For the total momentum $\mathbf{P}$ of the atmosphere, one has
\[ \begin{align} \mathbf{P} = \int_A\rho\mathbf{v}d^3r. \end{align} \]
According to Newton's Second Axiom only the external forces have to be taken into account for the time derivative of this quantity:
\[ \begin{align} \frac{d\mathbf{P}}{dt} = \int_A\rho\mathbf{f}_{\text{ext}}d^3r. \end{align} \]
$\mathbf{P}$ is therefore never conserved, since external forces always act at least through the surface and gravity.
For the total angular momentum $\mathbf{L}$ of the atmosphere, one has
\[ \begin{align} \mathbf{L} = \int_A\mathbf{r} \times \rho\mathbf{v}d^3r \Rightarrow \frac{d\mathbf{L}}{dt} = \int_A\mathbf{r} \times \rho\mathbf{f}_{\text{ext}}d^3r. \end{align} \]
The angular momentum is conserved exactly when all external forces act on the radial axis, i.e. only in stationary coordinates over a planet with radially symmetrical mass distribution inside.
For the total entropy $S$ of an atmosphere, up to a constant, one has
\[ \begin{align} S = \int_A\newtilde{s}d^3r \end{align} \]
with $\newtilde{s}$ as defined in Eq. (9.33). The prognostic equation of this quantity is Eq. (9.34):
\[ \begin{align} \frac{\partial\newtilde{s}}{\partial t} + \nabla\cdot\left(\newtilde{s}\mathbf{v}\right) &= c^{(V)}\frac{\rho}{T}q_T \end{align} \]
Global integration yields
\[ \begin{align} \frac{dS}{dt} = \int_Ac^{(V)}\frac{\rho}{T}q_Td^3r.\tag{14.6}\label{eq:entropy_balance_global_pre} \end{align} \]
This integral is nonnegative because the heat flux is positive in colder regions and negative in warmer regions. Thus one has
\[ \begin{align} \frac{dS}{dt} & \geq 0. \end{align} \]
In an ideal adiabatic single-phase atmosphere, the processes are reversible.
The prognostic equation of the specific kinetic energy $k = \frac{1}{2}\mathbf{v}^2$ is obtained by multiplying the momentum equation in the form Eq. (8.101) from the left with $\mathbf{v}\cdot $:
\[ \begin{align} \md{k} = \frac{\partial k}{\partial t} + \mathbf{v}\cdot\nabla k = -\frac{1}{\rho}\mathbf{v}\cdot\nabla p - wg + \mathbf{v}\cdot\mathbf{f}_R.\tag{14.8}\label{eq:e_kin_prog} \end{align} \]
Multiplying this equation by $\rho$ and adding the product of the continuity equation with $k$, one obtains
as a prognostic equation for the kinetic energy density $K = \frac{1}{2}\rho \mathbf{v}^2$. As the prognostic equation for the specific geopotential energy $\phi$, one obtains
\[ \begin{align} \md{\phi} = \mathbf{v}\cdot\nabla\phi = wg. \end{align} \]
Multiplying this equation by $\rho$ and adding the product of the continuity equation with $\phi$, one obtains
as a prognostic equation for the geopotential energy density $P = \rho\phi$. As the prognostic equation for the specific internal energy $i = c^{(V)}T$, one obtains
\[ \begin{align} \md{i} = \frac{\partial i}{\partial t} + \mathbf{v}\cdot\nabla i = -p\md{\alpha} + \epsilon = -p\alpha\nabla\cdot\mathbf{v} + \epsilon. \end{align} \]
Multiplying this equation by $\rho$ and adding the product of the continuity equation with $i$, one obtains
as a prognostic equation for the internal energy density $\newtilde{I} \coloneqq \rho i$. The energy density $e$ is the sum of kinetic, potential and internal energy per volume,
\[ \begin{align} e = K + P + \newtilde{I}. \end{align} \]
The total derivative of this is obtained by summing the equations just derived, Eqs. (14.9) - (14.13):
\[ \begin{align} \md{K} = \frac{\partial K}{\partial t} + \mathbf{v}\cdot\nabla K &= \frac{\partial K}{\partial t} + \mathbf{v}\cdot\nabla K = -\mathbf{v}\cdot\nabla p \textcolor{blue}{-\rho wg} - K\nabla\cdot\mathbf{v} + \rho\mathbf{v}\cdot\mathbf{f}_R\nonumber\\ + \md{P} = \frac{\partial P}{\partial t} + \mathbf{v}\cdot\nabla P &= \frac{\partial P}{\partial t} + \mathbf{v}\cdot\nabla P = \textcolor{blue}{\rho wg} - P\nabla\cdot\mathbf{v}\nonumber\\ + \md{\newtilde{I}} = \frac{\partial\newtilde{I}}{\partial t} + \mathbf{v}\cdot\nabla\newtilde{I} &= \frac{\partial\newtilde{I}}{\partial t} + \mathbf{v}\cdot\nabla\newtilde{I} = -p\nabla\cdot\mathbf{v} - \newtilde{I}\nabla\cdot\mathbf{v} + \epsilon\nonumber\\ \\[-0.2em] \overline{\hphantom{MMMMMMMMMMMMMMMMMMMM}} & \overline{\hphantom{MMMMMMMMMMMMMMMMMMMMMMMM}}\\[-0.2em]\\ \frac{\partial}{\partial t}\left(K + P + \newtilde{I}\right) + \mathbf{v}\cdot\nabla\left(K + P + \newtilde{I}\right) &= -\left(K + P + \newtilde{I}\right)\nabla\cdot\mathbf{v} - \mathbf{v}\cdot\nabla p - p\nabla\cdot\mathbf{v} + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon \end{align} \]
The terms marked blue are the gravity terms, these cancel each other out: if gravity produces kinetic energy, this comes at the expense of potential energy. Thus one obtains
\[ \begin{align} \frac{\partial e}{\partial t} + \mathbf{v}\cdot\nabla e &= -e\nabla\cdot\mathbf{v} - \nabla\cdot\left(p\mathbf{v}\right) + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon.\tag{14.16}\label{eq:bernoulli_pre} \end{align} \]
It thus follows
\[ \begin{align} \frac{\partial e}{\partial t} &= -\nabla\cdot\left(p\mathbf{v}\right) - \nabla\cdot\left(e\mathbf{v}\right) \Leftrightarrow\frac{\partial e}{\partial t} + \nabla\cdot\left(e\mathbf{v}\right) = -\nabla\cdot\left(p\mathbf{v}\right) + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon.\tag{14.17}\label{eq:poynting_met} \end{align} \]
This is also referred to as the Poynting theorem of meteorology in analogy to the Poynting theorem of ED Eq. (3.47). As a special case of Eq. (14.16), one obtains, for stationary flows of incompressible ideal media, because of
\[ \begin{align} \nabla\cdot\mathbf{v} &= 0,\\ \md{p} &= \mathbf{v}\cdot\nabla p \end{align} \]
the equation
\[ \begin{align} \md{}\left(K + P + p\right) = 0, \end{align} \]
which is called the Bernoulli equation. Further forms of this statement are
\[ \begin{align} \frac{1}{2}\rho\mathbf{v}^2 + \rho gz + p &= \text{const. along streamlines},\\ \frac{1}{2}\mathbf{v}² + gz + \frac{p}{\rho} &= \text{const. along streamlines}. \end{align} \]
The total energy $E$ of the atmosphere $A$ is
\[ \begin{align} E = \int_{A}^{}ed^3r. \end{align} \]
If one assumes $A$ to be constant, then by Gauss's theorem one has
\[ \begin{align} \frac{dE}{dt} = \int_{A}^{}\frac{\partial e}{\partial t}d^3r = \int_{A}^{} - \nabla\cdot\left[\left(e + p\right)\mathbf{v}\right] + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon d^3r. \end{align} \]
Owing to Eq. (8.96), one has
\[ \begin{align} \int_{A}\rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon d^3r = 0. \end{align} \]
It thus follows
\[ \begin{align} \frac{dE}{dt} = -\int_{O}^{}\left(e + p\right)\mathbf{v}\cdot d\mathbf{n} - \int_{v}^{}\left(e + p\right)\mathbf{v}\cdot d\mathbf{n}, \end{align} \]
Here $O$ is the top of the atmosphere and $U$ is the earth's surface. With the kinematic boundary condition, one thus has
\[ \begin{align} E &= \text{const}. \end{align} \]
For the time derivative of the internal energy of a single-phase atmosphere, with Eq. (14.13), one has
\[ \begin{align} \frac{d}{dt}\int_A\newtilde{I}d^3r = \frac{d}{dt}\int_A-p\nabla\cdot\mathbf{v} - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r = \int_A\mathbf{v}\cdot\nabla p - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r. \end{align} \]
The enthalpy balance is obtained by multiplying this equation by the adiabatic exponent $\kappa = \frac{c^{(p)}}{c^{(V)}}$. Analogously, for the kinetic and potential energy, respectively, one obtains
\[ \begin{align} \frac{d}{dt}\int_AKd^3r & \stackrel{\href{#eq:kin_energy_density_flux_form}{\text{Eq. (14.9)}}}{=} \int_A-\mathbf{v}\cdot\nabla p - \rho wg + \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r,\\ \frac{d}{dt}\int_APd^3r & \stackrel{\href{#eq:pot_energy_density_flux_form}{\text{Eq. (14.11)}}}{=} \int_A\rho wgd^3r. \end{align} \]
It thus follows
\[ \begin{align} \frac{d}{dt}\int_A\newtilde{I} + Pd^3r &= D - C,\\ \frac{d}{dt}\int_A K d^3r &= C - D \end{align} \]
with the adiabatic conversion
\[ \begin{align} C &\coloneqq \int_A-\mathbf{v}\cdot\nabla p - \rho wgd^3r \end{align} \]
and the dissipation
\[ \begin{align} D &\coloneqq \int_A - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r \stackrel{\href{ch-07-momentum-equation.html#eq:friction_work_sign}{\text{Eq. (8.87)}}}{\geq} 0. \end{align} \]
The adiabatic conversion stands for the conversion of potential and internal energy into kinetic energy. This is done by the pressure gradient and gravity doing work. The assignment can therefore be made in the momentum equation
\[ \begin{align} \underbrace{\overbrace{\frac{\partial\mathbf{v}}{\partial t}}^{\text{local time change}} = \overbrace{-\left(\mathbf{v}\cdot\nabla\right)\mathbf{v} - \mathbf{f}\times\mathbf{v}}^{\text{advection}} \overbrace{- \frac{1}{\rho}\nabla p + \mathbf{g}}^{\text{adiabatic conversion}}}_{\text{reversible}} + \underbrace{\overbrace{\mathbf{f}_R}^{\text{dissipation}}}_{\text{irreversible}} \end{align} \]
or
\[ \begin{align} \underbrace{\overbrace{\frac{\partial\mathbf{v}}{\partial t}}^{\text{local time change}} = \overbrace{\mathbf{v}\times\etabi - \nabla k}^{\text{advection}} \overbrace{- \frac{1}{\rho}\nabla p - \nabla\phi}^{\text{adiabatic conversion}}}_{\text{reversible}} + \underbrace{\overbrace{\mathbf{f}_R}^{\text{dissipation}}}_{\text{irreversible}} \end{align} \]
The kinetic energy $\KE$ of a column of air in a hydrostatic atmosphere is given by
\[ \begin{align} \KE = \int_0^{\infty}\frac{1}{2}\rho\mathbf{v}_h^2dz = \int_{p\left(0\right)}^{p\left(\infty\right)}\frac{1}{2}\rho\mathbf{v}_h^2\frac{\partial z}{\partial\rho}dp \stackrel{\partial z/\partial\rho = -1/\left(g\rho\right)}{=} -\frac{1}{g}\int_{p_S}^0\frac{1}{2}\mathbf{v}_h^2dp = \frac{1}{g}\int_0^{p_S}\frac{1}{2}\mathbf{v}_h^2dp \end{align} \]
where $p_S$ is the pressure at the earth's surface. Analogously, for the internal energy, one has
\[ \begin{align} \IE = \int_0^\infty\rho c^{(V)}Tdz = \frac{1}{g}\int_{0}^{p_S}c^{(V)}Tdp. \end{align} \]
For the potential energy $\PE$ one obtains with partial integration
\[ \begin{align} \PE &= \int_0^\infty\rho gzdz = \int_0^{p_S}z\left(p\right)dp = \int_0^{p_S}1\cdot z\left(p\right)dp = \left[pz\left(p\right)\right]_0^{p_S} - \int_0^{p_S}p\frac{\partial z}{\partial p}dp\nonumber\\ &= p_Sz_S + \frac{1}{g}\int_0^{p_S}\frac{p}{\rho}dp = p_Sz_S + \frac{1}{g}\int_0^{p_S}R_dTdp \end{align} \]
with $z_S$ as orography. One defines the total potential energy $\TPE$ by
\[ \begin{align} \TPE &\coloneqq \PE + \IE. \end{align} \]
This form of energy is often simply referred to as „potential energy“. One has
\[ \begin{align} \TPE &= p_Sz_S + g^{-1}\int_0^{p_S}\left(c^{(V)} + R_d\right)Tdp = p_Sz_S + g^{-1}\int_0^{p_S}hdp\nonumber\\ &= p_Sz_S + \frac{c^{(p)}}{g}\int_0^{p_S}\theta\left(\frac{p}{p_0}\right)^{R_d/c^{(p)}}dp\tag{14.41}\label{eq:tpe_id_1} \end{align} \]
with $h = c^{(p)}T$ as enthalpy. One estimates
\[ \begin{align} \frac{\KE}{\IE} \sim \frac{\newoverline{v}_h^2}{2c^{(V)}\newoverline{T}} \sim 0, 03\%. \end{align} \]
So there is much more internal than kinetic energy in the atmosphere.
Since the gravity potential could also be shifted by an arbitrary constant, one introduces the concept of available potential energy $\APE$. To this end, one first transforms Eq. (14.41) into the $\theta-$system:
\[ \begin{align} \TPE = p_Sz_S + \frac{c^{(p)}}{gp_0^{R_d/c^{(p)}}}\int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta \end{align} \]
Using partial integration one obtains
\[ \begin{align} \int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta &= \left[\theta p^{1 + \frac{R_d}{c^{(p)}}}\right]_\infty^{\theta_S} - \int_\infty^{\theta_S}p\left(\theta\right)\frac{d}{d\theta}\left(\theta p\left(\theta\right)^{R_d/c^{(p)}}\right)d\theta\nonumber\\ \Rightarrow\int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta &= \theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}} - \int_\infty^{\theta_S}p\left(\theta\right)\left(p\left(\theta\right)^{R_d/c^{(p)}} + \theta\frac{R_d}{c^{(p)}}\frac{\partial p}{\partial\theta}p\left(\theta\right)^{R_d/c^{(p)} - 1}\right)d\theta\nonumber\\ \Rightarrow\int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta &= \theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}} - \int_\infty^{\theta_S}p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}} + \frac{R_d}{c^{(p)}}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta\nonumber \end{align} \] \[ \begin{align} \Rightarrow\frac{c^{(p)} + R_d}{c^{(p)}}\int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta &= \theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}} - \int_\infty^{\theta_S}p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}d\theta\nonumber\\ \Rightarrow\int_\infty^{\theta_S}\theta p\left(\theta\right)^{R_d/c^{(p)}}\frac{\partial p}{\partial\theta}d\theta &= \frac{c^{(p)}}{c^{(p)} + R_d}\theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}} + \frac{c^{(p)}}{c^{(p)} + R_d}\int_{\theta_S}^\infty p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}d\theta\nonumber\\ \Rightarrow\TPE &= p_Sz_S + \frac{c^{(p)2}\theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)} + \frac{c^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\theta_S}^\infty p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}d\theta. \end{align} \]
To obtain the total potential energy contained in an atmosphere over an area $A$, one calculates
\[ \begin{align} \int_A\TPE dA &= \int_Ap_Sz_SdA + \frac{c^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_A\theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}}dA + \frac{c^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_A\int_{\theta_S}^\infty p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}d\theta dA\nonumber\\ &= A\newoverline{p_Sz_S} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\newoverline{\theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}}} + \frac{c^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\newoverline{\int_{\theta_S}^\infty p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}d\theta}\nonumber\\ &\approx A\newoverline{p_Sz_S} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\newoverline{\theta_Sp_S^{1 + \frac{R_d}{c^{(p)}}}} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}}d\theta.\tag{14.45}\label{eq:ape_deriv_1} \end{align} \]
According to Eq. (14.41),
\[ \begin{align} \int_A\TPE + \KE dA \end{align} \]
is conservative if no mass flux occurs across the lateral boundaries of $A$. The $\APE$ is the fraction of the $\TPE$ that could be converted into kinetic energy, i.e.,
\[ \begin{align} \APE \coloneqq \int_A\TPE - \TPE_{\text{min}}dA, \end{align} \]
where what is meant is the minimum of the potential energy of all hydrostatic states that can be reached from the initial state by adiabatic rearrangement without producing superadiabatic gradients. Here, the particles move along the isentropes and after such a rearrangement, all state variables only depend on the vertical coordinate, since otherwise pressure gradients would exist that would work towards this (the Coriolis force does no work, therefore has no energetic relevance and therefore does not have to be taken into account for this consideration). If one denotes the fields of the state of minimum potential energy by primed quantities, one can write
\[ \begin{align} \int_A\TPE_\text{min}dA &\approx A\newoverline{p'\left(z_S\right)z_S} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\newoverline{\theta'\left(z_S\right)p'\left(z_S\right)^{1 + \frac{R_d}{c^{(p)}}}} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}}d\theta.\tag{14.48}\label{eq:ape_deriv_2} \end{align} \]
The first two terms on the right-hand sides of Eqs. (14.45) and (14.48) refer exclusively to the surface; their difference is abbreviated as $C$. One thus obtains
\[ \begin{align} \APE \approx C + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} - \newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}}d\theta. \end{align} \]
The mass $M$ above the isentrope $\theta$ is conservative, hence
\[ \begin{align} \newoverline{p'\left(\theta\right)} = p'\left(\theta\right) = \newoverline{p\left(\theta\right)} \Rightarrow \newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} = \newoverline{p\left(\theta\right)}^{1 + \frac{R_d}{c^{(p)}}}, \end{align} \]
from which follows
\[ \begin{align} \APE &\approx C + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} - \newoverline{p\left(\theta\right)}^{1 + \frac{R_d}{c^{(p)}}}d\theta\nonumber\\ &= C + \frac{Ac^{(p)}}{gp_0^{R_d/c^{(p)}}\left(1 + \frac{R_d}{c^{(p)}}\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} - \newoverline{p\left(\theta\right)}^{1 + \frac{R_d}{c^{(p)}}}d\theta\nonumber\\ &= C + \frac{A}{\Gamma_dp_0^{\chi}\left(1 + \chi\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \chi}} - \newoverline{p\left(\theta\right)}^{1 + \chi}d\theta \end{align} \]
with the dry adiabatic temperature gradient $\Gamma_d = g/c^{(p)}$ and the abbreviation $\chi \coloneqq \frac{R_d}{c^{(p)}}$. One writes $p = \newoverline{p} + p'$; the Taylor expansion then holds
\[ \begin{align} p^{1 + \chi} = \newoverline{p}^{1 + \chi} + \left(1 + \chi\right)\newoverline{p}^{\chi}p' + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi-1}p'^2 + \dotsc \end{align} \]
From this it follows
\[ \begin{align} p^{1 + \chi} - \newoverline{p}^{1 + \chi} &= \left(1 + \chi\right)\newoverline{p}^{\chi}p' + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi-1}p'^2 + \dotsc\nonumber\\ \Leftrightarrow \newoverline{p^{1 + \chi}} - \newoverline{p}^{1 + \chi} &= \left(1 + \chi\right)\newoverline{p}^{\chi}\newoverline{p'} + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi - 1}\newoverline{p'^2} + \dotsc\nonumber\\ \Leftrightarrow \newoverline{p^{1 + \chi}} - \newoverline{p}^{1 + \chi} &= \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi - 1}\newoverline{p'^2} + \dotsc \end{align} \]
One thus obtains
\[ \begin{align} \APE & \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{1 + \chi}\newoverline{\left(\frac{p'}{\newoverline{p}}\right)^2}d\theta. \end{align} \]
Now this is transformed back into the p-system. For this, one uses
\[ \begin{align} p = p\left(\theta\left(p\right)\right) & \approx \newoverline{p}\left(\theta\left(p\right)\right) \Rightarrow p' \approx \theta'\frac{\partial\newoverline{p}}{\partial\theta}. \end{align} \]
Thus one obtains
\[ \begin{align} \APE & \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{\chi - 1}\newoverline{\left(\theta'\frac{\partial\newoverline{p}}{\partial\theta}\right)^2}d\theta = C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{\chi - 1}\newoverline{\theta'^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2dp\nonumber\\ &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p}\left(\newoverline{\theta_S}\right)}^{\newoverline{p}\left(\newoverline{\infty}\right)}\newoverline{p}^{\chi - 1}\newoverline{\theta'^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2\frac{\partial\theta}{\partial\newoverline{p}}dp = C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\frac{\newoverline{\theta'^2}}{\newoverline{\theta}^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2\frac{\partial\theta}{\partial\newoverline{p}}dp\nonumber\\ &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\frac{\newoverline{\theta'^2}}{\newoverline{\theta}^2}\frac{\partial\newoverline{p}}{\partial\theta}dp \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp\nonumber \end{align} \] \[ \begin{align} &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp = C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\left(\frac{\newoverline{p}}{p_0}\right)^{\chi}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp\nonumber\\ &\approx C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\frac{\newoverline{T}}{\newoverline{\theta}}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp = C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\frac{\newoverline{T}}{\newoverline{p}}\newoverline{\theta}\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp. \end{align} \]
With
\[ \begin{align} \theta' = \frac{\partial\theta}{\partial T}T' = \frac{\theta}{T}T' \approx \frac{\newoverline{\theta}}{\newoverline{T}}T' \end{align} \]
one can further transform this to
\[ \begin{align} \APE &\approx C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\frac{1}{\newoverline{p}}\newoverline{\theta}\newoverline{T}\newoverline{\left(\frac{T'}{\newoverline{T}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp\nonumber\\ &= C + \frac{A}{2}\int_{\newoverline{p_S}}^0\frac{\chi\newoverline{\theta}}{\Gamma_d\newoverline{p}}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}\newoverline{T}\newoverline{\left(\frac{T'}{\newoverline{T}}\right)^2}dp = C - \frac{A}{2}\int_0^{\newoverline{p_S}}\frac{\chi\newoverline{\theta}}{\Gamma_d\newoverline{p}}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}\newoverline{T}\newoverline{\left(\frac{T'}{\newoverline{T}}\right)^2}dp. \end{align} \]
According to Eq. (9.48), one has
\[ \begin{align} \frac{\Gamma_d\newoverline{p}}{\chi\newoverline{\theta}}\frac{\partial\newoverline{\theta}}{\partial p} \approx \newoverline{\frac{\Gamma_dp}{\chi\theta}\frac{\partial\theta}{\partial p}} = -\left(\Gamma_d - \newoverline{\Gamma}\right), \end{align} \]
it follows from this
\[ \begin{align} \APE \approx C + \frac{A}{2}\int_0^{\newoverline{p_S}}\frac{\newoverline{T}}{\Gamma_d - \newoverline{\Gamma}}\newoverline{\left(\frac{T'}{\newoverline{T}}\right)^2}dp. \end{align} \]
If one takes a climatological value $\newoverline{\Gamma} \approx \frac{2}{3}\Gamma_d$ and neglects $C$ to make an estimate, it follows
\[ \begin{align} \APE \approx \frac{3Ac^{(p)}}{2g}\int_{\newoverline{p_S}}^0\newoverline{T}\newoverline{\left(\frac{T'}{\newoverline{T}}\right)^2}dp. \end{align} \]
For $A$, one chooses the entire earth's surface and sets $T' \sim 15$ K, $\newoverline{T} \sim 270$ K; then one obtains
\[ \begin{align} \frac{\APE/A}{\KE} &\sim \frac{3\newoverline{T'^2}c^{(p)}}{\newoverline{T}\mathbf{v}_h^2} \sim 25. \end{align} \]
It remains an open question in this section why most of the available potential energy is not converted into kinetic energy.