D-dimensional developed MHD turbulence: Double expansion model

Abstract

Developed magnetohydrodynamic turbulence near two dimensions $d$ up to three dimensions has been investigated by means of renormalization group approach and double expansion regularization. A modification of standard minimal subtraction scheme has been used to analyze the stability of the Kolmogorov scaling regime which is governed by the renormalization group fixed point. The exact analytical expressions have been obtained for the fixed points. The continuation of the universal value of the inverse Prandtl number $u=1.562$ determined at $d=2$ up to $d=3$ restores the value of $u=1.393$ which is known in the kinetic fixed point from usual $ε$-expansion. The magnetic stable fixed point has been calculated and its stability region has been also examined. This point losses stability: (1) below critical value of dimension $d_c=2.36$ (independently on the $a$-parameter of a magnetic forcing) and, (2) below the value of $a_c=0.146$ (independently on the dimension).