The continuity equation is the balance equation of the mass. Let $\Omega\subseteq\mathbb{R}^3$ be an open subset, $\rho$ a density and $\mathbf{j}$ the corresponding flux density. Then using Gauss's theorem and a source density $Q$
\[ \begin{align} \frac{\partial}{\partial t}\int_\Omega\rho d^3r &= -\int_{\partial \Omega}\mathbf{j}\cdot d\mathbf{n} + \int_\Omega Qd^3r = -\int_\Omega\nabla\cdot\mathbf{j} - Q d^3r\nonumber\\ \Rightarrow \int_\Omega\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} - Qd^3r &= 0.\tag{7.1}\label{eq:deriv_cont_1} \end{align} \]
Since the integrand is continuous, it is already homogeneous and constantly equal to zero. If $\left|\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} - Q\right|>\epsilon>0$ holds at a point $\mathbf{r}_0\in\Omega$, then there exists an open environment $\omega\subseteq\Omega$ with $\mathbf{r}_0\in\omega$ and $\int_\omega\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} - Qd^3r \not= 0$ in contradiction to Eq. (7.1). It therefore applies
\[ \begin{align} \frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} = Q.\tag{7.2}\label{eq:cont_general} \end{align} \]
In specific volume terms $\alpha$ this can be written as
\[ \begin{align} -\frac{1}{\alpha^2}\frac{\partial\alpha}{\partial t} - \frac{1}{\alpha^2}\mathbf{v}\cdot\nabla\alpha + \frac{1}{\alpha}\nabla\cdot\mathbf{v} = 0 & {} & \Leftrightarrow \frac{\partial\alpha}{\partial t} = \alpha\nabla\cdot\mathbf{v} - \mathbf{v}\cdot\nabla\alpha.\tag{7.3}\label{eq:cont_spec_volume} \end{align} \]
note down. This is now noted separately for all components of the air:
\[ \begin{align} \frac{\partial\rho_d}{\partial t} + \nabla\cdot\mathbf{j}_d &= Q_d\tag{7.4}\label{eq:kont_d_pre}\\ \frac{\partial\rho_v}{\partial t} + \nabla\cdot\mathbf{j}_v &= Q_v\tag{7.5}\label{eq:kont_v_pre}\\ \frac{\partial\rho_i}{\partial t} + \nabla\cdot\mathbf{j}_i &= Q_i\tag{7.6}\label{eq:kont_i_pre} \end{align} \]
This applies
\[ \begin{align} \mathbf{j}_d = \rho_d\mathbf{v} - D_d\nabla\rho_d, & {} & \mathbf{j}_v = \rho_v\mathbf{v} - D_v\nabla\rho_v. \end{align} \]
The $D_d, D_v$ are the diffusion coefficients for dry and moist air, respectively. You can at first
\[ \begin{align} Q_d = 0 \end{align} \]
hold onto.
Now the $Q_x$ should be specified in more detail in the equations (7.4) - (7.6). There are five processes that contribute to them:
The last point will be ignored in the rest of the section; this will only be about the condensates. Diffusion only occurs with the gaseous components, so in $Q_d$ there is a term $D_d\Delta\rho_d$ and in $Q_v$ there is a term $D_v\Delta\rho_v$, here $D_d$ and $D_v$ are the diffusion coefficients of dry air and water vapor in air, respectively. Dry air is not affected by the other processes, so
\[ \begin{align} Q_d = D_d\Delta\rho_d \end{align} \]
applies.
If two particles of the condensate classes $j$ and $k$ meet, one can specify a probability $0\leq P_{j, k}\leq 1$ that they will then form a particle. This has a well-defined class $R_{j, k}$. Furthermore, let $\sigma_{j, k}$ be the cross section of the collisions of particles of classes $j$ and $k$. For example, if $j$ are spheres with radius $r_j$ and $k$ are spheres with radius $r_k$, then $\sigma_{j, k} = \pi\left(r_j + r_k\right)^2$. Let the average relative velocity between the particles of classes $j$ and $k$ be $\newoverline{v_{\text{rel}}}\left(j, k\right)$. Then a particle of class $j$ flies through a volume $\sigma_{j, k}\newoverline{v_{\text{rel}}}\left(j, k\right)t$ in time $t$ relative to the particles of class $k$. The average collision time $\tau _{j, k}$ is defined by requiring that exactly one particle of class $k$ is in this volume, i.e
\[ \begin{align} n_k\sigma_{j, k}\newoverline{v_{\text{rel}}}\left(j, k\right)\tau_{j, k} &\hastobe 1. \end{align} \]
All particles of group $j$ participate in collision processes, therefore the total collision rate of particles $j$ with particles $k$ is given by
\[ \begin{align} \frac{n_j}{\tau_{j, k}} &= n_jn_k\sigma_{j, k}\newoverline{v_{\text{rel}}}\left(j, k\right). \end{align} \]
If $\newtilde{m}_i$ is the mass of the particles of the group $i = R_{j, k}$, this results in a corresponding source strength
\[ \begin{align} \newtilde{m}_i\sigma_{j, k}n_jn_k\newoverline{v_{\text{rel}}}\left(j, k\right)P_{j, k}. \end{align} \]
Furthermore, a negative proportion of the swelling strength must also be taken into account, which occurs because particles of their original condensate class are lost when they combine with other particles. This leads to
\[ \begin{align} Q_i^{(\text{Kollisionen})} = \newtilde{m}_i\sum_{j}\sum_{k\leq j}^{}\sigma_{j, k}n_jn_k\newoverline{v_{\text{rel}}}\left(j, k\right)P_{j, k}\left(\delta_{i, R_{j, k}} - \delta_{j, i} - \delta_{k, i}\right). \end{align} \]
Because of mass conservation, this requires that the particle masses $\newtilde{m}_i$ are integer multiples of a minimum mass so that no mass artificially disappears or arises during collisions.
Decay processes are now being investigated. The following applies to the change in particle density $n_j$ due to decay
\[ \begin{align} dn_j = -n_j\lambda_jdt\Leftrightarrow\frac{dn_j}{dt} = -\lambda_j n_j, \end{align} \]
here $\lambda_j$ is the decay constant. The product of the decay is a particle of class $k$ and one of class $l$ with probability $Z_{j, k, l}$; the normalization applies here
\[ \begin{align} 1 = \sum_{k}\sum_{l\leq k}^{}Z_{j, k, l}, \end{align} \]
where $Z_{j, j, l} = Z_{j, k, j} = 0$, that is, the product of a decay must not be equal to the reactant. Positive and negative terms must be taken into account for each condensate class $i$, so that:
\[ \begin{align} Q_i^{\left(\text{Zerfälle}\right)} = \newtilde{m}_i\sum_{j}\sum_{k}\sum_{l\leq k}^{}\lambda_j n_j Z_{j, k, l}\left(\delta_{i, k} + \delta_{i, l} - \delta_{j, i}\right). \end{align} \]
Finally, phase transitions are discussed. These can take place in two ways.
First, condensation and resublimation can lead directly to the creation of new particles of class $i$, and evaporation and sublimation can lead to their destruction, respectively. The corresponding mass flux densities are denoted by $\newtilde{q}_{v, i}'$ in the first case and $\newtilde{q}_{i, v}'$ in the second case. Through this process you get terms
\[ \begin{align} Q_v^{\left(\pm\text{Entstehung}\right)} &= \sum_{j}^{}\newtilde{q}_{j, v}' - \newtilde{q}_{v, j}',\\ Q_i^{\left(\pm\text{Entstehung}\right)} &= \newtilde{q}_{v, i}' - \newtilde{q}_{i, v}'. \end{align} \]
Large condensation products such as hailstones do not form instantaneously, but rather grow, among other things. through condensation or resublimation of water vapor on smaller particles. They can also disappear again in the opposite way. During a time interval $\Delta t$, condense or resublimate a mass $m_i$ of particles of class $i$ within a volume $\Delta V$. This causes them to grow and become heavier, but it was discussed in Chap. 6 specifies that all particles of class $i$ should have the same mass $\newtilde{m}_i$. If this is the only process that takes place, it is due to conservation of mass
\[ \begin{align} \rho_{g\left(i\right)}\left(t\right)\Delta V + m_i = \rho_{g\left(i\right)}\left(t + \Delta t\right)\Delta V. \end{align} \]
Here $g\left(i\right)$ is the condensate class that arises from $i$ through growth. This follows
\[ \begin{align} \frac{\partial\rho_{g\left(i\right)}}{\partial t} = \frac{1}{\Delta V}\frac{dm_i}{dt}\eqqcolon \newtilde{q}_{v, i}''. \end{align} \]
This is initially confusing because condensation on particles $i$ changes the density $\rho_{g\left(i\right)}$; This results from the fact that no new particles of class $i$ are formed during this type of phase transition. In the case of evaporation or sublimation one obtains
\[ \begin{align} \frac{\partial\rho_{d\left(i\right)}}{\partial t}\eqqcolon\newtilde{q}_{i, v}'', \end{align} \]
where $d\left(i\right)$ is the condensate class formed from $i$ by shrinkage.
Thirdly, condensates can transform into one another by freezing or melting; the corresponding swelling strengths are denoted by $\newtilde{q}_{j, k}'''$. You receive
\[ \begin{align} Q_i^{\left(\text{Umwandlung}\right)} = \sum_{j}\newtilde{q}_{j, i}''' - \newtilde{q}_{i, j}'''. \end{align} \]
The source strengths are summarized as follows:
\[ \begin{align} Q_d &= D_d\Delta\rho_d\tag{7.23}\label{eq:quelle_trocken_masse}\\ Q_v &= \sum_{i}^{}\left(\newtilde{q}_{i, v}' - \newtilde{q}_{v, i}' + \newtilde{q}_{i, v}'' - \newtilde{q}_{v, i}''\right) + D_v\Delta\rho_v\tag{7.24}\label{eq:quelle_wasserdampf_masse}\\ Q_i &= \newtilde{m}_i\sum_{j}\sum_{k\leq j}^{}\sigma_{j, k}n_jn_k\newoverline{v_{\text{rel}}}\left(j, k\right)P_{j, k}\left(\delta_{i, R_{j, k}} - \delta_{j, i} - \delta_{k, i}\right) + \newtilde{q}_{v, i}' - \newtilde{q}_{i, v}'\nonumber\\ & + \newtilde{m}_i\sum_{j}\sum_{k}\sum_{l\leq k}^{}\lambda_j n_j Z_{j, k, l}\left(\delta_{i, k} + \delta_{i, l} - \delta_{j, i}\right) + \left(\newtilde{q}_{g\left(i\right), v}'' + \newtilde{q}_{v, d\left(i\right)}''\right) + \sum_{j}\newtilde{q}_{j, i}''' - \newtilde{q}_{i, j}'''\tag{7.25}\label{eq:quelle_kondensat_masse} \end{align} \]
The exact calculation of the $P_{j, k}, \lambda_j$ and $Z_{j, k, l}$ as well as the phase transition rates is the task of smaller-scale numerical methods and would exceed the aim of this book; this is analogous to the spectra.
Now the mass flux densities $\mathbf{j}_i$ have to be specified. Let $v_i$ be the equilibrium sink speed of the particles $i$, then one can from
\[ \begin{align} \mathbf{j}_i = \rho_i\mathbf{v} - \mathbf{k}\rho_iv_i \end{align} \]
go out. For the precipitation $P_i$ of the condensate class $i$ applies
\[ \begin{align} P_i = -j_i\left(z_{\text{SFC}}\right), \end{align} \]
where SFC refers to the earth's surface. These equations are also applicable to aerosols.
The specific humidity $q = \rho_i/\rho$ is also often used (def. see equation (5.167)), since it is a conserved quantity in the case of a component $i$ advected by the wind field:
\[ \begin{align} \md{q} = -\frac{q}{\rho}\md{\rho} + \frac{1}{\rho}\md{\rho_i} = q\nabla\cdot\mathbf{v} - q\nabla\cdot\mathbf{v} = 0\tag{7.28}\label{eq:q_conservative} \end{align} \]
The source terms of $q$ are those of $\rho_i$ divided by the density of the medium $\rho$.