source: palm/trunk/TUTORIAL/SOURCE/cloud_physics.tex @ 1105

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1% $Id: cloud_physics.tex 1105 2013-02-26 11:14:51Z maronga $
2\input{header_tmp.tex}
3
4\usepackage[utf8]{inputenc}
5\usepackage{ngerman}
6\usepackage{pgf}
7\usetheme{Dresden}
8\usepackage{subfigure}
9\usepackage{units}
10\usepackage{multimedia}
11\usepackage{hyperref}
12\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
13\usepackage{xmpmulti}
14\usepackage{tikz}
15\usetikzlibrary{shapes,arrows,positioning,decorations.pathreplacing}
16\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
17
18%---------- neue Pakete
19\usepackage{amsmath}
20\usepackage{amssymb}
21\usepackage{multicol}
22\usepackage{pdfcomment}
23\usepackage{xcolor}
24\usepackage{siunitx}
25\sisetup{%
26        mode = math, detect-family, detect-weight,     
27        exponent-product = \cdot,
28        number-unit-separator=\text{\,},
29        output-decimal-marker={,},
30}
31
32\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
33\date{last update: \today}
34\event{PALM Seminar}
35\setbeamertemplate{navigation symbols}{}
36\setbeamersize{text margin left=.5cm,text margin right=.2cm}
37\setbeamertemplate{footline}
38  {%
39    \begin{beamercolorbox}[rightskip=-0.1cm]&
40     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
41    \end{beamercolorbox}
42    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,%
43      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}%
44      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}%
45    \end{beamercolorbox}%
46%    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}%
47%    \end{beamercolorbox}
48  }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}}
49
50\title[PALM - Cloud Physics]{PALM - Cloud Physics}
51\author{Siegfried Raasch}
52
53% Notes:
54% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
55% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
56\begin{document}
57
58% Folie 1
59\begin{frame}
60\titlepage
61\end{frame}
62
63% Folie 2
64\begin{frame}
65   \frametitle{Contents}
66   
67   \begin{itemize}
68      \item<1->{Motivation}
69      \item<1->{Approach}
70      \item<1->{Extension if basic equations and SGS-model}
71      \item<1->{Additional Sources / Sinks in prognostic equations}
72      \item<1->{Control parameters}
73      \item<1->{Example of shallow cumulus clouds}
74   \end{itemize} 
75\end{frame}
76
77% Folie 3
78\begin{frame}
79   \frametitle{Why simulating clouds?}
80   
81   \begin{itemize}
82      \item<2->{Atmospheric boundary layers are usually covered with shallow clouds like cumulus or stratocumulus which are the inherent characteristic of more realistic boundary layers.}\\ \par\medskip
83      \item<3->{Optional feature to account for:}\\ \par\medskip
84      \begin{itemize}
85         \item<4->{Microphysical processes}
86         \begin{itemize}
87            \item<4->{Evaporation / condensation of cloud droplets}
88            \item<4->{Precipitation}
89            \item<4->{Transport of humidity and liquid water}\\ \par\medskip
90         \end{itemize}
91         \item<5->{Radiation processes}
92         \begin{itemize}
93            \item<5->{Short-wave radiation}
94            \item<5->{Long-wave radiation}
95         \end{itemize} 
96      \end{itemize} 
97   \end{itemize}
98\end{frame}
99
100% Folie 4
101\begin{frame}
102   \frametitle{Approach}
103   
104   \begin{itemize}
105      \item<1->{One-moment bulk model $\Rightarrow$ in contrast to PALM's Lagrangian cloud model (LCM) (see also particle\_model\_cloud\_physics.pdf, Riechelmann et al., 2012)}
106      \item<2->{Dynamics like advection and diffusion are covered by Navier-Stokes equations (see basic\_equations.pdf)}
107      \item<3->{Thermodynamics are considered by parameterizations $\Rightarrow$ non explicit treatment of microphysical processes}
108      \item<4->{Total water specific humidity $q$ is prognosed as an additional variable $\Rightarrow$ one-moment}
109      \item<5->{Liquid water specific humidity $q_l$ is determined diagnostically}
110   \end{itemize}
111   \uncover<6->{PALM's basic equations are extended to account for cloud microphysics}
112\end{frame}
113
114% Folie 5
115\begin{frame}
116   \frametitle{Definitions (I)}
117   
118   \begin{itemize}
119      \item<1->{Liquid water potential temperature $\theta_{l}$ (defined by Betts, 1973)\\
120      \begin{minipage}[c][1.5cm][c]{0.38\textwidth}
121            \qquad$\theta_{l}=\theta -\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}$
122      \end{minipage}
123      \begin{minipage}[c][1.5cm][c]{0.52\textwidth}
124         {\scriptsize $L_{v}$: latent heat of vaporization; $L_{v}=\SI{2,5e6}{J/kg}$\\
125         $c_{p}$: specific heat of dry air; $c_{p}=\SI{1005}{J/kg K}$}
126      \end{minipage}\\
127      is the potential temperature of an air parcel if all its liquid water evaporates due to an reversible moist adiabatic descent.}
128      \item<2->{Total water specific humidity $q$\\
129      \begin{minipage}[c][1.5cm][c]{0.38\textwidth}
130            \qquad$q = q_{v} + q_{l}$
131      \end{minipage}
132      \begin{minipage}[c][1.5cm][c]{0.52\textwidth}
133         {\scriptsize $q_{v}$: specific humidity\\
134         $q_{l}$: liquid water speciffic humidity}
135      \end{minipage}\\
136      }
137      \item<3->{$\theta_{l}$ and $q$ are the prognostic variables when using PALM's cloud physics model}
138   \end{itemize}
139\end{frame}
140
141% Folie 6
142\begin{frame}
143   \frametitle{Definitions (II)}
144   
145   
146   \begin{itemize}
147      \item<1->{Why using $\theta_{l}$ and $q$?}\\ \par\medskip
148      \begin{itemize}
149         \item<2->{$\theta_{l}$ and $q$ are conservative quantities in the absence of precipitation, radiation and freezing processes.}
150         \item<3->{Phase transitions do not have to be described explicitly in the prognostic equations.}
151         \item<4->{In case of dry convection (no condensation): $\theta_{l} \rightarrow \theta$ and $q \rightarrow q_{v}$}
152         \item<5->{Parameterizations of SGS-fluxes can be retained.}
153         \item<6->{...$\rightarrow$ see also Deardorff, 1976}\\ \par\medskip
154      \end{itemize}
155      \item<7->{Virtual potential temperature $\theta_{l}$\\
156      \begin{minipage}[c][1.5cm][c]{0.65\textwidth}
157            \qquad$\theta_{v}=\left[\theta_{l} +\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}\right] \left(1+0,61 q - 1,61q_{l}\right)$
158      \end{minipage}
159      \begin{minipage}[c][1.5cm][c]{0.22\textwidth}
160      \end{minipage}\\
161      }
162   \end{itemize}
163\end{frame}
164
165% Folie 7
166\begin{frame}
167   \frametitle{Extension of basic equations (I)}
168   
169   \begin{itemize}
170      \item<1->{First principle is solved for $\theta_{l}$ (instead of $\theta$)\\
171      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
172            \qquad$\frac{\partial\bar{\theta}_{l}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{\theta_{l}}}{\partial x_{k}}- \frac{\partial H_{k}}{\partial x_{k}} + Q_{\theta}$
173      \end{minipage}
174      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
175         {\scriptsize SGS flux: $H_{k}=\overline{u_{k} \theta_{l}} - \bar{u}_{k}\bar{\theta}_{l}$}
176      \end{minipage}\\ \par\medskip
177      }
178      \item<2->{Conservation equation for total water specific humidity $q$ (instead of $q_{v}$)\\
179      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
180            \qquad$\frac{\partial\bar{q}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{q}}{\partial x_{k}}- \frac{\partial W_{k}}{\partial x_{k}} + Q_{\theta}$
181      \end{minipage}
182      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
183         {\scriptsize SGS flux: $W_{k}=\overline{u_{k} q} - \bar{u}_{k}\bar{q}$}
184      \end{minipage}\\
185      }
186   \end{itemize}
187\end{frame}
188
189% Folie 8
190\begin{frame}
191   \frametitle{Extension of basic equations (II)}
192   
193   \begin{itemize}
194      \item<1->{Sources / Sinks due to radiation (RAD) and precipitation (PREC)
195      \begin{minipage}[c][3.0cm][c]{0.46\textwidth}
196      \begin{align*}
197         Q_{\theta} &= \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} + \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}}\\
198         Q_{W} &= \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}}
199      \end{align*}
200      \end{minipage}\\ \par\medskip
201      }
202      \item<2->{Diagnostic approach for $\bar{q}_{l}$ (all-or-nothing schema)
203      \begin{minipage}[c][1.5cm][c]{0.44\textwidth}
204      \begin{align*}
205         \bar{q}_{l} =
206         \begin{cases}
207         \bar{q}-\bar{q}_{s} & \text{if } \bar{q} > \bar{q}_{s} \\
208         0 & \text{if } otherwise
209         \end{cases}
210      \end{align*}
211      \end{minipage}\\ \par\medskip
212      $\bar{q}_{s}$ is the saturation value of the specific humidity which is determined based on Sommeria and Deardorff, 1977 and further described in cloud\_physics.pdf
213      }
214   \end{itemize}
215\end{frame}
216
217% Folie 9
218\begin{frame}
219   \frametitle{Extension of SGS model (I)}
220   
221   \begin{itemize}
222      \item<1->{SGS fluxes are modelled by means of a down-gradient approximation
223      \begin{minipage}[c][1.5cm][c]{0.6\textwidth}
224      \begin{equation*}
225         H_{k} = - K_{h} \frac{\partial\bar{\theta}_{l}}{\partial x_{k}} \qquad \text{;} \qquad W_{k} = - K_{h} \frac{\partial\bar{q}}{\partial x_{k}}
226      \end{equation*}
227      \end{minipage}\\ \par\medskip
228      }
229      \item<2->{SGS flux of potential temperature $\overline{u_{3}' \theta'}$ in prognostic equation of the SGS-TKE $\bar{e}$ is replaced by the flux of the virtual potential temperature $\overline{u_{3}' \theta_{v}'}$ which is modelled according to Deardorff, 1980 as:
230      \begin{minipage}[c][1.2cm][c]{0.44\textwidth}
231      \begin{equation*}
232         \overline{u_{3}' \theta_{v}'} = K_{1} \cdot H_{3} + K_{2} \cdot W_{3}
233      \end{equation*}
234      \end{minipage}\\ \par\medskip
235      }
236   \end{itemize}
237\end{frame}
238
239% Folie 10
240\begin{frame}
241   \frametitle{Extension of SGS model (II)}
242   
243   \begin{itemize}
244      \item<1->{The coefficients $K_{1}$ and $K_{2}$ depend on the saturation state of the grid volume (see also Cuijpers u. Duynkerke, 1993)\\ \par\medskip
245      \begin{itemize}
246         \item<2->{Unsaturated grid box ($\bar{q}_{l} = 0$)\\
247         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
248         \begin{align*}
249            K_{1} &= 1,0 + 0,61\cdot\bar{q}\\
250            K_{2} &=  0,61\cdot\bar{\theta}
251         \end{align*}
252         \end{minipage}\\ \par\medskip
253         }
254         \item<3->{Saturated grid box ($\bar{q}_{l} \neq 0$)\\ \par\medskip
255         \begin{minipage}[c][2.2cm][c]{0.64\textwidth}
256         \begin{align*}
257            K_{1} &= \frac{1,0 - \bar{q} + 1,61\cdot\bar{q}_{s}\left(1,0 + 0,622\frac{L_{v}}{R T}\right)}{1,0 + 0,622\frac{L_{v}}{R T}\frac{L_{v}}{c_{p} T} \bar{q}_{s}}\\
258            K_{2} &\theta \left(\frac{L_{v}}{c_{p} T}\cdot K_{1} -1,0\right)
259         \end{align*}
260         \end{minipage}
261         }
262      \end{itemize}
263      }
264   \end{itemize}
265\end{frame}
266
267% Folie 11
268\begin{frame}
269   \frametitle{Sources / Sinks (I)}
270   
271   \begin{itemize}
272      \item<1->{Radiation model (based on Cox, 1976) $\Rightarrow$ scheme of effective emissivity\\ \par\medskip
273      \begin{itemize}
274         \item<2->Very simple, accounts only for absorbtion and emission of long-wave radiation due to water vapour and cloud droplets and neglects horizontal divergences of radiation\\
275         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
276         \begin{equation*}
277            \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} = \left(\frac{\theta}{T}\right) \frac{1}{\varrho c_{p} \Delta z}\left[ \Delta F(z^{+}) - \Delta F(z^{-}) \right]
278         \end{equation*}
279         \end{minipage}\\ \par\medskip
280         \begin{tabbing}
281            $\Delta F$: \qquad \=Difference between upward and downward irradiance at\\
282            \>grid points above ($z^{+}$) and below ($z^{-}$) the level in\\
283            \>which $\bar{\theta}_{l}$ is defined.
284         \end{tabbing}
285         Further information: cloud\_physics.pdf
286         
287      \end{itemize}
288      }
289   \end{itemize}
290\end{frame}
291
292% Folie 12
293\begin{frame}
294   \frametitle{Sources / Sinks (II)}
295   
296   \begin{itemize}
297      \item<1->{Precipitation model (based on Kessler, 1969)\\ \par\medskip
298      \begin{itemize}
299         \item<2->{Simplified scheme which accounts only for the process of autoconversion for the formation of rain water.\\
300         \begin{minipage}[c][1.5cm][c]{0.44\textwidth}
301         \begin{align*}
302            \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} =
303            \begin{cases}
304            (\bar{q}_{l}-\bar{q}_{l_{\text{crit}}})/\tau & \text{if } \bar{q}_{l} > \bar{q}_{l_{\text{crit}}} \\
305            0 & \text{if } \bar{q}_{l} \leq \bar{q}_{l_{\text{crit}}}
306            \end{cases}
307         \end{align*}
308         \end{minipage}\\ \par\medskip
309         }
310         \item<3->{precipitation leaves grid box immediately if the threshold $\bar{q}_{l_{\text{crit}}} = \SI{0,5}{g/kg}$ is exceeded.}\\ \par\medskip
311         \item<4->{Timescale $\tau = \SI{1000}{s}$.}
312         \item<5->{
313         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
314         \begin{equation*}
315            \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}} = \frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right) \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}}
316         \end{equation*}
317         \end{minipage}
318         }
319      \end{itemize}
320      }
321   \end{itemize}
322\end{frame}
323
324% Folie 13
325\begin{frame}
326   \frametitle{Control parameters}
327   
328   \begin{itemize}
329      \item<1->{The following settings in the parameter file enable the use of the bulk cloud model:}\\ \par\medskip 
330      \scriptsize
331      \begin{itemize}\scriptsize
332         \item<2->{
333               $\left.
334               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
335               \text{humidity = .TRUE.}\qquad\qquad\qquad
336               \end{array}
337               \right\}:
338               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
339               \text{prognostic equations for specific} \\  \text{specific humidity } \bar{q} \text{ is solved}
340               \end{array}
341               $
342         }\\ \par\medskip 
343         \item<3->{
344               $\left.
345               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
346               \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.}
347               \end{array}
348               \right\}:
349               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
350               \text{prognostic equations for liquid water} \\  \text{potential temperature } \bar{\theta}_{l} \text{ and total water} \\ \text{specific humidity } \bar{q} \text{ are solved}
351               \end{array}
352               $
353         }\\ \par\medskip 
354         \item<4->{
355               $\left.
356               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
357               \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} \\ \text{precipitation = .TRUE.} \\ \text{radiation = .TRUE.}
358               \end{array}
359               \right\}:
360               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
361               \text{Kessler precipitation scheme and} \\  \text{radiation model are solved}
362               \end{array}
363               $
364         }
365      \end{itemize}
366   \end{itemize}
367\end{frame}
368
369% Folie 12
370\begin{frame}
371   \frametitle{Example - Setup for a cloudy boundary layer}
372   
373   \begin{figure}[H]
374      \begin{minipage}[c][6.5cm][c]{.50\linewidth}
375         \centering
376         CBL with shallow cumulus clouds:\\ \par\bigskip
377         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/cbl5_preview.png}
378      \end{minipage} 
379      \begin{minipage}[c][6.5cm][t]{.40\linewidth}
380         \centering
381         \includegraphics[width=0.9\linewidth]{cloud_physics_figures/param_clouds.png}
382      \end{minipage}
383   \end{figure}
384\end{frame}
385
386% Folie 13
387\begin{frame}
388   \frametitle{Example - Model output}
389   
390   \begin{figure}[H]
391      \begin{minipage}[c][6cm][c]{.45\linewidth}
392         \centering
393         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/profiles_cbl_cloud.png}
394      \end{minipage} 
395      \begin{minipage}[c][6cm][c]{.45\linewidth}
396         \centering
397         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/ql_xy_cbl_cloud.png}
398      \end{minipage}
399   \end{figure}
400\end{frame}
401
402% Folie 1$
403\begin{frame}
404   \frametitle{Bibliography}
405   
406   \tiny
407   \begin{thebibliography}{}
408      \bibitem[1]{betts1973}
409         \textsc{Betts, A.K., 1973:} \emph{Non-precipitating cumulus convection and its parameterization.}
410         \newblock Quart. J. Roy. Meteor. Soc., \textbf{99}, 178-196.
411      \bibitem[2]{cox1976}
412         \textsc{Cox, S. K., 1976:} \emph{Observations of cloud infrared effective emissivity.}
413         \newblock J. Atmos. Sci., \textbf{33}, 287-289.
414      \bibitem[3]{cuijpers1993}
415         \textsc{Cuijpers, J.W.M., P.G. Duynkerke, 1993:} \emph{Large eddy simulation of trade wind cumulus clouds.}
416         \newblock J. Atmos. Sci., \textbf{50}, 3894-3908.
417      \bibitem[4]{deardorff1976}
418         \textsc{Deardorff, J. W., 1976:} \emph{Usefullness of liquid-water potential temperature in shallow-cloud model.}
419         \newblock J. Appl. Meteor., \textbf{15}, 98-102.
420      \bibitem[5]{deardorff1980}
421         \textsc{Deardorff, J. W., 1980:} \emph{Stratocumulus-capped mixed layers derived from a three-dimensional model}.
422         \newblock Bondary-Layer Meteor., \textbf{18}, 495-527.
423      \bibitem[6]{kessler1969}
424         \textsc{Kessler, E., 1969:} \emph{On the distribution and continuity of water substance in atmospheric circulations.}
425         \newblock Meteor. Monogr., \textbf{32}, 84 pp.
426      \bibitem[7]{riechelmann2012}
427         \textsc{Riechelmann, T., Y. Noh, S. Raasch, 2012:} \emph{A new method for large-eddy simulations of clouds with Lagrangian droplets including the effects of turbulent collision.}
428         \newblock New J. Phys., \textbf{14}, 27.
429      \bibitem[8]{sommeria1977}
430         \textsc{Sommeria, G., J. W. Deardorff, 1977:} \emph{Subgrid-scale condensation in models of nonprecipitating clouds.}
431         \newblock J. Atmos. Sci., \textbf{34}, 344-355.
432      \bibitem[9]{cloudphys}
433         \textsc{cloud\_physics.pdf:} \emph{Introduction to the cloud physics model of PALM.}
434         \newblock {\tt trunk/DOC/tec/methods/cloud\_physics/cloud\_physics.pdf}.
435   \end{thebibliography}
436\end{frame}
437
438\end{document}
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