Baroclinic instability

Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

For a disturbance proportional to $\mathrm{Re}[a(z)e^{i\varphi (z)}{e}^{ik(x-ct)}]$, westward phase tilt means ${\varphi }_z>0$ for $k>0$. With $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$, a zonal average gives the displayed positive flux in the northern hemisphere, up to any meridional amplitude and time-growth factors. When the background buoyancy decreases northward, this flux extracts mean available potential energy and supports growing eddies.

For a uniform buoyancy frequency, constant Coriolis parameter and zonal shear $U=\Lambda z$ between rigid lids separated by $H$, a zero-interior-PV normal mode has ${\Psi }^{\prime \prime }-K^2\Psi =0$. Adiabatic no-normal-flow conditions give $c=\Lambda H/2\pm \Lambda H\sqrt{D(\eta )}/\eta$, $\eta =KH$. Growth requires $D<0$, or a negative squared phase speed relative to the mean flow, rather than a literal $c^2<0$ in an arbitrary moving frame. The cutoff is ${\eta }_c\simeq 2.39936$; for zero meridional wavenumber maximum growth occurs at $k{L}_D\simeq 1.60612$, with rate $0.30982\Lambda f_0/N$.

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.