% $Id: cloud_physics.tex 1515 2015-01-02 11:35:51Z boeske $ \input{header_tmp.tex} \usepackage[utf8]{inputenc} \usepackage{ngerman} \usepackage{pgf} \usetheme{Dresden} \usepackage{subfigure} \usepackage{units} \usepackage{multimedia} \usepackage{hyperref} \newcommand{\event}[1]{\newcommand{\eventname}{#1}} \usepackage{xmpmulti} \usepackage{tikz} \usetikzlibrary{shapes,arrows,positioning,decorations.pathreplacing} \def\Tiny{\fontsize{4pt}{4pt}\selectfont} %---------- neue Pakete \usepackage{amsmath} \usepackage{amssymb} \usepackage{multicol} \usepackage{pdfcomment} \usepackage{xcolor} \usepackage{siunitx} \sisetup{% mode = math, detect-family, detect-weight, exponent-product = \cdot, number-unit-separator=\text{\,}, output-decimal-marker={,}, } \institute{Institute of Meteorology and Climatology, Leibniz Universität Hannover} \selectlanguage{english} \date{last update: \today} \event{PALM Seminar} \setbeamertemplate{navigation symbols}{} \setbeamersize{text margin left=.5cm,text margin right=.2cm} \setbeamertemplate{footline} {% \begin{beamercolorbox}[rightskip=-0.1cm]& {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} \end{beamercolorbox} \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,% leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}% {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}% \end{beamercolorbox}% % \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}% % \end{beamercolorbox} }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}} \title[PALM - Cloud Physics]{PALM - Cloud Physics} \author{PALM group} % Notes: % jede subsection bekommt einen punkt im menu (vertikal ausgerichtet. % jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet) \begin{document} % Folie 1 \begin{frame} \titlepage \end{frame} % Folie 2 \begin{frame} \frametitle{Contents} \begin{itemize} \item<1->{Motivation} \item<1->{Approach} \item<1->{Extension if basic equations and SGS-model} \item<1->{Additional Sources / Sinks in prognostic equations} \item<1->{Control parameters} \item<1->{Example of shallow cumulus clouds} \end{itemize} \end{frame} % Folie 3 \begin{frame} \frametitle{Why simulating clouds?} \begin{itemize} \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 \item<3->{Optional feature to account for:}\\ \par\medskip \begin{itemize} \item<4->{Microphysical processes} \begin{itemize} \item<4->{Evaporation / condensation of cloud droplets} \item<4->{Precipitation} \item<4->{Transport of humidity and liquid water}\\ \par\medskip \end{itemize} \item<5->{Radiation processes} \begin{itemize} \item<5->{Short-wave radiation} \item<5->{Long-wave radiation} \end{itemize} \end{itemize} \end{itemize} \end{frame} % Folie 4 \begin{frame} \frametitle{Approach} \begin{itemize} \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)} \item<2->{Dynamics like advection and diffusion are covered by Navier-Stokes equations (see basic\_equations.pdf)} \item<3->{Thermodynamics are considered by parameterizations $\Rightarrow$ non explicit treatment of microphysical processes} \item<4->{Total water specific humidity $q$ is prognosed as an additional variable $\Rightarrow$ one-moment} \item<5->{Liquid water specific humidity $q_l$ is determined diagnostically} \end{itemize} \uncover<6->{PALM's basic equations are extended to account for cloud microphysics} \end{frame} % Folie 5 \begin{frame} \frametitle{Definitions (I)} \begin{itemize} \item<1->{Liquid water potential temperature $\theta_{l}$ (defined by Betts, 1973)\\ \begin{minipage}[c][1.5cm][c]{0.38\textwidth} \qquad$\theta_{l}=\theta -\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}$ \end{minipage} \begin{minipage}[c][1.5cm][c]{0.52\textwidth} {\scriptsize $L_{v}$: latent heat of vaporization; $L_{v}=\SI{2,5e6}{J/kg}$\\ $c_{p}$: specific heat of dry air; $c_{p}=\SI{1005}{J/kg K}$} \end{minipage}\\ is the potential temperature of an air parcel if all its liquid water evaporates due to an reversible moist adiabatic descent.} \item<2->{Total water specific humidity $q$\\ \begin{minipage}[c][1.5cm][c]{0.38\textwidth} \qquad$q = q_{v} + q_{l}$ \end{minipage} \begin{minipage}[c][1.5cm][c]{0.52\textwidth} {\scriptsize $q_{v}$: specific humidity\\ $q_{l}$: liquid water speciffic humidity} \end{minipage}\\ } \item<3->{$\theta_{l}$ and $q$ are the prognostic variables when using PALM's cloud physics model} \end{itemize} \end{frame} % Folie 6 \begin{frame} \frametitle{Definitions (II)} \begin{itemize} \item<1->{Why using $\theta_{l}$ and $q$?}\\ \par\medskip \begin{itemize} \item<2->{$\theta_{l}$ and $q$ are conservative quantities in the absence of precipitation, radiation and freezing processes.} \item<3->{Phase transitions do not have to be described explicitly in the prognostic equations.} \item<4->{In case of dry convection (no condensation): $\theta_{l} \rightarrow \theta$ and $q \rightarrow q_{v}$} \item<5->{Parameterizations of SGS-fluxes can be retained.} \item<6->{...$\rightarrow$ see also Deardorff, 1976}\\ \par\medskip \end{itemize} \item<7->{Virtual potential temperature $\theta_{l}$\\ \begin{minipage}[c][1.5cm][c]{0.65\textwidth} \qquad$\theta_{v}=\left[\theta_{l} +\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}\right] \left(1+0,61 q - 1,61q_{l}\right)$ \end{minipage} \begin{minipage}[c][1.5cm][c]{0.22\textwidth} \end{minipage}\\ } \end{itemize} \end{frame} % Folie 7 \begin{frame} \frametitle{Extension of basic equations (I)} \begin{itemize} \item<1->{First principle is solved for $\theta_{l}$ (instead of $\theta$)\\ \begin{minipage}[c][1.5cm][c]{0.46\textwidth} \qquad$\frac{\partial\bar{\theta}_{l}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{\theta_{l}}}{\partial x_{k}}- \frac{\partial H_{k}}{\partial x_{k}} + Q_{\theta}$ \end{minipage} \begin{minipage}[c][1.5cm][c]{0.46\textwidth} {\scriptsize SGS flux: $H_{k}=\overline{u_{k} \theta_{l}} - \bar{u}_{k}\bar{\theta}_{l}$} \end{minipage}\\ \par\medskip } \item<2->{Conservation equation for total water specific humidity $q$ (instead of $q_{v}$)\\ \begin{minipage}[c][1.5cm][c]{0.46\textwidth} \qquad$\frac{\partial\bar{q}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{q}}{\partial x_{k}}- \frac{\partial W_{k}}{\partial x_{k}} + Q_{\theta}$ \end{minipage} \begin{minipage}[c][1.5cm][c]{0.46\textwidth} {\scriptsize SGS flux: $W_{k}=\overline{u_{k} q} - \bar{u}_{k}\bar{q}$} \end{minipage}\\ } \end{itemize} \end{frame} % Folie 8 \begin{frame} \frametitle{Extension of basic equations (II)} \begin{itemize} \item<1->{Sources / Sinks due to radiation (RAD) and precipitation (PREC) \begin{minipage}[c][3.0cm][c]{0.46\textwidth} \begin{align*} Q_{\theta} &= \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} + \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}}\\ Q_{W} &= \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} \end{align*} \end{minipage}\\ \par\medskip } \item<2->{Diagnostic approach for $\bar{q}_{l}$ (all-or-nothing schema) \begin{minipage}[c][1.5cm][c]{0.44\textwidth} \begin{align*} \bar{q}_{l} = \begin{cases} \bar{q}-\bar{q}_{s} & \text{if } \bar{q} > \bar{q}_{s} \\ 0 & \text{if } otherwise \end{cases} \end{align*} \end{minipage}\\ \par\medskip $\bar{q}_{s}$ is the saturation value of the specific humidity which is determined based on Sommeria and Deardorff, 1977 and further described in cloud\_physics.pdf } \end{itemize} \end{frame} % Folie 9 \begin{frame} \frametitle{Extension of SGS model (I)} \begin{itemize} \item<1->{SGS fluxes are modelled by means of a down-gradient approximation \begin{minipage}[c][1.5cm][c]{0.6\textwidth} \begin{equation*} 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}} \end{equation*} \end{minipage}\\ \par\medskip } \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: \begin{minipage}[c][1.2cm][c]{0.44\textwidth} \begin{equation*} \overline{u_{3}' \theta_{v}'} = K_{1} \cdot H_{3} + K_{2} \cdot W_{3} \end{equation*} \end{minipage}\\ \par\medskip } \end{itemize} \end{frame} % Folie 10 \begin{frame} \frametitle{Extension of SGS model (II)} \begin{itemize} \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 \begin{itemize} \item<2->{Unsaturated grid box ($\bar{q}_{l} = 0$)\\ \begin{minipage}[c][1.5cm][c]{0.35\textwidth} \begin{align*} K_{1} &= 1,0 + 0,61\cdot\bar{q}\\ K_{2} &= 0,61\cdot\bar{\theta} \end{align*} \end{minipage}\\ \par\medskip } \item<3->{Saturated grid box ($\bar{q}_{l} \neq 0$)\\ \par\medskip \begin{minipage}[c][2.2cm][c]{0.64\textwidth} \begin{align*} 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}}\\ K_{2} &= \theta \left(\frac{L_{v}}{c_{p} T}\cdot K_{1} -1,0\right) \end{align*} \end{minipage} } \end{itemize} } \end{itemize} \end{frame} % Folie 11 \begin{frame} \frametitle{Sources / Sinks (I)} \begin{itemize} \item<1->{Radiation model (based on Cox, 1976) $\Rightarrow$ scheme of effective emissivity\\ \par\medskip \begin{itemize} \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\\ \begin{minipage}[c][1.5cm][c]{0.35\textwidth} \begin{equation*} \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] \end{equation*} \end{minipage}\\ \par\medskip \begin{tabbing} $\Delta F$: \qquad \=Difference between upward and downward irradiance at\\ \>grid points above ($z^{+}$) and below ($z^{-}$) the level in\\ \>which $\bar{\theta}_{l}$ is defined. \end{tabbing} Further information: cloud\_physics.pdf \end{itemize} } \end{itemize} \end{frame} % Folie 12 \begin{frame} \frametitle{Sources / Sinks (II)} \begin{itemize} \item<1->{Precipitation model (based on Kessler, 1969)\\ \par\medskip \begin{itemize} \item<2->{Simplified scheme which accounts only for the process of autoconversion for the formation of rain water.\\ \begin{minipage}[c][1.5cm][c]{0.44\textwidth} \begin{align*} \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} = \begin{cases} (\bar{q}_{l}-\bar{q}_{l_{\text{crit}}})/\tau & \text{if } \bar{q}_{l} > \bar{q}_{l_{\text{crit}}} \\ 0 & \text{if } \bar{q}_{l} \leq \bar{q}_{l_{\text{crit}}} \end{cases} \end{align*} \end{minipage}\\ \par\medskip } \item<3->{precipitation leaves grid box immediately if the threshold $\bar{q}_{l_{\text{crit}}} = \SI{0,5}{g/kg}$ is exceeded.}\\ \par\medskip \item<4->{Timescale $\tau = \SI{1000}{s}$.} \item<5->{ \begin{minipage}[c][1.5cm][c]{0.35\textwidth} \begin{equation*} \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}} \end{equation*} \end{minipage} } \end{itemize} } \end{itemize} \end{frame} % Folie 13 \begin{frame} \frametitle{Control parameters} \begin{itemize} \item<1->{The following settings in the parameter file enable the use of the bulk cloud model:}\\ \par\medskip \scriptsize \begin{itemize}\scriptsize \item<2->{ $\left. \begin{array}{ll} % für mehrzeiligen Text nötig \text{humidity = .TRUE.}\qquad\qquad\qquad \end{array} \right\}: \begin{array}{ll} % für mehrzeiligen Text nötig \text{prognostic equations for specific} \\ \text{specific humidity } \bar{q} \text{ is solved} \end{array} $ }\\ \par\medskip \item<3->{ $\left. \begin{array}{ll} % für mehrzeiligen Text nötig \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} \end{array} \right\}: \begin{array}{ll} % für mehrzeiligen Text nötig \text{prognostic equations for liquid water} \\ \text{potential temperature } \bar{\theta}_{l} \text{ and total water} \\ \text{specific humidity } \bar{q} \text{ are solved} \end{array} $ }\\ \par\medskip \item<4->{ $\left. \begin{array}{ll} % für mehrzeiligen Text nötig \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} \\ \text{precipitation = .TRUE.} \\ \text{radiation = .TRUE.} \end{array} \right\}: \begin{array}{ll} % für mehrzeiligen Text nötig \text{Kessler precipitation scheme and} \\ \text{radiation model are solved} \end{array} $ } \end{itemize} \end{itemize} \end{frame} % Folie 12 \begin{frame} \frametitle{Example - Setup for a cloudy boundary layer} \begin{figure}[H] \begin{minipage}[c][6.5cm][c]{.50\linewidth} \centering CBL with shallow cumulus clouds:\\ \par\bigskip \includegraphics[width=0.95\linewidth]{cloud_physics_figures/cbl5_preview.png} \end{minipage} \begin{minipage}[c][6.5cm][t]{.40\linewidth} \centering \includegraphics[width=0.9\linewidth]{cloud_physics_figures/param_clouds.png} \end{minipage} \end{figure} \end{frame} % Folie 13 \begin{frame} \frametitle{Example - Model output} \begin{figure}[H] \begin{minipage}[c][6cm][c]{.45\linewidth} \centering \includegraphics[width=0.95\linewidth]{cloud_physics_figures/profiles_cbl_cloud.png} \end{minipage} \begin{minipage}[c][6cm][c]{.45\linewidth} \centering \includegraphics[width=0.95\linewidth]{cloud_physics_figures/ql_xy_cbl_cloud.png} \end{minipage} \end{figure} \end{frame} % Folie 1$ \begin{frame} \frametitle{Bibliography} \tiny \begin{thebibliography}{} \bibitem[1]{betts1973} \textsc{Betts, A.K., 1973:} \emph{Non-precipitating cumulus convection and its parameterization.} \newblock Quart. J. Roy. Meteor. Soc., \textbf{99}, 178-196. \bibitem[2]{cox1976} \textsc{Cox, S. K., 1976:} \emph{Observations of cloud infrared effective emissivity.} \newblock J. Atmos. Sci., \textbf{33}, 287-289. \bibitem[3]{cuijpers1993} \textsc{Cuijpers, J.W.M., P.G. Duynkerke, 1993:} \emph{Large eddy simulation of trade wind cumulus clouds.} \newblock J. Atmos. Sci., \textbf{50}, 3894-3908. \bibitem[4]{deardorff1976} \textsc{Deardorff, J. W., 1976:} \emph{Usefullness of liquid-water potential temperature in shallow-cloud model.} \newblock J. Appl. Meteor., \textbf{15}, 98-102. \bibitem[5]{deardorff1980} \textsc{Deardorff, J. W., 1980:} \emph{Stratocumulus-capped mixed layers derived from a three-dimensional model}. \newblock Bondary-Layer Meteor., \textbf{18}, 495-527. \bibitem[6]{kessler1969} \textsc{Kessler, E., 1969:} \emph{On the distribution and continuity of water substance in atmospheric circulations.} \newblock Meteor. Monogr., \textbf{32}, 84 pp. \bibitem[7]{riechelmann2012} \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.} \newblock New J. Phys., \textbf{14}, 27. \bibitem[8]{sommeria1977} \textsc{Sommeria, G., J. W. Deardorff, 1977:} \emph{Subgrid-scale condensation in models of nonprecipitating clouds.} \newblock J. Atmos. Sci., \textbf{34}, 344-355. \bibitem[9]{cloudphys} \textsc{cloud\_physics.pdf:} \emph{Introduction to the cloud physics model of PALM.} \newblock {\tt trunk/DOC/tec/methods/cloud\_physics/cloud\_physics.pdf}. \end{thebibliography} \end{frame} \end{document}