= Solution
Choose $y$ northward and $z$ upward in the Northern Hemisphere. In a locally linear <Boussinesq approximation>, write the background <buoyancy> as $\bar b=b_*+By+N^2z$, with horizontal <buoyancy gradient> $B<0$ and positive squared <buoyancy frequency> $N^2$. The <stable density stratification> resists a purely vertical <displacement>, but the <isopycnals> slope upward poleward:
$$
s_\rho=\left.\frac{dz}{dy}\right|_{\bar b}=-\frac{B}{N^2}>0.
$$
The <geopotential surfaces> are horizontal in this local approximation. Their inclination differs from that of the <isopycnals>, which is the relevant <baroclinicity>.
Exchange two equal small <fluid elements>, each of volume $V$, between positions separated by $(\delta y,\delta z)$. The <adiabatic parcel exchange> has no thermal diffusion, so each parcel retains its <buoyancy>. The background <buoyancy> difference is $\Delta\bar b=B\delta y+N^2\delta z$. Because the variable-density contribution to <gravitational potential energy> is $-\rho_0 Vbz$, the change in the pair's <gravitational potential energy> is
$$
\Delta E_p=\rho_0V\,\Delta\bar b\,\delta z
=\rho_0V\,\delta z(B\delta y+N^2\delta z).
$$
For a poleward-rising exchange, set $\delta z=s\delta y$ with $s>0$. Then
$$
\boxed{\Delta E_p<0\quad\Longleftrightarrow\quad 0<s<s_\rho.}
$$
The trajectory lies between a <geopotential surface> and an <isopycnal>. The poleward-rising parcel starts more buoyant than the parcel it replaces, and the equatorward-sinking parcel starts less buoyant: the <buoyancy> does positive work on both. This releases <available potential energy> into <kinetic energy>, allowing <eddies> to amplify if the full rotating dynamics supports the displacement and dissipation does not remove the gain.
A trajectory steeper than the <isopycnals>, $s>s_\rho$, instead costs <gravitational potential energy>; an equatorward-rising trajectory also costs it. Motion exactly along a <geopotential surface> or an <isopycnal> gives zero change in this ideal exchange. At fixed $\delta y$, the largest release occurs at $s=s_\rho/2$, since $N^2s(s-s_\rho)$ is minimized there.
\Image[/past-exam-of-the-geophysical-fluid-dynamics-course-of-the-university-of-oxford/2023-sloping-parcel-exchange.png]
{title=Two <adiabatic parcel exchanges>: a shallow poleward rise releases <potential energy>, whereas a rise steeper than the <isopycnals> costs it. Slopes are exaggerated.}
This is an energetic possibility argument for <baroclinic instability>, not a sufficient <normal mode> instability criterion. In particular, unrestricted parcel motion must not be assumed in a rotating fluid: <conservation of angular momentum> can impose an additional restoring effect. <Symmetric instability> has a separate <Ertel potential vorticity> criterion; ordinary <Eady instability> can occur without violating that criterion.
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