source: palm/trunk/TUTORIAL/SOURCE/canopy_model.tex @ 1379

%$Id: canopy_model.tex 1205 2013-07-15 10:55:47Z raasch $
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\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
\date{last update: \today}
\event{PALM Seminar}
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\title[PALM's Canopy Model]{PALM's Canopy Model}
\author{Siegfried Raasch}


% Notes:
% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
\begin{document}

\begin{frame}
\titlepage
\end{frame}

\section{PALM's Canopy Model}
\subsection{The embedded Canopy Model}

%Folie 01
\begin{frame}
   \frametitle{Overview}
   \begin{itemize}
      \item<1->{The canopy model embedded in PALM can be used to study the effect of a plant canopy on e.g.:}
      \begin{itemize}
         \item<2->{mean flow field,}
         \item<3->{development of coherent turbulence structures,}
         \item<4->{scalar exchange processes between canopy and atmosphere.}
      \end{itemize}
      \item<5->{Within the canopy model, the plant canopy acts as a sink for momentum and as a source/sink for active (e.g. temperature) and passive (e.g. tracer) scalars.}
      \item<6->{The canopy model does not account for each plant element, but rather accounts for a volume averaged effect on the flow and scalar concentration, depending on:}
      \begin{itemize}
         \item<7->{leaf area density,}
         \item<8->{drag coefficient.}
      \end{itemize}
   \end{itemize}
\end{frame}



%Folie 02
\begin{frame}
   \frametitle{Theory (I)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{A plant canopy affects the flow by acting as a momentum sink due to form and viscous drag forces.}
      \item<2->{The effectiveness of momentum absorption depends on the amount of leaf area per unit volume and the aerodynamic drag.}
      \item<3->{Due to the aerodynamic drag, the flow is decelerated within the canopy, leading to an inflection point in the vertical profile of the horizontal velocity at the canopy top.
         \begin{center}
            \includegraphics[width=0.5\textwidth]{canopy_model_figures/abb1.png}
         \end{center}
      }
   \end{itemize}
   \end{footnotesize}
\end{frame}



%Folie 03
\begin{frame}
   \frametitle{Theory (II)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{The inflection point in the velocity profile introduces instabilities to the flow, leading to the formation of Kelvin-Helmholtz waves near the canopy top (\textcircled{{\tiny 1}}).}
      \item<2->{Wave breaking induces further instabilities, whereby a longitudinal component is added to the developing turbulence structures (\textcircled{{\tiny 2}} \& \textcircled{{\tiny 3}}).}
      \item<3->{Due to the persistent instabilities the turbulence structures develop a distinct three-dimensionality (\textcircled{{\tiny 4}}).}
      \item<4->{The large turbulence structures developing due to the inflection point instability significantly contribute to the vertical mixing of in-canopy and above-canopy air.
         \begin{center}
            \includegraphics[width=0.5\textwidth]{canopy_model_figures/abb2.png}
         \end{center}
      }
   \end{itemize}
   \end{footnotesize}
\end{frame}



%Folie 04
\begin{frame}
   \frametitle{Methods (I)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{The canopy model in PALM is based on the models used by Shaw and Schumann (1992) and Watanabe (2004).}
      \item<2->{The aerodynamic effect of the canopy on the turbulent flow is accounted for by an additional term in the momentum equations:\\
         \begin{align*}
            \frac{\partial \bar{u}_{i}}{\partial t} = \text{...} - c_{d} a U \bar{u}_{i}
         \end{align*}
      }
      \begin{itemize}
         \item<3->{ $c_{d}$ : drag coefficient}
         \item<4->{ $a $ : leaf area density $[m^{2}m^{-3}]$}
         \item<5->{ $U$ : $(u^{2} + v^{2} + w^{2})^{1/2}$ $[m s^{-1}]$}
         \item<6->{ $u_{i}$ : velocity component ($u_{1}=u$, $u_{2}=v$, $u_{3}=w$)}
      \end{itemize}
   \end{itemize}
   \uncover<7->{Note: The canopy model does not resolve the effect of single plant elements.}
   \end{footnotesize}
\end{frame}



%Folie 05
\begin{frame}
   \frametitle{Methods (II)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{The effect of the canopy on the subgrid scale turbulence is accounted for by adding a sink term to the prognostic equation for the subgrid scale turbulent kinetic energy:\\
         \begin{align*}
            \frac{\partial e}{\partial t} = \text{...} - 2 c_{d} a U e
         \end{align*}
      }
      \item<2->{It is assumed that the subgrid scale turbulent kinetic energy is dissipated by the canopy due to the rapid dissipation of wake turbulence in the lee of canopy elements (e.g. Watanabe, 2004).}
   \end{itemize}
   \end{footnotesize}
\end{frame}



%Folie 06
\begin{frame}
   \frametitle{Methods (III)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{If desired, the effect of the canopy on the sensible heat transport can be considered. A source term is added to the prognostic equation for potential temperature:\\
         \begin{align*}
            \frac{\partial \bar{\theta}}{\partial t} = \text{...} +  S_{\theta}
         \end{align*}
      }
      \item<2->{It is assumed that the foliage is warmed by the penetrating solar radiation and, in turn, warms the surrounding air.}
      \item<3->{The source strength $S_{\theta}$ is defined as the vertical derivative of the upward kinematic vertical heat flux $Q_{\theta}$, given by (Shaw and Schumann, 1992):\\
         \begin{align*}
            Q_{\theta}(z) = Q_{\theta}(h) exp(-\alpha F) \text{ , } Q_{\theta}(h) \text{ : Heat flux at canopy top}
         \end{align*}
      }
      \begin{itemize}
         \item<4->{ $\alpha = 0.6$ (extinction coefficient)}
         \item<5->{ $F = \int\limits_{z}^{h} a \: dz$ (downward cumulative leaf area index)}
      \end{itemize}
   \end{itemize}
   \end{footnotesize}
\end{frame}



%Folie 07
\begin{frame}
   \frametitle{Methods (IV)}
   \begin{footnotesize}
   \begin{itemize}
      \item<1->{The canopy might act as a sink or source for other scalars $q$ (e.g. humidity, passive tracer). Therefore, an additional term is added to the scalar transport equation:\\
         \begin{align*}
            \frac{\partial \bar{q}}{\partial t} = \text{...} - c_{q} a U (\bar{q} - q_{c})
         \end{align*}
      }
      \begin{itemize}
         \item<2->{ $c_{q}$ : scalar exchange coefficient}
         \item<3->{ $q_{c}$ : scalar concentration at leaf surface}
      \end{itemize}
   \end{itemize}
   \end{footnotesize}
\end{frame}



%Folie 08
\begin{frame}
   \frametitle{Basics (I)}
   \begin{itemize}
      \item<1->{The canopy model is switched on by setting the parameter {\small \texttt{plant\_canopy = .TRUE.}} within the \&inipar {\small \texttt{NAMELIST}} in the parameter file ({\small \texttt{PARIN}}).}
      \item<2->{All parameters for steering the canopy model are described in:\\
      {\scriptsize Documentation $\rightarrow$ Model steering $\rightarrow$ Parameters $\rightarrow$ Initialization $\rightarrow$ Canopy}\\
      (http://palm.muk.uni-hannover.de)
      }
      \item<3->{The following slides will describe how to set up a simulation with a simple horizontally homogeneous canopy block covering the entire model domain surface. In this case, {\small \texttt{canopy\_mode = 'block'}} must be set in \&inipar {\small \texttt{NAMELIST}}.}
   \end{itemize}
\end{frame}



%Folie 09
\begin{frame}
   \frametitle{Basic canopy parameter (I)}
   The parameters for steering the canopy model have to be added to the \&inipar {\small \texttt{NAMELIST}} in the parameter file ({\small \texttt{PARIN}}).\\
    \begin{itemize}
       \item<1->{Step I: Define the upper boundary of the plant canopy layer using the parameter {\small \texttt{pch\_index (grid point index, default 0)}}. {\small \texttt{pch\_index}} specifies the number of grid points resolving the canopy layer in the vertical direction.}
    \end{itemize}
   \vspace{10pt}

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   \tikzstyle{Key3} = [rectangle, text width=3.0cm, minimum size=16pt, font=\tiny]

\begin{tikzpicture}[>=latex']
        %%% Edit the following coordinate to change the shape of your
        %%% cuboid

        %% Vanishing points for perspective handling
        \coordinate (P1) at (-4cm,1.5cm); % left vanishing point (To pick)
        \coordinate (P2) at (10cm,1.5cm); % right vanishing point (To pick)

        %% (A1) and (A2) defines the 2 central points of the cuboid
        \coordinate (A1) at (0cm,0cm); % central top point (To pick)
        \coordinate (A2) at (0cm,-2cm); % central bottom point (To pick)
        
        %% (A3) to (A8) are computed given a unique parameter (or 2) .8
        % You can vary .8 from 0 to 1 to change perspective on left side
        \coordinate (A3) at ($(P1)!.8!(A2)$); % To pick for perspective 
        \coordinate (A4) at ($(P1)!.8!(A1)$);

        % You can vary .8 from 0 to 1 to change perspective on right side
        \coordinate (A7) at ($(P2)!.7!(A2)$);
        \coordinate (A8) at ($(P2)!.7!(A1)$);

        %% Automatically compute the last 2 points with intersections
        \coordinate (A5) at
          (intersection cs: first line={(A8) -- (P1)},
                            second line={(A4) -- (P2)});
        \coordinate (A6) at
          (intersection cs: first line={(A7) -- (P1)}, 
                            second line={(A3) -- (P2)});

        %% Drawing the canopy layer
        \coordinate (A9) at (0em,-1.7cm); % central bottom point (To pick)
        \coordinate (A10) at ($(P2)!.7!(A9)$);
        \coordinate (A12) at ($(P1)!.8!(A9)$); % To pick for perspective 
        \coordinate (A11) at
          (intersection cs: first line={(A10) -- (P1)}, 
                            second line={(A12) -- (P2)});

        %%% Depending of what you want to display, you can comment/edit
        %%% the following lines

        {\node [background, right=-0.8cm of A12] (background) {};}

        %% Possibly draw back faces

        \fill[gray!70] (A2) -- (A3) -- (A6) -- (A7) -- cycle; % face 6
        \node at (barycentric cs:A2=1,A3=1,A6=1,A7=1) {\tiny };
        
        \fill[gray!30] (A3) -- (A4) -- (A5) -- (A6) -- cycle; % face 3
        \node at (barycentric cs:A3=1,A4=1,A5=1,A6=1) {\tiny };
        
        \fill[gray!10] (A5) -- (A6) -- (A7) -- (A8) -- cycle; % face 4
        \node at (barycentric cs:A5=1,A6=1,A7=1,A8=1) {\tiny };
        
        
        \fill[green!90] (A9) -- (A10) -- (A11) -- (A12) -- cycle; % face 7
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        \fill[green!100] (A9) -- (A12) -- (A3) -- (A2) -- cycle; % face 8
        \node at (barycentric cs:A9=1,A12=1,A3=1,A2=1) {\tiny };
        
        \fill[green!100] (A9) -- (A10) -- (A7) -- (A2) -- cycle; % face 9
        \node at (barycentric cs:A9=1,A10=1,A7=1,A2=1) {\tiny };
        
        \draw[thin,dashed] (A5) -- (A6);
        \draw[thin,dashed] (A3) -- (A6);
        \draw[thin,dashed] (A7) -- (A6);

        \draw[thin,dashed] (A11) -- (A12);
        \draw[thin,dashed] (A10) -- (A11);
        
        %% Possibly draw front faces

        % \fill[orange] (A1) -- (A8) -- (A7) -- (A2) -- cycle; % face 1
        % \node at (barycentric cs:A1=1,A8=1,A7=1,A2=1) {\tiny f1};
        \fill[gray!50,opacity=0.2] (A1) -- (A2) -- (A3) -- (A4) -- cycle; % f2
        \node at (barycentric cs:A1=1,A2=1,A3=1,A4=1) {\tiny };
        \fill[gray!90,opacity=0.2] (A1) -- (A4) -- (A5) -- (A8) -- cycle; % f5
        \node at (barycentric cs:A1=1,A4=1,A5=1,A8=1) {\tiny };

        %% Possibly draw front lines
        \draw[thin] (A1) -- (A2);
        \draw[thin] (A3) -- (A4);
        \draw[thin] (A7) -- (A8);
        \draw[thin] (A1) -- (A4);
        \draw[thin] (A1) -- (A8);
        \draw[thin] (A2) -- (A3);
        \draw[thin] (A2) -- (A7);
        \draw[thin] (A4) -- (A5);
        \draw[thin] (A8) -- (A5);
        
        \draw[thin] (A9) -- (A10);
        \draw[thin] (A9) -- (A12);
        
        % Possibly draw points
        % (it can help you understand the cuboid structure)
        \foreach \i in {1,2,...,12}
        {
        %  \draw[fill=black] (A\i) circle (0.05em)
        %    node[above right] {\tiny \i};
        }
        % \draw[fill=black] (P1) circle (0.1em) node[below] {\tiny p1};
        % \draw[fill=black] (P2) circle (0.1em) node[below] {\tiny p2};
        
        %Key    
        \coordinate (K1) at (0.5cm,-2.3cm);
        \coordinate (K2) at (0.5cm,-2.7cm);
        
        
        \node [Key1, right=0.0cm of K1] (key1) {};
        \node [Key2, right=0.0cm of K2] (key2) {};
        \node [Key3, right=0.4cm of K1] (key11) {Total model domain};
        \node [Key3, right=0.4cm of K2] (key21) {Plant canopy volume};
        
        %% frame specific elements
        \node [Key3, right=0.45cm of A10] (label1) {\texttt{zw(pch\_index)} = canopy top};
        \node [Key3, right=0.05cm of A10] (label0) {};
        
        \draw[<-] (label0) -- (label1);
\end{tikzpicture}


\end{frame}



%Folie 10
\begin{frame}
   \frametitle{Basic canopy parameter (II)}
    \begin{itemize}
       \item<1->{ Step II: Construct the vertical profile of the leaf area density (lad) to prescribe the distribution of leaf area within the plant canopy volume.\\
       The canopy top is located between {\small \texttt{zu(pch\_index)}} and {\small \texttt{zu(pch\_index + 1)}} because this is the transition between the in-canopy grid point and the above-canopy grid point.
       }
    \end{itemize}
   \vspace{10pt}

   \tikzstyle{background} = [rectangle, fill=gray!10, text width=1\textwidth, text centered, rounded corners, minimum height=10em]
   \tikzstyle{Key1} = [rectangle, draw, fill=gray!70, text width=0.05, minimum size=0.05, font=\tiny]
   \tikzstyle{Key2} = [rectangle, draw, fill=green!90, text width=0.05, minimum size=0.05, font=\tiny]
   \tikzstyle{Key3} = [rectangle, text width=4.5cm, minimum size=16pt, font=\tiny]
   \tikzstyle{label} = [rectangle, text width=2.0cm, align=center, minimum size=16pt, font=\tiny]

\begin{tikzpicture}[>=latex']
        %%% Edit the following coordinate to change the shape of your
        %%% cuboid

        %% Vanishing points for perspective handling
        \coordinate (P1) at (-4cm,1.5cm); % left vanishing point (To pick)
        \coordinate (P2) at (10cm,1.5cm); % right vanishing point (To pick)

        %% (A1) and (A2) defines the 2 central points of the cuboid
        \coordinate (A1) at (0cm,0cm); % central top point (To pick)
        \coordinate (A2) at (0cm,-2cm); % central bottom point (To pick)
        
        %% (A3) to (A8) are computed given a unique parameter (or 2) .8
        % You can vary .8 from 0 to 1 to change perspective on left side
        \coordinate (A3) at ($(P1)!.8!(A2)$); % To pick for perspective 
        \coordinate (A4) at ($(P1)!.8!(A1)$);

        % You can vary .8 from 0 to 1 to change perspective on right side
        \coordinate (A7) at ($(P2)!.7!(A2)$);
        \coordinate (A8) at ($(P2)!.7!(A1)$);

        %% Automatically compute the last 2 points with intersections
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                            second line={(A4) -- (P2)});
        \coordinate (A6) at
          (intersection cs: first line={(A7) -- (P1)}, 
                            second line={(A3) -- (P2)});

        %% Drawing the canopy layer
        \coordinate (A9) at (0em,-1.7cm); % central bottom point (To pick)
        \coordinate (A10) at ($(P2)!.7!(A9)$);
        \coordinate (A12) at ($(P1)!.8!(A9)$); % To pick for perspective 
        \coordinate (A11) at
          (intersection cs: first line={(A10) -- (P1)}, 
                            second line={(A12) -- (P2)});

        %%% Depending of what you want to display, you can comment/edit
        %%% the following lines

        {\node [background, right=-0.8cm of A12] (background) {};}

        %% Possibly draw back faces

        \fill[gray!70] (A2) -- (A3) -- (A6) -- (A7) -- cycle; % face 6
        \node at (barycentric cs:A2=1,A3=1,A6=1,A7=1) {\tiny };
        
        \fill[gray!30] (A3) -- (A4) -- (A5) -- (A6) -- cycle; % face 3
        \node at (barycentric cs:A3=1,A4=1,A5=1,A6=1) {\tiny };
        
        \fill[gray!10] (A5) -- (A6) -- (A7) -- (A8) -- cycle; % face 4
        \node at (barycentric cs:A5=1,A6=1,A7=1,A8=1) {\tiny };
        
        
        \fill[green!90] (A9) -- (A10) -- (A11) -- (A12) -- cycle; % face 7
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        \fill[green!100] (A9) -- (A12) -- (A3) -- (A2) -- cycle; % face 8
        \node at (barycentric cs:A9=1,A12=1,A3=1,A2=1) {\tiny };
        
        \fill[green!100] (A9) -- (A10) -- (A7) -- (A2) -- cycle; % face 9
        \node at (barycentric cs:A9=1,A10=1,A7=1,A2=1) {\tiny };
        
        \draw[thin,dashed] (A5) -- (A6);
        \draw[thin,dashed] (A3) -- (A6);
        \draw[thin,dashed] (A7) -- (A6);

        \draw[thin,dashed] (A11) -- (A12);
        \draw[thin,dashed] (A10) -- (A11);
        
        %% Possibly draw front faces

        % \fill[orange] (A1) -- (A8) -- (A7) -- (A2) -- cycle; % face 1
        % \node at (barycentric cs:A1=1,A8=1,A7=1,A2=1) {\tiny f1};
        \fill[gray!50,opacity=0.2] (A1) -- (A2) -- (A3) -- (A4) -- cycle; % f2
        \node at (barycentric cs:A1=1,A2=1,A3=1,A4=1) {\tiny };
        \fill[gray!90,opacity=0.2] (A1) -- (A4) -- (A5) -- (A8) -- cycle; % f5
        \node at (barycentric cs:A1=1,A4=1,A5=1,A8=1) {\tiny };

        %% Possibly draw front lines
        \draw[thin] (A1) -- (A2);
        \draw[thin] (A3) -- (A4);
        \draw[thin] (A7) -- (A8);
        \draw[thin] (A1) -- (A4);
        \draw[thin] (A1) -- (A8);
        \draw[thin] (A2) -- (A3);
        \draw[thin] (A2) -- (A7);
        \draw[thin] (A4) -- (A5);
        \draw[thin] (A8) -- (A5);
        
        \draw[thin] (A9) -- (A10);
        \draw[thin] (A9) -- (A12);
        
        % Possibly draw points
        % (it can help you understand the cuboid structure)
        \foreach \i in {1,2,...,12}
        {
        %  \draw[fill=black] (A\i) circle (0.05em)
        %    node[above right] {\tiny \i};
        }
        % \draw[fill=black] (P1) circle (0.1em) node[below] {\tiny p1};
        % \draw[fill=black] (P2) circle (0.1em) node[below] {\tiny p2};
        
        %Key    
        \coordinate (K1) at (0.5cm,-2.3cm);
        \coordinate (K2) at (0.5cm,-2.7cm);
        
        
        \node [Key1, right=0.0cm of K1] (key1) {};
        \node [Key2, right=0.0cm of K2] (key2) {};
        \node [Key3, right=0.4cm of K1] (key11) {Total model domain};
        \node [Key3, right=0.4cm of K2] (key21) {Plant canopy volume};
        
        %% frame specific elements
        
        %% boundary layer profile 2D
        \coordinate (B1) at (4.0cm,0.3cm);
        \coordinate (B2) at (4.0cm,-2.5cm);
        
        \coordinate (B3) at (4.0cm,0.6cm);
        \coordinate (B4) at (6.6cm,-2.5cm);
        
        \draw[thin, dotted] (A10) -- (B1);
        \draw[thin, dotted] (A7) -- (B2);
        \draw[<-] (B3) -- (B2);
        \draw[<-] (B4) -- (B2);
        
        \coordinate (xlab) at (5.3cm,-2.5cm);
        \coordinate (ylab) at (4.0cm,-1.1cm);
        \node [label, below=0.0cm of xlab] (xlabel) {lad $[m^{2}m^{-3}]$};
        \node [label, rotate=90, above=0.0cm of ylab] (ylabel) {z$[m]$};
        
        
        \coordinate (B5) at (4.5cm,-2.5cm);
%       \draw [-,color=red] (B1) to [out=-10,in=90,looseness=2.0, relative=false] .. controls (4,0) and (5,0) .. (B5);
        \draw {(B1) .. controls (8.0,-0.2) and (4.5,-1.0) .. (B5)};
        
        \draw [decorate, decoration={markings, 
                mark=at position 00mm with{\draw[fill=black] (0,0) circle (0.1em);},
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                mark=at position 08mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 12mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 16mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 20mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 24mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 28mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 32mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 36mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 40mm with{\draw[fill=black] (0,0) circle (0.1em);},
                mark=at position 44mm with{\draw[fill=black] (0,0) circle (0.1em);},
                }] 
                {(B5) .. controls (4.5,-1.0) and (8.0,-0.2) .. (B1)};

        \node [Key3, right=1.1cm of B1] (label1) {\texttt{zu(pch\_index + 1)}: lad = 0.0 (default)};
        \node [Key3, left=0.1cm of B1] (label0) {};
        \draw[-, dashed] (label0) -- (label1);
        
        \coordinate (curvelabel00) at (4.65cm,-2.45cm);
        \coordinate (curvelabel01) at (5.0cm,-2.25cm);
        \draw[<-] (curvelabel00) -- (curvelabel01);
        \node [Key3, right=0.0cm of curvelabel01] (curvelabel02) {\texttt{lad\_surface} (default 0.0)};
        
        \coordinate (curvelabel10) at (5.4cm,-1.3cm);
        \coordinate (curvelabel11) at (5.6cm,-1.6cm);
        \draw[<-] (curvelabel10) -- (curvelabel11);
        \node [Key3, right=0.0cm of curvelabel11] (curvelabel12) {\texttt{lad\_vertical\_gradient\_level} (5)};
        
        \coordinate (curvelabel10) at (5.47cm,-1.05cm);
        \coordinate (curvelabel11) at (5.8cm,-1.3cm);
        \draw[<-] (curvelabel10) -- (curvelabel11);
        \node [Key3, right=0.0cm of curvelabel11] (curvelabel12) {\texttt{lad\_vertical\_gradient} (5)};
        
        \coordinate (curvelabel10) at (5.7cm,-0.93cm);
        \coordinate (curvelabel11) at (5.9cm,-1.0cm);
        \draw[<-] (curvelabel10) -- (curvelabel11);
        \node [Key3, right=0.0cm of curvelabel11] (curvelabel12) {\texttt{lad\_vertical\_gradient\_level} (6)};
\end{tikzpicture}


\end{frame}



%Folie 11
\begin{frame}
   \frametitle{Basic canopy parameter (III)}
   \begin{footnotesize}
    \begin{itemize}
       \item<1->{Step III: Prescribe a value for the parameter {\small \texttt{drag\_coefficient (default 0.0)}}. The drag coefficient is a dimensionless factor describing the magnitude of the form drag by the canopy working against the flow. A larger form drag results in a greater momentum reduction.}
    \end{itemize}
   

   \begin{minipage}{0.47\textwidth}
      \begin{center}
         \includegraphics[width=1\textwidth]{canopy_model_figures/large.png}\\
      \end{center}
   \end{minipage}
   \hfill
   \begin{minipage}{0.47\textwidth}
      \begin{center}
         \includegraphics[width=1\textwidth]{canopy_model_figures/small.png}\\
      \end{center}
   \end{minipage}
   
   \begin{minipage}{0.47\textwidth}
      \begin{center}
         Strong trees offer a larger form drag to the flow.
      \end{center}
   \end{minipage}
   \hfill
   \begin{minipage}{0.47\textwidth}
      \begin{center}
         Young / small trees offer a smaller form drag to the flow because they are more flexible.
      \end{center}
   \end{minipage}
   
   \end{footnotesize}

\end{frame}



%Folie 12
\begin{frame}
   \frametitle{Basic canopy parameter (IV)}
   \begin{itemize}
      \item<1->{For steering the effect of the canopy sensible heat transfer, prescribe a value for the sensible heat flux at the canopy top, using the parameter \texttt{cthf} (see Methods (III)).}
      \item<2->{The sink/source effect of the canopy on other scalar quantities, such as humidity or a passive tracer can be steered by the parameters \texttt{leaf\_surface\_concentration} and \texttt{scalar\_exchange\_coefficient} (see Methods (IV)).}
   \end{itemize}
\end{frame}



%Folie 13
\begin{frame}
   \frametitle{User-defined canopy}
   \begin{footnotesize}
   Do you want to simulate a more customized canopy, which e.g. covers only half the model surface?\\
    \begin{itemize}
       \item<2->{Step I: Copy the file \texttt{user\_init\_plant\_canopy.f90} from {\small \texttt{trunk/SOURCE}} to the directory {\small \texttt{\$Home/palm/current\_version/USER\_CODE/<enter job name>}} and make the desired changes for {\small \texttt{CASE ('user\_defined\_canopy\_1')}}.}
       \item<3->{Step II: In your parameter file set: {\scriptsize \texttt{canopy\_mode = 'user\_defined\_canopy\_1'}}}
    \end{itemize}
   \end{footnotesize}
   \vspace{7pt}

   \tikzstyle{background} = [rectangle, fill=gray!10, text width=1\textwidth, text centered, rounded corners, minimum height=10em]
   \tikzstyle{Key1} = [rectangle, draw, fill=gray!70, text width=0.05, minimum size=0.05, font=\tiny]
   \tikzstyle{Key2} = [rectangle, draw, fill=green!90, text width=0.05, minimum size=0.05, font=\tiny]
   \tikzstyle{Key3} = [rectangle, text width=2.0cm, minimum size=4pt, font=\tiny]
   \tikzstyle{text1} = [rectangle, text width=0.4\textwidth, minimum height=10em]
\begin{tikzpicture}[>=latex']
        %%% Edit the following coordinate to change the shape of your
        %%% cuboid
      
        %% Vanishing points for perspective handling
        \coordinate (P1) at (-4cm,1.5cm); % left vanishing point (To pick)
        \coordinate (P2) at (10cm,1.5cm); % right vanishing point (To pick)

        %% (A1) and (A2) defines the 2 central points of the cuboid
        \coordinate (A1) at (0cm,0cm); % central top point (To pick)
        \coordinate (A2) at (0cm,-2cm); % central bottom point (To pick)

        %% (A3) to (A8) are computed given a unique parameter (or 2) .8
        % You can vary .8 from 0 to 1 to change perspective on left side
        \coordinate (A3) at ($(P1)!.8!(A2)$); % To pick for perspective 
        \coordinate (A4) at ($(P1)!.8!(A1)$);

        % You can vary .8 from 0 to 1 to change perspective on right side
        \coordinate (A7) at ($(P2)!.7!(A2)$);
        \coordinate (A8) at ($(P2)!.7!(A1)$);

        %% Automatically compute the last 2 points with intersections
        \coordinate (A5) at
          (intersection cs: first line={(A8) -- (P1)},
                            second line={(A4) -- (P2)});
        \coordinate (A6) at
          (intersection cs: first line={(A7) -- (P1)}, 
                            second line={(A3) -- (P2)});

        %% Drawing the canopy layer
        \coordinate (A9) at (0em,-1.7cm); % central bottom point (To pick)
        \coordinate (A10) at ($(P2)!.7!(A9)$);
        \coordinate (A12) at ($(P1)!.8!(A9)$); % To pick for perspective 
        \coordinate (A11) at
          (intersection cs: first line={(A10) -- (P1)}, 
                            second line={(A12) -- (P2)});
        
        \coordinate (A13) at ($(A9)!.54!(A10)$);
        \coordinate (A14) at ($(A2)!.54!(A7)$);
        
        \coordinate (A15) at 
                  (intersection cs: first line={(A13) -- (P1)}, 
                                    second line={(A12) -- (P2)});
        \coordinate (A16) at 
                  (intersection cs: first line={(A14) -- (P1)}, 
                                    second line={(A3) -- (P2)});
        %%% Depending of what you want to display, you can comment/edit
        %%% the following lines
        
        {\node [background, right=-0.8cm of A12] (background) {};}

        %% Possibly draw back faces

        \fill[gray!70] (A2) -- (A3) -- (A6) -- (A7) -- cycle; % face 6
        \node at (barycentric cs:A2=1,A3=1,A6=1,A7=1) {\tiny };
        
        \fill[gray!30] (A3) -- (A4) -- (A5) -- (A6) -- cycle; % face 3
        \node at (barycentric cs:A3=1,A4=1,A5=1,A6=1) {\tiny };
        
        \fill[gray!10] (A5) -- (A6) -- (A7) -- (A8) -- cycle; % face 4
        \node at (barycentric cs:A5=1,A6=1,A7=1,A8=1) {\tiny };
        
        
        \fill[green!90] (A13) -- (A10) -- (A11) -- (A15) -- cycle; % face 4
        \node at (barycentric cs:A9=1,A10=1,A11=1,A12=1) {\tiny };
        
        \fill[green!100] (A13) -- (A15) -- (A16) -- (A14) -- cycle; % face 4
        \node at (barycentric cs:A9=1,A12=1,A3=1,A2=1) {\tiny };
        
        \fill[green!100] (A13) -- (A10) -- (A7) -- (A14) -- cycle; % face 4
        \node at (barycentric cs:A9=1,A10=1,A7=1,A2=1) {\tiny };
        
        \draw[thin,dashed] (A5) -- (A6);
        \draw[thin,dashed] (A3) -- (A6);
        \draw[thin,dashed] (A7) -- (A6);

        \draw[thin,dashed] (A11) -- (A15);
        \draw[thin,dashed] (A10) -- (A11);
        
        %% Possibly draw front faces

        % \fill[orange] (A1) -- (A8) -- (A7) -- (A2) -- cycle; % face 1
        % \node at (barycentric cs:A1=1,A8=1,A7=1,A2=1) {\tiny f1};
        \fill[gray!50,opacity=0.2] (A1) -- (A2) -- (A3) -- (A4) -- cycle; % f2
        \node at (barycentric cs:A1=1,A2=1,A3=1,A4=1) {\tiny };
        \fill[gray!90,opacity=0.2] (A1) -- (A4) -- (A5) -- (A8) -- cycle; % f5
        \node at (barycentric cs:A1=1,A4=1,A5=1,A8=1) {\tiny };

        %% Possibly draw front lines
        \draw[thin] (A1) -- (A2);
        \draw[thin] (A3) -- (A4);
        \draw[thin] (A7) -- (A8);
        \draw[thin] (A1) -- (A4);
        \draw[thin] (A1) -- (A8);
        \draw[thin] (A2) -- (A3);
        \draw[thin] (A2) -- (A7);
        \draw[thin] (A4) -- (A5);
        \draw[thin] (A8) -- (A5);
        
        \draw[thin] (A13) -- (A10);
        \draw[thin] (A13) -- (A15);
        \draw[thin] (A14) -- (A16);
        \draw[thin] (A13) -- (A14);
        \draw[thin] (A15) -- (A16);
        
        % Possibly draw points
        % (it can help you understand the cuboid structure)
        \foreach \i in {1,2,...,16}
        {
        %  \draw[fill=black] (A\i) circle (0.05em)
        %    node[above right] {\tiny \i};
        }
        % \draw[fill=black] (P1) circle (0.1em) node[below] {\tiny p1};
        % \draw[fill=black] (P2) circle (0.1em) node[below] {\tiny p2};
        
        %Key    
        \coordinate (K1) at (0.5cm,-2.3cm);
        \coordinate (K2) at (0.5cm,-2.7cm);
        
        
        \node [Key1, right=0.0cm of K1] (key1) {};
        \node [Key2, right=0.0cm of K2] (key2) {};
        \node [Key3, right=0.4cm of K1] (key11) {Total model domain};
        \node [Key3, right=0.4cm of K2] (key21) {Plant canopy volume};
        
        % % Text
        \uncover<4->{\node [text1, right=0.7cm of A7] (text1) {{\footnotesize Note: You might have to make changes in other parts of your {\small \texttt{USER\_CODE}} according to the changes / used parameters in: \texttt{user\_init\_plant\_canopy.f90}}};}
\end{tikzpicture}


\end{frame}

\end{document}