= Adiabatic parcel exchange
{title2=$\Delta E_p=\rho_0V\,\delta z\,\Delta b$}
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 $\rho_0V\delta z\Delta b$. 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.
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