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Turbulence, magnetic fields, and plasma physics in clusters of galaxiesa)
1.The cool cores occupy roughly the central , compared with the overall cluster size of order . Such a structure is, in fact, characteristic of the older type of clusters known historically as cooling-flow clusters (the more observationally popular members of this group are Hydra A, Centaurus, and Perseus), while younger, less dynamically relaxed clusters (e.g., Coma) have flatter density and temperature profiles. A good summary of the parameters in a number of cooling-flow clusters and the relevant references can be found in Ref. 21.
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23.These doubts are reinforced by the similarity between the inferred flow patterns associated with rising bubbles and known such patterns in viscous, rather than turbulent, fluid flows (Ref. 11).
24.An estimate of resistivity based on the standard Spitzer formula leads to the enormous values of the magnetic Reynolds number, , given in Table I.
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37.This is only valid if the characteristic parallel scales of all fields are larger than . In the collisionless regime, , we may assume that the pressure anisotropy is relaxed in the time particles streaming along the field cover the distance : this entails replacing in Eq. (3) by .
38.The anisotropy is small: . It turns out that is the natural small parameter that can be used to develop a reduced kinetic theory for the cluster plasma (Ref. 36).
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44.In space physics, where the importance of the plasma instabilities driven by pressure anisotropies of what in their case is an almost entirely collisionless plasma has long been understood (Ref. 79), a vast but inconclusive literature exists on this subject.
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50.Several authors have argued that no amplification is, in fact, necessary either because the seed fields may have already been strong enough to account, after compression of the cosmological plasma into a cluster, for the observed magnitude of the cluster field (Ref. 49), or because the field could be generated in AGNs and then ejected into the ICM (Ref. 80). We forego the discussion of these possibilities, primarily because we find the idea that the ICM turbulence will produce the right amount of magnetic energy independently of its history or external circumstances more appealing on a fundamental physics level and, indeed, intuitively supported by the fact that the energy density of the observed field is close to the energy density of the fluid motions.
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55.Note that turbulence in the sense of a broad inertial range is not required for the fluctuation dynamo. The viscous-scale motions, which dominantly stretch the field, are random but spatially smooth. Whether the viscous scale is much smaller than the outer scale is unimportant as long as the motions are random—not a problem in clusters, given the vigorous random stirring. Thus, the Reynolds number in all of our calculations need not be large. Numerical simulations of a randomly stirred magnetohydrodynamic (MHD) fluid with Re in the range confirm the insensitivity of the dynamo effect to the value of Re (Ref. 69 and 81).
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67.This, effectively, is the regime assumed by Sharma et al. (Ref. 82) in their numerical simulations of the magnetorotational instability in a collisionless plasma. Their set of fluid-like equations incorporates anisotropic pressure and an effective collision rate that enforces the marginal state of the plasma instabilities, in our notation.
69.A. A. Schekochihin, S. C. Cowley, S. F. Taylor, J. L. Maron, and J. C. MacWilliams, Astrophys. J.0004-637X 612, 276 (2004).
70.This means, in particular, that young clusters should already have dynamically significant magnetic fields.
74.This does not mean that cluster fields must be volume-filling. Indeed, a real cluster is probably a patchwork of turbulent and quiescent regions (or rather regions with widely varying rms rates of strain) rather than a volume homogeneously filled with turbulence. Furthermore, the turbulent dynamo only produces magnetic fields everywhere on the average (in time), while any particular snapshot is very intermittent (Ref. 69), see, e.g., Fig. 4.
75.Just such a structure appears to be evidenced by the polarized emission map from a radio relic in cluster A2256 (Ref. 33).
78.A. A. Schekochihin and S. C. Cowley, in Magnetohydrodynamics: Historical Evolution and Trends, edited by S. Molokov, R. Moreau, and H. K. Moffatt (Springer, Berlin, in press), e-print astro-ph/0507686.
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82.P. Sharma, G. W. Hammett, E. Quataert, and J. M. Stone, Astrophys. J.0004-637X 637, 952 (2006).
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