Any process that violates adiabaticity, i.e. any heat or mass flux across a system boundary, is diabatic. Heat power densities consist of:
heat conduction,
heat transfer,
latent heat,
radiation and
dissipation.
Mass source strengths arise from
diffusion,
phase transitions,
collisions and
decay,
Heat conduction occurs only in the gas phase and, according to Eq. (5.220), leads 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 $\kappa_h$ denoting the thermal diffusivity of moist air.
If evaporation, melting, or sublimation occurs, the required heat is extracted from the reactant. If the phase transition proceeds in the opposite direction, the heat acts on the product. In Chap. 7, three types of phase transition were distinguished. In the first case, new particles of class $i$ are created, and
\[ \begin{align} q_i' = c_{i, v}\left(\newtilde{q}_{v, i}' - \newtilde{q}_{i, v}'\right) \end{align} \]
this gives the heat power per volume acting on component $i$; here, $c_{i, v}$ is the phase-transition 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 converts particles into another condensate class $j$. However, the latent heat still acts on particles of component $i$:The heat transferred in this process within a time interval $\Delta t$ is split between the two condensate classes in proportion to mass; as $\Delta t$ tends to zero, the transferred heat per mass tends to zero, but the converted mass also tends to zero, so the heat acting on component $j$ vanishes to second order in 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 one another. In this case,
\[ \begin{align} q_i''' = \sum_{j}^{}c_{i, j}\newtilde{q}_{j, i}''' - c_{i, j}\newtilde{q}_{i, j}'''. \end{align} \]
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 $c_{i, j} = 0$, because in this case the latent heat acts on the solid phase $j$, as noted above. For the total heat source strength due to phase transitions, $q_i$, one obtains
\[ \begin{align} q_i = q_i' + q_i'' + q_i'''. \end{align} \]
It should be emphasized once more that no latent heat fluxes act directly on the gas phase; they act only on the condensates. The energy is then redistributed by heat transfer, including transfer to the gas phase.
If the condensates have temperatures different from that of the moist air, this is accompanied by heat transfer (diffusive heat flux across an interface). One starts from
\[ \begin{align} s = \xi\Delta T \end{align} \]
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 acting on 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 number density of condensates $i$, $A_i$ is their surface area, and $\xi_i$ is the heat transfer coefficient relative to moist air. For 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, the power density $q_{\mathrm{diss}}$ is assumed to be distributed among the components in proportion to their masses:
\[ \begin{align} q_{\text{diss}, i} = -\rho_i\mathbf{v}\cdot\mathbf{f}_R \end{align} \]
The total power density is then:
\[ \begin{align} q_{\text{diss}} = q_{h} + \sum_{i}^{}q_{\text{diss}, i} = -\left[\rho_h + \sum_{i}^{}\rho_i\right]\mathbf{v}\cdot\mathbf{f}_R = -\rho\mathbf{v}\cdot\mathbf{f}_R, \end{align} \]
which again corresponds to Eq. (8.66).
The Poynting theorem, Eq. (3.47), is a continuity equation for the radiative flux density $\mathbf{S}$. The derivation is classical: $-\mathbf{j}\cdot\mathbf{E}$ is the power delivered by the field to moving charges. The equation can be written as
\[ \begin{align} \frac{1}{v}_h\left(P_{\text{field}} + P_{\text{charges}}\right) = -\nabla\cdot\mathbf{S}, \end{align} \]
where $P_{\text{field}} = \frac{\partial w}{\partial t}$ is the local time derivative of the energy density, and $\mathbf{j}\cdot\mathbf{E}$ is the power delivered to charge carriers. The charges are generally not free but organized into atoms and molecules, and $P_{\text{charges}}$ can produce more general quantum-mechanical excitations. The equation can therefore be written as
\[ \begin{align} \frac{1}{v}_h\left(P_{\text{field}} + P_{\text{matter}}\right) = -\nabla\cdot\mathbf{S}. \end{align} \]
Since $P_{\text{field}}\ll P_{\text{matter}}$, one obtains, to good approximation,
\[ \begin{align} \frac{1}{v}_hP_{\text{matter}} = -\nabla\cdot\mathbf{S}. \end{align} \]
Planck's radiation law, Eq. (5.322), was previously formulated as a function of the angular frequency $\omega$. For spectra, however, one usually uses the wavelength $\lambda$ as the independent quantity. One has
\[ \begin{align} c = \frac{\lambda}{T}\Rightarrow\lambda = cT = \frac{2\pi c}{\frac{2\pi}{T}} = \frac{2\pi c}{\omega}\Rightarrow\omega = \frac{2\pi c}{\lambda}. \end{align} \]
With the requirement
\[ \begin{align} u\left(\omega\right)d\omega \hastobe u\left(\lambda\right)d\lambda \end{align} \]
it 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 $L_B$, it 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 $q_j$. One quickly realizes that, on passing through matter, for the change in spectral radiance one has
\[ \begin{align} dL\left(\Omega\right) \propto \rho_j, & {} & dL\left(\Omega\right) \propto ds, \end{align} \]
here $\Omega$ is a solid angle element. From this one can conclude:
\[ \begin{align} dL\left(\Omega\right) &= -\overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}\rho_jk_jLds}^{\text{Absorption}} + \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}\rho_jk_jL_Bds}^{\text{Emission}} + \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}ds\rho_j \int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega'}^{\text{in-scattering from other directions}} - \overbrace{\vphantom{-\rho_jk_jLds + \rho_jk_jL_Bds + ds\rho_j\int_{4\pi}s_j\left(\Omega', \Omega\right)L\left(\Omega'\right)d\Omega' - ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}ds\rho_j L\left(\Omega\right)\int_{4\pi}s_j\left(\Omega, \Omega'\right)d\Omega'}^{\text{out-scattering into other directions}}\nonumber \end{align} \]
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 radiative transfer equation. For the heating rate acting on component $i$, one then has
\[ \begin{align} q_j = \int_{0}^\infty\left(\nabla\cdot\mathbf{S}_\lambda\right)_jd\lambda. \end{align} \]
On the surface of the earth, one has
\[ \begin{align} \mathbf{j}_v\cdot\mathbf{n} = E + S - R - C, \end{align} \]
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 radiative transfer equation are
\[ \begin{align} \mathbf{S}_{\text{in}} &= S_0\mathbf{e}_{\text{Sun $\to$ Earth}} \end{align} \]
at the upper boundary, as well as
\[ \begin{align} L\left(\lambda, \vartheta, \varphi\right) &= \epsilon\left(\lambda, \vartheta, \varphi, T\right)L_B\left(\lambda, \vartheta, \varphi, T\right) \end{align} \]
Furthermore,
\[ \begin{align} \xi_{\text{SFC}}\left(T_{\text{SFC}} - T\right) \end{align} \]
is the heat flux density that occurs as a boundary condition at the surface in the first law for moist air.
\[ \begin{align} \forall\left(i \in \left\lbrace\text{gaseous components of air}\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{condensate classes}\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{condensate classes}\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{tracer classes}\right\rbrace\right)\frac{\partial\rho_i}{\partial t} + \nabla\cdot\mathbf{j}_i &= Q_i\nonumber\\ \forall\left(i \in \left\lbrace\text{condensate classes}\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{Boundary conditions}}& \nonumber\\ \mathbf{v}\cdot\mathbf{n} &= 0\text{ at the upper boundary}\nonumber\\ \rho_i &= 0\text{ at the upper boundary}\nonumber\\ \mathbf{v}\cdot\mathbf{n} &= 0\text{ at the lower boundary}\nonumber\\ \mathbf{S}_{\text{in}} &= S_0\mathbf{e}_{\text{Sun $\to$ Earth}}\text{ at the upper boundary}\nonumber\\ \mathbf{j}_v\cdot\mathbf{n} &= E + S - R - C\text{ at the lower boundary}\nonumber\\ L\left(\lambda, \vartheta, \varphi, T\right) &= \epsilon\left(\lambda, \vartheta, \varphi, T\right)L_B\left(\lambda, \vartheta, \varphi, T\right)\text{ at the upper boundary}\nonumber\\ \tau_{\text{SFC}}\left(T_{\text{SFC}} - T\right) &= \text{heat transfer at the lower boundary}\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.