Basic setup of the simulations

Our goal is to study the evolution of a buoyant radio lobe in the ICM of a galaxy cluster, including the effects of kinematic viscosity. Following the approach of Bruggen & Kaiser (2001), we simulate only the phase of the evolution after the radio-loud AGN has inflated a low-density bubble which has expanded to achieve pressure equilibrium with the ICM. At some point during this process, the AGN activity is assumed to cease. Thus, our simulation follows the evolution of a low-density bubble, initially in pressure equilibrium, placed in the central regions of an ICM atmosphere.

Even though real radio lobes will have rapid and complex internal flows (induced by the AGN jet during their inflation), we shall assume that both the ICM and the bubble interior are initially static. We stress that this is an important simplification in our models and must be kept in mind when interpreting the results. For example, our calculations will not be meaningful for addressing the issue of shocks and sound waves driven into the ICM by the radio-galaxy, since these phenomena are almost certainly dominated by the jet-driven inflation phase of the bubble which we are not modeling. Furthermore, the precise growth rates of the RT and Kelvin-Helmholtz (KH) will be influenced by the internal motions within the bubble left over from its inflation phase. However, our set-up allows a qualitative investigation of the hydrodynamics of buoyantly rising ICM bubbles.

Our detailed set-up is as follows. The undisturbed ICM atmosphere is given a density profile described by $\rho(r)=\rho_0\left[1+(r/r_0)^2\right]^{-0.75}$, where we choose units of mass and length such that $\rho_0=1$ and $r_0=1$. This atmosphere is assumed to be initially static and isothermal, with an adiabatic sound speed of $c_s=1$. The gravitational potential, $\Phi$, is assumed to be dominated by dark matter and, hence, is assumed to be fixed throughout the simulation. This gravitational potential is determined by the condition that the initial ICM atmosphere is in hydrostatic equilibrium, $\nabla p=-\rho\nabla\Phi$, where $p$ is the pressure. In this ICM atmosphere, we carve out a spherical bubble with density $\rho_{\rm bub}=0.01$ and radius $r_{\rm bub}=1/4$. This bubble is displaced from the center of the ICM atmosphere by $\Delta r=1/4$ (i.e., the boundary of the bubble touches the center of the ICM atmosphere). The initial pressure of the bubble is set equal to the initial ICM pressure at that radius (i.e., this is a hot, low-density bubble is in local pressure equilibrium with the ICM). We note that this is a very simplified form for the cluster density profile and gravitational potential. However, we feel that a more sophisticated cluster profile (e.g., using a Navarro, Frenk & White [1997] profile, and including the potential of the cD galaxy) would be unwarranted for the qualitative nature of the current investigation.

This initial state is evolved using the equations of 3-dimensional viscous hydrodynamics,

where ${\bf v}$ is the velocity field, $\epsilon$ is the internal energy density, and ${\bf\Pi}$ is the viscous stress tensor,

$$\Pi_{ik}=\mu \left(\frac{\partial v_i}{\partial x_k} + \frac... ...\frac{2}{3}\delta_{ik}\frac{\partial v_j}{\partial x_j}\right)$$ 3

Note that we only include ``pdV'' and viscous dissipation terms in the energy equation, eqn. 4. In particular, we neglect the radiative cooling which is unlikely to be relevant for the dynamical timescale processes modeled here. These equations are solved using the ZEUS-MP MHD code (Stone & Norman 1992a, 1992b), operating in Cartesian co-ordinates $(x,y,z)$. ZEUS is a fixed-grid, time-explicit Eulerian code which uses an artificial viscosity to handle shocks. When operated using van Leer advection, as in this work, it is formally of second order spatial accuracy. We model one half of the ICM atmosphere using a simulation volume of a cubic box of length $2r_0$ on a side. The cluster center ($r=0$) is placed in the center of one face of this cube, and reflecting boundary conditions are imposed on this face in order to account for the unmodelled half of the atmosphere. Outflow boundary conditions are imposed on all other faces.

The effects of viscosity are introduced into the basic ZEUS-MP code by adding explicit source terms to the momentum and energy equations (eqns. 4 and 5, respectively). To ensure numerical stability, the time-step on which the equations were evolved was constrained to be

$$dt=(dt_{\rm visc}^{-2}+dt_{\rm cour}^{-2})^{-1/2},$$ 4

where $dt_{\rm cour}$ is the usual CFL timestep and

$$dt_{\rm visc}={\cal C}_{\rm visc} \min\left[\frac{dx_i^2}{\mu}\right]$$ 5

is the relevant viscous timestep. In this last expression, $dx_i$ is the size of a grid cell in the $i$-th direction, and the minimization occurs over all grid cells. On the basis of a standard stability analysis, we choose ${\cal C}_{\rm visc}=0.1$.

Table 1: Basic information for the set of simulations presented in this paper.

$$\mu/10^{-3}$$ Run Re Simulation grid

$$\sim 2\times 10^3 200^3$$ 1

$$200^3$$ 2 1 500

$$200^3$$ 3 2 250

$$200^3$$ 4 4 125

$$200^3$$ 5 10 50

$$200^3$$ 6 20 25

In real systems, the interior of the ICM cavities is filled with extremely tenuous and strongly magnetized relativistic plasma; the coefficient of viscosity of this material is likely to be negligible. However, our (one-fluid) simulations model these structures simply as bubbles of very hot gas (with initial conditions of $T_{\rm bubble}=100T_{\rm ambient}$). If we were to include the temperature dependence of $\mu$ in our calculations (eqn. 1), the coefficient of viscosity inside the bubble would be three orders of magnitude greater than in the ambient ICM. This is unphysical. We address this problem by fixing $\mu$ to be constant in both space and time throughout a given simulation. This choice suppresses the viscosity of the simulated hot gas inside the bubble. We note that this method avoids the need to artificially truncate the viscosity within the bubble (e.g., see Ruszkowski, Brüggen & Begelman 2004), leading to greater robustness and numerical stability. We have, however, performed additional simulations in which we do truncate the viscosity within the bubble (using a density threshold) in order to assess the impact of this assumption. Comparing those simulations with the ones presented in this paper, we confirmed that the dynamics of the buoyant bubble considered here are not qualitatively effected by the constant $\mu$ assumption.

Using the same definition as Section 2, the Reynolds number characterizing the hydrodynamics of the simulated bubble is

$$Re\equiv \frac{vL\rho}{\mu}=50\left(\frac{v}{2c_s}\right)\le... ...o}{1\,{\rm unit}}\right)\left(\frac{\mu}{10^{-2}}\right)^{-1},$$ 6

where $v$ and $L$ are the characteristic velocity and size, respectively, of the bubble. We perform a set of simulations to explore a range of Reynolds numbers; see Table 1 for details of the runs performed.