The areas between the Tropics, which lie at latitudes $\pm\beta$, are called Tropics, the areas outside this strip as Extratropics. The areas at latitudes beyond the Arctic Circles that are located at latitudes $\pm\left(\frac{\pi}{2} - \beta\right)$ are called high latitudes. The areas between the tropics and the high latitudes are called middle latitudes. The dynamics of the extratropics differ significantly from those of the tropics, since in the former the diabatic fluxes are smaller due to weaker convection and radiation and the geostrophic assumption is more justified. The dynamics of the tropics therefore contain certain difficulties and are therefore discussed separately in Chap. 20 investigated.
Geostrophy applies to zero order on the synoptic scale in the extratropics, consequences of which were discussed in Sect. 13.10.1. In Section 13.11 the quasi-geostrophic equation system, consisting of the trend equation Eq. (13.226) and $\omega-$equation Eq. (13.231), which can be applied to zonal channels in the extratropics whose meridional extent allows application of the $\beta $plane approximation. In Section 16.7.4 it was shown that the baroclinic waves found as solutions of a quasi-geostrophic linearized two-layer model are unstable. The process triggered by this is cyclogenesis, which will be discussed in more detail here.
First you define 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) was
\[ \begin{align} \frac{\partial}{\partial t}\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) + \mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) &= 0\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) &= -\mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\psi &= -\mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\psi &= -v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right). \end{align} \]
Multiplying this by $f_0$ gives
\[ \begin{align} \frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\left(f_0\psi\right) &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left[\Delta\left(f_0\psi\right) + \frac{f_0^2}{\sigma}\frac{\partial^2\left(f_0\psi\right)}{\partial p^2}\right]. \end{align} \]
Due to Eq. (13.221) is
\[ \begin{align} \psi = \frac{\Phi}{f_0}, \end{align} \]
so it applies
\[ \begin{align} \frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\Phi &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi + \frac{f_0^2}{\sigma}\frac{\partial^2\Phi}{\partial p^2}\right)\nonumber\\ \Leftrightarrow\frac{\partial}{\partial t}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\Phi &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \mathbf{v}_{h, g}\cdot\nabla\left(\frac{f_0^2}{\sigma}\frac{\partial^2\Phi}{\partial p^2}\right). \end{align} \]
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) follows
\[ \begin{align} \nabla\frac{\partial\Phi}{\partial p}\cdot\frac{\partial\mathbf{v}_{h, g}}{\partial p} = \nabla\frac{\partial\Phi}{\partial p}\cdot\left(\mathbf{k}\times\frac{1}{f_0}\nabla\frac{\partial\Phi}{\partial p}\right) = \frac{1}{f_0}\nabla\frac{\partial\Phi}{\partial p}\cdot\left(\mathbf{k}\times\nabla\frac{\partial\Phi}{\partial p}\right) = 0, \end{align} \]
therefore you can too
\[ \begin{align} \Leftrightarrow\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\mathbf{v}_{h, g}\cdot\frac{\partial}{\partial p}\left(\nabla\frac{\partial\Phi}{\partial p}\right) - \frac{f_0^2}{\sigma}\left(\nabla\frac{\partial\Phi}{\partial p}\right)\cdot\frac{\partial\mathbf{v}_{h, g}}{\partial p}\nonumber\\ \Leftrightarrow\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial\Phi}{\partial t} &= -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right) \end{align} \]
write. With Eq. (18.1) you can put this in the form
\[ \begin{align} \underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)}\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi}_{=:A} = \underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -f_0v_g\beta}_{=:B}\underbrace{\vphantom{\left(\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -\mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right)}_{=:C} \underbrace{\vphantom{\left[\Delta + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right]\chi = -f_0v_g\beta - \mathbf{v}_{h, g}\cdot\nabla\left(\Delta\Phi\right) - \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)} -\frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left(\mathbf{v}_{h, g}\cdot\nabla\frac{\partial\Phi}{\partial p}\right)}_{=:D}\tag{18.9}\label{eq:tendency_eq_cyclogen} \end{align} \]
bring.
This equation will now be examined conceptually. Assuming that $\chi$ consists of only a single Fourier component in three-dimensional space, it follows
\[ \begin{align} \text{A} \sim -\chi \sim \text{Zyklogenese}, \end{align} \]
whereby cyclogenesis means a decrease in the pressure surface. In the extratropics, meridional waves are often found, which are superimposed on a zonal base current $U$. Therefore, the approach is made for the geopotential $\Phi$
\[ \begin{align} \Phi\left(x, y, p\right) = -Uf_0y + \frac{\newhat{v}f_0L}{2\pi}\sin\left(\frac{2\pi}{L}x\right)\sin\left(ap + b\right) + \Phi_0\left(p\right), \end{align} \]
where $\newhat{v}$ denotes the velocity amplitude of the meridional oscillation. The linear function $ap + b$ allows the disturbance to phase shift with height; $\Phi_0$ represents a background state.
In this case one finds for the term $B$
\[ \begin{align} B = -f_0\beta\newhat{v}\cos\left(\frac{2\pi}{L}x\right). \end{align} \]
This term therefore leads to cyclogenesis wherever $v_g$ is negative, i.e. upstream of the trough and downstream of the ridge. So it contributes to retrograde progression. Term $B$ is the advection of planetary vorticity.
For the expression C, which is the advection of relative vorticity, one first calculates
\[ \begin{align} \Delta\Phi = -\frac{2\pi}{L}\newhat{v}f_0\sin\left(\frac{2\pi}{L}x\right), \end{align} \]
whereby it should be remembered that $\Delta\Phi$ here only refers to the horizontal. From this it follows
\[ \begin{align} C = U\frac{4\pi^2}{L^2}\newhat{v}f_0\cos\left(\frac{2\pi}{L}x\right) = -\frac{2\pi U}{L\beta}B. \end{align} \]
This term therefore leads to cyclogenesis downstream of the trough and upstream of the ridge, and to anticyclogenesis elsewhere. So it contributes to normal progression and decreases with wavelength. At a cutoff wavelength
\[ \begin{align} \newtilde{L} = \frac{2\pi U}{\beta} \end{align} \]
$B$ and $C$ compensate each other, the wave is stationary. Even longer waves propagate retrogradely. Both terms $B$ and $C$ disappear at the axes of the pressure structures, so they only contribute to their displacement, but not to their deepening. The term $D$ is responsible for this. The dot product $\mathbf{v}_h\cdot\nabla\frac{\partial\Phi}{\partial p}$ is the horizontal temperature advection, as can be seen from the basic hydrostatic equation in the form Eq. (13.124) results. Therefore $D$ is also referred to as negative differential temperature advection. It applies
\[ \begin{align} D = \begin{cases} \text{positiv, falls Kaltluftadvektion mit der Tiefe zunimmt, }\\ \text{negativ, falls Warmluftadvektion mit der Tiefe zunimmt.} \end{cases} \end{align} \]
Warm air advection at altitude and cold air advection at depth therefore lead to cyclogenesis, while cold air advection at altitude and warm air advection at depth lead to anticyclogenesis. This is illustrative because warmer air has a larger specific volume.