are necessary to make the model multiplicatively renormalizable. Model (7) corresponds to a standard Feynman diagrammatic technique (see, e.g., [12,15] for details) and the standard analysis of canonical dimensions then shows which one-irreducible Green functions can possess UV superficial divergences.…”

“…Here we only conclude that using the RG analysis leads to the following result: possible scaling regimes are given by the IR stable fixed points of the system of five nonlinear RG differential equations (flow equations or also known as Gell-Mann-Low equations) for five scale dependent effective variables (charges)C = {ḡ,ū,χ 1 ,χ 2 ,χ 3 } of the model which are functions of the dimensionless scale parameter t = k/Λ [5,15]. The system of the flow equations is defined by the so-called β-functions of the model (they are functions of the charges, anisotropy parameters, space dimension, and parameters ε, η) and it has the following form…”

Stability of scaling regimes ind>~2developed turbulence with weak anisotropy

“…[8], in the anisotropic magnetohydrodynamic developed turbulence a stable regime generally does not exist. In [7,9], d-dimensional models with d > 2 were investigated for two cases: weak anisotropy [7] and strong one [9], and it has been shown that the stability of the isotropic fixed point is lost for dimensions d < d c = 2.68. It has also been shown that stability of the fixed point even for dimension d = 3 takes place only for sufficiently weak anisotropy.…”

“…It has also been shown that stability of the fixed point even for dimension d = 3 takes place only for sufficiently weak anisotropy. The only problem in these investigations is that it is impossible to use them in the case d = 2 because new ultra-violet (UV) divergences appear in the Green functions when one considers d = 2, and they were not taken into account in [7,9].…”