  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.

Astrophysics of Galaxies;High Energy Astrophysical Phenomena