The governing equations form the system that determines the temporal evolution of the atmosphere. First, however, its state must be described.
A particle is a small air volume $\Delta V$, chosen large enough that, in the long-term mean, it contains sufficiently many gas molecules for a continuum description of matter to be justified. The atmosphere $A\subseteq\mathbb{R}^3$ is the gaseous envelope of the Earth. The openness of $A$ is useful, since it allows directional derivatives in every direction at every point. The definition of the upper boundary depends on the problem under consideration. At the lower boundary, the atmosphere is bounded by the Earth's surface, including the motions taking place there; therefore the set $A$ is time-dependent, $A = A\left(t\right)$. All meteorological fields are assumed to be differentiable as often as required for the calculations. Air consists of dry air and tracers. These include:
water vapor
liquid or solid water
aerosols that do not consist of water
gaseous components present in variable concentrations
Moist air is a mixture of dry air and water vapor. The number of tracer classes considered, in particular the condensate classes, is regarded as sufficient once incorrect predictions are no longer attributable to coarse approximations in the governing equation system used for the forecast, but rather to errors in the initial and/or boundary conditions. The following state variables are defined:
The density of dry air $\rho_d:A\to\mathbb{R}$ is defined by
\[ \begin{align} \rho_d \coloneqq \frac{1}{\Delta V}\sum_{i = 1}^{N_d}m_i^{(d)}. \end{align} \]
Here, $N_d$ is the number of gas atoms, excluding H$_2$O and other trace gases treated separately, present in the volume $\Delta V$, and $m_i^{(d)}$ are their masses.
The wind vector $\mathbf{v}:A\to \mathbb{R}^3$ is defined by
\[ \begin{align} \mathbf{v} \coloneqq \frac{1}{\sum_{j = 1}^{N_g}m_j^{(g)}}\sum_{i = 1}^{N_g}m_i^{(g)}\mathbf{v}_i^{(g)},\tag{6.2}\label{eq:def_windvector} \end{align} \]
where $\mathbf{v}_i^{(g)}$ is the velocity of the $i$-th molecule of the gaseous fraction of air in $\Delta V$.
The temperature $T:A\to\mathbb{R}$ of the moist air.
Let $p:A\to\mathbb{R}$ denote the pressure.
The derivation of the Navier-Stokes equations is based on a continuum conception of matter, motivated by a continuum transition. The continuum transition is a heuristic idea based on the assumption that matter appears continuous once one zooms out far enough for the granular structure formed by atoms and molecules to blur. Mathematically, this means describing the system by continuously differentiable fields rather than by a collection of trajectories.
A similar problem arises when phase transitions and condensates are included in the description. One could describe condensates by trajectories while continuing to treat the gas phase as a fluid. This has two disadvantages: first, such a formulation requires a technical criterion for deciding from which size onward something is treated as condensate; second, the associated computational cost is enormous. Instead, one applies the so-called Second phase transition, in which the mass densities and other properties of condensates are likewise represented as continuously differentiable functions. This is, of course, not exact in the way Maxwell's equations are exact, but the underlying issue is already inherent in the Navier-Stokes equations themselves.
For condensates, which from this point onward are also referred to as tracers, the following additional state variables are introduced:
The densities of the tracer classes $\rho_i:A\to \mathbb{R}$ are defined analogously to $\rho_d$.
Let the $Q_i:A \to \mathbb{R}$ be the charge densities of the tracers.
The mass flux density $\mathbf{j}_i:A\to\mathbb{R}^3$ of the $i$-th tracer class is defined by
\[ \begin{align} \mathbf{j}_i \coloneqq\rho_i\sum_{k = 1}^{N_i}\mathbf{v}_k^{(i)}, \end{align} \]
where $\mathbf{v}_k^{(i)}$ is the velocity of the $k$-th particle of the corresponding tracer class in $\Delta V$.
Let the temperature $T_i:A\to \mathbb {R}$ be that of the $i$-th tracer class; all particles within a class are assumed to have the same temperature. Gaseous tracers have the temperature of moist air.
Furthermore, consider the field of spectral radiance $L_\lambda:A\times\left[0, \infty\right)\times\left[0, \pi\right]\times\left[0, 2\pi\right)\to\mathbb{R}$.
The electric field $\mathbf{E}$ and the magnetic flux density $\mathbf{B}$ must also be taken into account (e.g. for thunderstorms and the ionosphere).
The totality of all these quantities is an atmospheric state $Z$. The aim of this chapter is to formulate prognostic equations in such a way that the state trajectory is determined by initial and boundary conditions. Each state is treated as a quasi-stationary state. Through thermodynamic equations of state or other diagnostic equations that arise from exact or approximate relations among the elements of $Z$, a state may already be determined by a proper subset of $Z$ or by a bijection on such a subset.
Let the $i$-th condensate class have the microscopic density $\rho_i'$, meaning, for example, the density of water $\sim$ $10^3$ kgm$^{-3}$, in contrast to the density averaged over a particle. The $\rho_i'$ are assumed to be constants that do not depend on external conditions; for them
\[ \begin{align} \rho_i' = \frac{m_i}{V_i}, \end{align} \]
where $m_i$ is the mass of component $i$ and $V_i$ is its occupied volume. The density of air is generally given by
\[ \begin{align} \rho = \frac{m_g + \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}m_i}{V_g + \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace} V_i}. \end{align} \]
In general, the following holds for the component density $\rho_i$:
\[ \begin{align} \rho_i = \frac{m_i}{v}_h = \frac{m_i}{V_g + \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}V_i}. \end{align} \]
For the density $\rho_g$ of the gaseous air, one has
\[ \begin{align} \rho_g = \frac{m_g}{v}_h = \frac{m_g}{V_g + \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}V_i}. \end{align} \]
For the microscopic density of gaseous air, one obtains
\[ \begin{align} \rho_g' &= \frac{m_g}{V_g} = \frac{m_g}{V - \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}V_i} = \frac{1}{\frac{V}{m_g} - \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}\frac{V_i}{m_g}}\nonumber\\ &= \frac{1}{\frac{1}{\rho_g} - \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}\frac{V_i}{m_i}\frac{m_i}{m_g}} = \frac{\rho_g}{1 - \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}^{}\frac{\rho_i}{\rho_i'}}. \end{align} \]
The microscopic density of water vapor $\rho_v'$ is also referred to as microscopic humidity. Analogously,
\[ \begin{align} \rho_v' = \frac{m_v}{V_g} = \frac{\rho_v}{1 - \sum_{i \in \left\lbrace\text{condensate classes}\right\rbrace}^{}\frac{\rho_i}{\rho_i'}}. \end{align} \]
Hence, for the equation of state of air
\[ \begin{align} p = T_gR_g\rho_g'. \end{align} \]
In statistical physics, one passes from the microstate to the macrostate and regards that description as statistically complete. In the same spirit, one may ask whether the description developed in the previous two sections is complete. Under the second continuum transition, this would only be the case if infinitely many tracer classes were introduced and a continuous temperature distribution were assumed within each particle class. Aside from the restrictive assumption that all particles in a tracer class have the same temperature, the above description therefore has the potential to be complete under suitable assumptions.
Phase transition rates are relevant to the first law of thermodynamics because they are associated with latent heat fluxes, and they are likewise relevant to the continuity equations because they represent mass fluxes. The first law, however, is formulated materially for a specific particle, rather than in local time as in the continuity equation. This raises the question of whether one must distinguish between total and local time derivatives for phase transition rates, since these quantities are themselves time derivatives. To answer this, consider a measuring device that records the mass $m\left(t\right)$ that has changed phase up to time $t>0$ in a stationary control volume $V$. For the phase transition rate $q$, one measures
\[ \begin{align} q = \frac{m\left(t\right)}{tV}. \end{align} \]
Now consider a second measuring device that records the mass $m'\left(t\right)$ that has changed phase up to time $t>0$ in a particle volume $V'\left(t\right)$ moving with the wind field. Let $V'\left(0\right) = V\left(0\right)$. Then this device measures the phase transition rate
\[ \begin{align} q' = \frac{m'\left(t\right)}{tV'\left(t\right)}. \end{align} \]
In the limit $t\to 0$, one has $V'\to V$. Therefore
\[ \begin{align} \lim\limits_{t \to 0}\left(q - q'\right) &= \lim\limits_{t \to 0}\left[\frac{m\left(t\right)}{Vt} - \frac{m'\left(t\right)}{V'\left(t\right)t}\right] = \lim_{t \to 0}\frac{m\left(t\right)}{Vt} - \lim\limits_{t\to 0}\frac{m'\left(t\right)}{V'\left(t\right)t}\nonumber\\ &= \frac{1}{V}\lim\limits_{t \to 0}\frac{m\left(t\right)}{t} - \lim\limits_{t\to0}\frac{1}{V'\left(t\right)}\lim\limits_{t \to 0}\frac{m'\left(t\right)}{t}\nonumber\\ &= \frac{1}{V}\lim\limits_{t \to 0}\frac{m\left(t\right)}{t} - \frac{1}{v}_h\lim\limits_{t \to 0}\frac{m'\left(t\right)}{t} = \frac{1}{v}_h\lim\limits_{t \to 0}\frac{m\left(t\right) - m'\left(t\right)}{t} \end{align} \]
according to the standard rules for functional limits. With the true space- and time-dependent condensation rate $q_r\left(\mathbf{r}, t\right)$, one can write
\[ \begin{align} m\left(t\right) = \int_{0}^t\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3rdt' \end{align} \]
and
\[ \begin{align} m'\left(t\right) = \int_{0}^t\int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3rdt' \end{align} \]
This yields
\[ \begin{align} m\left(t\right) - m'\left(t\right) &= \int_{0}^t\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3rdt' - \int_{0}^t\int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3rdt'\nonumber\\ &= \int_{0}^t\left[\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3r - \int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3r\right]dt'. \end{align} \]
By the fundamental theorem of differential and integral calculus, it follows that
\[ \begin{align} \lim\limits_{t \to 0}\frac{m\left(t\right) - m'\left(t\right)}{t} &= \frac{1}{V}\lim\limits_{t \to 0}\left[\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3r - \int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3r\right] = 0. \end{align} \]
Hence $q - q' = 0$ in the limit $t\to 0$. The phase transition rates measured in the moving particle system are therefore the same as those measured in the resting system, so no distinction between total and local time derivatives arises here. The same applies analogously to all other source terms and forcings.