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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 Rm exceeds a modest critical value Rm,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 Rm 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 rho A l is constant, where rho is the local density. So B propto rho l. For a nearly incompressible fluid, or a flow with small changes in rho, one will obtain B propto 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 v0 / l0 ~ eta / lB2, where v0 is the typical velocity variation on scale l0 and lB is the scale of the magnetic field. This gives lB = leta ~ l0 / Rm1/2, the resistive scale leta for the flow, where Rm = v0 l0 / eta. 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 Rm > Rm,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 l0 / v0 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 leta (e.g., [1113]). In Kolmogorov turbulence, where the velocity variations on a scale l is vl propto l1/3, the e-folding time is shorter at smaller scales, l / vl propto l2/3, and so smaller eddies amplify the field faster. If Pm = Rm / 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 Rm geq 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 l0 ~ 100kpc and turbulent velocity v0 ~ 300 km s-1, leading to a growth time tau0 ~ l0 / v0 ~ 3 × 108 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 l0 ~ 100pc, v0 ~ 10km s-1 one gets tau0 ~ 107 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 lB appeq l0 Rm,cr-1/2 instead of the resistive scale leta. 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 leta [15]. Since the cluster Rm ~ 1029 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 Rm, 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 Rm ~ 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 2563 mesh points. The computational box contained just a few turbulent cells on the forcing scales. The Faraday rotation measure K integ ne Bdl for 2562 lines of sight through the computational domain was calculated, and various properties of its distribution were determined including the normalized standard deviation sigmaRM, normalized by K bar{n}e Brms l0 (Here ne and bar{n}e are the actual and average electron densities respectively, Brms the RMS value of the magnetic field in the box, l0 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 sigmaRM, which depends on the degree of the field intermittency. If the field saturated with many resistive scale reversals one expects sigmaRM << 1, whereas if lB appeq l0 Rm,cr-1/2 decides the coherence scale of the field, one expects sigmaRM of order unity. The simulations found the normalized sigmaRM ~ 0.5 for Pm = 1, Re = Rm approx 420 and sigmaRM ~ 0.3 for a Pm = 30, Re approx 44, Rm approx 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 Rm,Re are required. Further, as emphasized by [17], during epochs when cluster turbulence decays, one can still get a significant sigmaRM, 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.

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