  In the intracluster, intragroup, and circumgalactic medium (ICM, IGrM, CGM),
turbulence plays a vital role in the self-regulated feedback and feeding cycle
of central supermassive black holes (SMBHs). Here we continue our systematic
dissection of the turbulent "weather" in high-resolution hydrodynamical
simulations of feedback driven by active galactic nuclei (AGN). In
non-barotropic and stratified atmospheres, baroclinicity is expected to
generate fresh turbulence via misaligned gradients of density and pressure -
such as in cyclonic storms on Earth. In this work, we dissect for the first
time baroclinicity and its components in the astrophysical halo weather. Over
the macro-scale galaxy cluster, baroclinicity tends to be dynamically
subdominant for the enstrophy amplification. However, at and below the meso
scale near the SMBH (r<10 kpc; t<20 Myr), baroclinicity is important to seed
the initial enstrophy during active periods of AGN jet feedback. We find that
baroclinicity shows stronger correlation with the density rather than pressure
gradients. Despite the density-pressure gradient misalignment being often below
45{\deg}, their amplitudes boosted by mechanical AGN feedback are sufficient to
enable key enstrophy/turbulence generation. Our study provides a novel step
forward in understanding astrophysical atmospheres toward a unified
BlackHoleWeather framework, akin to the complexity of Earth's weather.
