Thermal conduction in galaxy clusters
The dominant baryonic component of a galaxy cluster is hot tenuous plasma that has accreted into the deep gravitational well formed by the dominant dark matter component. This makes galaxy clusters unique laboratories for a variety of plasma phenomena on an extremely wide range of scales. Intricate plasma processes on microscales, more than ten orders of magnitude smaller than the size of the cluster, affect the large-scale properties of the cluster; for example modifying particle transport influences the temperature profile. Many puzzling features of galaxy clusters, such as the stability of cool cores, sharp local gradients or the substructure seen in temperature maps, are closely tied to the problem of thermal conduction in the intracluster medium (ICM).
From X-ray observations it is now clear that the ICM demonstrates a variety of violent physical processes, such as cluster mergers, infalling galaxies, shock waves, and active galactic nuclei. These naturally render the plasma turbulent. In addition, radio observations show evidence that the ICM is pervaded by magnetic fields. The field magnitude is sufficient to confine the motion of charged particles to spiralling around field lines with a tiny Larmor radius, much smaller than the mean free paths of the particles. This effectively shuts down particle transport perpendicular to field lines. Moreover, such a plasma turns out to be unstable to pressure anisotropies that are easily generated by turbulent motions. These instabilities then grow rapidly on Larmor scales.
Spatial structure of the mirror instability after one shear time in the PIC simulations by Matt Kunz (Princeton). The magnetic-field strength is indicated by colour, where the colour scale is in the units of the initial value of the magnetic field. Field lines are shown by contours. The mirror fluctuations are elongated along the field lines.
When studying thermal conduction, the mirror instability is of particular interest. In this case, the magnetic field strength is perturbed with a significant amplitude on the order of the local mean magnetic field. The correlation length of mirror perturbations is only two orders of magnitude longer than the electron Larmor radius, but about ten orders of magnitude smaller that the collisional mean free path. This means that such perturbations are capable of magnetically mirroring the electrons: a charged particle spiralling along a field line is reflected from a region with a strong magnetic field. If perturbations of the magnetic field are generated by turbulence on scales above the collisional mean free path, magnetic trapping is ineffective. The mirror fluctuations, in contrast, are at the scales comparable to the ion Larmor radius, where magnetic mirrors can suppress electron transport considerably.
The scale of mirror fluctuations is far smaller than the current observational limits. Instead, one has to turn to numerical simulations. Only recently have particle-in-cell codes become capable of studying micro-instabilities driven by pressure anisotropies. In these simulations, a region of plasma (with a linear size of the order of a few hundreds of the ion Larmor radius) is subjected to a shear, stretching the magnetic field lines, producing pressure anisotropy, and triggering the instability.
Suppression factor of the electron diffusivity in mirror fluctuations (solid line) as a function of the ratio of the mean free path to the correlation length of the fluctuations. At large mean free paths, the suppression stays above a certain lower limit. For comparison, the dashed line shows suppression for a lognormal distribution of magnetic fluctuations with a similar width of the probability density.
Scientists at MPA have used the results of these simulations to investigate the motion of electrons in mirror fluctuations (shown in Fig. 1). By applying a Monte Carlo approach, the diffusion and thermal conduction coefficients have been estimated for a representative field line extracted from the simulation domain. The probability distribution function of the magnetic field strength along the field line turns out to have a cut-off at a field strength of several times the initial value. This leads to only a moderate amount of particle transport suppression. In the limit where the collisional mean free path is much larger than the correlation length of the mirror fluctuations, diffusion is suppressed by a factor of ~10 (Fig. 2). This value then has to be converted into the suppression of thermal conduction.
Due to the additional presence of diffusion in energy space the thermal conductivity is suppressed by approximately a factor of two less effectively than the particle transport. The resulting suppression by a factor of ~5 appears to depend only very weakly on macroscopic parameters of the ICM as long as the ion Larmor radius remains much smaller than the correlation scale of the mirror perturbations, which is indeed well satisfied in the ICM. The effect operates on top of other suppression mechanisms and independently of them.
