Turbulence in the intracluster, intragroup, and circumgalactic medium plays a crucial role in the self-regulated feeding and feedback loop of central supermassive black holes. We dissect the three-dimensional turbulent `weather in a high-resolution Eulerian simulation of active galactic nucleus (AGN) feedback, shown to be consistent with multiple multi-wavelength observables of massive galaxies. We carry out post-processing simulations of Lagrangian tracers to track the evolution of enstrophy, a proxy of turbulence, and its related sinks and sources. This allows us to isolate in depth the physical processes that determine the evolution of turbulence during the recurring strong and weak AGN feedback events, which repeat self-similarly over the Gyr evolution. We find that the evolution of enstrophy/turbulence in the gaseous halo is highly dynamic and variable over small temporal and spatial scales, similar to the chaotic weather processes on Earth. We observe major correlations between the enstrophy amplification and recurrent AGN activity, especially via its kinetic power. While advective and baroclinc motions are always sub-dominant, stretching motions are the key sources of the amplification of enstrophy, in particular along the jet/cocoon, while rarefactions decrease it throughout the bulk of the volume. This natural self-regulation is able to preserve, as ensemble, the typically-observed subsonic turbulence during cosmic time, superposed by recurrent spikes via impulsive anisotropic AGN features (wide outflows, bubbles, cocoon shocks). This study facilitates the preparation and interpretation of the thermo-kinematical observations enabled by new revolutionary X-ray IFU telescopes, such as XRISM and Athena.
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