Baroclinic instability

Definition

Consider the quasi-geostrophic equations on a beta plane with Coriolis parameter $f_0 + \beta y$ and buoyancy frequency $N(z) > 0$. The domain is periodic in $x$, with $0 \le z \le H$ and walls or decay in $y$. Let $U(y, z)$ be a steady zonal basic flow, in thermal wind balance with a basic buoyancy $B$ such that $f_0 \, \partial_z U = -\partial_y B$. The basic flow is unstable if the linearised equations admit a normal mode $\psi' = \operatorname{Re}[\hat\psi(y, z) e^{i k (x - c t)}]$ with $k$ real and $\operatorname{Im} c > 0$. The instability is baroclinic when the growth of such modes is supplied by the vertical shear $\partial_z U$. Then the eddies gain energy from the available potential energy of the basic state. They do so through a buoyancy flux $\overline{v' b'}$ directed down the basic buoyancy gradient.