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Cold gas in cluster cores: Global stability analysis and non-linear simulations of thermal instability

We perform global linear stability analysis and idealized numerical simulations in global thermal balance to understand the condensation of cold gas from hot/virial atmospheres (coronae), in particular the intracluster medium (ICM). We pay particular attention to geometry (e.g., spherical versus plane-parallel) and the nature of the gravitational potential. Global linear analysis gives a similar value for the fastest growing thermal instability modes in spherical and Cartesian geometries. Simulations and observations suggest that cooling in halos critically depends on the ratio of the cooling time to the free-fall time ($t_{cool}/t_{ff}$). Extended cold gas condenses out of the ICM only if this ratio is smaller than a threshold value close to 10. Previous works highlighted the difference between the nature of cold gas condensation in spherical and plane-parallel atmospheres; namely, cold gas condensation appeared easier in spherical atmospheres. This apparent difference due to geometry arises because the previous plane-parallel simulations focussed on {\em in situ} condensation of multiphase gas but spherical simulations studied condensation {\em anywhere} in the box. Unlike previous claims, our nonlinear simulations show that there are only minor differences in cold gas condensation, either in situ or anywhere, for different geometries. The amount of cold gas condensing depends on the shape of the gravitational potential well; gas has more time to condense if gravitational acceleration decreases toward the center. In our idealized simulations with heating balancing cooling in each layer, there can be significant mass/energy/momentum transfer across layers that can trigger condensation and drive $t_{cool}/t_{ff}$ far beyond the critical value close to 10. Triggered condensation is very prominent in plane-parallel simulations, in which a large amount of cold gas condenses out.