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1\documentclass[11pt,a4paper]{scrartcl}
2\usepackage[latin9]{inputenc}
3\usepackage[T1]{fontenc}
4\usepackage[english]{babel}
5\usepackage[a4paper,top=3.cm,bottom=3.5cm,outer=3cm,inner=3.cm]{geometry} 
6\usepackage{booktabs,longtable,tabularx} 
7\usepackage{amsmath,amssymb,textcomp}
8\usepackage{scrpage2}
9\usepackage[bookmarks=true,bookmarksopen=false,bookmarksnumbered=true,colorlinks=false]{hyperref}
10\setlength{\parindent}{0pt}
11
12\pagestyle{scrheadings}
13\clearscrheadfoot
14\cfoot{{\small\sf \thepage}}
15
16%Adapting the references
17\newenvironment{bibliographie}[1]
18{\begin{thebibliography}{0000}{}
19\leftskip=5mm \setlength{\itemindent}{-5mm}#1} 
20{\end{thebibliography}}
21\makeatletter
22\renewcommand\@biblabel[1]{\setlength\labelsep{0pt}} 
23\renewcommand\@cite[2]{{#1\if@tempswa , #2\fi}} 
24\makeatother
25
26
27\begin {document}
28
29\begin{center}
30{\LARGE\bf\textsf{Introduction to the cloud physics module of PALM}}
31\vspace{3.0mm}
32\linebreak
33{\Large\bf\textsf{\textendash Amendments to the dry version of PALM\textendash}}
34\linebreak
35\linebreak
36 Michael Schr\"{o}ter
37\linebreak
3813.3.2000
39\linebreak
40translated and adapted by
41\linebreak
42Rieke Heinze
43\linebreak
4415.07.2010
45\end{center}
46
47\section{Introduction}
48The dry version of PALM does not contain any cloud physics. It has been extended
49to account for a nearly complete water cycle and radiation processes:
50\vspace{0.2cm}
51\newline
52{\bf\textsf{Water cycle}}
53\begin{itemize}
54 \item evaporation/condensation
55 \item precipitation
56 \item transport of humidity and liquid water
57\end{itemize}
58{\bf\textsf{Radiation processes}}
59\begin{itemize}
60 \item short-wave radiation
61 \item long-wave radiation
62\end{itemize}
63The dynamical processes are covered by advection and diffusion and they are described by the implemented methods. For the consideration of the
64thermodynamical processes modifications are necessary in the thermodynamics of PALM . In doing so evaporation and condensation are treated as
65adiabatic processes whereas precipitation and radiation are treated as diabatic processes. In the dry version of PALM the thermodynamic variable
66is the potential temperature $\theta$. The first law of thermodynamics provides the prognostic equation
67for $\theta$. The system of thermodynamic variables has to be extended to deal with phase transitions:
68\begin{eqnarray*}
69 q_{v} & = &\textnormal{specific humidity to deal with water vapour} \\
70 q_{l} & = &\textnormal{liquid water content to deal with the liquid phase}
71\end{eqnarray*}
72Additionally, dependencies between these variables have to be introduced to describe the changes of state (condensation scheme).
73\newline
74In introducing the two variables liquid water potential temperature $\theta_{l}$ and total liquid water content $q$ the treatment of the
75thermodynamics is simplified. The liquid water potential temperature $\theta_{l}$ is defined by \cite{betts73} and represents the potential
76temperature attained by evaporating all the liquid water in an air parcel through reversible wet adiabatic descent. In a linearized version
77it is defined as
78\begin{eqnarray}
79 \theta_{l} & = & \theta -\frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)q_{l}.
80 \label{eq:theta_l}
81\end{eqnarray}
82For the total water content it is valid:
83\begin{eqnarray}
84 q & = & q_{v}+q_{l}.
85 \label{eq:q}
86\end{eqnarray}
87The usage of $\theta_{l}$ and $q$ as thermodynamic variables is based on the work of \cite{ogura63} and \cite{orville65}. The advantages of the
88$\theta_l$-$q$ system are discussed by \cite{deardorff76}:
89\begin{itemize}
90 \item Without precipitation, radiation and freezing processes $\theta_{l}$ and $q$ are conservative quantities (for the whole system).
91 \item Therewith, the treatment of grid volumes in which only a fraction is saturated is simplified (sub-grid scale condensation scheme).
92 \item Parameterizations of the sub-grid scale fluxes are retained.
93 \item The liquid water content is not a separate variable (storage space is saved).
94 \item For dry convection $\theta_{l}$ matches the potential temperature and $q$ matches the specific humidity when condensation is disabled.
95 \item Phase transitions do not have to be described as additional terms in the prognostic equations.
96\end{itemize}
97
98\section{Model equations}
99In combining the prognostic equations for dry convection with the processes for cloud physics the following set of prognostic and diagnostic
100model equations is gained:
101\newline
102\newline
103Equation of continuity
104\begin{eqnarray}
105 \frac{\partial\overline u_{j}}{\partial x_{j}} & = & 0
106 \label{eq:conti}
107\end{eqnarray}
108Equations of motion
109\begin{eqnarray}
110 \frac{\partial\overline u_{i}}{\partial t} & = &
111  -\frac{\partial \left(\overline u_{j} \overline u_{i}\right)}{\partial x_{j}} 
112  -\frac{1}{\rho_{0}}\frac{\partial \overline \pi^{\ast}}{\partial x_{i}} 
113  - \varepsilon_{ijk}f_{j}\overline u_{k} + \varepsilon_{i3k}f_{3}u_{\mathrm{g}_{k}}
114  + g\frac{\overline\theta_{v}-\langle\overline\theta_{v}\rangle}{\theta_{0}}\delta_{i3} 
115  -\frac{\partial\,\tau_{ij}}{\partial x_{j}}
116 \label{eq:motion}
117\end{eqnarray}
118with
119\begin{eqnarray}
120 \label{eq:pres}
121 \overline \pi^{\ast} & = & \overline p^{\ast} + \frac{2}{3}\rho_{0}\,\overline e \\
122 \label{eq:tau}
123 \tau_{ij} & = & \overline{u_{j}^{'}u_{i}^{'}} - \frac{2}{3}\overline e\,\delta_{ij} 
124\end{eqnarray}
125First law of thermodynamics
126\begin{eqnarray}
127 \frac{\partial\overline \theta_{l}}{\partial t} & = & 
128  -\frac{\partial \left(\overline u_{j} \overline \theta_{l}\right)}{\partial x_{j}}
129  -\frac{\partial\, \overline{u_{j}^{'}\theta_{l}^{'}}}{\partial x_{j}} 
130  +\left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{RAD}}
131  +\left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{PREC}}
132 \label{eq:theta}
133\end{eqnarray}
134Conservation equation for the total water content
135\begin{eqnarray}
136 \frac{\partial\overline q}{\partial t} & = & 
137  -\frac{\partial \left(\overline u_{j} \overline q\right)}{\partial x_{j}}
138  -\frac{\partial\, \overline{u_{j}^{'} q^{'}}}{\partial x_{j}} 
139  +\left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}}
140 \label{eq:total_water}
141\end{eqnarray}
142Conservation equation for the sub-grid scale turbulent kinetic energy $\overline{e}=\frac{1}{2}\overline{u_{i}^{'2}}$
143\begin{eqnarray}
144 \frac{\partial \overline e}{\partial t} & = & 
145  -\frac{\partial \left(\overline u_{j}\overline e\right)}{\partial x_{j}} 
146  -\overline{u_{j}^{'} u_{i}^{'}} \frac{\partial\overline u_{i}}{\partial x_{j}}
147  + \frac{g}{\theta_{0}}\overline{u^{'}_{3}\theta_{v}^{'}} 
148  -\frac{\partial}{\partial x_{j}}\left\{\overline{u^{'}_{j}\left(e^{'} + \frac{p^{'}}{\rho_{0}}\right)} \right\}- \epsilon
149 \label{eq:sgsTKE}
150\end{eqnarray}
151The virtual potential temperature is needed in equation (\ref{eq:motion}) to calculate the buoyancy term. It is defined by 
152e.g. \cite{sommeria77} as
153\begin{eqnarray}
154 \overline \theta_{v} &=&
155  \left(\overline \theta_{l} + \frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)\overline q_{l}\right)
156  \left(1 + 0.61\,\overline q - 1.61\,\overline q_{l}\right).
157 \label{eq:theta_v}
158\end{eqnarray}
159Therewith, the influence of changing in density due to condensation is considered in the buoyancy term.
160\newline
161The closure of the model equations is based on the approaches of \cite{deardorff80}:
162\begin{eqnarray}
163 \label{eq:ujui}
164 \overline{u_{j}^{'}u_{i}^{'}} & = & 
165  -K_{m}\left(\frac{\partial \overline u_{i}}{\partial x_{j}} 
166  + \frac{\partial \overline u_{j}}{\partial x_{i}} \right)
167  + \frac{2}{3}\overline e \,\delta_{ij} \\
168 \label{eq:ujtheta}
169 \overline{u_{j}^{'}\theta_{l}^{'}} & = & 
170  -K_{h}\left(\frac{\partial \overline \theta_{l}}{\partial x_{j}}\right) \\
171 \label{eq:ujq}
172 \overline{u_{j}^{'} q{'}} & = & 
173  -K_{h}\left(\frac{\partial \overline q}{\partial x_{j}}\right) \\
174 \label{eq:ujp}
175 \overline{u^{'}_{j}\left(e^{'} + \frac{p^{'}}{\rho_{0}}\right)} & = &
176  -2K_{m}\frac{\partial \overline e}{\partial x_{j}} \\
177 \label{eq:u3theta_v}
178 \overline{u_{3}^{'}\theta_{v}^{'}} & = &
179  K_{1} \,\overline{u_{3}^{'}\theta_{l}^{'}}
180  + K_{2} \,\overline{u_{3}^{'} q^{'}} \\
181 \label{eq:km} 
182  K_{m} & = & 0.1\,l\,\sqrt{\overline e} \\
183 \label{eq:kh} 
184  K_{h} & = & \left(1+2\frac{l}{\Delta}\right)K_{m} \\
185 \label{eq:epsilon} 
186  \epsilon & = & \left(0.19 + 0.74\,\frac{l}{\Delta}\right)\,\frac{\overline e^{\frac{3}{2}}}{l}
187\end{eqnarray}
188with
189\begin{eqnarray}
190 l = \begin{cases}
191  \min\left(\Delta,\,  0.7\,d,\, 0.76\, \sqrt{\overline e}\,\left(\frac{g}{\theta_{0}}\frac{\partial \overline\theta_{v}}
192  {\partial z}\right)^{-\frac{1}{2}}\right) & , \quad \frac{\partial \overline\theta_{v}}{\partial z} > 0\\
193  \min\left(\Delta,\, 0.7\, d\right)   & , \quad \frac{\partial \overline\theta_{v}}{\partial z} \leq 0
194  \end{cases}
195 \label{eq:l}
196\end{eqnarray}
197and
198\begin{eqnarray}
199 \Delta & = & \left(\Delta x \Delta y \Delta z\right)^{1/3}
200 \label{eq:delta}
201\end{eqnarray}
202At the lower boundary Monin-Obukhov similarity theory is valid ( $\overline{w^{'}q^{'}}=u_{\ast}q_{\ast}$).
203\newline
204\cite{cuijpers93} for example define the coefficients $K_{1}$ and $K_{2}$ as follows:\newline
205{\bf\textsf{in unsaturated air:}}
206\begin{eqnarray}
207 \label{eq:K_1}
208  K_{1} & = & 1.0 + 0.61\, \overline q \\
209 \label{eq:K_2}
210  K_{2} & = &  0.61\, \overline{\theta}
211\end{eqnarray}
212{\bf\textsf{in saturated air:}}
213\begin{eqnarray}
214 \label{eq:K_1_sat}
215  K_{1} & = & \frac{1.0-\overline q + 1.61\,\overline q_{s}\left(1.0 + 0.622\frac{L_{v}}{RT}\right)}
216  {1.0 + 0.622\frac{L_{v}}{RT}\,\frac{L_{v}}{c_{p}T}\overline q_{s}}  \\
217 \label{eq:K_2_sat}
218  K_{2} & = &  \overline{\theta}\left(\left(\frac{L_{v}}{c_{p}T}\right)K_{1}-1.0\right)
219\end{eqnarray}
220The saturation value of the specific humidity comes from the truncated Taylor expansion of $q_{s}(T)$:
221\begin{eqnarray}
222 q_{s}(T) = q_{s} = q_{s}\left(T_{l}\right)
223 + \left(\frac{\partial q_{s}}{\partial T}\right)_{T=T_{l}} (T-T_{l}).
224 \label{eq:qs1}
225\end{eqnarray}
226Using the Clausius-Clapeyron equation
227\begin{eqnarray}
228 \left(\frac{\partial q_{s}}{\partial T}\right)_{T=T_{l}} & = & 
229  0.622\frac{L_{v}q_{s}(T_{l})}{R\,T_{l}^{2}}
230 \label{eq:clausius}
231\end{eqnarray}
232with
233\begin{eqnarray}
234  T = T_{l} + \frac{L_{v}}{c_{p}}q_{l} \qquad \textnormal{respectively}  \qquad q_{l} = q - q_{s}
235 \label{eq:T}
236\end{eqnarray}
237gives
238\begin{eqnarray}
239  \overline q_{s} = \overline q_{s}(\overline T_{l})\frac{\left(1.0+\beta\,\overline q\right)}
240    {1.0 + \beta\, \overline q_{s}(\overline{T_{l}})}.
241 \label{eq:qs2}
242\end{eqnarray}
243Whereas
244\begin{eqnarray}
245  \overline q_{s}(\overline T_{l}) = 0.622\frac{\overline e_{s}(\overline T_{l})}
246    {p_{0}(z)-0.378\,\overline e_{s}(\overline T_{l})}
247 \label{eq:qs3}
248\end{eqnarray}
249and
250\begin{eqnarray}
251 \beta = 0.622\left(\frac{L_{v}}{R\,\overline T_{l}}\right) \left(\frac{L_{v}}{c_{p}\,\overline T_{l}}\right).
252 \label{eq:beta}
253\end{eqnarray}
254The actual liquid water temperature is defined as
255\begin{eqnarray}
256 \overline T_{l} = \left(\frac{p_{0}(z)}{p_{0}(z=0)}\right)^{\kappa} \overline\theta_{l}
257 \label{eq:T_l}
258\end{eqnarray}
259with $p_{0}(z=0) = 1000$\,hPa. The value of the saturation vapour pressure at the temperature $\overline T_{l}$ is
260calculated in the same way as in \cite{bougeault82}:
261\begin{eqnarray}
262 \overline e_{s}(\overline T_{l}) = 610.78 \exp\left(
263  17.269\frac{\overline T_{l}-273.16}{\overline T_{l}-35.86}\right).
264 \label{eq:es}
265\end{eqnarray}
266The hydrostatic pressure $p_{0}(z)$ is given by \cite{cuijpers93}:
267\begin{eqnarray}
268 p_{0}(z) = p_{0}(z=0)\left(\frac{T_{\mathrm{ref}}(z)}{T_{0}}\right)^{c_{p}/R}
269 \label{eq:p_0}
270\end{eqnarray}
271with
272\begin{eqnarray}
273 T_{\mathrm{ref}}(z) = T_{0} - \frac{g}{c_{p}} z.
274 \label{eq:T_ref}
275\end{eqnarray}
276The pressure is calculated once at the beginning of a simulation and remains unchanged. For the reference temperature at the earth surface $T_{0}$ 
277the initial surface temperature is applied. The ratio of the potential and the actual temperature is given by:
278\begin{eqnarray}
279 \frac{\theta}{T} = \left(\frac{p_{0}(z=0)}{p_{0}(z)}\right)^{\kappa}.
280 \label{eq:ratio}
281\end{eqnarray}
282The liquid water content $q_{l}$ is needed for the calculation of the virtual potential temperature (eq. (\ref{eq:theta_v})). It is
283calculated from the difference of the total water content at a single grid point and the saturation value at this grid point:
284\begin{eqnarray}
285 \overline q_{l} =
286 \begin{cases}
287   \overline q - \overline q_{s} & 
288   \textnormal{if} \quad \overline q > \overline q_{s} \\
289   0  & \textnormal{else}
290  \end{cases}
291 \label{eq:q_l}
292\end{eqnarray}
293With this approach a grid volume is either completely saturated or completely unsaturated. The values of the cloud cover of a grid volume
294can only become 0 or 1 (\textsl{0\%-or100\% scheme}).
295
296\section{Parameterization of the source terms in the conservation equations}
297\subsection{Radiation model}
298The source term for radiation processes is parameterized via the scheme of effective emissivity which is based on \cite{cox76}:
299\begin{eqnarray}
300 \left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{RAD}} & = &
301  -\frac{\theta}{T}\frac{1}{\rho \,c_{p}\,\Delta z}\left[\Delta F(z^{+})-\Delta F(z^{-})\right]
302 \label{eq:radiation_term}
303\end{eqnarray}
304$\Delta F$ describes the difference between upward and downward irradiance at the grid point above ($z^{+}$) and below ($z^{-}$)
305the level in which $\theta_{l}$ is defined.
306\newline
307The upward and downward irradiance $F\textnormal{\textuparrow}$ and $F\textnormal{\textdownarrow}$ are defined as follows:
308\begin{eqnarray}
309 \label{eq:F_up}
310  F\textnormal{\textuparrow}(z) & = &
311  B(0) + \varepsilon\textnormal{\textuparrow}(z,0)\left(B(z)-B(0)\right) \\
312 \label{eq:F_down} 
313  F\textnormal{\textdownarrow}(z) & = &
314  F\textnormal{\textdownarrow}(z_{\mathrm{top}})
315  + \varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}})\left(B(z)-F\textnormal{\textdownarrow}(z_{\mathrm{top}})\right)
316\end{eqnarray}
317$F\textnormal{\textdownarrow}(z_{\mathrm{top}})$ describes the impinging irradiance at the upper boundary of the model domain which has to be
318prescribed. $B(0)$ and $B(z)$ represent the black body emission at the ground and the height $z$ respectively.
319$\varepsilon\textnormal{\textuparrow}(z,0)$ and $\varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}})$ stand for the effective
320cloud emissivity of the liquid water between the ground and the level $z$ and between $z$ and the upper boundary of the model domain
321$z_{\mathrm{top}}$ respectively. They are defined as
322\begin{eqnarray}
323 \label{eq:epsilon_up}
324  \varepsilon\textnormal{\textuparrow}(z,0) & = & 
325   1- \exp\left(-a\cdot LWP(0,z)\right)\\
326 \label{eq:epsilon_down} 
327  \varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}}) & = & 
328   1- \exp\left(-b\cdot LWP(z,z_{\mathrm{top}})\right)
329\end{eqnarray}
330$LWP(z_{1},z_{2})$ describes the liquid water path which is the vertically added content of liquid water above each grid column:
331\begin{eqnarray}
332 LWP(z_{1},z_{2}) & = & \int_{z_{1}}^{z_{2}}\mathrm{dz}\,\rho\cdot\overline q_{l}.
333 \label{eq:LWP}
334\end{eqnarray}
335$a$ and $b$ are called mass absorption coefficients. Their empirical values are based on \linebreak
336\cite{stephans78} with $a=130\,\mathrm{m^{2}kg^{-1}}$ and $b=158\,\mathrm{m^{2}kg^{-1}}$.
337\newline
338The assumptions for the validity of this parameterization are:
339\begin{itemize}
340 \item Horizontal divergences in radiation are neglected.
341 \item Only absorption and emission of long-wave radiation due to water vapour and cloud droplets is considered.
342 \item The atmosphere is assumed to have constant in-situ temperature above and below the regarded level except for the earth surface.
343\end{itemize}
344
345\subsection{Precipitation model}
346The source term for precipitation processes is parameterized via a simplified scheme of \cite{kessler69}:
347\begin{eqnarray}
348 \left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}} & = &
349  \begin{cases}
350   \left(\overline q_{l}-\overline q_{l_{\mathrm{crit}}}\right)/ \tau & 
351   \quad\overline q_{l} > \overline q_{l_{\mathrm{crit}}} \\
352   0  & \quad\overline q_{l} \leq \overline q_{l_{\mathrm{crit}}}
353  \end{cases}
354 \label{precip_term_q}
355\end{eqnarray}
356The precipitation leaves the grid volume immediately if the threshold of the liquid water content
357$\overline q_{l_{\mathrm{crit}}}=0.5\,\mathrm{g/kg}$ is exceeded. Hence, evaporation of the rain drops does not occur.
358$\tau$ is a retarding time scale with a value of 1000\,s.
359\newline
360The influence of the precipitation on the temperature is as follows:
361\begin{eqnarray}
362 \left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{PREC}} & = &
363  -\frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)\left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}}
364 \label{precip_term_pt}
365\end{eqnarray}
366
367\section*{List of symbols}
368\setlength{\extrarowheight}{0.8mm}
369\begin{longtable}{p{2.5cm} p{9.0cm} p{2.5cm}} 
370\toprule
371\addlinespace
372\textbf{Variable} & \textbf{Description} &\textbf{Value} \\
373\midrule
374 $B$ & black body radiation & \\
375 $c_{p}$ & heat capacity for dry air with p=const  & $1005\,\mathrm{J\,K^{-1}kg^{-1}}$  \\
376 $d$ & normal distance to the nearest solid surface & \\
377 $\overline e$ & sub-grid scale turbulent kinetic energy  & \\
378 $e_{s}$ & saturation vapour pressure  & \\
379 $f_{i}$ & Coriolis parameter $i\in\{1,2,3\}$ & \\
380 $F\textnormal{\textuparrow}$ & upward irradiance  & \\
381 $F\textnormal{\textdownarrow}$ & downward irradiance  & \\
382 $i$, $j$, $k$  & integer indices & \\
383 $K_{h}$ & turbulent diffusion coefficient for momentum & \\
384 $K_{m}$ & turbulent diffusion coefficient for heat & \\
385 $K_{1}$ & coefficient & \\
386 $K_{2}$ & coefficient & \\
387 $l$ & mixing length  & \\
388 $L_{v}$ & heat of evaporation & $2.5\cdot 10^{6}\,\mathrm{J\,kg^{-1}}$ \\
389 $LWP$ & liquid water path & \\
390 $R$ & gas constant for dry air & $287\,\mathrm{J\,K^{-1}kg^{-1}}$\\
391 $T$ & actual temperature & \\
392 $T_{l}$ & actual liquid water temperature & \\
393 $u$, $v$, $w$, $u_{i}$  & velocity components, $i\in\{1,2,3\}$ & \\
394 $p_{0}$  & hydrostatic pressure & \\
395 $q$ & total water content & \\
396 $q_{l}$ & liquid water content & \\
397 $q_{l_{\mathrm{crit}}}$ & threshold for the formation of precipitation & \\
398 $q_{s}$ & specific humidity in case of saturation & \\
399 $q_{v}$ & specific humidity & \\ 
400 $x$, $y$, $z$, $x_{i}$  & Cartesian coordinates, $i\in\{1,2,3\}$ & \\
401 $\Delta$ & characteristic grid length & \\
402 $\epsilon$ & dissipation of sub-grid scale turbulent kinetic energy & \\
403 $\varepsilon\textnormal{\textuparrow}$ & upward effective cloud emissivity  & \\
404 $\varepsilon\textnormal{\textdownarrow}$ & downward effective cloud emissivity  & \\
405 $\kappa$ &$R/c_{p}$  & 0.286 \\
406 $\rho$ & air density  & \\
407 $\tau$ & time scale for the Kessler scheme & \\
408 $\theta$ & potential temperature & \\
409 $\theta_{l}$ & liquid water potential temperature & \\
410 $\theta_{v}$ & virtual potential temperature &  \\
411 $\theta_{0}$ & reference value for the potential temperature & \\
412 $\overline\psi$ & resolved scale variable & \\
413 $\psi^{'}$ & sub-grid scale variable & \\
414 $\psi^{\ast}$ & departure from the basic state (Boussinesq approximation) &  \\
415 $\langle\psi\rangle$ & horizontal mean \\
416\addlinespace
417\bottomrule
418\end{longtable}
419
420\setlength\labelsep{0pt} 
421\begin{bibliographie}{}
422 \bibitem[Betts (1973)]{betts73}
423   \textbf{Betts, A. K., 1973:} Non-precipitating cumulus convection and its parameterization.
424   \textit{Quart. J. R. Meteorol. Soc.}, \textbf{99}, 178-196.
425 \bibitem[Bougeault (1982)]{bougeault82}
426   \textbf{Bougeault, P., 1982:} Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations
427   against numerical data. \textit{J. Atmos. Sci.}, \textbf{38}, 2414-2428.
428 \bibitem[Cox (1976)]{cox76}
429   \textbf{Cox, S. K., 1976:} Observations of cloud infrared effective emissivity.
430   \textit{J. Atmos. Sci.}, \textbf{33}, 287-289.
431 \bibitem[Cuijpers and Duynkerke (1993)]{cuijpers93}
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458\end{bibliographie}
459
460\end {document}
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