For the total momentum $\mathbf{P}$ of the atmosphere applies
\[ \begin{align} \mathbf{P} = \int_A\rho\mathbf{v}d^3r. \end{align} \]
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 applies
\[ \begin{align} \mathbf{L} = \int_A\mathbf{r} \times \rho\mathbf{v}d^3r \Rightarrow \frac{d\mathbf{L}}{dt} = \int_A\mathbf{r} \times \rho\mathbf{f}_{\text{ext}}d^3r. \end{align} \]
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.
The total entropy $S$ of an atmosphere has all but one constant
\[ \begin{align} S = \int_A\newtilde{s}d^3r \end{align} \]
with $\newtilde{s}$ as in Eq. (9.33) is defined. 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 delivers
\[ \begin{align} \frac{dS}{dt} = \int_Ac^{(V)}\frac{\rho}{T}q_Td^3r.\tag{14.6}\label{eq:entropy_balance_global_pre} \end{align} \]
This integral is nonnegative because the heat flow is positive in colder areas and negative in warmer areas. Therefore applies
\[ \begin{align} \frac{dS}{dt} & \geq 0. \end{align} \]
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 $:
\[ \begin{align} \md{k} = \frac{\partial k}{\partial t} + \mathbf{v}\cdot\nabla k = -\frac{1}{\rho}\mathbf{v}\cdot\nabla p - wg + \mathbf{v}\cdot\mathbf{f}_R.\tag{14.8}\label{eq:e_kin_prog} \end{align} \]
If you multiply this equation by $\rho$ and add the product of the continuity equation by $k$, you get
\[ \begin{align} \md{K} = \frac{\partial K}{\partial t} + \mathbf{v}\cdot\nabla K = -\mathbf{v}\cdot\nabla p - \rho wg - K\nabla\cdot\mathbf{v} + \rho\mathbf{v}\cdot\mathbf{f}_R\tag{14.9}\label{eq:kin_energy_density_flux_form} \end{align} \]
as a prognostic equation for the kinetic energy density $K = \frac{1}{2}\rho \mathbf{v}^2$. The prognostic equation for the specific geopotential energy $\phi$ is obtained
\[ \begin{align} \md{\phi} = \mathbf{v}\cdot\nabla\phi = wg. \end{align} \]
If you multiply this equation by $\rho$ and add the product of the continuity equation by $\phi$, you get
\[ \begin{align} \md{P} = \frac{\partial P}{\partial t} + \mathbf{v}\cdot\nabla P = \rho wg - P\nabla\cdot\mathbf{v}\tag{14.11}\label{eq:pot_energy_density_flux_form} \end{align} \]
as a prognostic equation for the geopotential energy density $P = \rho\phi$. The prognostic equation for the specific internal energy is $i = c^{(V)}T$
\[ \begin{align} \md{i} = \frac{\partial i}{\partial t} + \mathbf{v}\cdot\nabla i = -p\md{\alpha} + \epsilon = -p\alpha\nabla\cdot\mathbf{v} + \epsilon. \end{align} \]
If you multiply this equation by $\rho$ and add the product of the continuity equation by $i$, you get
\[ \begin{align} \md{\newtilde{I}} = \frac{\partial\newtilde{I}}{\partial t} + \mathbf{v}\cdot\nabla\newtilde{I} = -\left(p + \newtilde{I}\right)\nabla\cdot\mathbf{v} + \rho\epsilon\tag{14.13}\label{eq:internal_energy_prognostic} \end{align} \]
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 derivation of this is obtained by summing the equations just derived (14.9) - (14.13):
\[ \begin{align} \md{K} = \frac{\partial K}{\partial t} + \mathbf{v}\cdot\nabla K &= \frac{\partial K}{\partial t} + \mathbf{v}\cdot\nabla K = -\mathbf{v}\cdot\nabla p \textcolor{blue}{-\rho wg} - K\nabla\cdot\mathbf{v} + \rho\mathbf{v}\cdot\mathbf{f}_R\nonumber\\ + \md{P} = \frac{\partial P}{\partial t} + \mathbf{v}\cdot\nabla P &= \frac{\partial P}{\partial t} + \mathbf{v}\cdot\nabla P = \textcolor{blue}{\rho wg} - P\nabla\cdot\mathbf{v}\nonumber\\ + \md{\newtilde{I}} = \frac{\partial\newtilde{I}}{\partial t} + \mathbf{v}\cdot\nabla\newtilde{I} &= \frac{\partial\newtilde{I}}{\partial t} + \mathbf{v}\cdot\nabla\newtilde{I} = -p\nabla\cdot\mathbf{v} - \newtilde{I}\nabla\cdot\mathbf{v} + \epsilon\nonumber\\ \cline{1-2} \frac{\partial}{\partial t}\left(K + P + \newtilde{I}\right) + \mathbf{v}\cdot\nabla\left(K + P + \newtilde{I}\right) &= -\left(K + P + \newtilde{I}\right)\nabla\cdot\mathbf{v} - \mathbf{v}\cdot\nabla p - p\nabla\cdot\mathbf{v} + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon \end{align} \]
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 you get
\[ \begin{align} \frac{\partial e}{\partial t} + \mathbf{v}\cdot\nabla e &= -e\nabla\cdot\mathbf{v} - \nabla\cdot\left(p\mathbf{v}\right) + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon.\tag{14.16}\label{eq:bernoulli_pre} \end{align} \]
This 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) is obtained for stationary flows of incompressible ideal media
\[ \begin{align} \nabla\cdot\mathbf{v} &= 0,\\ \md{p} &= \mathbf{v}\cdot\nabla p \end{align} \]
the equation
\[ \begin{align} \md{}\left(K + P + p\right) = 0, \end{align} \]
which is called Bernoulli equation. Other forms of this statement are:
\[ \begin{align} \frac{1}{2}\rho\mathbf{v}^2 + \rho gz + p &= \text{const. entlang Stromlinien},\\ \frac{1}{2}\mathbf{v}² + gz + \frac{p}{\rho} &= \text{const. entlang Stromlinien}. \end{align} \]
The total energy $E$ of the atmosphere is $A$
\[ \begin{align} E = \int_{A}^{}ed^3r. \end{align} \]
If one assumes $A$ to be constant, Gauss's theorem applies
\[ \begin{align} \frac{dE}{dt} = \int_{A}^{}\frac{\partial e}{\partial t}d^3r = \int_{A}^{} - \nabla\cdot\left[\left(e + p\right)\mathbf{v}\right] + \rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon d^3r. \end{align} \]
Due to Eq. (8.96) applies
\[ \begin{align} \int_{A}\rho\mathbf{v}\cdot\mathbf{f}_R + \epsilon d^3r = 0. \end{align} \]
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. So with the kinematic boundary condition applies:
\[ \begin{align} E &= \text{const}. \end{align} \]
For the time derivative of the internal energy of a single-phase atmosphere, Eq. (14.13)
\[ \begin{align} \frac{d}{dt}\int_A\newtilde{I}d^3r = \frac{d}{dt}\int_A-p\nabla\cdot\mathbf{v} - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r = \int_A\mathbf{v}\cdot\nabla p - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r. \end{align} \]
The enthalpy balance is obtained by multiplying this equation by the adiabatic exponent $\kappa = \frac{c^{(p)}}{c^{(V)}}$. Analogously one obtains for the kinetic or potential energy
\[ \begin{align} \frac{d}{dt}\int_AKd^3r & \stackrel{\href{#eq:kin_energy_density_flux_form}{\text{Glg. (14.9)}}}{=} \int_A-\mathbf{v}\cdot\nabla p - \rho wg + \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r,\\ \frac{d}{dt}\int_APd^3r & \stackrel{\href{#eq:pot_energy_density_flux_form}{\text{Glg. (14.11)}}}{=} \int_A\rho wgd^3r. \end{align} \]
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
\[ \begin{align} D &\coloneqq \int_A - \rho\mathbf{v}\cdot\mathbf{f}_Rd^3r \stackrel{\href{ch-07-momentum-equation.html#eq:friction_work_sign}{\text{Glg. (8.87)}}}{\geq} 0. \end{align} \]
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
\[ \begin{align} \underbrace{\overbrace{\frac{\partial\mathbf{v}}{\partial t}}^{\text{lokalzeitl. Änderung}} = \overbrace{-\left(\mathbf{v}\cdot\nabla\right)\mathbf{v} - \mathbf{f}\times\mathbf{v}}^{\text{Advektion}} \overbrace{- \frac{1}{\rho}\nabla p + \mathbf{g}}^{\text{adiabatische Konversion}}}_{\text{reversibel}} + \underbrace{\overbrace{\mathbf{f}_R}^{\text{Dissipation}}}_{\text{irreversibel}} \end{align} \]
or
\[ \begin{align} \underbrace{\overbrace{\frac{\partial\mathbf{v}}{\partial t}}^{\text{lokalzeitl. Änderung}} = \overbrace{\mathbf{v}\times\etabi - \nabla k}^{\text{Advektion}} \overbrace{- \frac{1}{\rho}\nabla p - \nabla\phi}^{\text{adiabatische Konversion}}}_{\text{reversibel}} + \underbrace{\overbrace{\mathbf{f}_R}^{\text{Dissipation}}}_{\text{irreversibel}} \end{align} \]
make.
The kinetic energy $\KE$ of a column of air in a hydrostatic atmosphere is by
\[ \begin{align} \KE = \int_0^{\infty}\frac{1}{2}\rho\mathbf{v}_h^2dz = \int_{p\left(0\right)}^{p\left(\infty\right)}\frac{1}{2}\rho\mathbf{v}_h^2\frac{\partial z}{\partial\rho}dp \stackrel{\partial z/\partial\rho = -1/\left(g\rho\right)}{=} -\frac{1}{g}\int_{p_S}^0\frac{1}{2}\mathbf{v}_h^2dp = \frac{1}{g}\int_0^{p_S}\frac{1}{2}\mathbf{v}_h^2dp \end{align} \]
given, where $p_S$ is the pressure at the earth's surface. The same applies to internal energy
\[ \begin{align} \IE = \int_0^\infty\rho c^{(V)}Tdz = \frac{1}{g}\int_{0}^{p_S}c^{(V)}Tdp. \end{align} \]
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“\. It applies
\[ \begin{align} \TPE &= p_Sz_S + g^{-1}\int_0^{p_S}\left(c^{(V)} + R_d\right)Tdp = p_Sz_S + g^{-1}\int_0^{p_S}hdp\nonumber\\ &= p_Sz_S + \frac{c^{(p)}}{g}\int_0^{p_S}\theta\left(\frac{p}{p_0}\right)^{R_d/c^{(p)}}dp\tag{14.41}\label{eq:tpe_id_1} \end{align} \]
with $h = c^{(p)}T$ as enthalpy. You estimate
\[ \begin{align} \frac{\KE}{\IE} \sim \frac{\newoverline{v}_h^2}{2c^{(V)}\newoverline{T}} \sim 0, 03\text{ \%}. \end{align} \]
So there is much more internal than kinetic energy in the atmosphere.
Since the gravity potential could also be shifted by an arbitrary constant, the concept of available potential energy $\APE$ is introduced. To do this, first transform 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) is
\[ \begin{align} \int_A\TPE + \KE dA \end{align} \]
conservative if there is no mass flow across the lateral edges 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 is meant the minimum potential energy of all hydrostatic states that can be reached from the initial state by adiabatic rearrangement without producing overadiabatic 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 you designate the fields of the state of minimum potential energy by primed quantities, you can make a note
\[ \begin{align} \int_A\TPE_\text{min}dA &\approx A\newoverline{p'\left(z_S\right)z_S} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\newoverline{\theta'\left(z_S\right)p'\left(z_S\right)^{1 + \frac{R_d}{c^{(p)}}}} + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}}d\theta.\tag{14.48}\label{eq:ape_deriv_2} \end{align} \]
The first two terms on the right-hand sides of the equations (14.45) and (14.48) refer exclusively to the surface; their difference is abbreviated as $C$. This gives you
\[ \begin{align} \APE \approx C + \frac{Ac^{(p)2}}{gp_0^{R_d/c^{(p)}}\left(c^{(p)} + R_d\right)}\int_{\newoverline{\theta_S}}^\infty\newoverline{p\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} - \newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}}d\theta. \end{align} \]
The mass $M$ over the isentrope $\theta$ is conservative, so
\[ \begin{align} \newoverline{p'\left(\theta\right)} = p'\left(\theta\right) = \newoverline{p\left(\theta\right)} \Rightarrow \newoverline{p'\left(\theta\right)^{1 + \frac{R_d}{c^{(p)}}}} = \newoverline{p\left(\theta\right)}^{1 + \frac{R_d}{c^{(p)}}}, \end{align} \]
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 notes $p = \newoverline{p} + p'$, so the Taylor expansion applies
\[ \begin{align} p^{1 + \chi} = \newoverline{p}^{1 + \chi} + \left(1 + \chi\right)\newoverline{p}^{\chi}p' + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi-1}p'^2 + \dotsc \end{align} \]
From this it follows
\[ \begin{align} p^{1 + \chi} - \newoverline{p}^{1 + \chi} &= \left(1 + \chi\right)\newoverline{p}^{\chi}p' + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi-1}p'^2 + \dotsc\nonumber\\ \Leftrightarrow \newoverline{p^{1 + \chi}} - \newoverline{p}^{1 + \chi} &= \left(1 + \chi\right)\newoverline{p}^{\chi}\newoverline{p'} + \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi - 1}\newoverline{p'^2} + \dotsc\nonumber\\ \Leftrightarrow \newoverline{p^{1 + \chi}} - \newoverline{p}^{1 + \chi} &= \frac{1}{2}\chi\left(1 + \chi\right)\newoverline{p}^{\chi - 1}\newoverline{p'^2} + \dotsc \end{align} \]
So you get
\[ \begin{align} \APE & \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{1 + \chi}\newoverline{\left(\frac{p'}{\newoverline{p}}\right)^2}d\theta. \end{align} \]
Now this is transformed back into the p-system. For this you use
\[ \begin{align} p = p\left(\theta\left(p\right)\right) & \approx \newoverline{p}\left(\theta\left(p\right)\right) \Rightarrow p' \approx \theta'\frac{\partial\newoverline{p}}{\partial\theta}. \end{align} \]
Thus you get
\[ \begin{align} \APE & \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{\chi - 1}\newoverline{\left(\theta'\frac{\partial\newoverline{p}}{\partial\theta}\right)^2}d\theta = C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{\theta_S}}^\infty\newoverline{p}^{\chi - 1}\newoverline{\theta'^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2dp\nonumber\\ &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p}\left(\newoverline{\theta_S}\right)}^{\newoverline{p}\left(\newoverline{\infty}\right)}\newoverline{p}^{\chi - 1}\newoverline{\theta'^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2\frac{\partial\theta}{\partial\newoverline{p}}dp = C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\frac{\newoverline{\theta'^2}}{\newoverline{\theta}^2}\left(\frac{\partial\newoverline{p}}{\partial\theta}\right)^2\frac{\partial\theta}{\partial\newoverline{p}}dp\nonumber\\ &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\frac{\newoverline{\theta'^2}}{\newoverline{\theta}^2}\frac{\partial\newoverline{p}}{\partial\theta}dp \approx C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi - 1}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp\nonumber \end{align} \] \[ \begin{align} &= C + \frac{A\chi}{2\Gamma_dp_0^{\chi}}\int_{\newoverline{p_S}}^0\newoverline{p}^{\chi}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp = C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\left(\frac{\newoverline{p}}{p_0}\right)^{\chi}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp\nonumber\\ &\approx C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\frac{\newoverline{T}}{\newoverline{\theta}}\frac{1}{\newoverline{p}}\newoverline{\theta}^2\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp = C + \frac{A\chi}{2\Gamma_d}\int_{\newoverline{p_S}}^0\frac{\newoverline{T}}{\newoverline{p}}\newoverline{\theta}\newoverline{\left(\frac{\theta'}{\newoverline{\theta}}\right)^2}\left(\frac{\partial\newoverline{\theta}}{\partial p}\right)^{-1}dp. \end{align} \]
With
\[ \begin{align} \theta' = \frac{\partial\theta}{\partial T}T' = \frac{\theta}{T}T' \approx \frac{\newoverline{\theta}}{\newoverline{T}}T' \end{align} \]
you 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) applies
\[ \begin{align} \frac{\Gamma_d\newoverline{p}}{\chi\newoverline{\theta}}\frac{\partial\newoverline{\theta}}{\partial p} \approx \newoverline{\frac{\Gamma_dp}{\chi\theta}\frac{\partial\theta}{\partial p}} = -\left(\Gamma_d - \newoverline{\Gamma}\right), \end{align} \]
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 $A$ you choose the entire earth's surface and set $T' \sim 15$ K, $\newoverline{T} \sim 270$ K, then follow
\[ \begin{align} \frac{\APE/A}{\KE} &\sim \frac{3\newoverline{T'^2}c^{(p)}}{\newoverline{T}\mathbf{v}_h^2} \sim 25. \end{align} \]
It remains an open question in this section why most of the available potential energy is not converted into kinetic energy.