source: palm/trunk/TUTORIAL/SOURCE/cloud_physics.tex @ 1727

Last change on this file since 1727 was 1515, checked in by boeske, 10 years ago

several updates in the tutorial

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1% $Id: cloud_physics.tex 1515 2015-01-02 11:35:51Z knoop $
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{Institute of Meteorology and Climatology, Leibniz UniversitÀt Hannover}
33\selectlanguage{english}
34\date{last update: \today}
35\event{PALM Seminar}
36\setbeamertemplate{navigation symbols}{}
37\setbeamersize{text margin left=.5cm,text margin right=.2cm}
38\setbeamertemplate{footline}
39  {%
40    \begin{beamercolorbox}[rightskip=-0.1cm]&
41     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
42    \end{beamercolorbox}
43    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,%
44      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}%
45      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}%
46    \end{beamercolorbox}%
47%    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}%
48%    \end{beamercolorbox}
49  }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}}
50
51\title[PALM - Cloud Physics]{PALM - Cloud Physics}
52\author{PALM group}
53
54% Notes:
55% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
56% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
57\begin{document}
58
59% Folie 1
60\begin{frame}
61\titlepage
62\end{frame}
63
64% Folie 2
65\begin{frame}
66   \frametitle{Contents}
67   
68   \begin{itemize}
69      \item<1->{Motivation}
70      \item<1->{Approach}
71      \item<1->{Extension if basic equations and SGS-model}
72      \item<1->{Additional Sources / Sinks in prognostic equations}
73      \item<1->{Control parameters}
74      \item<1->{Example of shallow cumulus clouds}
75   \end{itemize} 
76\end{frame}
77
78% Folie 3
79\begin{frame}
80   \frametitle{Why simulating clouds?}
81   
82   \begin{itemize}
83      \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
84      \item<3->{Optional feature to account for:}\\ \par\medskip
85      \begin{itemize}
86         \item<4->{Microphysical processes}
87         \begin{itemize}
88            \item<4->{Evaporation / condensation of cloud droplets}
89            \item<4->{Precipitation}
90            \item<4->{Transport of humidity and liquid water}\\ \par\medskip
91         \end{itemize}
92         \item<5->{Radiation processes}
93         \begin{itemize}
94            \item<5->{Short-wave radiation}
95            \item<5->{Long-wave radiation}
96         \end{itemize} 
97      \end{itemize} 
98   \end{itemize}
99\end{frame}
100
101% Folie 4
102\begin{frame}
103   \frametitle{Approach}
104   
105   \begin{itemize}
106      \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)}
107      \item<2->{Dynamics like advection and diffusion are covered by Navier-Stokes equations (see basic\_equations.pdf)}
108      \item<3->{Thermodynamics are considered by parameterizations $\Rightarrow$ non explicit treatment of microphysical processes}
109      \item<4->{Total water specific humidity $q$ is prognosed as an additional variable $\Rightarrow$ one-moment}
110      \item<5->{Liquid water specific humidity $q_l$ is determined diagnostically}
111   \end{itemize}
112   \uncover<6->{PALM's basic equations are extended to account for cloud microphysics}
113\end{frame}
114
115% Folie 5
116\begin{frame}
117   \frametitle{Definitions (I)}
118   
119   \begin{itemize}
120      \item<1->{Liquid water potential temperature $\theta_{l}$ (defined by Betts, 1973)\\
121      \begin{minipage}[c][1.5cm][c]{0.38\textwidth}
122            \qquad$\theta_{l}=\theta -\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}$
123      \end{minipage}
124      \begin{minipage}[c][1.5cm][c]{0.52\textwidth}
125         {\scriptsize $L_{v}$: latent heat of vaporization; $L_{v}=\SI{2,5e6}{J/kg}$\\
126         $c_{p}$: specific heat of dry air; $c_{p}=\SI{1005}{J/kg K}$}
127      \end{minipage}\\
128      is the potential temperature of an air parcel if all its liquid water evaporates due to an reversible moist adiabatic descent.}
129      \item<2->{Total water specific humidity $q$\\
130      \begin{minipage}[c][1.5cm][c]{0.38\textwidth}
131            \qquad$q = q_{v} + q_{l}$
132      \end{minipage}
133      \begin{minipage}[c][1.5cm][c]{0.52\textwidth}
134         {\scriptsize $q_{v}$: specific humidity\\
135         $q_{l}$: liquid water speciffic humidity}
136      \end{minipage}\\
137      }
138      \item<3->{$\theta_{l}$ and $q$ are the prognostic variables when using PALM's cloud physics model}
139   \end{itemize}
140\end{frame}
141
142% Folie 6
143\begin{frame}
144   \frametitle{Definitions (II)}
145   
146   
147   \begin{itemize}
148      \item<1->{Why using $\theta_{l}$ and $q$?}\\ \par\medskip
149      \begin{itemize}
150         \item<2->{$\theta_{l}$ and $q$ are conservative quantities in the absence of precipitation, radiation and freezing processes.}
151         \item<3->{Phase transitions do not have to be described explicitly in the prognostic equations.}
152         \item<4->{In case of dry convection (no condensation): $\theta_{l} \rightarrow \theta$ and $q \rightarrow q_{v}$}
153         \item<5->{Parameterizations of SGS-fluxes can be retained.}
154         \item<6->{...$\rightarrow$ see also Deardorff, 1976}\\ \par\medskip
155      \end{itemize}
156      \item<7->{Virtual potential temperature $\theta_{l}$\\
157      \begin{minipage}[c][1.5cm][c]{0.65\textwidth}
158            \qquad$\theta_{v}=\left[\theta_{l} +\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}\right] \left(1+0,61 q - 1,61q_{l}\right)$
159      \end{minipage}
160      \begin{minipage}[c][1.5cm][c]{0.22\textwidth}
161      \end{minipage}\\
162      }
163   \end{itemize}
164\end{frame}
165
166% Folie 7
167\begin{frame}
168   \frametitle{Extension of basic equations (I)}
169   
170   \begin{itemize}
171      \item<1->{First principle is solved for $\theta_{l}$ (instead of $\theta$)\\
172      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
173            \qquad$\frac{\partial\bar{\theta}_{l}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{\theta_{l}}}{\partial x_{k}}- \frac{\partial H_{k}}{\partial x_{k}} + Q_{\theta}$
174      \end{minipage}
175      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
176         {\scriptsize SGS flux: $H_{k}=\overline{u_{k} \theta_{l}} - \bar{u}_{k}\bar{\theta}_{l}$}
177      \end{minipage}\\ \par\medskip
178      }
179      \item<2->{Conservation equation for total water specific humidity $q$ (instead of $q_{v}$)\\
180      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
181            \qquad$\frac{\partial\bar{q}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{q}}{\partial x_{k}}- \frac{\partial W_{k}}{\partial x_{k}} + Q_{\theta}$
182      \end{minipage}
183      \begin{minipage}[c][1.5cm][c]{0.46\textwidth}
184         {\scriptsize SGS flux: $W_{k}=\overline{u_{k} q} - \bar{u}_{k}\bar{q}$}
185      \end{minipage}\\
186      }
187   \end{itemize}
188\end{frame}
189
190% Folie 8
191\begin{frame}
192   \frametitle{Extension of basic equations (II)}
193   
194   \begin{itemize}
195      \item<1->{Sources / Sinks due to radiation (RAD) and precipitation (PREC)
196      \begin{minipage}[c][3.0cm][c]{0.46\textwidth}
197      \begin{align*}
198         Q_{\theta} &= \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} + \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}}\\
199         Q_{W} &= \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}}
200      \end{align*}
201      \end{minipage}\\ \par\medskip
202      }
203      \item<2->{Diagnostic approach for $\bar{q}_{l}$ (all-or-nothing schema)
204      \begin{minipage}[c][1.5cm][c]{0.44\textwidth}
205      \begin{align*}
206         \bar{q}_{l} =
207         \begin{cases}
208         \bar{q}-\bar{q}_{s} & \text{if } \bar{q} > \bar{q}_{s} \\
209         0 & \text{if } otherwise
210         \end{cases}
211      \end{align*}
212      \end{minipage}\\ \par\medskip
213      $\bar{q}_{s}$ is the saturation value of the specific humidity which is determined based on Sommeria and Deardorff, 1977 and further described in cloud\_physics.pdf
214      }
215   \end{itemize}
216\end{frame}
217
218% Folie 9
219\begin{frame}
220   \frametitle{Extension of SGS model (I)}
221   
222   \begin{itemize}
223      \item<1->{SGS fluxes are modelled by means of a down-gradient approximation
224      \begin{minipage}[c][1.5cm][c]{0.6\textwidth}
225      \begin{equation*}
226         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}}
227      \end{equation*}
228      \end{minipage}\\ \par\medskip
229      }
230      \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:
231      \begin{minipage}[c][1.2cm][c]{0.44\textwidth}
232      \begin{equation*}
233         \overline{u_{3}' \theta_{v}'} = K_{1} \cdot H_{3} + K_{2} \cdot W_{3}
234      \end{equation*}
235      \end{minipage}\\ \par\medskip
236      }
237   \end{itemize}
238\end{frame}
239
240% Folie 10
241\begin{frame}
242   \frametitle{Extension of SGS model (II)}
243   
244   \begin{itemize}
245      \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
246      \begin{itemize}
247         \item<2->{Unsaturated grid box ($\bar{q}_{l} = 0$)\\
248         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
249         \begin{align*}
250            K_{1} &= 1,0 + 0,61\cdot\bar{q}\\
251            K_{2} &=  0,61\cdot\bar{\theta}
252         \end{align*}
253         \end{minipage}\\ \par\medskip
254         }
255         \item<3->{Saturated grid box ($\bar{q}_{l} \neq 0$)\\ \par\medskip
256         \begin{minipage}[c][2.2cm][c]{0.64\textwidth}
257         \begin{align*}
258            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}}\\
259            K_{2} &\theta \left(\frac{L_{v}}{c_{p} T}\cdot K_{1} -1,0\right)
260         \end{align*}
261         \end{minipage}
262         }
263      \end{itemize}
264      }
265   \end{itemize}
266\end{frame}
267
268% Folie 11
269\begin{frame}
270   \frametitle{Sources / Sinks (I)}
271   
272   \begin{itemize}
273      \item<1->{Radiation model (based on Cox, 1976) $\Rightarrow$ scheme of effective emissivity\\ \par\medskip
274      \begin{itemize}
275         \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\\
276         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
277         \begin{equation*}
278            \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]
279         \end{equation*}
280         \end{minipage}\\ \par\medskip
281         \begin{tabbing}
282            $\Delta F$: \qquad \=Difference between upward and downward irradiance at\\
283            \>grid points above ($z^{+}$) and below ($z^{-}$) the level in\\
284            \>which $\bar{\theta}_{l}$ is defined.
285         \end{tabbing}
286         Further information: cloud\_physics.pdf
287         
288      \end{itemize}
289      }
290   \end{itemize}
291\end{frame}
292
293% Folie 12
294\begin{frame}
295   \frametitle{Sources / Sinks (II)}
296   
297   \begin{itemize}
298      \item<1->{Precipitation model (based on Kessler, 1969)\\ \par\medskip
299      \begin{itemize}
300         \item<2->{Simplified scheme which accounts only for the process of autoconversion for the formation of rain water.\\
301         \begin{minipage}[c][1.5cm][c]{0.44\textwidth}
302         \begin{align*}
303            \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} =
304            \begin{cases}
305            (\bar{q}_{l}-\bar{q}_{l_{\text{crit}}})/\tau & \text{if } \bar{q}_{l} > \bar{q}_{l_{\text{crit}}} \\
306            0 & \text{if } \bar{q}_{l} \leq \bar{q}_{l_{\text{crit}}}
307            \end{cases}
308         \end{align*}
309         \end{minipage}\\ \par\medskip
310         }
311         \item<3->{precipitation leaves grid box immediately if the threshold $\bar{q}_{l_{\text{crit}}} = \SI{0,5}{g/kg}$ is exceeded.}\\ \par\medskip
312         \item<4->{Timescale $\tau = \SI{1000}{s}$.}
313         \item<5->{
314         \begin{minipage}[c][1.5cm][c]{0.35\textwidth}
315         \begin{equation*}
316            \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}}
317         \end{equation*}
318         \end{minipage}
319         }
320      \end{itemize}
321      }
322   \end{itemize}
323\end{frame}
324
325% Folie 13
326\begin{frame}
327   \frametitle{Control parameters}
328   
329   \begin{itemize}
330      \item<1->{The following settings in the parameter file enable the use of the bulk cloud model:}\\ \par\medskip 
331      \scriptsize
332      \begin{itemize}\scriptsize
333         \item<2->{
334               $\left.
335               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
336               \text{humidity = .TRUE.}\qquad\qquad\qquad
337               \end{array}
338               \right\}:
339               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
340               \text{prognostic equations for specific} \\  \text{specific humidity } \bar{q} \text{ is solved}
341               \end{array}
342               $
343         }\\ \par\medskip 
344         \item<3->{
345               $\left.
346               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
347               \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.}
348               \end{array}
349               \right\}:
350               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
351               \text{prognostic equations for liquid water} \\  \text{potential temperature } \bar{\theta}_{l} \text{ and total water} \\ \text{specific humidity } \bar{q} \text{ are solved}
352               \end{array}
353               $
354         }\\ \par\medskip 
355         \item<4->{
356               $\left.
357               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
358               \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} \\ \text{precipitation = .TRUE.} \\ \text{radiation = .TRUE.}
359               \end{array}
360               \right\}:
361               \begin{array}{ll} % fÃŒr mehrzeiligen Text nötig
362               \text{Kessler precipitation scheme and} \\  \text{radiation model are solved}
363               \end{array}
364               $
365         }
366      \end{itemize}
367   \end{itemize}
368\end{frame}
369
370% Folie 12
371\begin{frame}
372   \frametitle{Example - Setup for a cloudy boundary layer}
373   
374   \begin{figure}[H]
375      \begin{minipage}[c][6.5cm][c]{.50\linewidth}
376         \centering
377         CBL with shallow cumulus clouds:\\ \par\bigskip
378         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/cbl5_preview.png}
379      \end{minipage} 
380      \begin{minipage}[c][6.5cm][t]{.40\linewidth}
381         \centering
382         \includegraphics[width=0.9\linewidth]{cloud_physics_figures/param_clouds.png}
383      \end{minipage}
384   \end{figure}
385\end{frame}
386
387% Folie 13
388\begin{frame}
389   \frametitle{Example - Model output}
390   
391   \begin{figure}[H]
392      \begin{minipage}[c][6cm][c]{.45\linewidth}
393         \centering
394         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/profiles_cbl_cloud.png}
395      \end{minipage} 
396      \begin{minipage}[c][6cm][c]{.45\linewidth}
397         \centering
398         \includegraphics[width=0.95\linewidth]{cloud_physics_figures/ql_xy_cbl_cloud.png}
399      \end{minipage}
400   \end{figure}
401\end{frame}
402
403% Folie 1$
404\begin{frame}
405   \frametitle{Bibliography}
406   
407   \tiny
408   \begin{thebibliography}{}
409      \bibitem[1]{betts1973}
410         \textsc{Betts, A.K., 1973:} \emph{Non-precipitating cumulus convection and its parameterization.}
411         \newblock Quart. J. Roy. Meteor. Soc., \textbf{99}, 178-196.
412      \bibitem[2]{cox1976}
413         \textsc{Cox, S. K., 1976:} \emph{Observations of cloud infrared effective emissivity.}
414         \newblock J. Atmos. Sci., \textbf{33}, 287-289.
415      \bibitem[3]{cuijpers1993}
416         \textsc{Cuijpers, J.W.M., P.G. Duynkerke, 1993:} \emph{Large eddy simulation of trade wind cumulus clouds.}
417         \newblock J. Atmos. Sci., \textbf{50}, 3894-3908.
418      \bibitem[4]{deardorff1976}
419         \textsc{Deardorff, J. W., 1976:} \emph{Usefullness of liquid-water potential temperature in shallow-cloud model.}
420         \newblock J. Appl. Meteor., \textbf{15}, 98-102.
421      \bibitem[5]{deardorff1980}
422         \textsc{Deardorff, J. W., 1980:} \emph{Stratocumulus-capped mixed layers derived from a three-dimensional model}.
423         \newblock Bondary-Layer Meteor., \textbf{18}, 495-527.
424      \bibitem[6]{kessler1969}
425         \textsc{Kessler, E., 1969:} \emph{On the distribution and continuity of water substance in atmospheric circulations.}
426         \newblock Meteor. Monogr., \textbf{32}, 84 pp.
427      \bibitem[7]{riechelmann2012}
428         \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.}
429         \newblock New J. Phys., \textbf{14}, 27.
430      \bibitem[8]{sommeria1977}
431         \textsc{Sommeria, G., J. W. Deardorff, 1977:} \emph{Subgrid-scale condensation in models of nonprecipitating clouds.}
432         \newblock J. Atmos. Sci., \textbf{34}, 344-355.
433      \bibitem[9]{cloudphys}
434         \textsc{cloud\_physics.pdf:} \emph{Introduction to the cloud physics model of PALM.}
435         \newblock {\tt trunk/DOC/tec/methods/cloud\_physics/cloud\_physics.pdf}.
436   \end{thebibliography}
437\end{frame}
438
439\end{document}
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