Some estimates for magnetic fluctuations in a turbulent medium

Abstract

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B0(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b⋅curlb〉=−B0⋅curlB0. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B0⋅ curlB0/B02. The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b2〉≃β/η B02 for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a2〉 of the vector potential, rather than 〈b2〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B0 quantities to each other.