Adiabatic parcel exchange ($\Delta E_p={\rho }_{0}V\delta z\Delta b$)

Exchange two equal fluid elements of volume $V$ without thermal diffusion in a Boussinesq approximation. Each retains its buoyancy. If the separation is $(\delta y,\delta z)$ and the background buoyancy difference is $\Delta b=B\delta y+N^2\delta z$, the pair's gravitational potential energy changes by ${\rho }_{0}V\delta z\Delta b$. For $B<0$, $N^2>0$ and poleward-rising slope $s=\delta z/\delta y$, this is negative exactly when $0<s<-B/N^2$. The minimum at fixed $\delta y$ is attained at half the isopycnal slope. This energetic test alone is not sufficient for dynamical baroclinic instability: rotation and constraints on parcel motion must also be considered.