FLUCTUATIONS IN MEAN-FIELD DYNAMOS

D.Sokoloff, Moscow State University, Russia

12.05.2014

A traditional approach in treatment of various physical problems in random media is based on a consideration of mean-field equation for the physical field under investigation. This approach in dynamo theory was developed about 50 years ago and is known as mean-field dynamo. Of course, it is

possible now to avoid mean-field approach and address a dynamo problem in the framework of direct numerical

simulations only. The point however is that the mean-field description of a dynamo problem remains important for

interpretation of results of direct numerical simulations. Particularly important roles plays mean-field dynamos in

astrophysical problems because our knowledge concerning internal structure of remote celestial bodies is

usually very limited and gives some mean quantities only.

A specific feature of mean-field dynamos is that the number of the random turbulent or convective shells which are

involved in the averaging in development of mean-field dynamo models is usually rather large (say, $N=104$) however

much lower then, say, Avogardo number which controls averaging in molecular physics. Correspondingly, fluctuations

in mean-field dynamos play a much more important role in mean-field dynamos rather in other domains of statistical physics.

The talk presents a review of a sequence of recent papers which consider the role of fluctuations in mean-field dynamos.

We argue that the most important kind of fluctuations are fluctuations of the $\alpha$-coefficient which plays a crucial role in many mean-field dynamos.

The point is that $\alpha$- effect is usually quite weak (a naive estimate is $\alpha \approx 0.1 v$ where $v$ is the rms velocity) and one can expect 10\% -- 20 \% fluctuations of $\alpha$ for $N=104$. We argue that this level of $\alpha$-fluctuations in solar dynamos is sufficient to explain such phenomena as Maunder minimum and other transient phenomena in solar cyclic activity. A possible role of $\alpha$-fluctuations in geodynamo is discussed.

We discuss a possible role of large-scale fluctuations of mean-field velocities and its possible importance for

advective dynamos as well as the large-scale manifestations of small- scale magnetic fluctuations in solar mean-field dynamos.