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\title{Ice documentation}
\author{Larour, Morlighem, Seroussi}
\date{2009}

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pdftitle={ISSM Validation},
pdfauthor={Mathieu Morlighem and Helene Seroussi},
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\begin{document}
\thispagestyle{empty}

\vspace{10cm}
\begin{center}
\bf \LARGE Ice Sheet System Model 2008\\ Code Validation Guide
\end{center}

\vspace{4cm}

\begin{center}
Authors:\\   
\vspace{.5cm}
{\bf \large H\'el\`ene Seroussi$^{2,3}$} \\
\smallskip
{\bf \large Mathieu Morlighem$^{2,3}$} \\
\smallskip
{\bf \large \'Eric Larour$^{1}$} \\
\vfill
%\vspace{4.5cm}
$^{1}$Division 35, Thermal and Cryogenics Section, \\
Mechanical Division, MS 157-316.  \\
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109.\\
\vspace{.2cm}
$^{2}$Division 33, Radar Science and Enginnering Section, \\
Communications, Tracking and Radar Division, MS 157-316.  \\
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109.\\
\vspace{.2cm}
$^{3}$Laboratoire de M\'ecanique des sols, Structures et  mat\'eriaux (MSSMat)\\
\'Ecole Centrale Paris, CNRS UMR 8579\\
Grande Voie des Vignes, 92295 Chtenay-Malabry Cedex, FRANCE\\

\vspace{0.5cm}
\today
\end{center}


\cleardoublepage
\pagestyle{headings}
\tableofcontents

\newpage

\chapter{EISMINT}

The tests described in this chapters follow the work of Vincent Rommeleare \cite{Rommelaere1996}.

\section{Test 1: Mass concervation}
\subsection{Setup of the experiment}
We use an idealized geometry of a square ice shelf  of length 200km. The velocity is imposed everywhere as $u_0=400$m/yr and the initial thickness is 500m.\\
\\
The boundary conditions are free radiation on all sides except on the left boundary where a thickness wave is imposed:

\begin{equation}
H(t)=H_0 + \dfrac{H_0}{5}\sin\left( \dfrac{2\pi t}{T}\right) 
\end{equation}
\\
We want to check if the conservation scheme is able to propagate the signal along the shelf. The analytical solution is:

\begin{equation}
H(x,y,t)=H_0 + \dfrac{H_0}{5}\sin\left( \dfrac{2\pi}{T}\left( t-\dfrac{x}{u_0}\right) \right) 
\end{equation}

\subsection{Results and comparisons}
\begin{figure}[H]
  \centering
    \subfloat[Vincent Rommeleare's results]{\includegraphics[width=6.5cm]{EISMINT/MassConcervation/thickness_th}}\hspace*{1em}
    \subfloat[]{\includegraphics[width=1.5cm]{EISMINT/MassConcervation/legend_th}}
    \\
    \subfloat[Thickness with artificial diffusivity]{\includegraphics[width=8cm]{EISMINT/MassConcervation/thickness_art}}\hspace*{1em}
    \subfloat[Thickness without artificial diffusivity]{\includegraphics[width=8cm]{EISMINT/MassConcervation/thickness}}
  \caption{Comparision of the ISSM results with Vincent Rommeleare's}
\end{figure}

\newpage
\section{Test 2: Diagnostic 1}
We use a square ice shelf with an imposed velocity everywhere except on the front where a dynamic boundary condition is imposed (water pressure). Here are the results:
\begin{figure}[H]
  \centering %\hspace*{-1.5cm}
    \subfloat[Vx (m/yr) computed by ISSM]{\includegraphics[width=9cm]{EISMINT/Diagnostic1/vx}}
    \subfloat[Vy (m/yr) computed by ISSM]{\includegraphics[width=9cm]{EISMINT/Diagnostic1/vy}}\\
    \subfloat[Vx (m/yr) found by Romelaere]{\includegraphics[width=7.5cm]{EISMINT/Diagnostic1/vx_th}}\hspace*{2em}
    \subfloat[Vy (m/yr) found by Romelaere]{\includegraphics[width=7.5cm]{EISMINT/Diagnostic1/vy_th}}
  \caption{Comparision of the ISSM results with Vincent Rommeleare's}
\end{figure}

\newpage
\section{Test 3: Diagnostic 2}
The geometry is exactly the same as the previous one except that the upper boundary condition is change to account for an ice stream entering the ice shelf. The y-velocity becomes:

\begin{equation}
v(x,(y=L))=400\left( \left[\dfrac{x-100}{25} \right]^2 -1\right)Heav\left( 1-\left[\dfrac{x-100}{25} \right]^2\right)  
\end{equation}
\begin{figure}[H]
  \centering %\hspace*{-1.5cm}
    \subfloat[Vx (m/yr) computed by ISSM]{\includegraphics[width=9cm]{EISMINT/Diagnostic2/vx}}
    \subfloat[Vy (m/yr) computed by ISSM]{\includegraphics[width=9cm]{EISMINT/Diagnostic2/vy}}\\
    \subfloat[Vx (m/yr) found by Romelaere]{\includegraphics[width=7.5cm]{EISMINT/Diagnostic2/vx_th}}\hspace*{2em}
    \subfloat[Vy (m/yr) found by Romelaere]{\includegraphics[width=7.5cm]{EISMINT/Diagnostic2/vy_th}}
  \caption{Comparision of the ISSM results with Vincent Rommeleare's}
\end{figure}
\newpage

\chapter{ISMIP-HOM}
The tests described in this chapters follow the works of Laura Perichon \cite{Perichon2007} and Frank Pattyn \cite{Pattyn2008}. For a comprehensie description of the experiment, please see \cite{Pattyn2006a}.

\section{Test A}
\setcounter{subfigure}{0}
\subsection{Geometry}

This is a 3d ice-sheet flow over a bumpy bed experiment. Periodic boundary conditions are applied. The geometry follows:
\begin{itemize}
\item[--] surface $s(x,y)=-x\tan(0.5^o)$
\item[--] bed $b(x,y)=s-1000+500\sin\left( \dfrac{2\pi}{L}x\right) \sin\left( \dfrac{2\pi}{L}y\right)$
\item[--] $5\;km\leqslant L \leqslant 160\;km$
\end{itemize}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{ISMIP/TestA/geo.png}
\caption{Test A geometry}
\end{center}
\end{figure}

\newpage
\subsection{Results}

\subsubsection{5km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestA/clvx5p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestA/clvx5p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx5p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy5p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz5p}}
\end{figure}
\newpage

\subsubsection{10km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestA/clvx10p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestA/clvx10p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx10p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy10p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz10p}}
\end{figure}
\newpage

\subsubsection{20km}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestA/clvx20p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestA/clvx20p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx20p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy20p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz20p}}
\end{figure}
\newpage

\subsubsection{40km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestA/clvx40p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestA/clvx40p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx40p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy40p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz40p}}
\end{figure}
\newpage

\subsubsection{80km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestA/clvx80p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestA/clvx80p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx80p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy80p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz80p}}
\end{figure}
\newpage

\subsubsection{160km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestA/clvx160p}}
     \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestA/clvx160p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vx160p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vy160p}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestA/vz160p}}
\end{figure}
\newpage

\subsubsection{global (comparison)}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestA/TestAMaxVx}}
    \subfloat[Maximum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestA/TestAMaxVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestA/TestAMinVx}}
    \subfloat[Minimum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestA/TestAMinVx_th}}
\end{figure}
\newpage

\section{Test B}
\setcounter{subfigure}{0}
\subsection{Geometry}
This is a 2d ice-sheet flow over a rippled bed (flowline). Periodic boundary conditions are applied. The geometry follows:
\begin{itemize}
\item[--] surface $s(x,y)=-x\tan(0.5^o)$
\item[--] bed $b(x,y)=s-1000+500\sin\left( \dfrac{2\pi}{L}x\right) $
\item[--] $5\;km\leqslant L \leqslant 160\;km$
\end{itemize}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{ISMIP/TestB/geo.png}
\caption{Test B geometry}
\end{center}
\end{figure}

\newpage
\subsection{Results}


\subsubsection{5km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx5p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=8cm]{ISMIP/TestB/clvx5p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx5p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz5p}}
    \end{figure}
\newpage

\subsubsection{10km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx10p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=8cm]{ISMIP/TestB/clvx10p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx10p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz10p}}
    \end{figure}
\newpage

\subsubsection{20km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx20p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=8cm]{ISMIP/TestB/clvx20p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx20p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz20p}}
\end{figure}
\newpage

\subsubsection{40km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx40p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=8cm]{ISMIP/TestB/clvx40p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx40p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz40p}}
\end{figure}
\newpage

\subsubsection{80km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx80p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestB/clvx80p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx80p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz80p}}
\end{figure}
\newpage

\subsubsection{160km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestB/clvx160p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestB/clvx160p_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vx160p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestB/vz160p}}
\end{figure}
\newpage

\subsubsection{global (comparison)}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestB/TestBMaxVx}}
    \subfloat[Maximum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestB/TestBMaxVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestB/TestBMinVx}}%
%    \subfloat[Minimum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestB/TestBMinVx_th}}
\end{figure}
\newpage

\section{Test C (detailed)}
\setcounter{subfigure}{0}
\subsection{Geometry}

This is a 3d ice-stream flow over a slippery bed. Periodic boundary conditions are applied. The geometry follows:
\begin{itemize}
\item[--] surface $s(x,y)=-x\tan(0.1^o)$
\item[--] bed $b(x,y)=s-1000$
\item[--] sliding $\alpha^2(x,y)=1000+1000\sin\left( \dfrac{2\pi}{L}x\right)\sin\left( \dfrac{2\pi}{L}y\right)$
\item[--] $5\;km\leqslant L \leqslant 160\;km$
\end{itemize}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{ISMIP/TestC/geo.png}
\caption{Test C geometry}
\end{center}
\end{figure}

\newpage
\subsection{Results}

\subsubsection{5km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/clvx5p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/clvx5p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz5p}}
    %\subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz5p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vx5p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vy5p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vz5p}}
\end{figure}
\begin{figure}[H]
    \centering 
	\subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/txz5p}}
	\subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/tyz5p}}
\end{figure}
\newpage

\subsubsection{10km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/clvx10p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/clvx10p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz10p}}
    %\subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz10p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vx10p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vy10p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vz10p}}
\end{figure}
\begin{figure}[H]
    \centering 
	\subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/txz10p}}
	\subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/tyz10p}}
\end{figure}
\newpage

\subsubsection{20km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/clvx20p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/clvx20p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz20p}}
    %\subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz20p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vx20p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vy20p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vz20p}}
\end{figure}
\begin{figure}[H]
    \centering 
	\subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/txz20p}}
	\subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/tyz20p}}
\end{figure}
\newpage

\subsubsection{40km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/clvx40p}}
     \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/clvx40p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz40p}}
    %\subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz40p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vx40p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vy40p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vz40p}}
\end{figure}
\begin{figure}[H]
    \centering 
	\subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/txz40p}}
	\subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/tyz40p}}
\end{figure}
\newpage

\subsubsection{80km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering %\hspace*{-1.5cm}
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestC/clvx80p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestC/clvx80p_th}}
\end{figure}
\begin{figure}[H]
  \centering
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz80p}}
    %\subfloat[Txz (kPa) compatison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz80p_th}}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.2cm]{ISMIP/TestC/vx80p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/vx80p_th}}
    \end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.2cm]{ISMIP/TestC/vy80p}}
    \subfloat[Vy (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/vy80p_th}}
\end{figure}
\begin{figure}[H]
  \centering
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.2cm]{ISMIP/TestC/vz80p}}
    \subfloat[Vz (m/yr) compatison]{\includegraphics[width=7cm]{ISMIP/TestC/vz80p_th}}
\end{figure}
\begin{figure}[H]
  \centering
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.2cm]{ISMIP/TestC/txz80p}}
    \subfloat[Txz (kPa) compatison]{\includegraphics[width=7cm]{ISMIP/TestC/txz80p_th}}
\end{figure}
\begin{figure}[H]
  \centering
    \subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.2cm]{ISMIP/TestC/tyz80p}}
    \subfloat[Tyz (kPa) compatison]{\includegraphics[width=7cm]{ISMIP/TestC/tyz80p_th}}
\end{figure}
\newpage

\subsubsection{160km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/clvx160p}}
     \subfloat[Vx (m/yr) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/clvx160p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestC/cltxz160p}}
    %\subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestC/cltxz160p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vx160p}}
    \subfloat[Vy (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vy160p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/vz160p}}
\end{figure}
\begin{figure}[H]
    \centering 
	\subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/txz160p}}
	\subfloat[Tyz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestC/tyz160p}}
\end{figure}
\newpage

\subsubsection{global (comparison)}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestC/TestCMaxVx}}
    \subfloat[Maximum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestC/TestCMaxVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestC/TestCMinVx}}
    \subfloat[Minimum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestC/TestCMinVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestC/TestCMaxTxz}}
    \subfloat[Maximum Txz (kPa) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestC/TestCMaxTxz_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestC/TestCMinTxz}}
    \subfloat[Minimum Txz (kPa) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestC/TestCMinTxz_th}}
\end{figure}
\newpage


\section{Test D}
\setcounter{subfigure}{0}
\subsection{Geometry}
This is a 2d ice-stream flow over a slippery bed (flowline). Periodic boundary conditions are applied. The geometry follows:
\begin{itemize}
\item[--] surface $s(x,y)=-x\tan(0.1^o)$
\item[--] bed $b(x,y)=s-1000$
\item[--] sliding $\alpha^2(x,y)=1000+1000\sin\left( \dfrac{2\pi}{L}x\right)$
\item[--] $5\;km\leqslant L \leqslant 160\;km$
\end{itemize}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{ISMIP/TestD/geo.png}
\caption{Test D geometry}
\end{center}
\end{figure}

\newpage
\subsection{Results}

\subsubsection{5km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx5p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx5p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz5p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz5p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx5p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz5p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz5p}}
\end{figure}
\newpage

\subsubsection{10km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx10p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx10p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz10p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz10p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx10p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz10p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz10p}}
\end{figure}
\newpage

\subsubsection{20km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx20p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx20p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz20p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz20p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx20p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz20p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz20p}}
\end{figure}
\newpage

\subsubsection{40km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx40p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx40p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz40p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz40p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx40p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz40p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz40p}}
\end{figure}
\newpage

\subsubsection{80km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx80p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx80p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz80p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz80p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx80p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz80p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz80p}}
\end{figure}
\newpage

\subsubsection{160km}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestD/clvx160p}}
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestD/clvx160p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.2cm]{ISMIP/TestD/cltxz160p}}
    \subfloat[Txz (kPa) comparison]{\includegraphics[width=7cm]{ISMIP/TestD/cltxz160p_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vx160p}}
    \subfloat[Vz (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/vz160p}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9cm]{ISMIP/TestD/txz160p}}
\end{figure}
\newpage

\subsubsection{global (comparison)}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestD/TestDMaxVx}}
    \subfloat[Maximum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestD/TestDMaxVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Vx (m/yr) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestD/TestDMinVx}}
    \subfloat[Minimum Vx (m/yr) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestD/TestDMinVx_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Maximum Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestD/TestDMaxTxz}}
    \subfloat[Maximum Txz (kPa) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestD/TestDMaxTxz_th}}
\end{figure}
\vspace*{-2em}
\begin{figure}[H]
    \centering 
    \subfloat[Minimum Txz (kPa) computed by ISSM (Pattyn)]{\includegraphics[width=9.5cm]{ISMIP/TestD/TestDMinTxz}}
    \subfloat[Minimum Txz (kPa) comparison]{\includegraphics[width=8.5cm]{ISMIP/TestD/TestDMinTxz_th}}
\end{figure}
\newpage

\section{Test E}
\setcounter{subfigure}{0}
\subsection{Geometry}
This experiment deals with the cross section of a real glacier: \emph{Haut Glacier d'Arolla}. Two cases are studied:
\begin{itemize}
\item[--] basal velocity = 0 on the bed
\item[--] sliding $\alpha^2(x,y)=0$ for $0.44\leqslant x \leqslant 0.5$
\end{itemize}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{ISMIP/TestE/geo.png}
\caption{Test E geometry}
\end{center}
\end{figure}

\newpage
\subsection{Results}

\subsubsection{Frozen Bed}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestE/case1_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestE/velfrp}}
    \subfloat[Vx (m/yr) computed by ISSM (Stokes) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestE/velfrs}}
\end{figure}

\newpage

\subsubsection{Sliding}
\setcounter{subfigure}{0}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) comparison]{\includegraphics[width=6.5cm]{ISMIP/TestE/case2_th}}
\end{figure}
\begin{figure}[H]
    \centering 
    \subfloat[Vx (m/yr) computed by ISSM (Pattyn) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestE/velslp}}
    \subfloat[Vx (m/yr) computed by ISSM (Stokes) on a cross line]{\includegraphics[width=9.5cm]{ISMIP/TestE/velsls}}
\end{figure}
\newpage

\bibliographystyle{plain}
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\bibitem{Pattyn2008}
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\end{thebibliography}

\end{document}
