A&A 370, 533-540 (2001)

DOI: 10.1051/0004-6361:20010076

## The estimations of magnetic field fluctuations in turbulent atmospheres

**N. A. Silant'ev**

^{1, 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 )

** Abstract **

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
and the mean magnetic field
.
We estimate
, where and are the ohmic
(molecular) and turbulent diffusivities, respectively.
The -coefficient describes the enhancement of the mean magnetic
field. We found that the exact
Seehafer formula
is also valid
for the locally stationary and homogeneous evolution of magnetic
fluctuations ( is the Euler turbulent velocity).
It is shown that the usual -coefficient, more correctly
the term
, presents with opposite signs in
equations for *B*_{0}^{2} and
, i.e. the rate of increase of the
total magnetic energy does not
depend directly on this coefficient. We show that at the moment
*t*_{b} of the maximum of magnetic fluctuations,
. 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
is
valid only at the maximum of the vector potential fluctuations.
This estimate is corrected such that at *t*_{b} it also
gives
. We also give the approximate
expression for the -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*