Everything that violates adiabaticness, i.e. heat and mass flows across system boundaries, is diabatic. The power densities of heat are made up of:
The swelling strengths of the mass arise from
see equations (7.23) - (7.25).
Heat conduction only occurs in the gas phase and, according to Eq. (5.220) to a temperature tendency
\[ \begin{align} \frac{\partial\left(\rho c_h^{(v)}T\right)}{\partial t} = \nabla\cdot\left(\rho c_h^{(v)}\kappa\nabla T\right) \end{align} \]
with $\kappa_h$ as the thermal diffusivity of moist air.
If evaporation, melting or sublimation occurs, the heat required for this is removed from the educt. If the phase transition takes place in the other direction, the heat affects the product. In Chap. 7 a distinction was made between three types of phase transitions. In the first case, new particles of class $i$ are created, it applies
\[ \begin{align} q_i' = c_{i, v}\left(\newtilde{q}_{v, i}' - \newtilde{q}_{i, v}'\right) \end{align} \]
for the heat output per volume acting on component $i$, here $c_{i, v}$ is the phase transformation enthalpy of $i$ into the gas phase.
In the second case, the phase transition occurs between the gas and existing particles of component $i$, which leads to the transformation of particles into another class of condensate $j$. However, the latent heat still acts on the particles of component $i$:The heat transferred in this process within a time interval $\Delta t$ is divided by mass between the two condensate classes; for $\Delta t$ the heat transferred per mass approaches zero, but the converted mass also approaches zero, so that the heat acting on component $j$ approaches zero in the second order of time.
\[ \begin{align} q_i'' = c_{i, v}\left(\newtilde{q}_{v, i}'' - \newtilde{q}_{i, v}''\right) \end{align} \]
In the third case, condensates transform into each other. This applies to this
\[ \begin{align} q_i''' = \sum_{j}^{}c_{i, j}\newtilde{q}_{j, i}''' - c_{i, j}\newtilde{q}_{i, j}'''. \end{align} \]
Here $c_{i, j}$ is the phase transition enthalpy acting on $i$ in the corresponding process. For example, if $i$ is liquid and $j$ is solid, then $c_{i, j} = 0$, since in this case the latent heat would act on the solid phase $j$ as noted above. For the total source strength of heat due to phase transitions $q_i$ applies
\[ \begin{align} q_i = q_i' + q_i'' + q_i'''. \end{align} \]
It should be emphasized again at this point that there are no latent heat flows affecting the gas phase, only the condensates. The energy is then distributed via heat transfer, including to the gas phase.
If the condensates have different temperatures than the moist air, this is accompanied by a heat transfer (diffusive heat flow through an interface). One goes from the equation
\[ \begin{align} s = \xi\Delta T \end{align} \]
where $\Delta T$ is the temperature difference between the two phases, $\xi$ is the heat transfer coefficient and $s$ is the heat flux density. For the corresponding power density $q_i$ in the fluid, which acts on the condensate class $i$, one has
\[ \begin{align} q_i = n_iA_i\xi_i\left(T - T_i\right). \end{align} \]
Here $n_i$ is the particle density of the condensates $i$, $A_i$ is their surface and $\xi_i$ is the heat transfer coefficient with respect to moist air. This applies to the air
\[ \begin{align} q_h = \sum_{i}^{}n_iA_i\xi_i\left(T_i - T\right). \end{align} \]
In a medium with condensation products, it is assumed that the power density $q_{\mathrm{diss}}$ is distributed among the components according to the masses:
\[ \begin{align} q_{\text{diss}, i} = -\rho_i\mathbf{v}\cdot\mathbf{f}_R \end{align} \]
The total power density then results in:
\[ \begin{align} q_{\text{diss}} = q_{h} + \sum_{i}^{}q_{\text{diss}, i} = -\left[\rho_h + \sum_{i}^{}\rho_i\right]\mathbf{v}\cdot\mathbf{f}_R = -\rho\mathbf{v}\cdot\mathbf{f}_R, \end{align} \]
what again Eq. (8.66) corresponds.
The Poynting theorem Eq. (3.47) is a continuity equation for the radiant flux density $\mathbf{S}$. The derivation is classic, $ - \mathbf{j}\cdot\mathbf{E}$ is the power that the field performs on moving charges. You can write down the equation as
\[ \begin{align} \frac{1}{v}_h\left(P_{\text{Feld}} + P_{\text{Ladungen}}\right) = -\nabla\cdot\mathbf{S}, \end{align} \]
where $P_{\mathrm{field}} = \frac{\partial w}{\partial t}$ is the local-time derivative of the energy density and $\mathbf{j}\cdot\mathbf{E}$ is the charge carrier power. The charges are generally not free, but structured into atoms and molecules, and $P_{\mathrm{charges}}$ can lead to quantum mechanical excitations of a more general nature. This means you can write down the equation as
\[ \begin{align} \frac{1}{v}_h\left(P_{\text{Feld}} + P_{\text{Materie}}\right) = -\nabla\cdot\mathbf{S}. \end{align} \]
It is $P_{\mathrm{Field}}\ll P_{\mathrm{Matter}}$ and therefore one can get a good approximation
\[ \begin{align} \frac{1}{v}_hP_{\text{Materie}} = -\nabla\cdot\mathbf{S}. \end{align} \]
set.
Planck's radiation law Eq. (5.316) was previously formulated as a function of the angular frequency $\omega$. However, when it comes to spectra, the wavelength $\lambda$ is usually used as the independent quantity. It applies
\[ \begin{align} c = \frac{\lambda}{T}\Rightarrow\lambda = cT = \frac{2\pi c}{\frac{2\pi}{T}} = \frac{2\pi c}{\omega}\Rightarrow\omega = \frac{2\pi c}{\lambda}. \end{align} \]
With the demand
\[ \begin{align} u\left(\omega\right)d\omega \hastobe u\left(\lambda\right)d\lambda \end{align} \]
follows
\[ \begin{align} u\left(\lambda\right) = u\left(\omega\left(\lambda\right)\right)\left|\frac{d\omega}{d\lambda}\right| = \frac{\hbar 8\pi^3 c^3}{\pi^2c^3 \lambda^3}\frac{1}{\exp\left(\frac{\hbar 2\pi c}{k_BT\lambda}\right) - 1}\frac{2\pi c}{\lambda^2} = \frac{8\pi ch}{\lambda^5}\frac{1}{\exp\left(\frac{hc}{k_BT\lambda}\right) - 1}. \end{align} \]
For the spectral radiance of black body radiation, $L_B$ follows
\[ \begin{align} L_B\left(\lambda, T\right) = \frac{2hc^2}{\lambda^5}\frac{1}{\exp\left(\frac{hc}{k_BT\lambda}\right) - 1}. \end{align} \]
The spectral radiance in the atmosphere depends on the location, direction and wavelength, i.e. $L = L\left(\mathbf{r}, \lambda, \vartheta, \varphi\right)$. This results in the field of spectral radiation flux density $\mathbf{S}_\lambda$
\[ \begin{align} \mathbf{S}_\lambda\left(\mathbf{r}, \lambda\right) = \int_{0}^{2\pi}\int_{0}^{\pi}L\left(\mathbf{r}, \lambda, \vartheta, \varphi\right)\mathbf{e}\left(\vartheta, \varphi\right)\sin\left(\vartheta\right)d\vartheta d\varphi \end{align} \]
with $\mathbf{e}\left(\vartheta, \varphi\right)$ as the unit vector pointing in the direction specified by $\vartheta$ and $\varphi$.
Let $j\in\left\lbrace d, v, i\right\rbrace$ denote a component of air, where $i$ stands for a condensate class. Each component has individual radiation properties and therefore has its own spectral power density $q_j$. It quickly becomes clear that when passing through matter, the change in spectral radiance applies
\[ \begin{align} dL\left(\Omega\right) \propto \rho_j, & {} & dL\left(\Omega\right) \propto ds, \end{align} \]
here $\Omega$ is a solid angle element. From this one can conclude:
\[ \begin{align} dL\left(\Omega\right) &= -\overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}\rho_jk_jLds}^{\text{Absorption}} + \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}\rho_jk_jL_Bds}^{\text{Emission}} + \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}ds\rho_j \int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega'}^{\text{Zustreuung aus anderen Raumrichtungen}} - \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}^{\text{Wegstreuung in andere Raumrichtungen}}\nonumber \end{align} \]
Here $k_j$ is the absorption coefficient and $s_j$ is the scattering cross section. This merely describes the change in spectral radiance due to component $i$. To describe the actual change, one must sum over all components:
\[ \begin{align} dL\left(\Omega\right) &= ds\left(L_B - L\left(\Omega\right)\right)\sum_{j}^{}\left(k_j\rho_j\right) + ds\sum_{i}^{}\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega'\nonumber\\ & - ds\sum_{j}^{}\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'.\tag{11.20}\label{eq:strahlungsuebertragungsgleichung} \end{align} \]
This is the radiation transfer equation. The heating rate acting on component $i$ then applies
\[ \begin{align} q_j = \int_{0}^\infty\left(\nabla\cdot\mathbf{S}_\lambda\right)_jd\lambda. \end{align} \]
On the surface of the earth applies
\[ \begin{align} \mathbf{j}_v\cdot\mathbf{n} = E + S - R - C, \end{align} \]
Here $\mathbf{n}$ stands for the normal vector of the earth's surface, $E$ for the evaporation rate, $S$ for the sublimation rate, $R$ for the resublimation rate and $C$ for the condensation rate, all with the dimension mass per area and time. This is a boundary condition on the flux density $\mathbf{j}_v$.
The boundary conditions in the radiation transfer equation are:
\[ \begin{align} \mathbf{S}_{\text{in}} &= S_0\mathbf{e}_{\text{Sonne $\to$ Erde}} \end{align} \]
at the top edge as well
\[ \begin{align} L\left(\lambda, \vartheta, \varphi\right) &= \epsilon\left(\lambda, \vartheta, \varphi, T\right)L_B\left(\lambda, \vartheta, \varphi, T\right) \end{align} \]
Furthermore is
\[ \begin{align} \xi_{\text{SFC}}\left(T_{\text{SFC}} - T\right) \end{align} \]
the heat flux density, which occurs as a boundary condition at the surface in the first law for moist air.
\[ \begin{align} \forall\left(i \in \left\lbrace\text{gasförmige Bestandteile der Luft}\right\rbrace\right)\md{\mathbf{v}_i} &= -\frac{1}{\rho}\nabla p + \mathbf{v}_i\times\mathbf{f} + \mathbf{g} + \nu\Delta\mathbf{v}_g\nonumber\\ \forall\left(i \in \left\lbrace\text{Kondensatklassen}\right\rbrace\right)\mathbf{j}_i &= \rho_i\mathbf{v} - \mathbf{k}\rho_iv_i\nonumber\\ p &= T_gR_g\rho_g'\nonumber\\ c_g^{(v)}\md{T} + p\md{}\left[\frac{1 - \sum_{j \in \left\lbrace\text{Kondensatklassen}\right\rbrace}^{}\frac{\rho_j}{\rho_j'}}{\rho_g}\right] &= \frac{q_g}{\rho_g} \nonumber\\ \frac{\partial\rho_d}{\partial t} + \nabla\cdot\mathbf{j}_d &= 0\nonumber\\ \forall\left(i \in \left\lbrace\text{Tracerklassen}\right\rbrace\right)\frac{\partial\rho_i}{\partial t} + \nabla\cdot\mathbf{j}_i &= Q_i\nonumber\\ \forall\left(i \in \left\lbrace\text{Kondensatklassen}\right\rbrace\right)c_i^{(V)}\md{T} &= \frac{q_i}{\rho_i} \nonumber\\ dL\left(\Omega\right) &= ds\left(L_B - L\left(\Omega\right)\right)\sum_{j\in\left\lbrace d, v, i\right\rbrace}^{}\left(k_j\rho_j\right)\nonumber\\ & + ds\sum_{j\in\left\lbrace d, v, i\right\rbrace}^{}\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega'\nonumber\\ & - ds\sum_{j\in\left\lbrace d, v, i\right\rbrace}^{}\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'\nonumber\\ \text{\underline{Randbedingungen}}& \nonumber\\ \mathbf{v}\cdot\mathbf{n} &= 0\text{ am Oberrand}\nonumber\\ \rho_i &= 0\text{ am Oberrand}\nonumber\\ \mathbf{v}\cdot\mathbf{n} &= 0\text{ am Unterrand}\nonumber\\ \mathbf{S}_{\text{in}} &= S_0\mathbf{e}_{\text{Sonne $\to$ Erde}}\text{ am Oberrand}\nonumber\\ \mathbf{j}_v\cdot\mathbf{n} &= E + S - R - C\text{ am Unterand}\nonumber\\ L\left(\lambda, \vartheta, \varphi, T\right) &= \epsilon\left(\lambda, \vartheta, \varphi, T\right)L_B\left(\lambda, \vartheta, \varphi, T\right)\text{ am Oberrand}\nonumber\\ \tau_{\text{SFC}}\left(T_{\text{SFC}} - T\right) &= \text{ Wärmedurchgang am Unterrand}\nonumber \end{align} \]
Not only is the determinism of these equations mathematically unclear, they are also incomplete because, as equations of statistical physics, they contain higher statistical moments. These are usually small, but due to the nonlinearity, the smallest inaccuracies lead to a major change in the solution after a time $t > t_{\text{crit}}$. A non-statistical prediction is therefore in principle only possible for a limited period of time.