  We present three-dimensional simulations of viscous dissipation of AGN
induced gas motions and waves in clusters of galaxies. These simulations are
motivated by recent detections of ripples in the Perseus and Virgo clusters.
Although the sound waves generated by buoyant bubbles decay with distance from
the cluster center, we show that these waves can contribute substantially to
offsetting the radiative cooling at distances significantly exceeding the
bubble size. The energy flux of the waves declines more steeply with radius
than the inverse-square law predicted by energy conservation, implying that
dissipation plays an important role in tapping the wave energy. We show that
such dispersing sound waves/weak shocks are detectable as ripples on
unsharp-masked X-ray cluster maps, and point out that the interfaces between
the intracluster medium and old bubbles are also clearly detectable in
unsharp-masked X-ray maps. This opens up the possibility of detecting fossil
bubbles that are difficult to detect in radio emission. This mode of heating is
consistent with other observational constraints, such as the presence of cool
rims around the bubbles and the absence of strong shocks. Thus, the mechanism
offers a way of heating clusters in a spatially distributed and gentle fashion.
We also discuss the energy transfer between the central AGN and the surrounding
medium. In our numerical experiments, we find that roughly 65 per cent of the
energy injected by the AGN is transferred to the intracluster medium and
approximately 25 percent of the injected energy is dissipated by viscous
effects and contributes to heating of the gas. The overall transfer of heat
from the AGN to the gas is comparable to the radiative cooling losses. The
simulations were performed with the FLASH adaptive mesh refinement code.
