source: palm/trunk/TUTORIAL/SOURCE/canopy_model.tex @ 1494

%$Id: canopy_model.tex 1493 2014-11-12 15:39:53Z maronga $
\input{header_tmp.tex}
%\input{../header_lectures.tex}

%\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage{ngerman}
\usepackage{pgf}
\usetheme{Dresden}
\usepackage{subfigure}
\usepackage{units}
\usepackage{amsmath}
\usepackage{multimedia}
\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
\usepackage{xmpmulti}
\usepackage{tikz}
\usepackage{pdfcomment}
\usetikzlibrary{shapes,arrows,positioning,calc,decorations.pathmorphing,decorations.pathreplacing,decorations.markings}
\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
\usepackage{listings}
\lstset{language=[90]Fortran,
  basicstyle=\ttfamily \tiny,
  keywordstyle=\color{black},
  commentstyle=\color{black},
  morecomment=[l]{!\ }% Comment only with space after !
}
\usepackage{tabto}

\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
\date{last update: \today}
\event{PALM Seminar}
\setbeamertemplate{navigation symbols}{}

\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.png}}

\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->{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 source/sink strength for momentum and scalars depends on e.g.:}
      \begin{itemize}
         \item<7->{leaf area density,}
         \item<8->{drag coefficient,}
         \item<9->{wind speed.}
      \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, creating 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->{ $c_{d}$\tabto{.5cm}: drag coefficient}
         \item<4->{ $a $\tabto{.5cm}:    leaf area density $[m^{2}m^{-3}]$}
         \item<5->{ $U$\tabto{.5cm}:     $(u^{2} + v^{2} + w^{2})^{1/2}$ $[m s^{-1}]$}
         \item<6->{ $u_{i}$\tabto{.5cm}: velocity component ($u_{1}=u$, $u_{2}=v$, $u_{3}=w$)}
      \end{itemize}
   \end{itemize}
   \uncover<7->{Note: The canopy model does not resolve single plant elements, it rather models a volume-averaged effect of the forest drag on the flow.}
   \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 plant 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. In turn, the heated foliage 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 enabled by adding {\small \texttt{NAMELIST}} group {\small \texttt{\&canopy\_par}} to the parameter file ({\small \texttt{PARIN}}), subsequently to {\small \texttt{NAMELIST}} group {\small \texttt{\&d3par}}.}
      \item<2->{All parameters for steering the canopy model are described in:\\
      {\scriptsize Documentation $\rightarrow$ Model steering $\rightarrow$ Parameters $\rightarrow$ Plant 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 {\small \texttt{\&canopy\_par}} {\small \texttt{NAMELIST}}.}
   \end{itemize}
\end{frame}



%Folie 09
\begin{frame}
   \frametitle{Basic canopy parameters (I)}
   The parameters for steering the canopy model have to be added to the {\small \texttt{\&canopy\_par}} {\small \texttt{NAMELIST}} in the parameter file ({\small \texttt{PARIN}}).\\
    \begin{itemize}
       \item<1->{Step I: Define the upper boundary of the plant canopy volume 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}

   \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=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
        \node at (barycentric cs:A9=1,A10=1,A11=1,A12=1) {\tiny };
        
        \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 parameters (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.\\
                  Two methods are available to prescribe an lad profile:}
       \begin{itemize}
          \item<2->{Create piecewise linear segments by prescribing vertical lad gradients,} 
          \item<3->{Prescribe a beta probability distribution.} 
       \end{itemize}

    \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
        \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
        \node at (barycentric cs:A9=1,A10=1,A11=1,A12=1) {\tiny };
        
        \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)};
        

\end{tikzpicture}


\end{frame}



%Folie 11
\begin{frame}
   \frametitle{Basic canopy parameters (II)}
    \begin{itemize}
       \item<1->{ Method I: Create lad profile from piecewise linear segments by prescribing vertical lad gradients.}
       \item[]{The canopy top is located at {\small \texttt{zw(pch\_index)}} (w-grid), which is in the center between {\small \texttt{zu(pch\_index)}} and {\small \texttt{zu(pch\_index + 1)}} (u-grid).}
    \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
        \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
        \node at (barycentric cs:A9=1,A10=1,A11=1,A12=1) {\tiny };
        
        \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);},
                mark=at position 04mm with{\draw[fill=black] (0,0) circle (0.1em);},
                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 12
\begin{frame}
   \frametitle{Basic canopy parameters (II)}
    \begin{itemize}
       \item<1->{ Method II: Create \textit{lad} profile by prescribing a distribution, following this beta probability density function:}
    \end{itemize}


   %equation for beta distribution:
   \vspace{-3pt}
   \scriptsize
   \begin{equation*}
    f_{PDF}(\frac{z}{H},\alpha,\beta) = \frac{(\frac{z}{H})^{\alpha-1}\;(1-\frac{z}{H})^{\beta-1}}{\int_{0}^{1}\;(\frac{z}{H})^{\alpha-1}\;(1-\frac{z}{H})^{\beta-1}\;d(\frac{z}{H})} 
   \end{equation*}
   \vspace{5pt}
   \begin{equation*}
    \rightarrow \hspace{5pt} lad(z) = LAI\cdot f_{PDF}(\frac{z}{H})\cdot H
   \end{equation*}
   \normalsize



   %draw figure:
   \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
        \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
        \node at (barycentric cs:A9=1,A10=1,A11=1,A12=1) {\tiny };
        
        \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)};
        
        %parameters to be prescribed:
        \node [Key3, right=2.1cm of B1] (label1) {Prescribe values for coefficients\\ $\alpha$, $\beta$ and for $LAI$:};

        \coordinate (curvelabel11) at (5.9cm,-0.5cm);
        \node [Key3, right=1.0cm of curvelabel11] (curvelabel12) {\texttt{alpha\_lad},\\ \texttt{beta\_lad},\\ \texttt{lai\_beta}.};
\end{tikzpicture}


\end{frame}



%Folie 13
\begin{frame}
   \frametitle{Basic canopy parameters (III)}
   \begin{footnotesize}
    \begin{itemize}
       \item<1->{Step III: Prescribe a value for the parameter {\small \texttt{canopy\_drag\_coeff (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 14
\begin{frame}
   \frametitle{Basic canopy parameters (IV)}
   \begin{itemize}
      \item<1->{For steering the effect of the sensible heat transfer inside the canopy, 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\_conc} and \texttt{leaf\_scalar\_exch\_coeff} (see Methods (IV)).}
   \end{itemize}
\end{frame}



%Folie 15
\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 in: \texttt{user\_init\_plant\_canopy.f90}}};}
\end{tikzpicture}


\end{frame}

\end{document}