In Sect. 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 one finds that there are latitudes of particularly strong baroclinicity at which a jet exists in the tropopause region. These frontal zones are accordingly classic regions of formation for low-pressure systems (see Chap. 18). It is unclear, however, why these air-mass boundaries exist at all. Intuitively, one would sooner expect such zones of strong gradients to be broken down by some mechanism, at the latest by diffusion. The aim of this chapter is to resolve this puzzle.
Statistical quantities of the Earth system are called climate, and compilations of such characteristics are called climatologies. Usually one forms 30-year means. This chapter is concerned with zonal means
\[ \begin{align} \newoverline{\psi}\left(\phi, z\right) \coloneqq \frac{1}{2\pi}\int_{\lambda = 0}^{2\pi}\psi\left(\phi, \lambda, t\right)d\lambda \end{align} \]
of quantities $\psi$, where $\lambda$ is the geographical longitude. These means satisfy 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 that one takes
\[ \begin{align} \frac{\partial\newoverline{\psi}}{\partial t} = 0 \end{align} \]
as the starting point. For deviations one writes $\psi' \coloneqq \psi - \newoverline{\psi}$, so that the complete field satisfies
\[ \begin{align} \psi\left(\phi, \lambda, z, t\right) = \newoverline{\psi}\left(\phi, \lambda\right) + \psi'\left(\phi, \lambda, z, t\right). \end{align} \]
These deviations satisfy
\[ \begin{align} \newoverline{\psi'} = \newoverline{\psi - \newoverline{\psi}} = \newoverline{\psi} - \newoverline{\newoverline{\psi}} \stackrel{\href{#eq_zonal_mean_rule_0}{\text{Eq. (19.5)}}}{=} \newoverline{\psi} - \newoverline{\psi} = 0. \end{align} \]
By comparison with Eqs. (17.4) - (17.6) one finds that zonal means are Reynolds means. Here one restricts oneself to $\psi = p, T, u, v, w$. Asymmetries in the properties of the Earth's surface (orography, roughness, radiative 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. In the short-wave range it would receive, in the tropics, an energy supply
\[ \begin{align} S_\text{in} = \frac{S_0}{4\pi}\cos\left(\phi\right) > 0. \end{align} \]
Here $S_\text{in}$ is the vertical radiative flux density in the short-wave range and $\phi$ is, as usual, the latitude. All quantities are considered as climatological averages. Exactly at the poles the vertical short-wave radiative flux density would be zero, if one disregards scattering and other weak effects. In the long-wave range the outgoing radiation $S_\text{out}$ at the poles would be weaker than in the tropics, but not zero. Since the global integral over the incoming short-wave radiation equals the global integral over the outgoing long-wave 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. Assuming a stationary temperature field, it follows for
\[ \begin{align} \text{convergence of the radiative flux density} &+ \text{convergence of the internal energy flux density} = 0\\ \Rightarrow\text{convergence of the radiative flux density} &= \text{divergence of the internal energy flux density},\\ \Rightarrow S_\text{in} - S_\text{out} &= \text{divergence of the internal energy flux density}\\ \Rightarrow S_\text{in} - S_\text{out} &= \frac{c^{(V)}}{a}\left[\frac{\partial V}{\partial\phi} - V\tan\left(\phi\right)\right].\tag{19.15}\label{eq:hadley_base} \end{align} \]
Here
\[ \begin{align} V \coloneqq \int_0^\infty\rho Tvdz \end{align} \]
is the vertically integrated meridional flux density of internal energy. Substituting into 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 obtains
\[ \begin{align} \frac{S_0}{4\pi}\cos\left(\phi\right) - \frac{S_0}{4} &= \frac{c^{(V)}}{a}\left[\frac{\partial V\left(\phi\right)}{\partial\phi} - V\left(\phi\right)\tan\left(\phi\right)\right]\nonumber\\ \Rightarrow\cos\left(\phi\right) - \pi &= \frac{4\pi c^{(V)}}{S_0 a}\left[\frac{\partial V\left(\phi\right)}{\partial\phi} - V\left(\phi\right)\tan\left(\phi\right)\right]. \end{align} \]
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 detour of the Earth's surface. In the tropics the atmosphere is thus warmed from below and thereby destabilized, while in the polar regions it is cooled from below and thereby 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 accordingly run towards the poles and in the lower troposphere towards the equator.
Assuming a homogeneous surface pressure, the higher temperature in the tropics leads to an upward doming of the geopotential in this region. This induces a geostrophic zonal flow that, in both hemispheres, is directed along the planet's rotation.
Such meridional circulations are called Hadley circulations. The three important points of the Hadley circulation are summarized once more:
Differential heating leads to destabilization in the tropics and stabilization in the polar regions.
This leads to a meridional circulation: the vertical velocity is positive in the tropics and negative at the poles. In the upper troposphere the air flows towards the poles and in the lower troposphere towards the equator.
This leads to a geostrophic zonal flow directed along the planet's rotation.
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. Thus 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 nevertheless be produced in the atmosphere and ocean. This happens via irreversible processes, especially friction.
The atmosphere is supplied with heat from space via radiation. Part of this heat is used to generate kinetic energy (wind). This energy is ultimately dissipated, whereby entropy is produced. The same amount of heat that the atmosphere has absorbed it finally releases back into space, only this time associated with a higher entropy flux. Thus space puts energy into the Earth, which produces kinetic energy, and receives the energy back in dissipated form. This corresponds to a heat engine.
The starting point is the linearized shallow water equations Eqs. (13.173) - (13.174). In components these read
\[ \begin{align} \frac{\partial u}{\partial t} &= -g\frac{\partial d}{\partial x} + fv,\\ \frac{\partial v}{\partial t} &= -g\frac{\partial d}{\partial y} - fu,\\ \frac{\partial d}{\partial t} &= -D\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} - v\frac{\tan\left(\phi\right)}{a}\right). \end{align} \]
Zonal means of zonal gradients vanish, so that 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 quantities, 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) it follows that $\newoverline{v} = 0$, so that of the entire system of equations only
\[ \begin{align} f\newoverline{u} &= -g\frac{\partial\newoverline{d}}{\partial y}. \end{align} \]
remains; $\newoverline{u}$ is thus geostrophically balanced.
Writing the momentum equation of the nonlinear shallow water equations Eq. (15.166) component-wise, one obtains
\[ \begin{align} \frac{\partial u}{\partial t} &= -\frac{\partial\left(gh + gb + k\right)}{\partial x} + qhv = -g\frac{\partial\left(h + b + k\right)}{a\cos\left(\phi\right)\partial\lambda} - \nabla k + \left(f + \zeta\right)v,\\ \frac{\partial v}{\partial t} &= -\frac{\partial\left(gh + gb + k\right)}{\partial y} - \left(f + \zeta\right)u. \end{align} \]
Zonal averaging leads to
\[ \begin{align} \frac{\partial\newoverline{u}}{\partial t} &= f\newoverline{v} + \newoverline{\zeta v} = \left(f + \newoverline{\zeta}\right)\newoverline{v} + \newoverline{\zeta'v'},\\ \frac{\partial\newoverline{v}}{\partial t} &= -\frac{\partial\left(g\newoverline{h} + g\newoverline{b} + \newoverline{k}\right)}{\partial y} - \left(f + \newoverline{\zeta}\right)\newoverline{u} - \newoverline{\zeta'u'}. \end{align} \]
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 reasons of mass conservation one has
\[ \begin{align} \newoverline{v} = 0,\tag{19.36}\label{eq:v_bar_0_swe} \end{align} \]
from which
\[ \begin{align} \newoverline{\zeta'v'} = 0 \end{align} \]
follows immediately. Within the framework of the shallow water equations there is thus no turbulent meridional transport of relative vorticity. Furthermore, from Eq. (19.36) it follows that
\[ \begin{align} \newoverline{\zeta} &= -\frac{\partial\newoverline{u}}{\partial y} + \frac{\newoverline{u}}{a}\tan\left(\phi\right),\\ \newoverline{k} = \frac{1}{2}\newoverline{u^2} + \frac{1}{2}\newoverline{v^2} \Rightarrow \frac{\partial\newoverline{k}}{\partial y} &= \newoverline{u\frac{\partial u}{\partial y}} + \newoverline{v\frac{\partial v}{\partial y}} = \newoverline{u}\frac{\partial\newoverline{u}}{\partial y} + \newoverline{v}\frac{\partial\newoverline{v}}{\partial y} + \newoverline{u'\frac{\partial u'}{\partial y}} + \newoverline{v'\frac{\partial v'}{\partial y}}\nonumber\\ &= \newoverline{u}\frac{\partial\newoverline{u}}{\partial y} + \newoverline{u'\frac{\partial u'}{\partial y}} + \newoverline{v'\frac{\partial v'}{\partial y}}. \end{align} \]
Putting this into Eq. (19.35), it 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
\[ \begin{align} \zeta' = \frac{\partial v'}{\partial x} - \frac{\partial u'}{\partial y} + \frac{v'}{a}\tan\left(\phi\right) \end{align} \]
this can be further transformed into
\[ \begin{align} g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y} &= -f\newoverline{u} - \frac{\newoverline{u}}{a}\tan\left(\phi\right)\newoverline{u} - \newoverline{u'\frac{\partial u'}{\partial y}} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{\left(\frac{\partial v'}{\partial x} - \frac{\partial u'}{\partial y} + \frac{v'}{a}\tan\left(\phi\right)\right)u'}\nonumber\\ \Leftrightarrow g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y} &= -f\newoverline{u} - \frac{\newoverline{u}}{a}\tan\left(\phi\right)\newoverline{u} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{\left(\frac{\partial v'}{\partial x} + \frac{v'}{a}\tan\left(\phi\right)\right)u'}\nonumber\\ \Leftrightarrow \textcolor{red}{g\frac{\partial\left(\newoverline{h} + \newoverline{b}\right)}{\partial y}} &= \textcolor{red}{-f\newoverline{u}} - \newoverline{v'\frac{\partial v'}{\partial y}} - \newoverline{u'\frac{\partial v'}{\partial x}} + \textcolor{blue}{\newoverline{\frac{u'v'}{a}\tan\left(\phi\right)} - \frac{\newoverline{u}^2}{a}\tan\left(\phi\right)}. \end{align} \]
The terms marked in red contain the geostrophic balance, the parts marked in blue are metric terms. The black terms are the turbulent momentum advection.
The application of quasi-geostrophic theory is restricted to the $\beta-$plane and is therefore not possible globally (see Sect. 13.11). Nevertheless, the quasi-geostrophic equations are also to 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.