The hot plasma permeating clusters of galaxies often shows a central peak in the X-ray surface brightness that is coincident with a drop in entropy. This is taken as evidence for a cooling flow where the radiative cooling in the central regions of a cluster causes a slow subsonic inflow (e.g. Fabian, 1994). Searches in all wavebands have revealed significantly less cool gas than predicted (Böhringer et al., 2002) indicating that the mass deposition rate of cooling flows is much lower than expected. However, most cooling flow clusters host an Active Galactic Nucleus (AGN) at their centres (Burns, 1990). AGN can inflate large bubbles of hot plasma that subsequently rise through the cluster atmosphere, thus stirring this gas (Churazov et al., 2000(Churazov et al., , 2001). Here we report on the results from highly resolved hydrodynamic simulations which for the first time show that buoyant bubbles increase the cooling time in the inner cluster regions and thereby significantly reduce the deposition of cold gas. This work demonstrates that the action of AGN in the centres of cooling flow clusters can explain the observed absence of large quantities of cooled gas.The explanation of the absence of cold gas in the gaseous atmospheres of cooling flow clusters strongly suggests the presence of an, at least periodically active, source of heat in their centres. Heating the gas at the cluster centre increases its entropy and thus its radiative cooling time. However, not all heating processes will explain the observations. Fully convective heating of the cluster gas would lead to the entropy being highest in the cluster centre, opposite to what is observed (Böhringer et al., 2002).It has been suggested that the removal of cold gas and the feedback of energy through star formation and related supernovae can explain some of the observed properties of clusters (e.g. Voit & Bryan, 2001). While this is a promising scenario for the initial formation of clusters within the framework of hierarchical structure formation in the universe, the level of star formation found in detailed investigations of present-day cooling flow clusters is insufficient to explain the mass deposition rates inferred from X-ray observations (Böhringer et al., 2002). Moreover, the amount of star formation during the evolution of clusters necessary to explain the distribution of gas observed in them today, is difficult to reconcile with the observed abundances of heavy elements (Wu et al., 2000).Radio-loud AGN drive strong outflows in the form of jets that inflate bubbles or lobes. The lobes are filled with hot plasma and can heat the cluster gas in various ways. In the case of very energetic jets the expansion of the lobes will be supersonic. The resulting strong shock will heat and compress the gas (Heinz et al., 1998;Kaiser & Alexander, 1999;Reynolds et al., 2001). However, there is mounting evidence that weaker jets, which are presumably much more common, do not lead to efficient shock heating . Nevertheless, they still produce pockets of low-dens...