# Effective Amp\ere force in developed magnetohydrodynamic turbulence Academic Article

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### abstract

• The interaction between a large-scale uniform weak magnetic field B and a developed small-scale magnetohydrodynamic (MHD) turbulence is studied. It is found that the effective mean Amp\ere force in the turbulence is given by ${\mathbf{F}}_{\mathit{m}}$=-\ensuremath{\nabla}(${\mathit{Q}}_{\mathit{p}}$${\mathbf{B}}^{2}$/8\ensuremath{\pi})+(B\ensuremath{\cdot}\ensuremath{\nabla})${\mathit{Q}}_{\mathit{s}}$B/4\ensuremath{\pi}. The turbulent magnetic coefficients ${\mathit{Q}}_{\mathit{p}}$ and ${\mathit{Q}}_{\mathit{s}}$ are drastically decreased at large magnetic Reynolds numbers, whereas in the absence of turbulence ${\mathit{Q}}_{\mathit{p}}$=${\mathit{Q}}_{\mathit{s}}$=1. This phenomenon arises due to a negative contribution of the MHD turbulence to the mean magnetic force. This is caused by the generation of magnetic fluctuations at the expense of fluctuations of the velocity field. This effect is nonlinear in terms of the large-scale magnetic field. It is shown here that in turbulence with a mean large-scale magnetic field, a universal ${\mathit{k}}^{\mathrm{\ensuremath{-}}1}$ spectrum of magnetic fluctuations exists; this spectrum is independent of the exponent of the spectrum of the turbulent velocity field. A variant of the renormalization group (RNG) method allows the derivation of the scaling and amplitude of the turbulent transport coefficients: turbulent viscosity, turbulent magnetic diffusion, and turbulent magnetic coefficients. A small parameter in the RNG method is the ratio \ensuremath{\varepsilon}=${\mathit{B}}^{2}$/(8\ensuremath{\pi}${\mathit{W}}_{\mathit{k}}$) of the large-scale magnetic energy density to the energy density ${\mathit{W}}_{\mathit{k}}$=(1/2)\ensuremath{\rho}〈${\mathit{u}}^{2}$〉 of the turbulent velocity field.

### publication date

• October 1, 1994