The regions between the two tropics, the parallels of latitude located at $\pm\beta$, are called the tropics, and the regions outside this band are called the extratropics. The regions at latitudes beyond the polar circles, which are located at latitudes $\pm\left(\frac{\pi}{2} - \beta\right)$, are called the high latitudes. The regions between the tropics and the high latitudes are called the middle latitudes. The dynamics of the extratropics differ substantially from those of the tropics, since in the former the diabatic fluxes are smaller owing to the weaker convection and radiation, and the geostrophic assumption is better justified. The dynamics of the tropics thus involve certain difficulties and are therefore examined separately in Chap. 20.
Geostrophy holds to zeroth order on the synoptic scale in the extratropics; consequences of this were discussed in Sect. 13.10.1. In Sect. 13.11 the quasi-geostrophic system of equations was already derived, consisting of the tendency equation Eq. (13.226) and the $\omega-$equation Eq. (13.231), which can be applied to zonal channels in the extratropics whose meridional extent permits the use of the $\beta-$plane approximation. In Sect. 16.7.4 it was shown that the baroclinic waves obtained as solutions of a quasi-geostrophic linearized two-layer model are unstable. The process triggered by this is cyclogenesis, which is discussed in more detail here.
One first defines the geopotential tendency by
\[ \begin{align} \chi \coloneqq \frac{\partial\Phi}{\partial t}.\tag{18.1}\label{sec:tendency_def} \end{align} \]
Eq. (13.226) read
\[ \begin{align} \frac{\partial}{\partial t}\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) + \mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) &= 0\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) &= -\mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\psi &= -\mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\psi &= -v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right). \end{align} \]
Multiplying this by $f_0$ gives
\[ \begin{align} \frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\left(f_0\psi\right) &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left[\Delta\left(f_0\psi\right) + \frac{f_0^2}{\sigma}\frac{\partial^2\left(f_0\psi\right)}{\partial p^2}\right]. \end{align} \]
By virtue of Eq. (13.211) one has
\[ \begin{align} \psi = \frac{\Phi}{f_0}, \end{align} \]
and therefore
\[ \begin{align} \frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\Phi &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi + \frac{f_0^2}{\sigma}\frac{\partial^2\Phi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\Phi &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \mathbf{v}_{h, g}\cdot\nabla\left(\frac{f_0^2}{\sigma}\frac{\partial^2\Phi}{\partial p^2}\right). \end{align} \]
Assuming a homogeneous static stability parameter $\sigma$, it follows
\[ \begin{align} \left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\mathbf{v}_{h, g}\cdot\nabla\frac{\partial^2\Phi}{\partial p^2}\nonumber\\ \Leftrightarrow\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\mathbf{v}_{h, g}\cdot\frac{\partial}{\partial p}\left(\nabla\frac{\partial\Phi}{\partial p}\right). \end{align} \]
From Eq. (13.191) it follows that
\[ \begin{align} \nabla\frac{\partial\Phi}{\partial p}\cdot\frac{\partial\mathbf{v}_{h, g}}{\partial p} = \nabla\frac{\partial\Phi}{\partial p}\cdot\left(\mathbf{k}\times\frac{1}{f_0}\nabla\frac{\partial\Phi}{\partial p}\right) = \frac{1}{f_0}\nabla\frac{\partial\Phi}{\partial p}\cdot\left(\mathbf{k}\times\nabla\frac{\partial\Phi}{\partial p}\right) = 0, \end{align} \]
so that one can also write
\[ \begin{align} \Leftrightarrow\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\mathbf{v}_{h, g}\cdot\frac{\partial}{\partial p}\left(\nabla\frac{\partial\Phi}{\partial p}\right) - \frac{f_0^2}{\sigma}\left(\nabla\frac{\partial\Phi}{\partial p}\right)\cdot\frac{\partial\mathbf{v}_{h, g}}{\partial p}\nonumber\\ \Leftrightarrow\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right) \end{align} \]
With Eq. (18.1) this can be cast into the form:
\[ \begin{align} \underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi}_{=:A} = \underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -f_0v_g\beta}_{=:B}\underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -\mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right)}_{=:C} \underbrace{\vphantom{\left[\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right]\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -\frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)}_{=:D}\tag{18.9}\label{eq:tendency_eq_cyclogen} \end{align} \]
This equation is now examined conceptually. Assuming that $\chi$ consists of only a single Fourier component in three-dimensional space, it follows that
\[ \begin{align} \text{A} \sim -\chi \sim \text{cyclogenesis}, \end{align} \]
where cyclogenesis here refers to a lowering of the pressure surface. In the extratropics one often finds meridional waves that are superimposed on a zonal basic flow $U$. Accordingly, one makes for the geopotential $\Phi$ the ansatz
\[ \begin{align} \Phi\left(x, y, p\right) = -Uf_0y + \frac{\newhat{v}f_0L}{2\pi}\sin\left(\frac{2\pi}{L}x\right)\sin\left(ap + b\right) + \Phi_0\left(p\right), \end{align} \]
where $\newhat{v}$ denotes the velocity amplitude of the meridional oscillation. The linear function $ap + b$ permits a phase shift of the disturbance with height; $\Phi_0$ represents a background state.
For the term $B$ one finds in this case
\[ \begin{align} B = -f_0\beta\newhat{v}\cos\left(\frac{2\pi}{L}x\right). \end{align} \]
This term therefore leads to cyclogenesis wherever $v_g$ is negative, i.e. upstream of the trough and downstream of the ridge. It thus contributes to retrograde progression. Term $B$ is the advection of planetary vorticity.
For the term $C$, which is the advection of relative vorticity, one first computes
\[ \begin{align} \Delta\Phi = -\frac{2\pi}{L}\newhat{v}f_0\sin\left(\frac{2\pi}{L}x\right), \end{align} \]
where it should be recalled that $\Delta\Phi$ here refers only to the horizontal. From this it follows that
\[ \begin{align} C = U\frac{4\pi^2}{L^2}\newhat{v}f_0\cos\left(\frac{2\pi}{L}x\right) = -\frac{2\pi U}{L\beta}B. \end{align} \]
This term therefore leads to cyclogenesis downstream of the trough and upstream of the ridge, and to anticyclogenesis elsewhere. It thus contributes to normal progression and decreases with wavelength. At a critical wavelength
\[ \begin{align} \newtilde{L} = \frac{2\pi U}{\beta} \end{align} \]
$B$ and $C$ compensate one another and the wave is stationary. Even longer waves propagate retrogradely. Both terms $B$ and $C$ vanish at the axes of the pressure systems, so that they contribute solely to their displacement, but not to their deepening. The term $D$ is responsible for the latter. The scalar product $\mathbf{v}_h\cdot\nabla\frac{\partial\Phi}{\partial p}$ is the horizontal temperature advection, as follows from the hydrostatic fundamental equation in the form Eq. (13.124). For this reason $D$ is also referred to as the negative differential temperature advection. One has
\[ \begin{align} D = \begin{cases} \text{positive if cold-air advection increases with depth, }\\ \text{negative if warm-air advection increases with depth.} \end{cases} \end{align} \]
Warm-air advection aloft and cold-air advection below therefore lead to cyclogenesis, while cold-air advection aloft and warm-air advection below lead to anticyclogenesis. This is intuitively clear, since warmer air has a larger specific volume.