In Section 16.7.4 it is shown that a strong thermal wind is a prerequisite for the formation of synoptic disturbances in the middle latitudes. In reality, it is found that there are latitudes with particularly strong baroclinicity in which a jet exists in the tropopause region. These frontal zones are therefore classic areas where low pressure areas form (see chapter 18). However, it is unclear why these air mass boundaries exist at all. Intuitively, it would be more obvious to assume that such zones of strong gradients are broken down by some mechanism, at the latest by diffusion. The aim of this chapter is to clear up this ambiguity.
Statistical variables of the Earth system are called climate, compilations of such characteristics are called climatologies. Typically, 30-year funds are formed. This chapter is about zonal resources
\[ \begin{align} \newoverline{\psi}\left(\phi, z\right) \coloneqq \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\psi\left(\phi, \lambda, t\right)d\lambda \end{align} \]
of sizes $\psi$, where $\lambda$ is the geographical longitude. These funds meet the rules
\[ \begin{align} \frac{\partial\newoverline{\psi}}{\partial t} &= \frac{1}{2\pi}\frac{\partial}{\partial t}\int_{\lambda = 0}^{2\pi}\psi d\lambda = \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\frac{\psi}{\partial t}d\lambda = \newoverline{\frac{\psi}{\partial t}},\\ \newoverline{\frac{\partial\psi}{\partial x}} &= \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\frac{\psi}{\partial x}d\lambda = \frac{1}{2\pi}\left[\psi\right]_0^{2\pi} = \frac{1}{2\pi}\left[\psi\right]_0^0 = 0,\\ \frac{\partial\newoverline{\psi}}{\partial y} &= \frac{1}{2\pi}\frac{\partial}{\partial y}\int_{\lambda = 0}^{2\pi}\psi d\lambda = \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\frac{\partial\psi}{\partial y}d\lambda = \newoverline{\frac{\partial\psi}{\partial y}},\\ \newoverline{\newoverline{\psi}} &= \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\newoverline{\psi}d\lambda = \newoverline{\psi}.\tag{19.5}\label{eq_zonal_mean_rule_0} \end{align} \]
They are assumed to be independent of time, so it is of
\[ \begin{align} \frac{\partial\newoverline{\psi}}{\partial t} = 0 \end{align} \]
went out. For deviations, write down $\psi' \coloneqq \psi - \newoverline{\psi}$, which therefore applies to the complete field
\[ \begin{align} \psi\left(\phi, \lambda, z, t\right) = \newoverline{\psi}\left(\phi, \lambda\right) + \psi'\left(\phi, \lambda, z, t\right). \end{align} \]
Meet these deviations
\[ \begin{align} \newoverline{\psi'} = \newoverline{\psi - \newoverline{\psi}} = \newoverline{\psi} - \newoverline{\newoverline{\psi}} \stackrel{\href{#eq_zonal_mean_rule_0}{\text{Glg. (19.5)}}}{=} \newoverline{\psi} - \newoverline{\psi} = 0. \end{align} \]
By comparing with the equations (17.4) - (17.6) one finds that zonal means are Reynolds means. Here one limits oneself to $\psi = p, T, u, v, w$. Asymmetries in the properties of the earth's surface (orography, roughness, radiation properties) lead to asymmetries in the climatology; such effects are not taken into account here.
Imagine an earth without obliquity, i.e. an earth without seasons. This would have an energy supply in the short wave range in the tropics
\[ \begin{align} S_\text{in} = \frac{S_0}{4\pi}\cos\left(\phi\right) > 0. \end{align} \]
Here $S_\text{in}$ is the vertical radiation flux density in the short-wave range and $\phi$ is, as usual, the width. All variables are considered as climatological averages. Exactly at the poles, the vertical shortwave radiation flux density would be zero, barring scattering and other weak effects. In the long-wave range, the radiation $S_\text{out}$ at the poles would be weaker than in the tropics, but not zero. Since the global integral over the short-wave incoming radiation is equal to the global integral over the long-wave outgoing radiation,
\[ \begin{align} &\int_{\phi = -\pi/2}^{\pi/2}\int_{\lambda = 0}^{2\pi}S_0\Theta\left[\cos\left(\lambda\right)\right]\cos\left(\phi\right)a^2\cos\left(\phi\right)d\lambda d\phi = a^2S_0\int_{\phi = -\pi/2}^{\pi/2}\cos\left(\phi\right)^2\int_{\lambda = 0}^{2\pi}\Theta\left[\cos\left(\lambda\right)\right]d\lambda d\phi\nonumber\\ &a^2S_0\int_{\phi = -\pi/2}^{\pi/2}\cos\left(\phi\right)^22d\phi = a^2S_02\int_{\phi = -\pi/2}^{\pi/2}\cos\left(\phi\right)^2d\phi = \pi a^2S_0\\ &\Rightarrow\int_{\phi = -\pi/2}^{\pi/2}\int_{\lambda = 0}^{2\pi}S_\text{out}\cos\left(\phi\right)d\phi d\lambda = \pi a^2S_0, \end{align} \]
The radiation balance is positive in the tropics and negative in the polar regions. This is called differential heating. If one assumes a stationary temperature field, it follows for
\[ \begin{align} \text{Konvergenz der Strahlungsflussdichte} &+ \text{Konvergenz der Flussdichte innerer Energie} = 0\\ \Rightarrow\text{Konvergenz der Strahlungsflussdichte} &= \text{Divergenz der Flussdichte innerer Energie},\\ \Rightarrow S_\text{in} - S_\text{out} &= \text{Divergenz der Flussdichte innerer Energie}\\ \Rightarrow S_\text{in} - S_\text{out} &= \frac{c^{(V)}}{a}\left[\frac{\partial V}{\partial\phi} - V\tan\left(\phi\right)\right].\tag{19.15}\label{eq:hadley_base} \end{align} \]
Here is
\[ \begin{align} V \coloneqq \int_0^\infty\rho Tvdz \end{align} \]
the vertically integrated meridional flux density of internal energy. If you put in Eq. (19.15)
\[ \begin{align} S_\text{in} = \frac{S_0}{4\pi}\cos\left(\phi\right) \end{align} \]
as well as the approximation
\[ \begin{align} S_\text{out} = \frac{S_0}{4} \end{align} \]
one, you get
\[ \begin{align} \frac{S_0}{4\pi}\cos\left(\phi\right) - \frac{S_0}{4} &= \frac{c^{(V)}}{a}\left[\frac{\partial V\left(\phi\right)}{\partial\phi} - V\left(\phi\right)\tan\left(\phi\right)\right]\nonumber\\ \Rightarrow\cos\left(\phi\right) - \pi &= \frac{4\pi c^{(V)}}{S_0 a}\left[\frac{\partial V\left(\phi\right)}{\partial\phi} - V\left(\phi\right)\tan\left(\phi\right)\right]. \end{align} \]
This is a differential equation for the function $V = V\left(\phi\right)$.
The interaction of radiation with the atmosphere essentially takes place via the earth's surface. The atmosphere is warmed from below in the tropics and thus becomes unstable, and in the polar regions it is cooled from below and thus stabilized. This leads to positive vertical motion in the tropics and negative vertical motion at the poles. For reasons of mass conservation, the flow in the upper troposphere must run towards the poles and in the lower troposphere towards the equator.
Assuming homogeneous ground pressure, the higher temperature in the tropics leads to an increase in the geopotential in this area. This induces a geostrophic zonal flow directed in the direction of the planet's rotation in both hemispheres.
Such meridional circulations are called Hadley circulation. The three important points of the Hadley circulation are summarized again:
In the climatological average, both the energy and the entropy of the atmosphere are constant. Radiation adds heat $q$ in the tropics over a climatological time interval, which is released again in the polar regions. Since the temperature at the poles is lower, the amount of entropy released is greater than that absorbed. The interaction with the Earth leads to an increase in entropy for the rest of the universe. In order for the entropy of the Earth itself not to decrease, entropy must be produced in the atmosphere and ocean. This happens via irreversible processes, especially friction.
The atmosphere is supplied with heat from space via radiation. Some of this heat is used to generate kinetic energy (wind). This energy is ultimately dissipated, producing entropy. The same amount of heat that the atmosphere has absorbed is finally released back into space, only this time this is associated with a higher entropy flow. So the universe puts energy into the earth, which produces kinetic energy, and gets the energy back in dissipated form. This corresponds to a heat engine.
First, the linearized shallow water equations Equations (13.173) - (13.174) are assumed. These are in components
\[ \begin{align} \frac{\partial u}{\partial t} &= -g\frac{\partial d}{\partial x} + fv,\\ \frac{\partial v}{\partial t} &= -g\frac{\partial d}{\partial y} - fu,\\ \frac{\partial d}{\partial t} &= -D\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} - v\frac{\tan\left(\phi\right)}{a}\right). \end{align} \]
Zonal means of zonal gradients disappear, thus one obtains
\[ \begin{align} \frac{\partial\newoverline{u}}{\partial t} &= f\newoverline{v},\\ \frac{\partial\newoverline{v}}{\partial t} &= -g\frac{\partial\newoverline{d}}{\partial y} - f\newoverline{u},\\ \frac{\partial\newoverline{d}}{\partial t} &= -D\left(\frac{\partial\newoverline{v}}{\partial y} - \newoverline{v}\frac{\tan\left(\phi\right)}{a}\right). \end{align} \]
If one assumes constant averaged sizes, it follows
\[ \begin{align} 0 &= f\newoverline{v},\tag{19.26}\label{eq:clim_deriv_0}\\ 0 &= -g\frac{\partial\newoverline{d}}{\partial y} - f\newoverline{u},\\ 0 &= -D\left(\frac{\partial\newoverline{v}}{\partial y} - \newoverline{v}\frac{\tan\left(\phi\right)}{a}\right). \end{align} \]
From Eq. (19.26) follows $\newoverline{v} = 0$, so all that remains of the entire system of equations is
\[ \begin{align} f\newoverline{u} &= -g\frac{\partial\newoverline{d}}{\partial y}. \end{align} \]
left, $\newoverline{u}$ is geostrophically balanced.
If you write down the momentum equation of the nonlinear shallow water equations Eq. (15.166) component-wise, you get
\[ \begin{align} \frac{\partial u}{\partial t} &= -\frac{\partial\left(gh + gb + k\right)}{\partial x} + qhv = -g\frac{\partial\left(h + b + k\right)}{a\cos\left(\phi\right)\partial\lambda} - \nabla k + \left(f + \zeta\right)v,\\ \frac{\partial v}{\partial t} &= -\frac{\partial\left(gh + gb + k\right)}{\partial y} - \left(f + \zeta\right)u. \end{align} \]
Zonal averaging performs
\[ \begin{align} \frac{\partial\newoverline{u}}{\partial t} &= f\newoverline{v} + \newoverline{\zeta v} = \left(f + \newoverline{\zeta}\right)\newoverline{v} + \newoverline{\zeta'v'},\\ \frac{\partial\newoverline{v}}{\partial t} &= -\frac{\partial\left(g\newoverline{h} + g\newoverline{b} + \newoverline{k}\right)}{\partial y} - \left(f + \newoverline{\zeta}\right)\newoverline{u} - \newoverline{\zeta'u'}. \end{align} \]
Zonal means are assumed to be constant, from which it follows
\[ \begin{align} 0 &= \left(f + \newoverline{\zeta}\right)\newoverline{v} + \newoverline{\zeta'v'},\\ 0 &= -\frac{\partial\left(g\newoverline{h} + g\newoverline{b} + \newoverline{k}\right)}{\partial y} - \left(f + \newoverline{\zeta}\right)\newoverline{u} - \newoverline{\zeta'u'}.\tag{19.35}\label{eq:swe_mean_deriv_0} \end{align} \]
For mass conservation reasons applies
\[ \begin{align} \newoverline{v} = 0,\tag{19.36}\label{eq:v_bar_0_swe} \end{align} \]
from which immediately
\[ \begin{align} \newoverline{\zeta'v'} = 0 \end{align} \]
follows. In the context of the shallow water equations, there is no turbulent meridional transport of relative vorticity. Furthermore, it follows from Eq. (19.36)
\[ \begin{align} \newoverline{\zeta} &= -\frac{\partial\newoverline{u}}{\partial y} + \frac{\newoverline{u}}{a}\tan\left(\phi\right),\\ \newoverline{k} = \frac{1}{2}\newoverline{u^2} + \frac{1}{2}\newoverline{v^2} \Rightarrow \frac{\partial\newoverline{k}}{\partial y} &= \newoverline{u\frac{\partial u}{\partial y}} + \newoverline{v\frac{\partial v}{\partial y}} = \newoverline{u}\frac{\partial\newoverline{u}}{\partial y} + \newoverline{v}\frac{\partial\newoverline{v}}{\partial y} + \newoverline{u'\frac{\partial u'}{\partial y}} + \newoverline{v'\frac{\partial v'}{\partial y}}\nonumber\\ &= \newoverline{u}\frac{\partial\newoverline{u}}{\partial y} + \newoverline{u'\frac{\partial u'}{\partial y}} + \newoverline{v'\frac{\partial v'}{\partial y}}. \end{align} \]
Putting this into Eq. (19.35), follows
\[ \begin{align} g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y} &= -f\newoverline{u} - \frac{\newoverline{u}}{a}\tan\left(\phi\right)\newoverline{u} - \newoverline{u'\frac{\partial u'}{\partial y}} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{\zeta'u'}. \end{align} \]
With
\[ \begin{align} \zeta' = \frac{\partial v'}{\partial x} - \frac{\partial u'}{\partial y} + \frac{v'}{a}\tan\left(\phi\right) \end{align} \]
Can this be further transformed to
\[ \begin{align} g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y} &= -f\newoverline{u} - \frac{\newoverline{u}}{a}\tan\left(\phi\right)\newoverline{u} - \newoverline{u'\frac{\partial u'}{\partial y}} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{\left(\frac{\partial v'}{\partial x} - \frac{\partial u'}{\partial y} + \frac{v'}{a}\tan\left(\phi\right)\right)u'}\nonumber\\ \Leftrightarrow g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y} &= -f\newoverline{u} - \frac{\newoverline{u}}{a}\tan\left(\phi\right)\newoverline{u} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{\left(\frac{\partial v'}{\partial x} + \frac{v'}{a}\tan\left(\phi\right)\right)u'}\nonumber\\ \Leftrightarrow \textcolor{red}{g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y}} &= \textcolor{red}{-f\newoverline{u}} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{u'\frac{\partial v'}{\partial x}} + \textcolor{blue}{\newoverline{\frac{u'v'}{a}\tan\left(\phi\right)} - \frac{\newoverline{u}^2}{a}\tan\left(\phi\right)}. \end{align} \]
The red marked terms contain the geostrophic balance, the blue marked parts are metric terms. The black terms are the turbulent momentum advection.
The application of the quasigeostrophic theory is limited to the $\beta $level and is therefore not possible globally (see section 13.11). Nevertheless, the quasi-geostrophic equations should also be zonally averaged here.
Frontal zones and jets can be better understood with the semi-geostrophic system of equations than with the quasi-geostrophic one.