Baroclinic instability arises when isopycnal (constant-density) surfaces are tilted relative to isobaric (constant-pressure) surfaces — i.e., when ∇ρ × ∇p ≠ 0.
In Earth's atmosphere, this occurs at mid-latitudes where the equator-pole temperature gradient meets the Coriolis effect. The result: extratropical cyclones, weather fronts, and the jet stream.
The Eady growth rate σ_max ≈ 0.31f(ΔU/H)/N characterizes how fast perturbations amplify. Waves extract available potential energy from the background shear, converting it to kinetic energy.