Magnetic fields in a conducting medium can be amplified by the inductive effects associated with the motions of the medium. In this process, generally referred to as a dynamo, the kinetic energy associated with the motions is tapped to amplify magnetic energy. The plasmas in galaxies and clusters are most often turbulent. The dynamo in this context is referred to as a `turbulent dynamo', and its analysis must rely on statistical methods or direct numerical simulations. Turbulent dynamos are conveniently divided into fluctuation (or small-scale) and mean-field (or large-scale) dynamos. The fluctuation dynamo produces magnetic fields that are correlated only on scales of the order of or smaller than the energy-carrying scale of the random motions. We will discuss this here and turn to mean-field dynamos in the next section.

The importance of fluctuation dynamos in cosmic objects obtains
because they are generic in any random flow
where *R*_{m} exceeds a modest critical value
*R*_{m,cr} ~ 100.
Fluid particles in such a flow randomly walk away from each other. A
magnetic field line frozen into such a fluid will then be extended by the
random stretching (if *R*_{m} is large enough).
Consider a small segment of a thin flux tube of length *l* and
cross-section *A*,
and magnetic field strength *B*, in a highly
conducting fluid. Then, as the fluid moves about, conservation
of flux implies *BA* is constant, and conservation of mass
implies
*A* *l* is constant, where
is the local
density. So *B*
*l*. For
a nearly incompressible fluid, or a flow with small changes in
, one
will obtain *B*
*l*. Any random shearing motion which increases
*l* will also amplify *B*; an increase in *l* leading to
a decrease in *A* (because of incompressibility) and hence
an increase in *B* (due to flux freezing).

Of course since the scale of
individual field structures decreases, (that is since *A* ~ 1 /
*B*), as the field strength increases,
the rate of Ohmic dissipation increases until it compensates the effect of
random stretching. For a single scale random flow, this happens
when *v*_{0} / *l*_{0} ~
/
*l*_{B}^{2}, where *v*_{0} is the
typical velocity variation on scale *l*_{0} and
*l*_{B} is the scale
of the magnetic field. This gives *l*_{B} =
*l*_{} ~ *l*_{0} /
*R*_{m}^{1/2}, the resistive scale
*l*_{} for the flow, where *R*_{m} =
*v*_{0} *l*_{0} /
.
What happens after this can only be addressed by
a quantitative calculation. For a random flow which is delta-correlated
in time, it was shown by Kazantsev
[12],
that magnetic field can grow provided *R*_{m} >
*R*_{m,cr} ~ 30-100, depending on the form of
the velocity correlation function.
In the kinematic regime the field grows exponentially
roughly on the eddy turnover time *l*_{0} /
*v*_{0} and is also predicted to be intermittent,
i.e., concentrated into structures whose size, in at least one dimension,
is as small as the resistive scale
*l*_{} (e.g.,
[1113]).
In Kolmogorov turbulence, where the velocity variations on a
scale *l* is *v*_{l}
*l*^{1/3}, the *e*-folding time is shorter at
smaller scales, *l* / *v*_{l}
*l*^{2/3}, and so smaller eddies amplify the field
faster. If *P*_{m} = *R*_{m} / *Re*
>> 1, as relevant for galactic
and cluster plasma, even the viscous scale
eddies can exponentially amplify the field.

The fluctuation dynamo has since been convincingly confirmed beyond
the limits of Kazantsev's theory by numerical simulations of forced and
flows
[14,
15],
especially when *R*_{m}
*Re*. Such simulations
are also able to follow the fluctuation dynamo
into the non-linear regime where the Lorentz forces becomes strong
enough to affect the flow as to saturate the growth of magnetic field.

In the context of galaxy clusters turbulence would mainly be driven by
the continuous and ongoing merging of subclusters to form larger and
larger mass objects. One typically expects the largest turbulent scales
of *l*_{0} ~ 100kpc and turbulent velocity
*v*_{0} ~ 300 km s^{-1}, leading to a growth time
_{0} ~
*l*_{0} / *v*_{0} ~ 3 × 10^{8} yr;
thus for a cluster lifetime of a few Gyr, one could then have significant
amplification by the fluctuation dynamo (cf.
[17]).
And in the case of galactic interstellar turbulence driven
by supernovae, if we adopt values corresponding to the local ISM of
*l*_{0} ~ 100pc, *v*_{0} ~ 10km s^{-1}
one gets
_{0} ~
10^{7} yr. Again the fluctuation dynamo would rapidly grow the
magnetic field even for very young high redshift protogalaxies.

The major uncertainty, in case we want to
use the fluctuation dynamo to explain observed Faraday rotation
measures in a galaxy clusters or that inferred in some high redshift
protogalaxies, is how intermittent the
field remains when the dynamo saturates.
A simple model exploring an ambipolar drift type nonlinearity
[16],
suggests that the
smallest scale of the magnetic structures will be renormalized in the
saturated state to become *l*_{B}
*l*_{0}
*R*_{m,cr}^{-1/2} instead of the resistive scale
*l*_{}.
This essentially happens via a renormalization
of the effective magnetic diffusivity in these models
[16],
such that the dynamo is
saturated via a reduction of the effective magnetic Reynolds
number down to its critical value for the dynamo action.
In this case one could indeed obtain significant Faraday rotation measure
through such a fluctuation dynamo generated field.
On the other hand it has been argued that
the fluctuation dynamo generated field remains highly intermittent
even at saturation, with field reversals typically occurring on the
resistive scale *l*_{}
[15].
Since the cluster *R*_{m} ~ 10^{29} if one uses
naively the collisional Spitzer resistivity, one hardly expects to
see any Faraday rotation from such a field.
Plasma effects would then important to renormalize the
effective *R*_{m}, much below this ridiculously large value
[3].
One needs a better theoretical understanding
of the non-linear saturation of fluctuation dynamos and the extent to
which plasma effects alter the resistivity of cluster plasma, to make
further progress.

Meanwhile direct simulations of the fluctuation dynamo at modest
values of *R*_{m} ~ 400-1000, to examine
the resulting Faraday rotation, have been carried out
[17],
following the methods of Haugen, Brandenburg and Dobler
[14].
The simulations use a Cartesian box on a
cubic grid with 256^{3} mesh points.
The computational box contained just a few turbulent cells
on the forcing scales.
The Faraday rotation measure *K*
*n*_{e} **B** ⋅ *d***l** for
256^{2} lines of sight through the computational domain was
calculated, and various properties of its distribution were determined
including the normalized standard deviation
_{RM},
normalized by *K*
_{e}
*B*_{rms} *l*_{0} (Here *n*_{e} and
_{e}
are the actual and average
electron densities respectively, *B*_{rms} the RMS
value of the magnetic field in the box, *l*_{0} the forcing
scale of the random flow, and *K*
is a constant of proportionality which converts the line
of sight integral into physical units of RM).
For a random field one expects a distribution of Faraday rotation
with mean zero and a standard deviation
_{RM},
which depends on the degree of the field intermittency.
If the field saturated with many resistive scale reversals
one expects _{RM}
<< 1, whereas if *l*_{B}
*l*_{0}
*R*_{m,cr}^{-1/2}
decides the coherence scale of the field,
one expects _{RM}
of order unity. The simulations found the normalized
_{RM} ~ 0.5 for
*P*_{m} = 1, *Re* = *R*_{m}
420 and
_{RM} ~ 0.3 for
a *P*_{m} = 30,
*Re* 44,
*R*_{m}
1300 case. Therefore the fluctuation dynamo does indeed seem to
be able to produce magnetic fields which will give significant
Faraday rotation measure, although simulations
at much higher values of *R*_{m},*Re* are required.
Further, as emphasized by
[17],
during epochs
when cluster turbulence decays, one can still get a significant
_{RM}, as it
decreases in time as a shallower power law than
the field itself, because of an increase in the field coherence scale.
Similar ideas involving cluster turbulence and fluctuation dynamos
have also been invoked by
[18]
to explain magnetic fields in cool cores of galaxy clusters.
The way the fluctuation dynamo saturates will be important
not only to decide if it can result in observable Faraday
rotation, but also to issues of Cosmic ray confinement in the
early protogalaxy.