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\institute{Institut für Meteorologie und Klimatologie, Leibniz Universität Hannover}
\date{last update: \today}
\event{PALM Seminar}
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\title[PALM - Cloud Physics]{PALM - Cloud Physics}
\author{Siegfried Raasch}

% Notes:
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\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}