Simulations including viscosity

Figure 3: Mid-plane density slices (upper panels) and simulated X-ray surface brightness maps (lower-panels) for the $Re=250$ case (Run 3) shown at three times; $t=1$ (left panels), $t=2$ (middle panels) and $t=4$ (right panels). Arrows indicating fluid velocity have been superposed on the density slices. Note how the viscosity stabilizes the bubble, allowing a flattened by intact buoyant ``cap'' to form. Both the X-ray surface brightness and H$\alpha$-inferred velocity field around the ghost cavity of Per-A can be qualitatively reproduced by this model.

Having described the inviscid ``control'' case, we now proceed to discuss the effect of viscosity on the buoyant evolution of radio-lobes. As in the inviscid case, the evolution is driven by the joint action of buoyancy and secondary KH instabilities. However, unlike the inviscid case where KT instabilities operate at the contact discontinuity on spatial scales down to the grid scale, viscosity suppresses the KH instability on small spatial scales. This has a profound effect on the evolution of the bubble; even a moderate amount of viscosity can prevent the shredding of the bubble, which can subsequently float out of the core being rather flattened but otherwise intact.

As a specific example, Fig. 3 shows the $Re=250$ (Run 3). This can be considered a model of the ghost cavities around Perseus-A if the ICM possesses a coefficient of viscosity of $\mu\approx 0.25\mu_{\rm um}$. As discussed in Section 2, this level of viscosity may be plausible even in the presence of tangled, chaotic magnetic fields. It can be seen from the mid-plane density and velocity fields (Fig. 3; upper panels) that the evolution of the bubble is driven by buoyancy, with secondary KH instabilities largely unable to overcome the action of the viscosity. As the bubble floats upwards, it flattens into a broad cap. The surface brightness maps associated with Run 3 (Fig. 3; lower panels) show that one does, indeed, produce a detached and flattened cavity in the ICM emission as observed in the Perseus cluster.

Figure 4: Mid-plane density slices (upper panels of each set) and simulated X-ray surface brightness maps (lower-panels of each set) for the six simulations presented in this paper (Runs 1-6 ordered from left to right). Results are shown for three times, $t=1$ (upper set of panels), $t=4$ (middle set of panels) and $t=8$ (lower set of panels). See text for a discussion of these results.

Figure 4 shows results for the full range of viscosity explored in this paper at three fixed times ($t=1, 4, 8$). It can be seen that the formation of an flattened but intact buoyant bubble occurs in all of our viscous simulations. However, the timescale on which the evolution proceeds is a strong function of the viscosity. For example, the $t=1$ density slice of Run 2 ($Re=500$) is very similar to the $t=4$ slice of Run 6 ($Re=50$). This fact may point to a solution of the ``shock problem'' noted in the introduction, an issue that we shall return to in Section 4.2.

Viscosity also has important implications for the flow pattern in the disturbed ICM. In principle, the presence of viscosity can facilitate the development of large scale vortex rings in the trailing region beneath the rising bubble. This phenomena is seen for our highest viscosity cases. For the levels of viscosity that are probably relevant to the Perseus cluster (${\rm Re}=100-200$), this effect is not seen. However, even for these levels of viscosity, our simulations show flow patterns that qualitatively match those inferred from the H$\alpha$-filament geometry in Perseus. In these cases, the flattened buoyant bubble undergoes a minor fragmentation due to the action of secondary KH instabilities. As a result of this fragmentation, a small torus of radio plasma is left in the trailing region behind the main buoyant bubble. The simulations show a strong ICM circulation around this trailing torus, producing a streamlines that resemble the H$\alpha$ filament geometry in Perseus. A prediction of this model is that sufficiently sensitive X-ray maps should reveal subtle depressions at the center of these vortices, and sufficiently sensitive and spatially resolved low-frequency radio maps should reveal a corresponding torus of aged radio plasma.

Figure 5: Volume integrated viscous dissipation rate for our $Re=250$ case, scaled with parameters relevant for the Perseus cluster (i.e., the simulation domain has a linear size of 40kpc in each dimension, one code unit of density is $0.03\hbox{${\rm cm}^{-3}\,$}$, and the initial ICM sound speed is $780\hbox{${\rm\thinspace km}{\rm\thinspace s}^{-1}\,$}$). One code unit of time is then 24Myr. As noted in the main text, the second and rather sharp peak in the dissipation rate is likely due to an interaction of the flow pattern with the boundaries of the computation and hence should not be considered physical.

An important consequence of ICM viscosity is that it provides an explicit mechanism by which the radio-galaxy induced disturbance can heat the ICM. While a full treatment of viscous ICM heating by radio-galaxies almost certainly requires following the jet-driven inflation of the bubbles (e.g., Reynolds, Heinz & Begelman 2002), as well as multiple epochs of activity (Ruzskowski, Brüggen & Begelman 2004), it is instructive to compute the viscous dissipation rate in our idealized simulation. Figure 5 shows the dissipation rate as a function of time for our canonical ``Perseus-like'' model, the $Re=250$ case. The dissipation rate and the time coordinate are given in physical units assuming parameters relevant to the Perseus cluster (see caption of Fig. 5). The dissipation rate dispays two peaks. The rather broad peak centered at about 90Myr coincides with the secondary KH instability entering the strongly non-linear regime, and the subsequent ``folding'' of the flattened bubble. The second and rather sharp peak corresponds to the venting of material out of the boundary of the simulation and, hence, should not be considered physical. In general, the dissipated power achieves the rather modest levels of $P\sim 10^{41}\hbox{${\rm\thinspace erg}{\rm\thinspace s}^{-1}\,$}$. However, this heat source continues to operate for a period of about 200Myr, an order of magnitude more than the plausible recurrence timescale of the radio-galaxy activity (Fabian et al. 2003a). Thus, the possibility of balancing the radiative losses with viscous dissipation from the combined effect of many bubbles remains open. Furthermore, it is likely that the dissipation of the fluid modes driven by the initial inflation of the bubble (and hence not modelled here) will deliver just as much energy, if not more, than the viscous dissipation during the buoyant phase.