We have seen that cluster magnetic field strengths can be calculated through their effects on the polarization properties of radio galaxies by using an analytical formulation (Sec. 3.5.2) based on the approximation that the magnetic field is tangled on a single scale. However, detailed observations of radio sources and MHD simulations [144, 145, 161] suggest that it is necessary to consider more realistic cluster magnetic fields which fluctuate over a wide range of spatial scales. To accomplish this, Murgia et al.  simulated random three-dimensional magnetic fields with a power-law power spectrum: | B|2 -n, where represents the wave number of the fluctuation scale. They investigated the effects of the expected Faraday rotation on the polarization properties of radio galaxies and radio halos, by analyzing the rotation measure effects produced by a magnetic field with a power spectrum which extends over a large range of spatial scales (6 - 770 kpc) and with different values of the spectral index (n = 2, 3, 4).
10.1. Simulated rotation measures
Fig. 10 (top) shows the simulated RM images obtained with different values of the index n for a typical cluster of galaxies (see caption for more details). Different power spectrum indexes will generate different magnetic field configurations and therefore will give rise to very different simulated RM images. Fig. 10 (bottom) shows the simulated profiles of RM, |<RM>|, and |<RM>| / RM (left, central and right panels, respectively), as a function of the projected distance from the cluster center. While both RM and |<RM>| increase linearly with the cluster magnetic field strength, the ratio |<RM>| / RM depends only on the magnetic field power spectrum slope, for a given range of fluctuation scales. Therefore the comparison between RM data of radio galaxies embedded in a cluster of galaxies and the simulated profiles, allows the inference of both the strength and the power spectrum slope of the cluster magnetic field.
Figure 10. Top: simulated RM images for magnetic field power spectrum spectral index n = 2, 3, 4. The electron gas density of the cluster follow a standard -model with a core radius rc = 400 kpc (indicated by a circle in the figure) a central density ne(0) = 10-3 cm-3 and = 0.6. The three power spectra are normalized to have the same total magnetic field energy which is distributed over the range of spatial scales from 6 kpc up to 770 kpc. The field at the cluster center is B0 = 1 µG and its energy density decreases from the cluster center according to B(r)2 ne(r). Bottom: radial profiles (RM, |<RM>| and |<RM>| / RM respectively) obtained from the RM simulations described above. The profiles have been obtained by averaging the simulated RM images in regions of 50 × 50 kpc2, which is a typical size for radio galaxies. The figure is from Murgia et al. .
Typical measured values of |<RM>| / RM for cluster radio galaxies are derived to be in the range 0.2 - 1.2, once the Galaxy contribution is subtracted . Thus the comparison of the observations with the simulations leads to a rather flat cluster magnetic field power spectrum, with a spectral index n 2. This indicates that most of the magnetic energy density is on the smaller scales.
Another result of the simulations is that, when a power spectrum of the magnetic field is assumed, the inferred magnetic field strength is about a factor of 2 lower than the value computed from Eq. 46 if the single scale c is taken to be equal to the smallest patchy structures detectable in the RM images, as frequently used. This implies that the magnetic fields derived from RM measurements may be overestimated (see Sec. 9).