If you introduce $N \geq 1$ tracers (i.e. parts of the air that do not belong to dry air), of which $1 \leq N_C \leq N - 1$ are condensed, you need $N$ additional continuity equations. If we make the assumption of local thermodynamic equilibrium (all condensates have the same temperature as the gas phase), there is no need to introduce additional first laws. The basic approach for the dynamic and thermodynamic coupling of the gaseous and condensed tracers to the equation system for dry air was defined in Section 10.3.
Let $\rho$ be a mass density. In preparation, the Crank-Nicolson method is noted with $n$ as the current time step:
\[ \begin{align} \rho^{(n + 1)} = \rho^{(n)} + \Delta t\frac{1}{2}\left(\frac{\partial\rho^{(n)}}{\partial t} + \frac{\partial\rho^{(n + 1)}}{\partial t}\right) \end{align} \]
At this point we are only concerned with vertical convergence; the other terms are calculated explicitly:
\[ \begin{align} \rho^{(n + 1)} = \rho^{(n)} + \Delta t\newdot{\rho}^{(n)}_{\text{expl}} - \Delta t\frac{1}{2}\left(\frac{\partial\left(\rho^{(n)}w^{(n + 1)}\right)}{\partial z} + \frac{\partial\left(\rho^{(n + 1)}w^{(n + 1)}\right)}{\partial z}\right) \end{align} \]
Only the generalized density is treated in the form of the Crank-Nicolson method; the speed is taken from the new time step for reasons of stability. The share
\[ \begin{align} -\frac{1}{2}\frac{\partial\left(\rho^{(n)}w^{(n + 1)}\right)}{\partial z} \end{align} \]
is absorbed into the explicit tendency, i.e. defined
\[ \begin{align} \newdot{\rho}^{(n)}_{\text{expl}} \to \newdot{\rho}^{(n)}_{\text{expl}} - \frac{1}{2}\frac{\partial\left(\rho^{(n)}w^{(n + 1)}\right)}{\partial z} \end{align} \]
around. Vertical discretization requires a layer index $1 \leq i \leq N_L$:
\[ \begin{align} \rho_i^{(n + 1)} = \rho_i^{(n)} + \Delta t\newdot{\rho_i}^{(n)}_{\text{expl}} - \frac{\Delta t}{2}\frac{A_{i - 1/2}\newoverline{\rho_k^{(n + 1)}}^{(i - 1/2)}w_{i - 1/2}^{(n + 1)} - A_{i + 1/2}\newoverline{\rho_k^{(n + 1)}}^{(i + 1/2)}w_{i + 1/2}^{(n + 1)}}{V_i}\tag{36.5}\label{eq:adv_gen_density_deriv_0} \end{align} \]
The vertical averages of the density are generally noted
\[ \begin{align} \newoverline{\rho_k}^{(i + 1/2)} = \alpha_{i + \frac{1}{2}, i}\rho_{i} + \alpha_{i + \frac{1}{2}, i + 1}\rho_{i + 1} \end{align} \]
with
\[ \begin{align} \alpha_{i + \frac{1}{2}, i} + \alpha_{i + \frac{1}{2}, i + 1} = 1. \end{align} \]
Putting this into Eq. (36.5), you get
\[ \begin{align} &\rho_i^{(n + 1)} = \rho_i^{(n)} + \Delta t\newdot{\rho_i}^{(n)}_{\text{expl}}\nonumber\\ &- \frac{\Delta t}{2}\left(\alpha_{i - \frac{1}{2}, i - 1}\rho_{i - 1}^{(n + 1)} + \alpha_{i - \frac{1}{2}, i}\rho_i^{(n + 1)}\right)\frac{A_{i - 1/2}w_{i - 1/2}^{(n + 1)}}{V_i} + \frac{\Delta t}{2}\left(\alpha_{i + \frac{1}{2}, i}\rho_i^{(n + 1)} + \alpha_{i + \frac{1}{2}, i + 1}\rho_{i + 1}^{(n + 1)}\right)\frac{A_{i + 1/2}w_{i + 1/2}^{(n + 1)}}{V_i}. \end{align} \]
You now define
\[ \begin{align} W_{i + 1/2} \coloneqq A_{i + 1/2}w_{i + 1/2}^{(n + 1)}, \end{align} \]
thus you get
\[ \begin{align} &\rho_i^{(n + 1)} = \rho_i^{(n)} + \Delta t\newdot{\rho_i}^{(n)}_{\text{expl}}\nonumber\\ &- \frac{\Delta t}{2}\left(\alpha_{i - \frac{1}{2}, i - 1}\rho_{i - 1}^{(n + 1)} + \alpha_{i - \frac{1}{2}, i}\rho_i^{(n + 1)}\right)\frac{W_{i - 1/2}^{(n + 1)}}{V_i} + \frac{\Delta t}{2}\left(\alpha_{i + \frac{1}{2}, i}\rho_i^{(n + 1)} + \alpha_{i + \frac{1}{2}, i + 1}\rho_{i + 1}^{(n + 1)}\right)\frac{W_{i + 1/2}^{(n + 1)}}{V_i}. \end{align} \]
At $i = 1$, $w_{i - 1/2} = 0$:
\[ \begin{align} &\rho_1^{(n + 1)} = \rho_1^{(n)} + \Delta t\newdot{\rho}^{(n)}_{1, \text{expl}} + \frac{\Delta t}{2}\left(\alpha_{\frac{3}{2}, 1}\rho_1^{(n + 1)} + \alpha_{\frac{3}{2}, 2}\rho_{2}^{(n + 1)}\right)\frac{W_{3/2}^{(n + 1)}}{V_1} \end{align} \]
If $i = N_L$ then $w_{i + 1/2} = 0$:
\[ \begin{align} \rho_{N_L}^{(n + 1)} = \rho_{N_L}^{(n)} + \Delta t\newdot{\rho}^{(n)}_{N_L, \text{expl}} - \frac{\Delta t}{2}\left(\alpha_{N_L - \frac{1}{2}, N_L - 1}\rho_{N_L - 1}^{(n + 1)} + \alpha_{N_L - \frac{1}{2}, N_L}\rho_{N_L}^{(n + 1)}\right)\frac{W_{N_L - 1/2}^{(n + 1)}}{V_{N_L}}. \end{align} \]
The vector $\mathbf{x}$ of the unknowns is defined by
\[ \begin{align} \mathbf{x} = \left(\begin{array}{c} \rho^{(n + 1)}_{1}\\ \rho^{(n + 1)}_{2}\\ \vdots\\ \rho^{(n + 1)}_{N_L} \end{array}\right), \end{align} \]
for this a linear system of equations applies
\[ \begin{align} A\cdot\mathbf{x} = \mathbf{r} \end{align} \]
with a matrix $A$ and a right-hand side $\mathbf{r}$. For this right side applies
\[ \begin{align} \mathbf{r} = \left(\begin{array}{c} \rho^{(n)}_{1} + \Delta t\newdot{\rho}^{(n)}_{1, \text{expl}}\\ \rho^{(n)}_{2} + \Delta t\newdot{\rho}^{(n)}_{2, \text{expl}}\\ \rho^{(n)}_{3} + \Delta t\newdot{\rho}^{(n)}_{3, \text{expl}}\\ \vdots\\ \rho^{(n)}_{N_L - 1} + \Delta t\newdot{\rho}^{(n)}_{N_L - 1, \text{expl}}\\ \rho^{(n)}_{N_L} + \Delta t\newdot{\rho}^{(n)}_{N_L, \text{expl}} \end{array}\right). \end{align} \]
For the matrix $A$ one obtains
\[ \begin{align} A &= \left(\begin{array}{cccc} d_1 & e_1 & \dots & 0 \\ b_1 & d_2 & e_2 \hspace{2 cm}\dots & 0 \\ \vdots & \hspace{2 cm}\ddots & \ddots & \vdots \\ 0 & \dots & b_{N_L - 1} & d_{N_L} \end{array}\right) \end{align} \]
with vectors $\mathbf{c}, \mathbf{e} \in \mathbb{R}^{N_L - 1}$, $\mathbf{e} \in \mathbb{R}^{N_L}$. For this you get
\[ \begin{align} c_i &= \frac{\Delta t}{2}\frac{\alpha_{i + \frac{1}{2}, i}W_{i + 1/2}^{(n + 1)}}{V_{i + 1}},\\ d_1 &= 1 - \frac{\Delta t}{2}\alpha_{\frac{3}{2}, 1}\frac{W_{3/2}^{(n + 1)}}{V_1},\\ d_i &= 1 + \frac{\Delta t}{2}\frac{\alpha_{i - \frac{1}{2}, i}W_{i - 1/2}^{(n + 1)} - \alpha_{i + \frac{1}{2}, i}W_{i + 1/2}^{(n + 1)}}{V_i},\\ d_{N_L} &= 1 + \frac{\Delta t}{2}\frac{\alpha_{N_L - \frac{1}{2}, N_L}W_{N_L - 1/2}^{(n + 1)}}{V_{N_L}},\\ e_i &= -\frac{\Delta t}{2}\alpha_{i + \frac{1}{2}, i + 1}\frac{W_{i + 1/2}^{(n + 1)}}{V_i}. \end{align} \]
Limiters serve the purpose of maintaining upper bounds for flows. These upper bounds usually refer to the fact that mass densities resulting from the flows cannot be negative, as this would be unphysical. For tracers with very variable densities, this can occur purely through advection. Since limiters are numerical interventions that cannot be derived purely from physical laws, the numerics must be constructed in such a way that they have to be called as rarely as possible and, if they do, the interventions are as small as possible.
After advection, diffusion and phase transitions, it is possible for negative densities to arise despite the limiter. These must be set to zero at the end of the time step. Although this violates conservation of mass, there is no alternative.
If the air is supersaturated at time $t$, a phase transition from gaseous to liquid or from gaseous to solid occurs until the air is saturated. If the air is undersaturated and cloud water or cloud ice is still present, the phase transition takes place in the other direction. For modeling it is assumed that after a time step of length $\Delta t$ the air is saturated, i.e
\[ \begin{align} p_v\left(t + \Delta t\right) = p_S\left(t + \Delta t\right),\tag{36.22}\label{eq:model_phase_trans_deriv_1} \end{align} \]
Here $p_v$ is the vapor pressure and $p_S$ is, as usual, the saturation vapor pressure. From this the density $\Delta\rho_v$ of water vapor should be determined, which must condense or resublimate to reach this state. This is not simply $\rho_S\left(t\right) - \rho_v\left(t\right),$ where $\rho_S$ is the saturation density, as this would leave the phase transition heat unaccounted for. From Eq. (36.22) follows
\[ \begin{align} \rho_v\left(t + \Delta t\right)R_vT\left(t + \Delta t\right) &= p_S\left[T\left(t + \Delta t\right)\right]\nonumber\\ \Leftrightarrow\left(\rho_v + \Delta\rho_v\right)R_vT\left(t + \Delta t\right) &= p_S\left[T\left(t + \Delta t\right)\right], \end{align} \]
Here quantities without explicitly stated time dependence denote quantities at time $t$, i.e. $\rho_v\coloneqq\rho_v\left(t\right).$ From this it follows that $\rho_t$ is the total density
\[ \begin{align} \left(\rho_v + \Delta\rho_v\right)R_v\left(T - \Delta\rho_v\frac{L_v}{\rho_tc_p}\right) &= p_S\left(T - \Delta\rho_v\frac{L_v}{\rho_tc_p}\right). \end{align} \]
If you multiply out the left side and do a first order Taylor expansion on the right side, you get
\[ \begin{align} p_v + \Delta\rho_vR_vT - R_v\Delta\rho_v\rho_v\frac{L_v}{\rho_tc_p} - \left(\Delta\rho_v\right)^2R_v\frac{L_v}{\rho_tc_p} &= p_S - p_S'\Delta\rho_v\frac{L_v}{\rho_tc_p}\nonumber\\ \Leftrightarrow -\left(\Delta\rho_v\right)^2R_v\frac{L_v}{\rho_tc_p} + \Delta\rho_v\left(R_vT - R_v\rho_v\frac{L_v}{\rho_tc_p} + p_S'\frac{L_v}{\rho_tc_p}\right) + p_v - p_S &= 0\nonumber\\ \Leftrightarrow a\left(\Delta\rho_v\right)^2 + b\Delta\rho_v + c &= 0\nonumber\\ \Leftrightarrow \left(\Delta\rho_v\right)^2 + p\Delta\rho_v + q &= 0 \end{align} \]
with
\[ \begin{align} a &\coloneqq -R_v\frac{L_v}{\rho_tc_p} < 0,\\ b &\coloneqq R_vT - R_v\rho_v\frac{L_v}{\rho_tc_p} + p_S'\frac{L_v}{\rho_tc_p},\\ c &\coloneqq p_v - p_S,\\ p &\coloneqq \frac{b}{a},\\ q &\coloneqq \frac{c}{a}. \end{align} \]
This implies two solutions
\[ \begin{align} \left(\Delta\rho_v\right)_{1,2} = -\frac{p}{2} \pm \sqrt{\frac{p^2}{4} - q}. \end{align} \]
Now it remains to be determined which of the two signs is the physically correct one. In the case of undersaturation, $p_v
\[ \begin{align} \Delta\rho_v = -\frac{p}{2} - \sqrt{\frac{p^2}{4} - q}. \end{align} \]
If phase transitions are calculated using the averaged thermodynamic variables, the subscale variability considered in Section 17.9.3 is neglected. Here the fundamental considerations made in Section 17.9.3 are to be made usable for a model. Let $T = T\left(\mathbf{r}\right)$ be the temperature field valid at a fixed time $t$ in a grid box $V\subseteq\mathbb{R}^3$. Let $T_\mathrm{min}$ and $T_\mathrm{max}$ be the minimum and maximum temperatures within this set. Then, due to the strict monotonicity of $p_S\left(T\right)$, the minimum and maximum saturation vapor pressure is derived:
\[ \begin{align} p_{S,\text{min}} &= p_S\left(T_\text{min}\right),\\ p_{S,\text{max}} &= p_S\left(T_\text{max}\right). \end{align} \]
Let $p_v$ be the average vapor pressure within the grid box under consideration. Now one makes the assumption that the vapor pressure $p_v$ is homogeneous within $V$. The basic idea is that condensation begins at $p_v>p_{S,\mathrm{min}}$, but initially in a very small volume fraction, the cloud-filled volume fraction of the grid box $c_f$. For $c_f$ you do the following approach
\[ \begin{align} c_f\left(p_v\leq p_{S,\text{min}}\right) &= 0,\\ c_f\left(p_v\geq p_{S,\text{max}}\right) &= 1. \end{align} \]
This is correct in any case. The simplest way to fulfill this for $p_{S,\mathrm{min}}\leq p_v\leq p_{S,\mathrm{max}}$ is a linear interpolation:
\[ \begin{align} c_f\left(p_v\right) &= \frac{p_v - p_{S,\text{min}}}{p_{S,\text{max}} - p_{S,\text{min}}}. \end{align} \]
This would be the case, for example, if the temperature were equally distributed between $T_\mathrm{min}$ and $T_\mathrm{max}$ and the saturation vapor pressure in this interval depended linearly on the temperature, which is true to the first order. The idea is now to use one approach to calculate the vapor pressure with which to calculate the phase transition rates in the case $p_v>p_{S,\mathrm{min}}$ (in which only cloud formation can take place).
\[ \begin{align} p_v^{(\text{eff})} = p_{S,\text{min}} + \xi\left(p_v - p_{S,\text{min}}\right)\tag{36.38}\label{eq:p_v_eff} \end{align} \]
to express. We now have to determine $\xi$. $\xi$ calculates over
\[ \begin{align} \xi = \frac{p_v^{(\text{eff})} - p_{S,\text{min}}}{p_v - p_{S,\text{min}}} = \frac{\int_V\text{max}\left(p_v - p_S\left(\mathbf{r}\right),0\right)d^3r}{V\left(p_v - p_{S,\text{min}}\right)}. \end{align} \]
Under the assumption already made above that $T$ and $p_S$ are equally distributed within $V$, in the case $p_{S,\mathrm{min}}\leq p_v\leq p_{S,\mathrm{max}}$ it follows
\[ \begin{align} \xi &= \frac{\int_{p_{S,\text{min}}}^{p_v}p_v - p_S'dp_S'}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)} = \frac{p_v\left(p_v - p_{S,\text{min}}\right) - \frac{1}{2}\left(p_v^2 - p_{S,\text{min}}^2\right)}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)}\nonumber\\ &= \frac{p_v\left(p_v - p_{S,\text{min}}\right) - \frac{1}{2}\left(p_v + p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)} = \frac{p_v - \frac{1}{2}\left(p_v + p_{S,\text{min}}\right)}{p_{S,\text{max}} - p_{S,\text{min}}} = \frac{p_v - p_{S,\text{min}}}{2\left(p_{S,\text{max}} - p_{S,\text{min}}\right)} = \frac{c_f}{2}. \end{align} \]
In the case $p_v>p_{S,\text{max}}$ applies
\[ \begin{align} c_f &= 1, \end{align} \]
the grid box is 100 percent covered with clouds. In this case it follows for $\xi$
\[ \begin{align} \xi &= \frac{\int_{p_{S,\text{min}}}^{p_{S,\text{max}}}p_v - p_S'dp_S'}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)} = \frac{p_v\left(p_{S,\text{max}} - p_{S,\text{min}}\right) - \frac{1}{2}\left(p_{S,\text{max}}^2 - p_{S,\text{min}}^2\right)}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)}\nonumber\\ &= \frac{p_v\left(p_{S,\text{max}} - p_{S,\text{min}}\right) - \frac{1}{2}\left(p_{S,\text{max}} + p_{S,\text{min}}\right)\left(p_{S,\text{max}} - p_{S,\text{min}}\right)}{\left(p_{S,\text{max}} - p_{S,\text{min}}\right)\left(p_v - p_{S,\text{min}}\right)} = \frac{p_v - \frac{1}{2}\left(p_{S,\text{max}} + p_{S,\text{min}}\right)}{p_v - p_{S,\text{min}}}. \end{align} \]
The previous derivation in this section refers to the formation of clouds (dealing with supersaturation, also known as saturation adjustment). The fact that only a certain proportion $c_f$ of the grid boxes is filled with clouds also has consequences for the interactions and evaporation of condensates. The approach is this:
The approach is used for the portion $c_p$ of the grid box filled with precipitation.
\[ \begin{align} c_p = \begin{cases} \frac{1}{3}, c_f = 0 \land \rho_s + \rho_r + \rho_g>0,\\ c_f, \:\text{sonst}. \end{cases} \end{align} \]
It is therefore assumed that the proportion filled with precipitation is equal to the proportion filled with clouds, unless there is only precipitation but no clouds in the grid box. This is typically the case under clouds from which precipitation falls.
This scheme takes the following four classes of condensate into account:
This scheme is adequate for global models. It is assumed that at and below $-35^\circ$ C the cloud water is completely ice-shaped and at and above $-10^\circ$ C the cloud water is completely liquid and there is a linear transition in between. This can be justified by observations [35].
The size distributions of the condensates are relevant to calculate the falling velocities, the interactions with the radiation field and the interactions with each other. Therefore, a formula for the effective radius must be specified for each condensate class.
\[ \begin{align} d_{\text{eff},i} &= \left(1,2351 + 0,0105\cdot\left(t - t_0\right)\right)\cdot\left(45,8966\cdot w^{0,2214} + 0,7957\cdot w^{0,2535}\cdot\left(t - 83,15\right)\right).\tag{36.44}\label{eq:sun_rikus} \end{align} \]
Here, $t\coloneqq T/\text{K}$ is the measure of the temperature in Kelvin, $t_0\coloneqq 273.15$, $w \coloneqq 1000\left(\rho_i + \rho_s\right)/\left(\text{kg}/\text{m}^3\right)$ is the measure of the cloud ice density (including snow) in $\text{g}/\text{m}^3$ and $d_\text{eff}$ the measure of the effective diameter in micrometers. The result of Eq. (36.44) is restricted in the following way:
\[ \begin{align} d_\text{eff} \to \text{max}\left(\text{min}\left(d_\text{eff},155\right),20 + 40\cdot\cos\left(\phi\right)\right) \end{align} \]
with $\phi$ as the geographical latitude. For the effective radius $r_\text{eff}$ in (36.44)
\[ \begin{align} r_{\text{eff},i} = \frac{3\sqrt{3}}{8}d_{\text{eff},i}\cdot 10^{-6}\text{m} \end{align} \]
derived. The fact that the factor is not simply 1/2 is because ice crystals i. A. are not spherical.
The assumption of Marshall-Palmer distributions implies that the process of breaking up condensates, which limits the spectrum upwards, does not have to be explicitly simulated, but is already parameterized via the distribution.
To determine the falling speeds of the condensates, one should refer to Section 23.2.3, but with some simplifications:
\[ \begin{align} v_S = a\cdot r^b = 9,356\frac{\text{m}^{0,7}}{\text{s}}\cdot\left(2\cdot r_\text{eff}\right)^{0,3} = 11,519\frac{\text{m}^{0,7}}{\text{s}}\cdot\left(r_\text{eff}\right)^{0,3}\tag{36.47}\label{eq:fall_snow} \end{align} \]
[16] angenommen.
If the falling speed of a precipitation class is $v_{D,P}\left(D\right)$, it has a volume mixing ratio $r_P$ and a size spectrum $n\left(D\right)$, a mass-weighted falling speed can be calculated
\[ \begin{align} v_P\left(D\right) = \frac{1}{r_P}\int_0^\infty\frac{\pi D^3}{6}v_{D,P}\left(D\right)n\left(D\right)dD \end{align} \]
calculate. Assume that $v_{D,P}$ has the form
\[ \begin{align} v_{D,P}\left(D\right) = aD^b \end{align} \]
as well as the Marshall-Palmer distribution Eq. (23.95) applies, follows
\[ \begin{align} v_P\left(D\right) &= \frac{1}{r_P}\int_0^\infty\frac{\pi D^3}{6}aD^bn_0e^{-\lambda D}dD = \frac{an_0\pi}{6r_P}\int_0^\infty D^{b + 3}e^{-\lambda D}dD = \frac{an_0\pi}{6r_P\lambda^{b + 4}}\int_0^\infty D^{b + 3}e^{-D}dD\nonumber\\ &= \frac{an_0\pi\Gamma\left(b + 4\right)}{6r_P\lambda^{b + 4}}. \end{align} \]
From Eq. (23.99) is obtained
\[ \begin{align} \frac{1}{\lambda^4} = \frac{r_P}{\pi n_0}. \end{align} \]
From this it follows
\[ \begin{align} v_P\left(D\right) &= \frac{a\Gamma\left(b + 4\right)}{6\lambda^b} \stackrel{\href{ch-22-cloud-microphysics.html#eq:marshall-palmer_d_lambda}{\text{Glg. (23.100)}}}{=} \frac{\Gamma\left(b + 4\right)}{6}aD^b. \end{align} \]
The simplest understanding of how rain occurs is to assume that it occurs when cloud water $\rho_w$ exceeds a critical mixing ratio $q_{w,\mathrm{krit}}$. A simple mathematical approach based on this for the source density of rainwater is:
\[ \begin{align} Q_r = \frac{\text{max}\left(\rho_w - q_{w,\text{krit}}\cdot\rho_h,0\right)}{\tau}. \end{align} \]
This is the so-called Kessler approach [25]. Typical values are $q_{w,\mathrm{crit}} = 0.2\cdot 10^{-3}\mathrm{ kg}/\mathrm{m}^3$, $\tau = 1000\mathrm{ s}$.
The Kessler approach presented in Sect. 36.5.3.1 can also be applied to the formation of snow, typical values are $q_{i,\text{krit}} = 1.0\cdot 10^{-3}\text{ kg}/\text{m}^3$, $\tau = 1000\text{ s}$.
The Kessler approach presented in Sect. 36.5.3.1 can also be applied to the formation of sleet, typical values are $q_{s,\text{krit}} = 0.6\cdot 10^{-3}\text{ kg}/\text{m}^3$, $\tau = 1000\text{ s}$.
In this scheme, a fifth condensate class is added to the four-condensate scheme, namely graupel. The five condensate classes used here are:
This scheme is adequate for regional models. It corresponds to the scheme presented in Sect. 36.5 with the following modifications:
In the schemes considered so far, each condensate class is described by a prognostic variable, namely the mass density or the specific density. This would only be a complete description if all condensates of the respective class had the same radius and temperature as the gas phase. However, neither is generally the case. Two extensions are available: