Changes between Initial Version and Version 1 of doc/app/bcmequ

Timestamp:

Feb 27, 2019 3:05:48 PM (6 years ago)

Author:

westbrink

Comment:

--

doc/app/bcmequ

@@ +1 @@
+== Overview ==
+[[TracNav(doc/app/partoc|nocollapse)]]
+[[TracNav(doc/tec/bcmtoc|nocollapse)]]
+
+[[NoteBox(note,This page is part of the ** Bulk Cloud Model** (BCM) documentation. \\ It contains a listing of physical basics and the equations used.. \\ For an overview of all BCM-related pages\, see the **[wiki:doc/tec/microphysics Bulk Cloud Model main page]**.)]]
+
+
+
+Therefore, PALM solves the prognostic equations for the total water mixing ratio
+{{{
+#!Latex
+\begin{align*}
+  & q = q_\mathrm{v} + q_\mathrm{l},
+\end{align*}
+}}}
+instead of ''q'',,v,,, and for a linear approximation of the liquid water potential temperature ([#emanuel1994 e.g., Emanuel, 1994])
+{{{
+#!Latex
+\begin{align*}
+  \theta_\mathrm{l} = \theta - \frac{L_\mathrm{V}}{c_p \Pi}
+  q_\mathrm{l}\,,
+\end{align*}
+}}}
+instead of ''θ'' as described in Sect. [wiki:doc/tec/gov governing equations]. Since ''q'' and ''θ'',,l,, are conserved quantities for wet adiabatic processes, condensation/evaporation is not considered for these variables.
+
+PALM offers three different schemes ([#kessler1969 Kessler (1969)], [#seifert2001 Seifert and Beheng (2001],[#seifert2006  2006)], [#morrison2007 Morrison et al. (2007)]) for the treatment of liquid phase microphysics. The [#kessler1969 Kessler (1969)] scheme provides a computational inexpensive way for the bulk microphysics. However, it only converts supersaturation into liquid water and considering autoconversion after a parameterization of [#kessler1969 Kessler (1969)].
+
+A more detailed parameterization is given by following the two-moment scheme of [#seifert2001 Seifert and Beheng (2001],[#seifert2006  2006)], which is based on the separation of the droplet spectrum into droplets with radii < 40 μm (cloud droplets) and droplets with radii ≥ 40 μm (rain droplets). Here, the model predicts the first two moments of these partial droplet spectra, namely cloud and rain
+droplet number concentration (''N'',,c,, and ''N'',,r,,, respectively) as well as cloud and rain water mixing ratio
+(''q'',,c,, and ''q'',,r,,, respectively). Consequently, ''q'',,l,, is the sum of both ''q'',,c,, and ''q'',,r,,. The moments' corresponding microphysical tendencies are derived by assuming the partial droplet spectra to follow a gamma distribution that can be described by the predicted quantities and empirical relationships for the distribution's slope and shape parameters. For a detailed derivation of these terms, see [#seifert2001 Seifert and Beheng (2001],[#seifert2006  2006)].
+
+We employ the computational efficient implementation of this scheme as used in the UCLA-LES ([#savic2008 Savic-Jovcic and Stevens, 2008]) and DALES ([#heus2010 Heus et al., 2010]) models. We thus solve only two additional prognostic equations for ''N'',,r,, and ''q'',,r,,:
+{{{
+#!Latex
+\begin{align*}
+ \frac{\partial N_\mathrm{r}}{\partial t} = - u_j \frac{\partial N_\mathrm{r}}{\partial x_j} - \frac{\partial}{\partial x_j}\left(\overline{u_j^{\prime\prime}N_\mathrm{r}^{\prime\prime}}\right) +   \Psi_{N_\mathrm{r}},\\
+ \frac{\partial q_\mathrm{r}}{\partial t} = - u_j
+ \frac{\partial q_\mathrm{r}}{\partial x_j} - \frac{\partial}{\partial
+   x_j}\left(\overline{u_j^{\prime\prime}q_\mathrm{r}^{\prime\prime}}\right)
+ + \Psi_{q_\mathrm{r}},
+\end{align*}
+}}}
+with the sink/source terms ''Ψ'',,Nr,, and ''Ψ'',,qr,,, and the SGS fluxes
+{{{
+#!Latex
+\begin{align*}
+  &  \overline{u_j^{\prime\prime}N_\mathrm{r}^{\prime\prime}} = -K_\mathrm{h} \:\frac{\partial q_\mathrm{r}} {\partial x_{i}}\,\\
+  & \overline{u_j^{\prime\prime}q_\mathrm{r}^{\prime\prime}} =
+  -K_\mathrm{h} \:\frac{\partial N_\mathrm{r}} {\partial
+    x_{i}}\,
+\end{align*}
+}}}
+with ''N'',,c,, and ''q'',,c,, being a fixed parameter and a diagnostic quantity, respectively.
+
+The [#morrison2007 Morrison et al. (2007)] microphysics scheme can be understood as an extension of the scheme of [#seifert2001 Seifert and Beheng (2001],[#seifert2006  2006)], where ''N'',,c,, and ''q'',,c,, are prognostic quantities as well. Moreover, using the [#morrison2007 Morrison et al. (2007)] scheme includes an explicit calculation of diffusional growth and an activation parameterization.
+
+In the next subsections we will describe the diagnostic/prognostic determination (in dependence of the chosen scheme) of ''q'',,c,,. From Sect. [wiki:doc/tec/microphysics#Autoconversion autoconversion] on, the microphysical processes considered in the sink/source terms of ''θ'',,l,,, ''q'', ''N'',,r,, and ''q'',,r,,, as well as ''N'',,c,, and ''q'',,c,, for the [#morrison2007 Morrison et al. (2007)] scheme.
+{{{
+#!Latex
+\begin{align*}
+  &  \Psi_{\theta_\mathrm{l}} = - \frac{L_\mathrm{v}}{c_p \Pi} \varphi_q,\\
+  &  \Psi_{q}  = \left.\frac{\partial q}{\partial t} \right|_\text{sed, c} + \left.\frac{\partial q}{\partial t} \right|_\text{sed, r},\\
+  &  \Psi_{N_\mathrm{r}} = \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_{\text{auto}}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{slf/brk}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_{\text{evap}}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{sed, r},\\
+  & \Psi_{q_\mathrm{r}} = \left.\frac{\partial
+      q_\mathrm{r}}{\partial t} \right|_{\text{auto}} +
+  \left.\frac{\partial q_\mathrm{r}}{\partial t}
+  \right|_{\text{accr}}+ \left.\frac{\partial q_\mathrm{r}}{\partial
+      t} \right|_{\text{evap}}+ \left.\frac{\partial
+      q_\mathrm{r}}{\partial t} \right|_\text{sed, r},\\
+  &  \Psi_{N_\mathrm{c}} = \left.\frac{\partial N_\mathrm{c}}{\partial t} \right|_{\text{acti}}+ \left.\frac{\partial N_\mathrm{c}}{\partial t} \right|_\text{auto}+ \left.\frac{\partial N_\mathrm{c}}{\partial t} \right|_{\text{evap}}+ \left.\frac{\partial N_\mathrm{c}}{\partial t} \right|_\text{sed, c},\\
+  & \Psi_{q_\mathrm{c}} = \left.\frac{\partial
+      q_\mathrm{c}}{\partial t} \right|_{\text{auto}} +
+  \left.\frac{\partial q_\mathrm{c}}{\partial t}
+  \right|_{\text{accr}}+ \left.\frac{\partial q_\mathrm{c}}{\partial
+      t} \right|_{\text{cond,evap}}+ \left.\frac{\partial
+      q_\mathrm{c}}{\partial t} \right|_\text{sed, c},
+\end{align*}
+}}}
+are used in the formulations of [#seifert2006 Seifert and Beheng (2006)] unless explicitly specified. Section [wiki:doc/tec/microphysics#Turbulenceclosure turbulence closure] gives an overview of the necessary changes for the turbulence closure [wiki:doc/tec/sgs#Turbulenceclosure (cf. Sect. turbulence closure)] using ''q'' and ''θ'',,l,, instead of ''q'',,v,, and ''θ'', respectively.
+
+[[Image(Table4.png,600px,border=1)]]
+
+== Activation of cloud droplets ==
+
+The use of the  [#morrison2007 Morrison et al. (2007)] scheme enables a prognostic equation for the cloud droplet number concentration. Here, it is assumed that cloud droplets are activated in dependence of the current supersaturation. This basic method is called Twomey activation scheme with the general form of
+{{{
+#!Latex
+\begin{align*}
+  & N_\mathrm{CCN}=N_\mathrm{0} \times S^\mathrm{k},
+\end{align*}
+}}}
+where ''N'',,CCN,, is the number of activated aerosols, ''N'',,0,, is the number concentration of dry aerosol, S is the supersaturation and k is power index between 0 and 1. In PALM the supersaturation is calculated explicitly by their thermodynamic fields of potential temperature and water vapor mixing ratio. However, curvature and solution effects can be considered with an analytical extension of the Twomey type activation scheme of [#khvorostyanov2006 Khvorostyanov and Curry (2006)]. By doing so, the number of activated aerosol is calculated by 
+{{{
+#!Latex
+\begin{align*}
+  & N_\mathrm{CCN}=\frac{N_\mathrm{0}}{2} [1-\text{erf}(u)];\hspace{1.5cm} u = \frac{\ln(S_\mathrm{0}/S)}{\sqrt{2} \ln \sigma_\mathrm{s}}
+\end{align*}
+}}}
+where erf is the Gaussian error function, and 
+{{{
+#!Latex
+\begin{align*}
+   S_\mathrm{0} & = r_\mathrm{d0}^{-(1+\beta)} \left(\frac{4A^3}{27b}\right)^{1/2},\\
+   \sigma_\mathrm{s} & = \sigma_\mathrm{d}^{1+\beta}.
+\end{align*}
+}}}
+Here A is the Kelivn parameter and b and ''β'' depend on the chemical composition and physical properties of the dry aerosol.
+Since aerosol is not predicted in this scheme, the number of aerosols previously activated is assumed to be equal to the number of droplets ''N'',,c,,.
+Therefore, the actual activation rate is given by  
+{{{
+#!Latex
+\begin{align*}
+    \left. \frac{\partial N_\mathrm{c}}{\partial t} \right|_{\text{acti}} = \text{max}\left(\frac{N_\mathrm{CCN}-N_\mathrm{c}}{\Delta t},0\right).
+\end{align*}
+}}}
+== Diffusional growth of cloud water ==
+
+By the usage of [#seifert2001 Seifert and Beheng (2001],[#seifert2006  2006)] scheme the diagnostic estimation of ''q'',,c,, is based on the assumption that water supersaturations are immediately removed by the diffusional growth of cloud droplets only. This can be justified since the bulk surface area of cloud droplets exceeds that of rain drops considerably ([#stevens2008 Stevens and Seifert, 2008]). Following this saturation adjustment approach, ''q'',,c,, is obtained by
+{{{
+#!Latex
+\begin{align*}
+  & q_\mathrm{c}=\max{\left(0, q - q_\mathrm{r} - q_\mathrm{s}
+    \right)},
+\end{align*}
+}}}
+where ''q'',,s,, is the saturation mixing ratio. Because ''q'',,s,, is a function of ''T'' (not predicted), ''q'',,s,, is computed from the liquid water temperature ''T'',,l,, = ''Π θ,,l,, in a first step:
+{{{
+#!Latex
+\begin{align*}
+ q_\mathrm{s}(T_\mathrm{l}) = \frac{R_\mathrm{d}}{R_\mathrm{v}}
+ \frac{p_\text{v, s}(T_\mathrm{l})}{p - p_\text{v, s}(T_\mathrm{l})},
+\end{align*}
+}}}
+using an empirical relationship for the saturation water vapor pressure ''p'',,v,s,, ([#bougeault1981 Bougeault, 1981]):
+{{{
+#!Latex
+\begin{align*}
+  & p_\text{v, s}(T_\mathrm{l}) = 610.78 \text{Pa} \cdot
+  \exp{\left(17.269\,\frac{T_\mathrm{l}-273.16\,\text{K}}{T_\mathrm{l}-35.86\,\text{K}}
+    \right)}.
+\end{align*}
+}}}
+''q'',,s,,(''T'') is subsequently calculated from a 1st-order Taylor series expansion of ''q'',,s,, at ''T'',,l,, ([#sommeria1977 Sommeria and Deardorff, 1977]):
+{{{
+#!Latex
+\begin{align*}
+  & q_\mathrm{s}(T)=q_\mathrm{s}(T_\mathrm{l})\frac{1+\beta\,q}{1+
+    \beta\,q_\mathrm{s}(T_\mathrm{l})},
+\end{align*}
+}}}
+with
+{{{
+#!Latex
+\begin{align*}
+  & \beta = \frac{L_\mathrm{v}^2}{R_\mathrm{v} c_p
+    T_\mathrm{l}^2}.
+\end{align*}
+}}}
+
+In contrast to that an explicit approach for the diffusional growth is applied in case of [#morrison2007 Morrison et al. (2007)]. The condensation rate is calculated following [#khairoutdinov2000 Khairoutdinov and Kogan (2000)] and given by
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial q_\mathrm{c}}{\partial t}
+  \right|_{\text{cond,evap}}= \frac{4
+  \pi\,G(T,p)}{\rho_\mathrm{a}}S\, R_\mathrm{c},
+\end{align*}
+}}}
+where S is the supersaturation, ''R'',,c,, the integral radius and G(T,p) a function of temperature and pressure considering heat conductivity and diffusion.
+Using this explicit approach the used timestep must fulfill a new criterion, since it is assumed that the supersaturation is constant during one timestep. The typical diffusion timescale is given by [#arnason1971 Arnason and Brown (1971)] with
+{{{
+#!Latex
+\begin{align*}
+  & \Delta t \leq 2 \tau
+\end{align*}
+}}}
+with
+{{{
+#!Latex
+\begin{align*}
+  & \tau = (4\, \pi\,D_\mathrm{v}\,\langle r_\mathrm{c}\rangle)^{-1}.
+\end{align*}
+}}}
+However, in PALM this criterion is not explicitly checked. Too ensure that unrealistic condensation or evaporation rates are avoided this scheme is limited to the value of the saturation-adjustment scheme. 
+== Autoconversion ==
+
+In the following Sects. [wiki:doc/tec/microphysics#Autoconversion Autoconversion] - [wiki:doc/tec/microphysics#Self-collectionandbreakup Self-collection and breakup] we describe collision and coalescence processes by applying the stochastic collection equation ([#pruppacher1997 e.g., Pruppacher and Klett, 1997, Chap. 15.3]) in the framework of the described two-moment scheme. As two species (cloud and rain droplets, hereafter also denoted as c and r, respectively) are considered only, there are three possible interactions affecting the rain quantities: autoconversion, accretion, and selfcollection. Autoconversion summarizes all merging of cloud droplets resulting in rain drops
+(c + c → r). Accretion describes the growth of rain drops by the collection of cloud droplets (r + c → r). Selfcollection denotes the merging of rain drops (r + r → r).
+
+The local temporal change of ''q'',,r,, due to autoconversion is
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial q_\mathrm{r}}{\partial t}
+  \right|_{\text{auto}}=\frac{K_{\text{auto}}}{20\,m_{\text{sep}}}\frac{(\mu_\mathrm{c} +2)
+    (\mu_\mathrm{c} +4)}{(\mu_\mathrm{c} + 1)^2} q_\mathrm{c}^2
+  m_\mathrm{c}^2 \cdot \left[1+
+    \frac{\Phi_{\text{auto}}(\tau_\mathrm{c})}{(1-\tau_\mathrm{c})^2}\right]
+  \rho_0.
+\end{align*}
+}}}
+Assuming that all new rain drops have a radius of 40 μm corresponding to the separation mass ''m'',,sep,, ''= 2.6 x 10^-10^'' kg, the local temporal change of ''N'',,r,, is
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial N_\mathrm{r}}{\partial t}
+  \right|_{\text{auto}}= \rho \left.\frac{\partial
+      q_\mathrm{r}}{\partial t} \right|_{\text{auto}}
+  \frac{1}{m_{\text{sep}}}.
+\end{align*}
+}}}
+Here, ''K'',,auto,, ''= 9.44 x 10^9^'' m^3^ kg^-2^ s^-1^ is the autoconversion kernel, ''μ'',,c,,'' = 1'' is the shape parameter of the cloud droplet gamma distribution and
+''m'',,c,, ''= ρ q'',,c,, ''/ N'',,c,, is the mean mass of cloud droplets. ''τ'',,c,, ''= 1 - q'',,c,,'' / (q'',,c,,'' + q'',,r,,) is a dimensionless timescale steering the autoconversion similarity function
+{{{
+#!Latex
+\begin{align*}
+  &
+  \Phi_{\text{auto}}=600\,\cdot\,\tau_\mathrm{c}^{0.68}\,\left(1-\tau_\mathrm{c}^{0.68}\right)^3.
+\end{align*}
+}}}
+The increase of the autoconversion rate due to turbulence can be considered optionally by an increased autoconversion kernel depending on the local kinetic energy dissipation rate after [#seifert2010 Seifert et al. (2010)].
+
+== Accretion ==
+
+The increase of ''q'',,r,, by accretion is given by:
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial q_\mathrm{r}}{\partial t}
+  \right|_{\text{accr}}=
+  K_{\text{accr}}\,q_\mathrm{c}\,q_\mathrm{r}\,\Phi_{\text{accr}}(\tau_\mathrm{c})
+  \left(\rho_0\,\rho \right)^{\frac{1}{2}},
+\end{align*}
+}}}
+with the accretion kernel ''K'',,accr,,'' = 4.33'' m^3^ kg^-1^ s^-1^ and the similarity function
+{{{
+#!Latex
+\begin{align*}
+  & \Phi_{\text{accr}}=\left(\frac{\tau_\mathrm{c}}{\tau_\mathrm{c} +
+      5 \times 10^{-5}}\right)^4.
+\end{align*}
+}}}
+Turbulence effects on the accretion rate can be considered after using the kernel after [#seifert2010 Seifert et al. (2010)].
+
+== Self-collection and breakup ==
+
+Selfcollection and breakup describe merging and splitting of rain drops, respectively, which affect the rain water drop number concentration only. Their combined impact is parametrized as
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial N_\mathrm{r}}{\partial t}
+  \right|_\text{slf/brk}=
+  -(\Phi_{\text{break}}(r)+1)\,\left.\frac{\partial
+      N_\mathrm{r}}{\partial t} \right|_{\text{self}},
+\end{align*}
+}}}
+with the breakup function
+{{{
+#!Latex
+\begin{align*}
+  & \Phi_{\text{break}} =
+  \begin{cases} 0 & \text{for~}  \widetilde{r_\mathrm{r}} < 0.15 \times 10^{-3}\,\mathrm{m},\\
+    K_{\text{break}} (\widetilde{r_\mathrm{r}}-r_{\text{eq}}) &
+    \text{otherwise},
+  \end{cases}
+\end{align*}
+}}}
+depending on the volume averaged rain drop radius
+{{{
+#!Latex
+\begin{align*}
+  &
+  \widetilde{r_\mathrm{r}}=\left(\frac{\rho\,q_\mathrm{r}}{\frac{4}{3}\,\pi\,\rho_{\mathrm{l},0}\,N_\mathrm{r}}
+  \right)^{\frac{1}{3}},
+\end{align*}
+}}}
+the equilibrium radius ''r'',,eq,, ''= 550 x 10^-6^'' m and the breakup kernel ''K'',,break,, ''= 2000'' m^-1^. The local temporal change of ''N'',,r,, due to selfcollection is
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial N_\mathrm{r}}{\partial t}
+  \right|_{\text{self}}= K_{\text{self}}\,N_\mathrm{r}\,q_\mathrm{r}
+  \left(\rho_0\,\rho \right)^{\frac{1}{2}},
+\end{align*}
+}}}
+with the selfcollection kernel ''K'',,self,, ''= 7.12'' m^3^ kg^-1^ s^-1^.
+
+== Evaporation of rainwater ==
+
+The evaporation of rain drops in subsaturated air (relative water supersaturation ''S < 0'') is parametrized following [#seifert2008 Seifert (2008)]:
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial q_\mathrm{r}}{\partial t}
+  \right|_{\text{evap}}= 2
+  \pi\,G\,S\,\frac{N_\mathrm{r}\,\lambda_\mathrm{r}^{\mu_\mathrm{r}+1}}{\Gamma(\mu_\mathrm{r}+1)}\,f_\mathrm{v}\,\rho,
+\end{align*}
+}}}
+where
+{{{
+#!Latex
+\begin{align*}
+  & G = \left[\frac{R_\mathrm{v}T}{K_\mathrm{v}p_\text{v, s}(T)} +
+    \left(\frac{L_\mathrm{V}}{R_\mathrm{v} T}-1\right)
+    \frac{L_\mathrm{V}}{\lambda_\mathrm{h}\,T}\right]^{-1},
+\end{align*}
+}}}
+with ''K'',,v,,'' = 2.3 x 10^-5^'' m^2^ s^-1^ being the molecular diffusivity water vapor in air and ''λ'',,h,,'' = 2.43 x 10^-2^'' W m^-1^ K^-1^ being the heat conductivity of air. Here, ''N'',,r,, ''λ'',,r,,^''μ'',,r,,''+1^ / Γ(μ'',,r,,+1) denotes the intercept parameter of the rain drop gamma distribution with ''Γ'' being the gamma-function. Following [#stevens2008 Stevens and Seifert (2008)], the slope parameter reads as
+{{{
+#!Latex
+\begin{align*}
+  & \lambda_\mathrm{r} = \frac{\left((\mu_\mathrm{r}+3)
+      (\mu_\mathrm{r}+2) (\mu_\mathrm{r}+1)\right)^{\frac{1}{3}}}{2
+    \cdot \widetilde{r_\mathrm{r}}},
+\end{align*}
+}}}
+with ''μ'',,r,, being the shape parameter, given by
+{{{
+#!Latex
+\begin{align*}
+  & \mu_\mathrm{r} = 10\,\cdot\,\left(1 +
+    \tanh{\left(1200\,\cdot\,\left(2 \cdot \widetilde{r_\mathrm{r}} -
+          0.0014 \right)\right)} \right).
+\end{align*}
+}}}
+In order to account for the increased evaporation of falling rain drops, the so-called ventilation effect, a ventilation factor ''f'',,v,, is calculated optionally by a series expansion considering the rain drop size distribution ([#seifert2008 Seifert, 2008, Appendix]).
+
+The complete evaporation of rain drops (i.e., their evaporation to a size smaller than the separation radius of 40 µm) is
+parametrized as
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial N_\mathrm{r}}{\partial t}
+  \right|_{\text{evap}}= \gamma\,\frac{N_\mathrm{r}}{\rho
+    q_\mathrm{r}}\,\left.\frac{\partial q_\mathrm{r}}{\partial t}
+  \right|_{\text{evap}},
+\end{align*}
+}}}
+with ''γ = 0.7'' (see also [#heus2010 Heus et al., 2010]).
+
+== Sedimentation of cloudwater ==
+
+As shown by [#ackerman Ackerman et al. (2009)], the sedimentation of cloud water has to be taken in account for the simulation of stratocumulus clouds. They suggest the cloud water sedimentation flux to be calculated as
+{{{
+#!Latex
+\begin{align*}
+  & F_{q_\mathrm{c}} = k \left(\frac{4}{3}
+    \pi\rho_\mathrm{l}N_\mathrm{c}\right)^{-2/3} \left(\rho
+    q_\mathrm{c}\right)^{\frac{5}{3}} \exp{\left(5
+      \ln^2{\sigma_\mathrm{g}}\right)},
+\end{align*}
+}}}
+based on a Stokes drag approximation of the terminal velocities of log-normal distributed cloud droplets.  Here, ''k = 1.2 x 10^8^'' m^-1^ s^-1^ is a parameter and ''σ'',,g,, ''= 1.3'' the geometric SD of the cloud droplet size distribution ([#geoffroy Geoffroy et al., 2010]). The tendency of ''q'' results from the sedimentation flux divergences and reads as
+{{{
+#!Latex
+\begin{align*}
+  & \left.\frac{\partial q}{\partial t} \right|_\text{sed, c}= -
+  \frac{\partial F_{q_\mathrm{c}}}{\partial z} \frac{1}{\rho}.
+\end{align*}
+}}}
+
+== Sedimentation of rainwater ==
+
+The sedimentation of rain water is implemented following [#stevens2008 Stevens and Seifert (2008)]. The sedimentation velocities are based on an empirical relation for the terminal fall velocity after [#rogers1993 Rogers et al. (1993)]. They are given by
+{{{
+#!Latex
+\begin{align*}
+  & w_{N_\mathrm{r}} = \left(9.65\,\text{m\,s}^{-1} -
+    9.8\,\text{m\,s}^{-1} \left(1+
+      600\,\text{m}/\lambda_\mathrm{r}\right)^{-(\mu_\mathrm{r} + 1)}
+  \right),
+\end{align*}
+}}}
+and
+{{{
+#!Latex
+\begin{align*}
+  & w_{q_\mathrm{r}} = \left(9.65\,\text{m\,s}^{-1} -
+    9.8\,\text{m\,s}^{-1} \left(1+
+      600\,\text{m}/\lambda_\mathrm{r}\right)^{-(\mu_\mathrm{r} + 4)}
+  \right).
+\end{align*}
+}}}
+The resulting sedimentation fluxes ''F'',,Nr,, and ''F'',,qr,, are calculated using a semi-Lagrangian
+scheme and a slope limiter (see [#stevens2008 Stevens and Seifert, 2008], their Appendix A). The resulting tendencies read as
+{{{
+#!Latex
+\begin{align*}
+ &
+ \left.\frac{\partial q_\mathrm{r}}{\partial t} \right|_\text{sed, r}= -\frac{\partial F_{q_\mathrm{r}}}{\partial z},~ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{sed, r}= -\frac{\partial F_{N_\mathrm{r}}}{\partial z},\;\text{and}~ \left.\frac{\partial q}{\partial t} \right|_\text{sed, r}=
+ \left.\frac{\partial q_\mathrm{r}}{\partial t} \right|_\text{sed, r}.
+\end{align*}
+}}}
+
+== Turbulence closure ==
+
+Using bulk cloud microphysics, PALM predicts liquid water temperature ''θ'',,l,, and total water mixing ratio ''q'' instead of ''θ''
+and ''q'',,v,,. Consequently, some terms in the Eq. for 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}{\theta_{\mathrm{v}}}^{\prime\prime}}$
+}}}
+of Sect. [wiki:/doc/tec/sgs turbulence closure] are unknown. We thus follow [#cuijpers1993 Cuijpers and Duynkerke (1993)] and calculate the SGS buoyancy flux from the known SGS fluxes
+{{{
+#!Latex 
+$\overline{w^{\prime\prime}{\theta_{\mathrm{l}}}^{\prime\prime}}$ 
+}}}
+and 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}{q}^{\prime\prime}}$.
+}}}
+In unsaturated air (''q'',,c,, = 0) the Eq. for 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}
+    {\theta_{\mathrm{v}}}^{\prime\prime}}$
+}}}
+of Sect. [wiki:/doc/tec/sgs turbulence closure] is then replaced by
+{{{
+#!Latex
+\begin{align*}
+  & \overline{w^{\prime\prime}
+    {\theta_{\mathrm{v}}}^{\prime\prime}}=K_1\,\cdot\,\overline{w^{\prime\prime}
+    {\theta_\mathrm{l}}^{\prime\prime}} +
+  K_2\,\cdot\,\overline{w^{\prime\prime} {q}^{\prime\prime}},
+\end{align*}
+}}}
+with
+{{{
+#!Latex
+\begin{align*}
+  &  K_1 = 1+\left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right)\,\cdot\,q,\\
+  & K_2 =
+  \left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right)\,\cdot\,\theta_\mathrm{l},
+\end{align*}
+}}}
+and in saturated air (''q'',,c,, > 0) by
+{{{
+#!Latex
+\begin{align*}
+  &
+  K_1 =\frac{1 - q + \frac{R_\mathrm{v}}{R_\mathrm{d}} (q-q_\mathrm{l}) \cdot \left(1 + \frac{L_\mathrm{V}}{R_\mathrm{v} T} \right)}{1 + \frac{L_\mathrm{V}^2}{R_\mathrm{v} c_p T^2} (q-q_\mathrm{l})},\\
+  & K_2 = \left(\frac{L_\mathrm{V}}{c_p T} K_1 - 1 \right)
+  \cdot \theta.
+\end{align*}
+}}}