Renormalization group in magnetohydrodynamic turbulence

The renormalization group (RNG) theory is applied to magnetohydrodynamic (MHD) equations written in Elshser variables, as done by Yakhot and Orszag for Navier-Stokes equations. As a result, a system of coupled nonlinear differential equations for the “effective” or turbulent “viscosities” is obtained. Without solving this system, it is possible to prove their exponential behavior at the “fixed point” and also determine the effective viscosity and resistivity. Strictly speaking, the results do not allow negative effective viscosity or resistivity, but in certain cases the effective resistivity can be continued to negative values, but not the effective viscosity. In other cases, the system tends to zero effective viscosity or resistivity. The range of possible values of the turbulent Prandtl number is also determined; the system tends to different values of this number, depending on the initial values of the viscosity and resistivity and the way the system is excited.