The governing equations form the system of equations that determine the temporal development of the atmosphere. First, however, their condition must be described.
A particle is a small volume of air $\Delta V$, which is so large that on a long-term average it contains such a large number of gas molecules that a continuous picture of matter can be assumed. The atmosphere $A\subseteq\mathbb{R}^3$ is the gas envelope of the Earth. The openness makes sense so that a directional derivation in every direction is possible at every point. The definition of the upper bound depends on the problem being addressed. At the lower limit, the atmosphere is limited by the earth's surface, including the movements that take place there, so the amount $A$ 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:
Moist air is a mixture of dry air and water vapor. The number of tracer classes taken into account, in particular the condensate classes, is considered sufficient when incorrect predictions are no longer due to coarseness in the equation system used for the prediction, but to errors in the initial and/or boundary conditions. The following state variables are defined:
\[ \begin{align} \rho_d \coloneqq \frac{1}{\Delta V}\sum_{i = 1}^{N_d}m_i^{(d)}. \end{align} \]
Here, $N_d$ is the number of gas atoms other than H$_2$O and other separately considered trace gases present in the volume $\Delta V$, and the $m_i^{(d)}$ are their masses.
\[ \begin{align} \mathbf{v} \coloneqq \frac{1}{\sum_{j = 1}^{N_g}m_j^{(g)}}\sum_{i = 1}^{N_g}m_i^{(g)}\mathbf{v}_i^{(g)},\tag{6.2}\label{eq:def_windvector} \end{align} \]
where $\mathbf{v}_i^{(g)}$ is the velocity of the $i-$th molecule of the gaseous portion of the air in $\Delta V$.
The derivation of the Navier-Stokes equations is based on a continuous conception of matter, which was motivated by a continuum transition. The continuum transition is a heuristic idea based on the assumption that matter will eventually become continuous if you just zoom out enough to blur the grittiness of matter in the form of atoms and molecules. Mathematically, this means using continuously differentiable fields to describe the system instead of a number of trajectories.
You are faced with a similar problem if you include phase transitions and condensates in the description. You could now describe the condensates using trajectory curves, while still treating the gas phase as a fluid. However, this has two disadvantages: First of all, for such a formulation you have to answer the technical question of what size something is considered to be condensate; In addition, the computational effort required for this is enormous. Instead, you do the so-called Second phase transition, which also considers the mass densities and other properties of the condensates as continuously differentiable functions. Of course, this is not exact in the sense that the Maxwell equations are exact, but the fundamental problem is already inherent in the Navier-Stokes equations themselves.
The following state variables are also introduced for the condensates, which from now on are also referred to as Tracer:
\[ \begin{align} \mathbf{j}_i \coloneqq\rho_i\sum_{k = 1}^{N_i}\mathbf{v}_k^{(i)}, \end{align} \]
where $\mathbf{v}_k^{(i)}$ is the velocity of the $k$th particle of the corresponding tracer class in $\Delta V$.
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 conditions and boundary conditions. Each state is considered a quasi-stationary state. Through thermodynamic equations of state or other diagnostic equations that arise from approximate or approximate relations between the elements of $Z$, a state can already be determined by a real subset of $Z$ or by a bijection on this subset.
The $i-$th condensate class has the microscopic density $\rho_i'$, which means, 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 the volume occupied by it. The density of air is generally given by:
\[ \begin{align} \rho = \frac{m_g + \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}m_i}{V_g + \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace} V_i}. \end{align} \]
The general rule applies to the density of the component $\rho_i$
\[ \begin{align} \rho_i = \frac{m_i}{v}_h = \frac{m_i}{V_g + \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}V_i}. \end{align} \]
The following applies to the density $\rho_g$ of the gaseous air
\[ \begin{align} \rho_g = \frac{m_g}{v}_h = \frac{m_g}{V_g + \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}V_i}. \end{align} \]
For the microscopic densities of gaseous air one obtains
\[ \begin{align} \rho_g' &= \frac{m_g}{V_g} = \frac{m_g}{V - \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}V_i} = \frac{1}{\frac{V}{m_g} - \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}\frac{V_i}{m_g}}\nonumber\\ &= \frac{1}{\frac{1}{\rho_g} - \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}\frac{V_i}{m_i}\frac{m_i}{m_g}} = \frac{\rho_g}{1 - \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}^{}\frac{\rho_i}{\rho_i'}}. \end{align} \]
The microscopic density of water vapor $\rho_v'$ is also referred to as microscopic humidity. The same applies to them
\[ \begin{align} \rho_v' = \frac{m_v}{V_g} = \frac{\rho_v}{1 - \sum_{i \in \left\lbrace\text{Kondensatklassen}\right\rbrace}^{}\frac{\rho_i}{\rho_i'}}. \end{align} \]
This therefore applies to the equation of state of air
\[ \begin{align} p = T_gR_g\rho_g'. \end{align} \]
In statistical physics, the description moves from the microstate to the macrostate and sees this description as statistically complete. Similarly, one may question whether the description developed in the previous two sections is complete. This would only be the case (assuming the second continuum transition) if an infinite number of tracer classes were introduced and a continuous temperature distribution was assumed in each particle class. Except for the restrictive assumption that all particles of a tracer class have the same temperature, the above description has the potential to be a complete description under certain assumptions.
Phase transition rates matter to the first law of thermodynamics because they are associated with latent heat flows, as well as to the continuity equations that they are mass flows. However, the first law is formulated materially for a specific particle and not in local time like the continuity equation. The question arises as to whether there is a distinction between total and local time derivatives in the phase transition rates, since these quantities are also time derivatives. To answer this question, imagine a measuring device that measures the mass $m\left(t\right)$, which has changed phase up to a time $t>0$ in a stationary control volume $V$. Measure for the phase transition rate $q$
\[ \begin{align} q = \frac{m\left(t\right)}{tV}. \end{align} \]
Imagine a second measuring device that measures the mass $m'\left(t\right)$, which has changed phase up to a time $t>0$ in a particle $V'\left(t\right)$ that moves with the wind field. Let $V'\left(0\right) = V\left(0\right)$. Then this meter measures the phase transition rate
\[ \begin{align} q' = \frac{m'\left(t\right)}{tV'\left(t\right)}. \end{align} \]
In the case $t\to 0$ goes $V'\to V$. Therefore applies
\[ \begin{align} \lim\limits_{t \to 0}\left(q - q'\right) &= \lim\limits_{t \to 0}\left[\frac{m\left(t\right)}{Vt} - \frac{m'\left(t\right)}{V'\left(t\right)t}\right] = \lim_{t \to 0}\frac{m\left(t\right)}{Vt} - \lim\limits_{t\to 0}\frac{m'\left(t\right)}{V'\left(t\right)t}\nonumber\\ &= \frac{1}{V}\lim\limits_{t \to 0}\frac{m\left(t\right)}{t} - \lim\limits_{t\to0}\frac{1}{V'\left(t\right)}\lim\limits_{t \to 0}\frac{m'\left(t\right)}{t}\nonumber\\ &= \frac{1}{V}\lim\limits_{t \to 0}\frac{m\left(t\right)}{t} - \frac{1}{v}_h\lim\limits_{t \to 0}\frac{m'\left(t\right)}{t} = \frac{1}{v}_h\lim\limits_{t \to 0}\frac{m\left(t\right) - m'\left(t\right)}{t} \end{align} \]
according to the calculation rules for functional limit values. With the true space- and time-dependent condensation rate $q_r\left(\mathbf{r}, t\right)$ one can
\[ \begin{align} m\left(t\right) = \int_{0}^t\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3rdt' \end{align} \]
as well as
\[ \begin{align} m'\left(t\right) = \int_{0}^t\int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3rdt' \end{align} \]
note down. This gives you
\[ \begin{align} m\left(t\right) - m'\left(t\right) &= \int_{0}^t\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3rdt' - \int_{0}^t\int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3rdt'\nonumber\\ &= \int_{0}^t\left[\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3r - \int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3r\right]dt'. \end{align} \]
According to the main theorem of differential and integral calculus follows
\[ \begin{align} \lim\limits_{t \to 0}\frac{m\left(t\right) - m'\left(t\right)}{t} &= \frac{1}{V}\lim\limits_{t \to 0}\left[\int_{V}q_{r}\left(\mathbf{r}, t\right)d^3r - \int_{V'\left(t\right)}q_{r}\left(\mathbf{r}, t\right)d^3r\right] = 0. \end{align} \]
So $q - q' = 0$ for the limit $t\to 0$. The phase transition rates in the particle system are measured in the same way as in the system at rest, so that the distinction between total and local time derivatives does not occur here. This applies analogously to all other source strengths and forcings.