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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
4414.12.2009
45\linebreak
46\linebreak
47last update
48\linebreak
49\today
50\end{center}
51
52\section{Introduction}
53The dry version of PALM does not contain any cloud physics. It has been extended
54to account for a nearly complete water cycle and radiation processes:
55\vspace{0.2cm}
56\newline
57{\bf\textsf{Water cycle}}
58\begin{itemize}
59 \item evaporation/condensation
60 \item precipitation
61 \item transport of humidity and liquid water
62\end{itemize}
63{\bf\textsf{Radiation processes}}
64\begin{itemize}
65 \item short-wave radiation
66 \item long-wave radiation
67\end{itemize}
68The dynamical processes are covered by advection and diffusion and they are described by the implemented methods. For the consideration of the
69thermodynamical processes modifications are necessary in the thermodynamics of PALM. In doing so evaporation and condensation are treated as
70adiabatic processes whereas precipitation and radiation are treated as diabatic processes. In the dry version of PALM the thermodynamic variable
71is the potential temperature $\theta$. The first law of thermodynamics provides the prognostic equation
72for $\theta$. The system of thermodynamic variables has to be extended to deal with phase transitions:
73\begin{eqnarray*}
74 q_{v} & = &\textnormal{specific humidity to deal with water vapour} \\
75 q_{l} & = &\textnormal{liquid water content to deal with the liquid phase}
76\end{eqnarray*}
77Additionally, dependencies between these variables have to be introduced to describe the changes of state (condensation scheme).
78\newline
79In introducing the two variables liquid water potential temperature $\theta_{l}$ and total liquid water content $q$ the treatment of the
80thermodynamics is simplified. The liquid water potential temperature $\theta_{l}$ is defined by \cite{betts73} and represents the potential
81temperature attained by evaporating all the liquid water in an air parcel through reversible wet adiabatic descent. In a linearized version
82it is defined as
83\begin{eqnarray}
84 \theta_{l} & = & \theta -\frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)q_{l}.
85 \label{eq:theta_l}
86\end{eqnarray}
87For the total water content it is valid:
88\begin{eqnarray}
89 q & = & q_{v}+q_{l}.
90 \label{eq:q}
91\end{eqnarray}
92The usage of $\theta_{l}$ and $q$ as thermodynamic variables is based on the work of \cite{ogura63} and \cite{orville65}. The advantages of the
93$\theta_l$-$q$ system are discussed by \cite{deardorff76}:
94\begin{itemize}
95 \item Without precipitation, radiation and freezing processes $\theta_{l}$ and $q$ are conservative quantities (for the whole system).
96 \item Therewith, the treatment of grid volumes in which only a fraction is saturated is simplified (sub-grid scale condensation scheme).
97 \item Parameterizations of the sub-grid scale fluxes are retained.
98 \item The liquid water content is not a separate variable (storage space is saved).
99 \item For dry convection $\theta_{l}$ matches the potential temperature and $q$ matches the specific humidity when condensation is disabled.
100 \item Phase transitions do not have to be described as additional terms in the prognostic equations.
101\end{itemize}
102
103\section{Model equations}
104In combining the prognostic equations for dry convection with the processes for cloud physics the following set of prognostic and diagnostic
105model equations is gained:
106\newline
107\newline
108Equation of continuity
109\begin{eqnarray}
110 \frac{\partial\overline u_{j}}{\partial x_{j}} & = & 0
111 \label{eq:conti}
112\end{eqnarray}
113Equations of motion
114\begin{eqnarray}
115 \frac{\partial\overline u_{i}}{\partial t} & = &
116  -\frac{\partial \left(\overline u_{j} \overline u_{i}\right)}{\partial x_{j}} 
117  -\frac{1}{\rho_{0}}\frac{\partial \overline \pi^{\ast}}{\partial x_{i}} 
118  - \varepsilon_{ijk}f_{j}\overline u_{k} - \varepsilon_{i3k}f_{3}u_{\mathrm{g}_{k}}
119  + g\frac{\overline\theta_{v}-\langle\overline\theta_{v}\rangle}{\theta_{0}}\delta_{i3} 
120  -\frac{\partial\,\tau_{ij}}{\partial x_{j}}
121 \label{eq:motion}
122\end{eqnarray}
123with
124\begin{eqnarray}
125 \label{eq:pres}
126 \overline \pi^{\ast} & = & \overline p^{\ast} + \frac{2}{3}\rho_{0}\,\overline e \\
127 \label{eq:tau}
128 \tau_{ij} & = & \overline{u_{j}^{'}u_{i}^{'}} - \frac{2}{3}\overline e\,\delta_{ij} 
129\end{eqnarray}
130First law of thermodynamics
131\begin{eqnarray}
132 \frac{\partial\overline \theta_{l}}{\partial t} & = & 
133  -\frac{\partial \left(\overline u_{j} \overline \theta_{l}\right)}{\partial x_{j}}
134  -\frac{\partial\, \overline{u_{j}^{'}\theta_{l}^{'}}}{\partial x_{j}} 
135  +\left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{RAD}}
136  +\left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{PREC}}
137 \label{eq:theta}
138\end{eqnarray}
139Conservation equation for the total water content
140\begin{eqnarray}
141 \frac{\partial\overline q}{\partial t} & = & 
142  -\frac{\partial \left(\overline u_{j} \overline q\right)}{\partial x_{j}}
143  -\frac{\partial\, \overline{u_{j}^{'} q^{'}}}{\partial x_{j}} 
144  +\left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}}
145 \label{eq:total_water}
146\end{eqnarray}
147Conservation equation for the sub-grid scale turbulent kinetic energy $\overline{e}=\frac{1}{2}\overline{u_{i}^{'2}}$
148\begin{eqnarray}
149 \frac{\partial \overline e}{\partial t} & = & 
150  -\frac{\partial \left(\overline u_{j}\overline e\right)}{\partial x_{j}} 
151  -\overline{u_{j}^{'} u_{i}^{'}} \frac{\partial\overline u_{i}}{\partial x_{j}}
152  + \frac{g}{\theta_{0}}\overline{u^{'}_{3}\theta_{v}^{'}} 
153  -\frac{\partial}{\partial x_{j}}\left\{\overline{u^{'}_{j}\left(e^{'} + \frac{p^{'}}{\rho_{0}}\right)} \right\}- \epsilon
154 \label{eq:sgsTKE}
155\end{eqnarray}
156The virtual potential temperature is needed in equation (\ref{eq:motion}) to calculate the buoyancy term. It is defined by 
157e.g. \cite{sommeria77} as
158\begin{eqnarray}
159 \overline \theta_{v} &=&
160  \left(\overline \theta_{l} + \frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)\overline q_{l}\right)
161  \left(1 + 0.61\,\overline q - 1.61\,\overline q_{l}\right).
162 \label{eq:theta_v}
163\end{eqnarray}
164Therewith, the influence of changing in density due to condensation is considered in the buoyancy term.
165\newline
166The closure of the model equations is based on the approaches of \cite{deardorff80}:
167\begin{eqnarray}
168 \label{eq:ujui}
169 \overline{u_{j}^{'}u_{i}^{'}} & = & 
170  -K_{m}\left(\frac{\partial \overline u_{i}}{\partial x_{j}} 
171  + \frac{\partial \overline u_{j}}{\partial x_{i}} \right)
172  + \frac{2}{3}\overline e \,\delta_{ij} \\
173 \label{eq:ujtheta}
174 \overline{u_{j}^{'}\theta_{l}^{'}} & = & 
175  -K_{h}\left(\frac{\partial \overline \theta_{l}}{\partial x_{j}}\right) \\
176 \label{eq:ujq}
177 \overline{u_{j}^{'} q{'}} & = & 
178  -K_{h}\left(\frac{\partial \overline q}{\partial x_{j}}\right) \\
179 \label{eq:ujp}
180 \overline{u^{'}_{j}\left(e^{'} + \frac{p^{'}}{\rho_{0}}\right)} & = &
181  -2K_{m}\frac{\partial \overline e}{\partial x_{j}} \\
182 \label{eq:u3theta_v}
183 \overline{u_{3}^{'}\theta_{v}^{'}} & = &
184  K_{1} \,\overline{u_{3}^{'}\theta_{l}^{'}}
185  + K_{2} \,\overline{u_{3}^{'} q^{'}} \\
186 \label{eq:km} 
187  K_{m} & = & 0.1\,l\,\sqrt{\overline e} \\
188 \label{eq:kh} 
189  K_{h} & = & \left(1+2\frac{l}{\Delta}\right)K_{m} \\
190 \label{eq:epsilon} 
191  \epsilon & = & \left(0.19 + 0.74\,\frac{l}{\Delta}\right)\,\frac{\overline e^{\frac{3}{2}}}{l}
192\end{eqnarray}
193with
194\begin{eqnarray}
195 l = \begin{cases}
196  \min\left(\Delta,\,  0.7\,d,\, 0.76\, \sqrt{\overline e}\,\left(\frac{g}{\theta_{0}}\frac{\partial \overline\theta_{v}}
197  {\partial z}\right)^{-\frac{1}{2}}\right) & , \quad \frac{\partial \overline\theta_{v}}{\partial z} > 0\\
198  \min\left(\Delta,\, 0.7\, d\right)   & , \quad \frac{\partial \overline\theta_{v}}{\partial z} \leq 0
199  \end{cases}
200 \label{eq:l}
201\end{eqnarray}
202and
203\begin{eqnarray}
204 \Delta & = & \left(\Delta x \Delta y \Delta z\right)^{1/3}
205 \label{eq:delta}
206\end{eqnarray}
207At the lower boundary Monin-Obukhov similarity theory is valid ( $\overline{w^{'}q^{'}}=u_{\ast}q_{\ast}$).
208\newline
209\cite{cuijpers93} for example define the coefficients $K_{1}$ and $K_{2}$ as follows:\newline
210{\bf\textsf{in unsaturated air:}}
211\begin{eqnarray}
212 \label{eq:K_1}
213  K_{1} & = & 1.0 + 0.61\, \overline q \\
214 \label{eq:K_2}
215  K_{2} & = &  0.61\, \overline{\theta}
216\end{eqnarray}
217{\bf\textsf{in saturated air:}}
218\begin{eqnarray}
219 \label{eq:K_1_sat}
220  K_{1} & = & \frac{1.0-\overline q + 1.61\,\overline q_{s}\left(1.0 + 0.622\frac{L_{v}}{RT}\right)}
221  {1.0 + 0.622\frac{L_{v}}{RT}\,\frac{L_{v}}{c_{p}T}\overline q_{s}}  \\
222 \label{eq:K_2_sat}
223  K_{2} & = &  \overline{\theta}\left(\left(\frac{L_{v}}{c_{p}T}\right)K_{1}-1.0\right)
224\end{eqnarray}
225The saturation value of the specific humidity comes from the truncated Taylor expansion of $q_{s}(T)$:
226\begin{eqnarray}
227 q_{s}(T) = q_{s} = q_{s}\left(T_{l}\right)
228 + \left(\frac{\partial q_{s}}{\partial T}\right)_{T=T_{l}} (T-T_{l}).
229 \label{eq:qs1}
230\end{eqnarray}
231Using the Clausius-Clapeyron equation
232\begin{eqnarray}
233 \left(\frac{\partial q_{s}}{\partial T}\right)_{T=T_{l}} & = & 
234  0.622\frac{L_{v}q_{s}(T_{l})}{R\,T_{l}^{2}}
235 \label{eq:clausius}
236\end{eqnarray}
237with
238\begin{eqnarray}
239  T = T_{l} + \frac{L_{v}}{c_{p}}q_{l} \qquad \textnormal{respectively}  \qquad q_{l} = q - q_{s}
240 \label{eq:T}
241\end{eqnarray}
242gives
243\begin{eqnarray}
244  \overline q_{s}(T) = \overline q_{s}(\overline T_{l})\frac{\left(1.0+\beta\,\overline q\right)}
245    {1.0 + \beta\, \overline q_{s}(\overline{T_{l}})}.
246 \label{eq:qs2}
247\end{eqnarray}
248Whereas
249\begin{eqnarray}
250  \overline q_{s}(\overline T_{l}) = 0.622\frac{\overline e_{s}(\overline T_{l})}
251    {p_{0}(z)-0.378\,\overline e_{s}(\overline T_{l})}
252 \label{eq:qs3}
253\end{eqnarray}
254and
255\begin{eqnarray}
256 \beta = 0.622\left(\frac{L_{v}}{R\,\overline T_{l}}\right) \left(\frac{L_{v}}{c_{p}\,\overline T_{l}}\right).
257 \label{eq:beta}
258\end{eqnarray}
259The actual liquid water temperature is defined as
260\begin{eqnarray}
261 \overline T_{l} = \left(\frac{p_{0}(z)}{p_{0}(z=0)}\right)^{\kappa} \overline\theta_{l}
262 \label{eq:T_l}
263\end{eqnarray}
264with $p_{0}(z=0) = 1000$\,hPa. The value of the saturation vapour pressure at the temperature $\overline T_{l}$ is
265calculated in the same way as in \cite{bougeault82}:
266\begin{eqnarray}
267 \overline e_{s}(\overline T_{l}) = 610.78 \exp\left(
268  17.269\frac{\overline T_{l}-273.16}{\overline T_{l}-35.86}\right).
269 \label{eq:es}
270\end{eqnarray}
271The hydrostatic pressure $p_{0}(z)$ is given by \cite{cuijpers93}:
272\begin{eqnarray}
273 p_{0}(z) = p_{0}(z=0)\left(\frac{T_{\mathrm{ref}}(z)}{T_{0}}\right)^{c_{p}/R}
274 \label{eq:p_0}
275\end{eqnarray}
276with
277\begin{eqnarray}
278 T_{\mathrm{ref}}(z) = T_{0} - \frac{g}{c_{p}} z.
279 \label{eq:T_ref}
280\end{eqnarray}
281The pressure is calculated once at the beginning of a simulation and remains unchanged. For the reference temperature at the earth surface $T_{0}$ 
282the initial surface temperature is applied. The ratio of the potential and the actual temperature is given by:
283\begin{eqnarray}
284 \frac{\theta}{T} = \left(\frac{p_{0}(z=0)}{p_{0}(z)}\right)^{\kappa}.
285 \label{eq:ratio}
286\end{eqnarray}
287The liquid water content $q_{l}$ is needed for the calculation of the virtual potential temperature (eq. (\ref{eq:theta_v})). It is
288calculated from the difference of the total water content at a single grid point and the saturation value at this grid point:
289\begin{eqnarray}
290 \overline q_{l} =
291 \begin{cases}
292   \overline q - \overline q_{s}(\overline T) & 
293   \textnormal{if} \quad \overline q > \overline q_{s}(\overline T) \\
294   0  & \textnormal{else}
295  \end{cases}
296 \label{eq:q_l}
297\end{eqnarray}
298With this approach a grid volume is either completely saturated or completely unsaturated. The values of the cloud cover of a grid volume
299can only become 0 or 1 (\textsl{0\%-or100\% scheme}).
300
301\section{Parameterization of the source terms in the conservation equations}
302\subsection{Radiation model}
303The source term for radiation processes is parameterized via the scheme of effective emissivity which is based on \cite{cox76}:
304\begin{eqnarray}
305 \left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{RAD}} & = &
306  -\frac{\theta}{T}\frac{1}{\rho \,c_{p}\,\Delta z}\left[\Delta F(z^{+})-\Delta F(z^{-})\right]
307 \label{eq:radiation_term}
308\end{eqnarray}
309$\Delta F$ describes the difference between upward and downward irradiance at the grid point above ($z^{+}$) and below ($z^{-}$)
310the level in which $\theta_{l}$ is defined.
311\newline
312The upward and downward irradiance $F\textnormal{\textuparrow}$ and $F\textnormal{\textdownarrow}$ are defined as follows:
313\begin{eqnarray}
314 \label{eq:F_up}
315  F\textnormal{\textuparrow}(z) & = &
316  B(0) + \varepsilon\textnormal{\textuparrow}(z,0)\left(B(z)-B(0)\right) \\
317 \label{eq:F_down} 
318  F\textnormal{\textdownarrow}(z) & = &
319  F\textnormal{\textdownarrow}(z_{\mathrm{top}})
320  + \varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}})\left(B(z)-F\textnormal{\textdownarrow}(z_{\mathrm{top}})\right)
321\end{eqnarray}
322$F\textnormal{\textdownarrow}(z_{\mathrm{top}})$ describes the impinging irradiance at the upper boundary of the model domain which has to be
323prescribed. $B(0)$ and $B(z)$ represent the black body emission at the ground and the height $z$ respectively.
324$\varepsilon\textnormal{\textuparrow}(z,0)$ and $\varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}})$ stand for the effective
325cloud emissivity of the liquid water between the ground and the level $z$ and between $z$ and the upper boundary of the model domain
326$z_{\mathrm{top}}$ respectively. They are defined as
327\begin{eqnarray}
328 \label{eq:epsilon_up}
329  \varepsilon\textnormal{\textuparrow}(z,0) & = & 
330   1- \exp\left(-a\cdot LWP(0,z)\right)\\
331 \label{eq:epsilon_down} 
332  \varepsilon\textnormal{\textdownarrow}(z,z_{\mathrm{top}}) & = & 
333   1- \exp\left(-b\cdot LWP(z,z_{\mathrm{top}})\right)
334\end{eqnarray}
335$LWP(z_{1},z_{2})$ describes the liquid water path which is the vertically added content of liquid water above each grid column:
336\begin{eqnarray}
337 LWP(z_{1},z_{2}) & = & \int_{z_{1}}^{z_{2}}\mathrm{dz}\,\rho\cdot\overline q_{l}.
338 \label{eq:LWP}
339\end{eqnarray}
340$a$ and $b$ are called mass absorption coefficients. Their empirical values are based on \linebreak
341\cite{stephans78} with $a=130\,\mathrm{m^{2}kg^{-1}}$ and $b=158\,\mathrm{m^{2}kg^{-1}}$.
342\newline
343The assumptions for the validity of this parameterization are:
344\begin{itemize}
345 \item Horizontal divergences in radiation are neglected.
346 \item Only absorption and emission of long-wave radiation due to water vapour and cloud droplets is considered.
347 \item The atmosphere is assumed to have constant in-situ temperature above and below the regarded level except for the earth surface.
348\end{itemize}
349
350\subsection{Precipitation model}
351The source term for precipitation processes is parameterized via a simplified scheme of \cite{kessler69}:
352\begin{eqnarray}
353 \left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}} & = &
354  \begin{cases}
355   \left(\overline q_{l}-\overline q_{l_{\mathrm{crit}}}\right)/ \tau & 
356   \quad\overline q_{l} > \overline q_{l_{\mathrm{crit}}} \\
357   0  & \quad\overline q_{l} \leq \overline q_{l_{\mathrm{crit}}}
358  \end{cases}
359 \label{precip_term_q}
360\end{eqnarray}
361The precipitation leaves the grid volume immediately if the threshold of the liquid water content
362$\overline q_{l_{\mathrm{crit}}}=0.5\,\mathrm{g/kg}$ is exceeded. Hence, evaporation of the rain drops does not occur.
363$\tau$ is a time scale with a value of 1000\,s.
364\newline
365The influence of the precipitation on the temperature is as follows:
366\begin{eqnarray}
367 \left(\frac{\partial \overline\theta_{l}}{\partial t}\right)_{\mathrm{PREC}} & = &
368  -\frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right)\left(\frac{\partial \overline q}{\partial t}\right)_{\mathrm{PREC}}
369 \label{precip_term_pt}
370\end{eqnarray}
371
372\section*{List of symbols}
373\setlength{\extrarowheight}{0.8mm}
374\begin{longtable}{p{2.5cm} p{9.0cm} p{2.5cm}} 
375\toprule
376\addlinespace
377\textbf{Variable} & \textbf{Description} &\textbf{Value} \\
378\midrule
379 $B$ & black body radiation & \\
380 $c_{p}$ & heat capacity for dry air with p=const  & $1005\,\mathrm{J\,K^{-1}kg^{-1}}$  \\
381 $d$ & normal distance to the nearest solid surface & \\
382 $\overline e$ & sub-grid scale turbulent kinetic energy  & \\
383 $e_{s}$ & saturation vapour pressure  & \\
384 $f_{i}$ & Coriolis parameter $i\in\{1,2,3\}$ & \\
385 $F\textnormal{\textuparrow}$ & upward irradiance  & \\
386 $F\textnormal{\textdownarrow}$ & downward irradiance  & \\
387 $i$, $j$, $k$  & integer indices & \\
388 $K_{h}$ & turbulent diffusion coefficient for momentum & \\
389 $K_{m}$ & turbulent diffusion coefficient for heat & \\
390 $K_{1}$ & coefficient & \\
391 $K_{2}$ & coefficient & \\
392 $l$ & mixing length  & \\
393 $L_{v}$ & heat of evaporation & $2.5\cdot 10^{6}\,\mathrm{J\,kg^{-1}}$ \\
394 $LWP$ & liquid water path & \\
395 $R$ & gas constant for dry air & $287\,\mathrm{J\,K^{-1}kg^{-1}}$\\
396 $T$ & actual temperature & \\
397 $T_{l}$ & actual liquid water temperature & \\
398 $u$, $v$, $w$, $u_{i}$  & velocity components, $i\in\{1,2,3\}$ & \\
399 $p_{0}$  & hydrostatic pressure & \\
400 $q$ & total water content & \\
401 $q_{l}$ & liquid water content & \\
402 $q_{l_{\mathrm{crit}}}$ & threshold for the formation of precipitation & \\
403 $q_{s}$ & specific humidity in case of saturation & \\
404 $q_{v}$ & specific humidity & \\ 
405 $x$, $y$, $z$, $x_{i}$  & Cartesian coordinates, $i\in\{1,2,3\}$ & \\
406 $\Delta$ & characteristic grid length & \\
407 $\epsilon$ & dissipation of sub-grid scale turbulent kinetic energy & \\
408 $\varepsilon\textnormal{\textuparrow}$ & upward effective cloud emissivity  & \\
409 $\varepsilon\textnormal{\textdownarrow}$ & downward effective cloud emissivity  & \\
410 $\kappa$ &$R/c_{p}$  & 0.286 \\
411 $\rho$ & air density  & \\
412 $\tau$ & time scale for the Kessler scheme & \\
413 $\theta$ & potential temperature & \\
414 $\theta_{l}$ & liquid water potential temperature & \\
415 $\theta_{v}$ & virtual potential temperature &  \\
416 $\theta_{0}$ & reference value for the potential temperature & \\
417 $\overline\psi$ & resolved scale variable & \\
418 $\psi^{'}$ & sub-grid scale variable & \\
419 $\psi^{\ast}$ & departure from the basic state (Boussinesq approximation) &  \\
420 $\langle\psi\rangle$ & horizontal mean \\
421\addlinespace
422\bottomrule
423\end{longtable}
424
425\setlength\labelsep{0pt} 
426\begin{bibliographie}{}
427 \bibitem[Betts (1973)]{betts73}
428   \textbf{Betts, A. K., 1973:} Non-precipitating cumulus convection and its parameterization.
429   \textit{Quart. J. R. Meteorol. Soc.}, \textbf{99}, 178-196.
430 \bibitem[Bougeault (1982)]{bougeault82}
431   \textbf{Bougeault, P., 1982:} Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations
432   against numerical data. \textit{J. Atmos. Sci.}, \textbf{38}, 2414-2428.
433 \bibitem[Cox (1976)]{cox76}
434   \textbf{Cox, S. K., 1976:} Observations of cloud infrared effective emissivity.
435   \textit{J. Atmos. Sci.}, \textbf{33}, 287-289.
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463\end{bibliographie}
464
465\end {document}
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