N. Leprovost, Berengere Dubrulle,  P-H. Chavanis, CEA Saclay

The recent success of two experimental fluid dynamo [1,2]
has renewed the interest in the mechanism of dynamo saturation,
and, thus, of equilibrium configuration in MHD. At the present time,
there is no general theory to tackle this problem, besides dimensional theory.
For example, in a conducting fluid with typical velocity $V$, density $\rho$ ,
Reynolds number $Re$ and magnetic Prandtl number $Pm$,
the typical level of magnetic field reached at saturation is necessarily [3]:

B^2=\mu_o\rho V^2 f(Re,Pm),

where $f$ is a priori an arbitrary function of $Re$ and $Pm$.
Many numerical simulation [4] lead to $f=1$, i.e. equipartition between the
magnetic and turbulent energy.
This is therefore often taken as a working tool in astrophysical or geophysical application. However,
this result is far from applying to any saturated dynamo.
Moreover, it does not give any information about possible anisotropy of the saturated field.
It would therefore be interesting to build robust algorithm to derive the function $f$.
By robust, we mean which depends on characteristic
global quantities of the system (like total energy)
but not necessarily on small-scale dissipation, or boundary conditions.

In this talk, we present strategies based upon extremization principles,
in the case of the axisymmetric MHD equations. 
We study the equilibrium shape by using a minimum
energy principle under the constraints of the MHD axisymmetric equations.
We also propose a numerical algorithm based
on a maximum energy dissipation principle to compute in a consistent way the equilibrium states.
Then, we develop the statistical mechanics
of such flows and recover the same equilibrium
states giving a justification of the minimum energy principle.
We find that fluctuations obey a Gaussian shape 
and we make the link between
the conservation of the Casimirs on the coarse-grained scale and the process of energy dissipation.


[1] A. Gailitis  et al., Phys. Rev. Lett.  86, 3024 (2001).
[2] R. Stieglitz and U. M"uller, Phys. Fluids  13, 561 (2001).
[3] F. P\'etr\'elis and S. Fauve, Eur. Phys. J. B,  22, 273 (2001).
[4] V. Archontis, PhD University of Copenhagen (2000).