= Cloud mircrophysics =

PALM offers an embedded bulk cloud microphysics representation that takes into account the liquid water specific humidity and warm (i.e., no ice) cloud-microphysical processes. Therefore, PALM solves the prognostic equations for the total water content
{{{
#!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.

Liquid phase microphysics are parametrized 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). 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 specific humidity
(''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.

In the next subsections we will describe the diagnostic determination 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,,,
{{{
#!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},
\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)]]
[[Image(Table5.png,600px,border=1)]]
[[Image(Table6.png,600px,border=1)]]

== Diffusional growth of cloud water ==

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 specific humidity. 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-\left(1-R_\mathrm{d}/R_\mathrm{v}\right)\,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*}
}}}

== 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 content ''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*}
}}}

== Recent applications ==

The two-moment cloud microphysics scheme has been used within the framework of the HD(CP)^2^ (High Definition Clouds and Precipitation for Climate Prediction, [http://www.hdcp2.eu]) project to produce LES simulation data for the evaluation and benchmarking of ICON-LES ([#dipankar2015 Dipankar et al., 2015]). Figure 8 contains a snapshot from such a benchmark simulation, where three continuous days were simulated. The figure shows the simulated clouds including precipitation events on 26 April 2013 during a frontal passage. Moreover, cloud microphysics have been recently used for investigations of shadow effects of shallow convective clouds on the ABL and their feedback on the cloud field ([#gronemeier2016 Gronemeier et al. 2016]).

[[Image(06.png,600px,border=1)]]

Figure 8: Snapshot of the cloud field from a PALM run for three continuous days of the HD(CP)^2^ Observational Prototype Experiment. Shown is the 3-D field of ''q'',,c,, (white to gray colors) as well as rain water (''q'',,r,,'' > 0'', blue) on 26 April 2013. The simulation had a grid-spacing of ''50 m'' on a ''50 x 50 km^2^'' domain. Large-scale advective tendencies for ''θ'',,l,, and ''q'' were taken from COSMOE-DE (regional model of German Meteorological Service, DWD) analyses. The copyright for the underlying satellite image is held by Cnes / Spot Image, Digitalglobe. 

== References ==
* [=#emanuel1994]'''Emanuel KA.''' 1994. Atmospheric Convection. Oxford University Press.

* [=#seifert2001]'''Seifert A, Beheng KD.''' 2001. A double-moment parameterization for simulating autoconversion, accretion and selfcollection. Atmos. Res. 59: 265–281.

* [=#seifert2006]'''Seifert A, Beheng KD.''' 2006. A two-moment cloud microphysics parameterization for mixed-phase clouds. Part 1: Model description. Meteorol. Atmos. Phys. 92: 45–66.

* [=#savic2008]'''Savic-Jovcic V, Stevens B.''' 2008. The structure and mesoscale organization of precipitating stratocumulus. J. Atmos. Sci. 65: 1587–1605. [http://dx.doi.org/10.1175/2007JAS2456.1 doi]

* [=#heus2010]'''Heus T, van Heerwaarden CC, Jonker HJJ, Pier Siebesma A, Axelsen S, van den Dries K, Geoffroy O, Moene AF, Pino D, de Roode SR, Vilà-Guerau de Arellano J.''' 2010. Formulation of the Dutch Atmospheric Large-Eddy Simulation (DALES) and overview of its applications. Geosci. Model Dev. 3: 415–444. [http://dx.doi.org/10.5194/gmd-3-415-2010 doi]

* [=#stevens2008]'''Stevens B, Seifert A.''' 2008. Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection. J. Meteor. Soc. Jpn. 86: 143–162.

* [=#bougeault1981]'''Bougeault, P.''' 1981. Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations against numerical data. J. Atmos. Sci. 38: 2414–2428.

* [=#sommeria1977]'''Sommeria G, Deardorff JW.''' 1977. Subgrid-scale condensation in models of nonprecipitating clouds. J. Atmos. Sci. 34: 344–355.

* [=#pruppacher1997]'''Pruppacher HR, and Klett JD.''' 1997. Microphysics of Clouds and Precipitation. 2nd Edn. Kluwer Academic Publishers. Dordrecht.

* [=#seifert2010]'''Seifert A, Nuijens L, Stevens B.''' 2010. Turbulence effects on warm-rain autoconversion in precipitating shallow convection. Q. J. Roy. Meteor. Soc. 136: 1753–1762.

* [=#seifert2008]'''Seifert A.''' 2008. On the parameterization of evaporation of raindrops as simulated by a one-dimensional rainshaft model. J. Atmos. Sci. 65: 3608–3619. [http://dx.doi.org/10.1175/2008JAS2586.1 doi].

* [=#ackerman]'''Ackerman, AS, vanZanten MC, Stevens B, Savic-Jovcic V, Bretherton CS, Chlond A, Golaz J-C, Jiang H, Khairoutdinov M, Krueger SK, Lewellen DC, Lock A, Moeng C-H, Nakamura K, Petters MD, Snider JR, Weinbrecht S, Zuluaf M.''' 2009. Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer, Mon. Weather Rev., 137: 1083–1110.

* [=#geoffroy]'''Geoffroy O, Brenguier J-L, Burnet F.''' 2010. Parametric representation of the cloud droplet spectra for LES warm bulk microphysical schemes. Atmos. Chem. Phys. 10: 4835–4848. [http://dx.doi.org/10.5194/acp-10-4835-2010 doi].

* [=#rogers1993]'''Rogers RR, Baumgardner D, Ethier SA, Carter DA, Ecklund WL.''' 1993. Comparison of raindrop size distributions measured by radar wind profiler and by airplane. J. Appl. Meteorol. 32: 694–699.

* [=#cuijpers1993]'''Cuijpers JWM, Duynkerke PG.''' 1993. Large eddy simulation of trade wind cumulus clouds. J. Atmos. Sci. 50: 3894–3908.

* [=#dipankar2015]'''Dipankar A, Stevens B, Heinze R, Moseley C, Zängl G, Giorgetta M, Brdar D.''' 2015. Large eddy simulation using the general circulation model ICON. J. Adv. Mod. Earth Syst. 07. [http://dx.doi.org/10.1002/2015MS000431 doi].

* [=#gronemeier2016]'''Gronemeier T, Kanani-Sühring F, Raasch S.''' 2016.  Do shallow cumulus clouds have the potential to trigger secondary circulations via shading? Boundary-Layer Meteorol. [http://dx.doi.org/10.1007/s10546-016-0180-7 doi].