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There is growing evidence, from observations as well as theory, that active galaxies can provide widespread mechanical heating of their environments. In clusters, this heating apparently occurs in a rather gentle fashion, and is driven by buoyancy. Simulations of buoyant plumes show significant and fast lateral spreading, generation of sound waves, cool rims of entrained gas surrounding the hot bubbles and a mismatch between X-ray and radio emission, resulting in `ghost cavities'. The evenness and spatial distribution of heating may be adequate to balance cooling globally, prevent cooling catastrophes, and thus quench so-called `cooling flows'. Energy spreading to larger radii during cluster assembly might be able to account for the observed entropy excesses in present-day clusters.

While the initial results are promising, numerical simulations have a long way to go before they can adequately the represent the physics of AGN heating. Three-dimensional simulations are already being done, but their resolution needs to be increased in order to study the effects of mixing and thermal instability. Magnetic fields, which have played little role in models to date, may have important effects on the dynamics, transport properties (viscosity and thermal conduction), and radio emissivity of clusters. At the microphysical level, we need to understand why the hot (relativistic?) gas injected by active galaxies appears to mix relatively little with the ICM. Indeed, we cannot exclude the possibility that some of the injected `cosmic rays' do stream through the thermal background. If they couple effectively to the ICM via hydromagnetic waves, they will heat the gas in much the same fashion as expanding bubbles, as they traverse the pressure gradient (Loewenstein et al. 1991).

Finally, we need to better understand what sets the efficiency of kinetic energy output from black hole accretion flows, the speed and degree of collimation of the output (winds vs. narrow jets), and feedback effects that couple the evolution of the ICM to the growth rate of the black hole. Whether (and how) such feedback fixes the MBH - Mbulge correlation by regulating the black hole mass, the galaxy mass, or both, remains to be seen.

Support for this work was provided by National Science Foundation grant AST-0307502 and the National Aeronautics and Space Administration through Chandra Fellowship Award Number PF3-40029 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the NASA under contract NAS8-39073. Some of the work reported here was done in collaboration with M. Brüggen (International University Bremen); S. Roychowdhury and B. Nath (Raman Research Institute, Bangalore); and C. Kaiser (University of Southampton).

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