The importance of viscosity in the ICM

Before discussing simulations of viscous systems, we shall address in brief the general issue of viscosity in the ICM. Initially suppose that the ICM can be described as a thermal fully-ionized plasma which is unmagnetized. The relevant coefficient of viscosity is given by Braginskii (1958) and Spitzer (1962) as

$$\mu_{\rm um}=2.2\times 10^{-15}\frac{T^{5/2}}{\ln\Lambda}\,{\rm g}\,{\rm cm}^{-1}\,{\rm s}^{-1},$$ 1

where $T$ is the temperature of the plasma measured in Kelvin and $\ln\Lambda$ is a Coulomb integral. Scaling to a temperature of $kT=5T_5{\rm\thinspace keV}$ and using $\ln\Lambda=30$ gives $\mu=1.88\times 10^3T_5\,{\rm g}\,{\rm cm}^{-1}\,{\rm s}^{-1}$.

It is customary to measure the importance of viscosity in a fluid of density $\rho$ through the Reynolds number, $Re=UL/\nu$, where $U$ and $L$ are characteristic velocities and length scales of the system and $\nu=\mu/\rho$. Of course, in a complicated system such as a radio-galaxy/ICM interaction, there is no unique velocity and length scale and so it is not possible to define a universal Reynolds numbers that characterizes the system. However, provided one uses some consistent choice for $U$ and $L$, the Reynolds number is still useful as a means to parameterize the relative importance of viscosity between different AGN/ICM systems. It also allows a means of matching the viscosity imposed in simulations with that expected in real systems.

We choose the maximum dimension of the bubble as our characteristic length scale, and half of the adiabatic sound speed (i.e. a typical buoyancy-induced rise velocity) as our characteristic velocity. Scaled to the NW ghost cavity of the Perseus cluster (with number density $n\approx 0.03\hbox{${\rm cm}^{-3}\,$}$, and $kT\approx 5{\rm\thinspace keV}$ which gives $c_s^2=780\hbox{${\rm\thinspace km}{\rm\thinspace s}^{-1}\,$}$), gives a Reynolds number of

$$Re=62\left(\frac{U}{390\hbox{${\rm\thinspace km}{\rm\thinspa... ...m}^{-3}\,$}}\right)\left(\frac{\mu}{\mu_{\rm um}}\right)^{-1}.$$ 2

We note that this Reynolds number is almost an order of magnitude smaller than the fiducial value of $Re=400$ quoted by Robinson et al. (2003), primarily due to our higher (and more realistic) fiducial temperature. Thus, restating the conclusion of Fabian et al. (2003a), viscosity may be relevant to the evolution of an AGN induced bubble in the ICM.

The major uncertainty is the effect that magnetic fields have on the macroscopic viscosity. The case of a uniform field is readily analyzed (Spitzer 1962). The proton gyro-radius corresponding to any non-negligible magnetic field is very small, leading to extremely efficient suppression of the local coefficient of viscosity perpendicular to the magnetic field; for typical ICM conditions, the perpendicular coefficient of viscosity is suppressed by the enormous factor of $\sim 10^{23}$ (Spitzer 1962). However, the effective macroscopic viscosity in the case of a realistic magnetic field configuration (which is almost certainly tangled, and may be chaotic) is an open question. A similar issue has recently been addressed in the context of thermal conduction. In that case, the local thermal conductivity is also suppressed by a very large factor perpendicular to the field. However, the exponential divergence of neighbouring field lines in a chaotic field structure results in an effective thermal conductivity, $\kappa$, that is suppressed below the unmagnetized value , $\kappa_{\rm um}$ by only a factor of $10^{-2}-0.2$ (depending on the spectrum of fluctuations in the field structure; Narayan & Medvedev 2001). While the tensorial nature of the viscous stress tensor prevents a precise mapping of the two problems, similar arguments may apply and we might expect the effective coefficient of viscosity to be suppressed below the unmagnetized value by some factor ranging from $10^{-2}$ to unity.

Clearly, the effective thermal conductivity and viscosity characterizing the ICM is still very much an open theoretical question, due to uncertainties in both the basic physics of transport processes in hot plasmas as well as the magnetic field structure present in the ICM. To make progress we must assume that certain conditions exist, compute the consequences for radio-galaxy/ICM interactions, and compare with the recent observations. This is the motivation for the rest of this paper. While the evidence for non-negligible ICM viscosity is still circumstantial, we show that the action of such a viscosity allows the morphology of ghost cavities (in particular, the NW ghost cavity in Perseus-A) to be reproduced, and may be an important mechanism for stabilizing the ICM core against radiative losses.