EDP Sciences
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Volume 370, Number 2, May I 2001
Page(s) 533 - 540
Section Stellar atmospheres
DOI http://dx.doi.org/10.1051/0004-6361:20010076

A&A 370, 533-540 (2001)
DOI: 10.1051/0004-6361:20010076

The estimations of magnetic field fluctuations in turbulent atmospheres

N. A. Silant'ev1, 2

1  Instituto Nacional de Astrofísica, Óptica y Electrónica, Apartado Postal 51 y 216, Z.P. 72000, Pue. México
2  Main Astronomical Observatory of Russian Academy of Sciences, 196140 St. Petersburg, Russia

(Received 18 May 2000 / Accepted 29 November 2000 )

We qualitatively analyze the equations of the magnetohydrodynamics for various quadratic fluctuations of the magnetic field and the vector potential. For the stationary inhomogeneous state, we have established the exact integral relations between the magnetic field fluctuations ${\vec b}({\vec r},t)$ and the mean magnetic field ${\vec B}_0({\vec r},t)$. We estimate $\alpha \approx (\eta+\beta){\vec B}_0\cdot
(\nabla \times {\vec B}_0)/B_0^2$, where $\eta$ and $\beta$ are the ohmic (molecular) and turbulent diffusivities, respectively. The $\alpha$ -coefficient describes the enhancement of the mean magnetic field. We found that the exact Seehafer formula ${\vec B}_{0}\cdot \langle {\vec u}\times {\vec b}\rangle =
-\eta \langle {\vec b} \cdot (\nabla \times {\vec b})\rangle $ is also valid for the locally stationary and homogeneous evolution of magnetic fluctuations (${\vec u}$ is the Euler turbulent velocity). It is shown that the usual $\alpha$-coefficient, more correctly the term $\alpha {\vec B}_0\cdot (\nabla \times {\vec B}_0)$, presents with opposite signs in equations for B02 and $\langle b^2\rangle$, i.e. the rate of increase of the total magnetic energy does not depend directly on this coefficient. We show that at the moment tb of the maximum of magnetic fluctuations, $\langle b^2\rangle \approx B_0^2$. The same also holds for the locally homogeneous and stationary regime if the spectrum of the turbulence has a slow decrease in the inertial region of wave numbers. The detailed analysis of two-dimensional turbulence has shown that the Zeldovich estimate $\langle b^2\rangle \approx \beta /\eta B_0^2$ is valid only at the maximum of the vector potential fluctuations. This estimate is corrected such that at tb it also gives $\langle b^2\rangle \approx B_0^2$. We also give the approximate expression for the $\alpha$-coefficient which takes into account the back reaction of the mean magnetic field onto the turbulence itself.

Key words: magnetic fields -- turbulence -- stars: magnetic fields -- Sun: magnetic fields

© ESO 2001