Index: sm/workshop/2012/Talks/07_Ice_flow_models/11_Ice_flow_models.tex =================================================================== --- /issm/workshop/2012/Talks/07_Ice_flow_models/11_Ice_flow_models.tex (revision 14034) +++ (revision ) @@ -1,791 +1,0 @@ -%Preamble {{{1 -\documentclass[8pt,notes=hide]{beamer} % Document class: beamer %8 pts is also a good option -\usetheme[life,headstyle=institute,fnolabel=,sidebar=.15\paperwidth,footstyle=low,footsep={ $\cdot$ },TPomitframeno]{Pasadena} -\useinnertheme{MLHacks} % ISSM bonuses (ghostframes for example) -\usepackage[T1]{fontenc} % T1 encoding is better for foreign langages -\usepackage{aeguill,aecompl} % Virtual fonts -\usepackage{amsmath,amssymb,amsfonts}% Mathematical symbols (useful for equations) -\usepackage{bm} % maths letters, real = IR -\usepackage{xcolor} % We can define new colors -\usepackage{graphicx} % Include pictures -\usepackage{times} % Use Times Font instead of default -\usepackage{multicol} -\usepackage{appendixnumberbeamer} -\usepackage[framed,numbered,autolinebreaks,useliterate]{mcode} -\usepackage{booktabs} %nice tables (toprule, bottomrule,...) -\graphicspath{ % directories where the pictures are located -{Images/} -} -\hypersetup{pdfstartview={Fit}} % Acrobat reader: open fit to screen - -\setbeamercovered{transparent} %invisible, transparent, dynamic, highly dynamic -%}}} -%Newcommands {{{1 -\newcommand*{\MAT}[1]{\ensuremath{\mathbf{#1}} } %Matrice name (Bold) -\newcommand*{\VEC}[1]{\ensuremath{\bm{#1}} } %Vector name (Bold Italic) -\newcommand*{\SET}[1]{\ensuremath{\mathcal{#1} }} %Set, Space (italic) -\newcommand*{\R}[0]{{\mathbb R}} %real set IR -\newcommand*{\N}[0]{{\mathbb N}} %natural set IN -\newcommand*{\Trace}[1]{\text{Tr}\left( #1\right)} %trace (Tr(A)) -\newcommand*{\abs}[1]{\left| #1\right|} %absolute value |x| -\newcommand*{\norm}[1]{\left\| #1 \right\|} %norm ||x|| -\newcommand*{\ent}[1]{\lfloor #1 \rfloor} %floor function [f] -\newcommand*{\grad}[1]{\nabla #1} -\newcommand*{\divergence}[1]{\nabla \cdot #1} -\newcommand*{\Divergence}[1]{\nabla \cdot #1} -\newcommand*{\eps}[0]{\dot{\bm \varepsilon}} %strain rate tensor -\newcommand*{\epseff}[0]{\ensuremath{\dot{\varepsilon}_{e} }} %effective strain rate -\newcommand*{\epscomp}[0]{\dot{\varepsilon}} %strain rate tensor component -\newcommand*{\stress}[0]{{\bm \sigma}} %stress tensor -\newcommand*{\devstress}[0]{\stress'} %deviatoric stress tensor -\newcommand*{\stresscomp}[0]{\ensuremath{\sigma}} %stress tensor component -\newcommand*{\vel}[0]{\VEC{v}} %velocity vector -\newcommand*{\velbar}[0]{\VEC{\bar{v} }} %velocity vector BAR -\newcommand*{\velobs}[0]{\ensuremath{\VEC{v}^{\text{obs}} }}%velocity obs -\newcommand*{\ubar}[0]{\ensuremath{\overline{v}_x}} %velocity vector BAR -\newcommand*{\vbar}[0]{\ensuremath{\overline{v}_y}} %velocity vector BAR -\newcommand*{\vx}[0]{\ensuremath{v_x}} %vx -\newcommand*{\vy}[0]{\ensuremath{v_y}} %vy -\newcommand*{\vz}[0]{\ensuremath{v_z}} %vz -\newcommand*{\vxobs}[0]{\ensuremath{v_x^{\text{obs}} }}%vx obs -\newcommand*{\vyobs}[0]{\ensuremath{v_y^{\text{obs}} }}%vy -\newcommand*{\pressure}[0]{p} %pressure -\newcommand*{\nx}[0]{\ensuremath{n_x}} %nx -\newcommand*{\ny}[0]{\ensuremath{n_y}} %ny -\newcommand*{\nz}[0]{\ensuremath{n_z}} %nz -\newcommand*{\Lagrangian}[0]{\ensuremath{\mathcal{L} }} -\newcommand*{\Ms}[0]{\ensuremath{\dot{M}_s}} -\newcommand*{\Mb}[0]{\ensuremath{\dot{M}_b}} -\newcommand*{\Omb}[0]{\color{blue}\Omega_1\color{black}} -\newcommand*{\Omr}[0]{\color{red}\Omega_2\color{black}} -\newcommand*{\Vb}[0]{\color{blue}V_1\color{black}} -\newcommand*{\Vr}[0]{\color{red}V_2\color{black}} -\newcommand*{\Vbt}[0]{\color{blue}V_{1}\color{black}} -\newcommand*{\Vrt}[0]{\color{red}V_{2}\color{black}} -\newcommand*{\rest}[2]{\ensuremath{{#1}_{\left. \right|_{#2}} }} -%%Automatically insert outline slide at the begining of each new section (very nice!) -%\AtBeginSection[]{ -%\begin{frame} -% \ghostframe %do not count slide -% \frametitle{Outline} % title of the slide -% \tableofcontents[currentsection] -%\end{frame} -%} -%}}} -%Presentation properties{{{1 -\title[Ice flow models]{Ice Sheet System model} -\subtitle{Ice flow models} -\author[Larour et al.]{ -Eric \textsc{Larour}\inst{1}, -Eric \textsc{Rignot}\inst{1,3}, -Mathieu \textsc{Morlighem}\inst{1,2}, -\textbf{H\'el\`ene \textsc{Seroussi}}\inst{1,2} -Chris \textsc{Borstad}\inst{1}, -Feras \textsc{Habbal}\inst{1,3}, -Daria \textsc{Halkides}\inst{1,4}, -Behnaz \textsc{Khakbaz}\inst{1}, -John \textsc{Schiermeier}\inst{1}, -Nicole \textsc{Schlegel}\inst{1} -\vspace{1em} -} -\institute[Jet Propulsion Laboratory]{ -\inst{1}Jet Propulsion Laboratory - California Institute of Technology\\ -\inst{2}Laboratoire MSSMat, \'Ecole Centrale Paris, France\\ -\inst{3}University of California, Irvine\\ -\inst{4}Joint Institute for Regional Earth System Science \& Engineering, UCLA -} -\conference[ISSM Workshop 2011]{ISSM Workshop 2011} -\date[]{\hspace{-15em}December 2011\hspace{3em}\copyright Copyright 2011. All rights reserved} -\logo{\includegraphics[width=8em]{ISSMlogo}} -%}}} -\begin{document} - -%Title slide %{{{1 -\begin{frame}[plain] %No headers or footers - \ghostframe - \titlepage -\end{frame} %}}} -\begin{frame}{Outline}%{{{1 - \tableofcontents -\end{frame}%}}} -\section{Ice flow equations} -\subsection{Approximations implemented} -\begin{frame} \frametitle{Ice Sheet flow equations} %{{{ - \begin{block}{Incompressibility} - \begin{equation} \everymath{\displaystyle} %pour mettre des displaystyle partout - \forall \VEC{x}\in\Omega \hspace{3em} \nabla \cdot {\vel} =\Trace{\eps} - =\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} - =0 - \end{equation} - \begin{itemize} - \item $\vel=\left(u,v,w\right)$ ice velocity (m/yr) - \item $\eps$ strain rate tensor (yr\textsuperscript{-1}) - \end{itemize} - \end{block} - - \begin{block}{Incompressible viscous fluid} - \begin{equation} - {\bm \sigma'}=2\mu\eps - \end{equation} - \begin{itemize} - \item ${\bm \sigma'}$ deviatoric stress - \item $\mu$ ice viscosity - \item $\eps$ strain rate tensor - \end{itemize} - \end{block} - - \begin{block}{Glen's flow law} - \begin{equation} - \mu = \frac{B}{2 \, \epseff^{\frac{n-1}{n}} } - \end{equation} - \begin{itemize} - \item $B$ ice hardness - \item $n$ Glen's law coefficient ($n=3$) - \item $\epseff$ effective strain rate (second invariant) - \end{itemize} - \end{block} - \note{ - Ice is treated as an incompressible material so the divergence of its velocity is equal to 0. It - is also considered as a purely viscous material, so the behavior law only includes the strain - rate tensor. As it is incompressible, only the deviatoric stress is involved. mu is the ice - viscosity estimated using Glen's flow law. This law was established in the 50's based on in-situ - measurements and lab experiments. Notice that the effective strain rate tensor is involved so it - is a non-linear material law. } -\end{frame} %}}} -\begin{frame} \frametitle{Ice Sheet flow equations} %{{{ - \vspace{-10pt} - \begin{block}{Conservation of momentum} - \begin{equation} - \forall \VEC{x}\in\Omega \hspace{3em} \nabla \cdot \devstress-\nabla P + \rho \VEC{g} = \VEC{0} - \end{equation} - Assumptions: - \begin{enumerate} - \item Stokes flow (quasi-static assumption) - \item Coriolis effect negligible - \end{enumerate} - \end{block} - \begin{block}{Boundary conditions} - \renewcommand\arraystretch{1.5} - \begin{tabular}[b]{lll} - Ice/Air interface: Free surface & $\Gamma_s$ & $\stress \cdot\VEC{n}=P_{atm}\;\VEC{n}\simeq\VEC{0}$\\ - Ice/Ocean interface: water pressure & $\Gamma_w$ & $\stress \cdot\VEC{n}=P_w\;\VEC{n}$\\ - Ice/Bedrock interface (1): lateral friction & $\Gamma_b$ & $ \left( \stress \cdot\VEC{n} + \beta\vel \right)_\parallel =\VEC{0}$ \\ - Ice/Bedrock interface (2): impenetrability & $\Gamma_b$ & $ \vel\cdot\VEC{n}=\VEC{0} $ \\ - Side boundaries: Dirichlet & $\Gamma_u$ & $\vel=\vel_{obs}$ - \end{tabular} - \end{block} - \note{ - Two assumptions are made in the conservation of the momentum. The first one is that we have a - quasi-static flow so the acceleration is neglected and we have Stokes equations. We second - assumption is that the Coriolis effect is negligible. The boundary conditions used for this - models are a free surface on the upper surface (ice/air interface), the air pressure is almost - equal to 0. Water pressure is applied at the ice/ocean interface. For the ice/bedrock interface, - the ice cannot penetrate the rock so the velocity normal to the interface is 0. Basal friction - applies tangential to the interface. Non homogeneous Dirichlet boundary conditions based on - observations of velocity are applied for the other lateral boundaries. - } -\end{frame} %}}} -\begin{frame} \frametitle{Models description} %{{{ - - \begin{columns} - \begin{column}{.6\linewidth} - Full-Stokes model: - \begin{itemize} - \item Momentum balance + incompressibility - \item 3D model - \item Four unknowns ($\vx,\vy,\vz,\pressure$) - \end{itemize} - \end{column} - \begin{column}{.4\linewidth} - \includegraphics[width=0.6\textwidth]{Stokes} - \end{column} - \end{columns} - - \begin{block}{Model equations} - \begin{align*} - \begin{array}{c} - \displaystyle \left\{ - \begin{array}{l} - \displaystyle - \frac{\partial}{\partial x} \left( 2\mu \frac{\partial \vx}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vx}{\partial y} + \mu \frac{\partial \vy}{\partial x} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vx}{\partial z} \color{black} - {\color{black} +\mu \frac{\partial \vz}{\partial x}} \color{black}\right)\color{black} - -\frac{\partial \pressure}{\partial x}=0\\ - \\ - \displaystyle - \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial y} +\mu \frac{\partial \vy}{\partial x} \right) - +\frac{\partial}{\partial y} \left( 2 \mu \frac{\partial \vy}{\partial y} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vy}{\partial z} \color{black} - {\color{black} +\mu \frac{\partial \vz}{\partial y}} \color{black}\right) \color{black} - -\frac{\partial \pressure}{\partial y}=0\\ - \\ - \displaystyle - {\color{black} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial z} +\mu \frac{\partial \vz}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vy}{\partial z} +\mu \frac{\partial \vz}{\partial y} \right)} + - \frac{\partial}{\partial z} \left( 2 \mu \frac{\partial \vz}{\partial z} \right) - -\frac{\partial \pressure}{\partial z}- \rho g=0 - \end{array} \right. \\ - \\ - \displaystyle \frac{\partial \vx}{\partial x}+\frac{\partial \vy}{\partial y}+\frac{\partial \vz}{\partial z}=0 - \end{array} - \end{align*} - \end{block} - - \note{All the ice flow models are derived from the momentum balance and incompressibility - equations. If we neglect the acceleration terms, we obtain the full-Stokes equations. It is a 3D - model with four unknowns (three components of velocity vx, vy and vz and the pressure P).\\ - A first approximation was introduced by Blatter in 1995 and improved by Pattyn in 2003. We call - it "higher order model" or Blatter/Pattyn model. It is also 3d but the horizontal gradients - of the vertical velocity are neglected compared to the vertical gradient of the horizontal - velocities (first two lines). The bridging effect is also neglected. So only the black and blue terms - remain. After some algebraic manipulation we can show that the equations are decoupled and the - horizontal and vertical components of velocity are solved successively. We have a system of 2 - equations with two unknowns.\\ - The last approximation we use here is the shallow-shelf or Shelfy-Stream approximation introduced - by MacAyeal in 1989. An additional assumption is made here: we neglect the vertical shear so the - ice moves like a block. The equations can be vertically integrated which gives a 2d model with 2 - unknowns. Only the black terms of the equations remain.} -\end{frame} %}}} -\begin{frame} \frametitle{Models description} %{{{ - - \begin{columns} - \begin{column}{.6\linewidth} - Higher-order model: - \begin{itemize} - \item \cite{Blatter1995,Pattyn2003} - \item 3D model - \item Horizontal and vertical velocity decoupled - \item 2 ($\vx,\vy$) + 1 ($\vz$) unknowns - \end{itemize} - \end{column} - \begin{column}{.4\linewidth} - \includegraphics[width=0.6\textwidth]{Pattyn} - \end{column} - \end{columns} - - \begin{block}{Model equations} - \begin{align*} - \begin{array}{c} - \displaystyle \left\{ - \begin{array}{l} - \displaystyle - \frac{\partial}{\partial x} \left( 2\mu \frac{\partial \vx}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vx}{\partial y} + \mu \frac{\partial \vy}{\partial x} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vx}{\partial z} \color{black} - {\color{red} +\mu \frac{\partial \vz}{\partial x}} \color{black}\right)\color{black} - -\frac{\partial \pressure}{\partial x}=0\\ - \\ - \displaystyle - \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial y} +\mu \frac{\partial \vy}{\partial x} \right) - +\frac{\partial}{\partial y} \left( 2 \mu \frac{\partial \vy}{\partial y} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vy}{\partial z} \color{black} - {\color{red} +\mu \frac{\partial \vz}{\partial y}} \color{black}\right) \color{black} - -\frac{\partial \pressure}{\partial y}=0\\ - \\ - \displaystyle - {\color{red} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial z} +\mu \frac{\partial \vz}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vy}{\partial z} +\mu \frac{\partial \vz}{\partial y} \right)} + - \frac{\partial}{\partial z} \left( 2 \mu \frac{\partial \vz}{\partial z} \right) - -\frac{\partial \pressure}{\partial z}- \rho g=0 - \end{array} \right. \\ - \\ - \displaystyle \frac{\partial \vx}{\partial x}+\frac{\partial \vy}{\partial y}+\frac{\partial \vz}{\partial z}=0 - \end{array} - \end{align*} - \end{block} - - \note{All the ice flow models are derived from the momentum balance and incompressibility - equations. If we neglect the acceleration terms, we obtain the full-Stokes equations. It is a 3D - model with four unknowns (three components of velocity vx, vy and vz and the pressure P).\\ - A first approximation was introduced by Blatter in 1995 and improved by Pattyn in 2003. We call - it "higher order model" or Blatter/Pattyn model. It is also 3d but the horizontal gradients - of the vertical velocity are neglected compared to the vertical gradient of the horizontal - velocities (first two lines). The bridging effect is also neglected. So only the black and blue terms - remain. After some algebraic manipulation we can show that the equations are decoupled and the - horizontal and vertical components of velocity are solved successively. We have a system of 2 - equations with two unknowns.\\ - The last approximation we use here is the shallow-shelf or Shelfy-Stream approximation introduced - by MacAyeal in 1989. An additional assumption is made here: we neglect the vertical shear so the - ice moves like a block. The equations can be vertically integrated which gives a 2d model with 2 - unknowns. Only the black terms of the equations remain.} -\end{frame} %}}} -\begin{frame} \frametitle{Models description} %{{{ - - \begin{columns} - \begin{column}{.6\linewidth} - Shelfy-stream approximation: - \begin{itemize} - \item \cite{MacAyeal1989} - \item 2D model - \item Horizontal and vertical velocity decoupled - \item 2 ($\vx,\vy$) + 1 ($\vz$) unknowns - \end{itemize} - \end{column} - \begin{column}{.4\linewidth} - \includegraphics[width=0.6\textwidth]{MacAyeal} - \end{column} - \end{columns} - - \begin{block}{Model equations} - \begin{align*} - \begin{array}{c} - \displaystyle \left\{ - \begin{array}{l} - \displaystyle - \frac{\partial}{\partial x} \left( 2\mu \frac{\partial \vx}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vx}{\partial y} + \mu \frac{\partial \vy}{\partial x} \right) - \color{blue} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vx}{\partial z} \color{black} - {\color{red} +\mu \frac{\partial \vz}{\partial x}} \color{blue}\right)\color{black} - -\frac{\partial \pressure}{\partial x}=0\\ - \\ - \displaystyle - \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial y} +\mu \frac{\partial \vy}{\partial x} \right) - +\frac{\partial}{\partial y} \left( 2 \mu \frac{\partial \vy}{\partial y} \right) - \color{blue} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vy}{\partial z} \color{black} - {\color{red} +\mu \frac{\partial \vz}{\partial y}} \color{blue}\right) \color{black} - -\frac{\partial \pressure}{\partial y}=0\\ - \\ - \displaystyle - {\color{red} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial z} +\mu \frac{\partial \vz}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vy}{\partial z} +\mu \frac{\partial \vz}{\partial y} \right)} + - \frac{\partial}{\partial z} \left( 2 \mu \frac{\partial \vz}{\partial z} \right) - -\frac{\partial \pressure}{\partial z} - \rho g=0 - \end{array} \right. \\ - \\ - \displaystyle \frac{\partial \vx}{\partial x}+\frac{\partial \vy}{\partial y}+\frac{\partial \vz}{\partial z}=0 - \end{array} - \end{align*} - \end{block} - - \note{All the ice flow models are derived from the momentum balance and incompressibility - equations. If we neglect the acceleration terms, we obtain the full-Stokes equations. It is a 3D - model with four unknowns (three components of velocity vx, vy and vz and the pressure P).\\ - A first approximation was introduced by Blatter in 1995 and improved by Pattyn in 2003. We call - it "higher order model" or Blatter/Pattyn model. It is also 3d but the horizontal gradients - of the vertical velocity are neglected compared to the vertical gradient of the horizontal - velocities (first two lines). The bridging effect is also neglected. So only the black and blue terms - remain. After some algebraic manipulation we can show that the equations are decoupled and the - horizontal and vertical components of velocity are solved successively. We have a system of 2 - equations with two unknowns.\\ - The last approximation we use here is the shallow-shelf or Shelfy-Stream approximation introduced - by MacAyeal in 1989. An additional assumption is made here: we neglect the vertical shear so the - ice moves like a block. The equations can be vertically integrated which gives a 2d model with 2 - unknowns. Only the black terms of the equations remain.} -\end{frame} %}}} -\begin{frame} \frametitle{Models description} %{{{ - - \begin{columns} - \begin{column}{.6\linewidth} - Shallow ice approximation: - \begin{itemize} - \item \cite{Hutter1983} - \item 3D analytical model - \item 2 unknowns ($\vx,\vy$) computed separately - \end{itemize} - \end{column} - \begin{column}{.4\linewidth} - \includegraphics[width=0.35\textwidth]{Hutter} - \end{column} - \end{columns} - - \begin{block}{Model equations} - \begin{align*} - \begin{array}{c} - \displaystyle \left\{ - \begin{array}{l} - \displaystyle - {\color{red} \frac{\partial}{\partial x} \left( 2\mu \frac{\partial \vx}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vx}{\partial y} + \mu \frac{\partial \vy}{\partial x} \right)} - +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vx}{\partial z} - {\color{red} +\mu \frac{\partial \vz}{\partial x}}\right) - -\frac{\partial \pressure}{\partial x}=0\\ - \\ - \displaystyle - {\color{red} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial y} +\mu \frac{\partial \vy}{\partial x} \right) - +\frac{\partial}{\partial y} \left( 2 \mu \frac{\partial \vy}{\partial y} \right) } - +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vy}{\partial z} - {\color{red}+\mu \frac{\partial \vz}{\partial y}} \right) - -\frac{\partial \pressure}{\partial y}=0\\ - \\ - \displaystyle - {\color{red} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial z} +\mu \frac{\partial \vz}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vy}{\partial z} +\mu \frac{\partial \vz}{\partial y} \right) + - \frac{\partial}{\partial z} \left( 2 \mu \frac{\partial \vz}{\partial z} \right) } - -\frac{\partial \pressure}{\partial z}- \rho g=0 - \end{array} \right. \\ - \\ - \displaystyle \frac{\partial \vx}{\partial x}+\frac{\partial \vy}{\partial y}+\frac{\partial \vz}{\partial z}=0 - \end{array} - \end{align*} - \end{block} - - \note{The last model is the shallow ice approximation, proposed in the 80's by Hutter based on - the shallow aspect ratio of ice (much thinner than larger). The assumptions made here are rather - different than for the previous three models as only the variations of pressure and vertical - variations of horizontal velocities are kept. This model is semi-analytical and is computed - easily. However it is very simple and cannot be use at hight resolution (less than 10 or 20 km), - for this reason it is not appropriate to accurately describe ice streams. We will not use it in - the rest of this presentation.} -\end{frame} %}}} -\begin{frame} \frametitle{Material non-linearity} %{{{ - - \begin{block}{Model equations} - \begin{align*} - \begin{array}{c} - \displaystyle \left\{ - \begin{array}{l} - \displaystyle - \frac{\partial}{\partial x} \left( 2\mu \frac{\partial \vx}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vx}{\partial y} + \mu \frac{\partial \vy}{\partial x} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vx}{\partial z} \color{black} - {\color{black} +\mu \frac{\partial \vz}{\partial x}} \color{black}\right)\color{black} - -\frac{\partial \pressure}{\partial x}=0\\ - \\ - \displaystyle - \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial y} +\mu \frac{\partial \vy}{\partial x} \right) - +\frac{\partial}{\partial y} \left( 2 \mu \frac{\partial \vy}{\partial y} \right) - \color{black} +\frac{\partial}{\partial z} \left( \mu \frac{\partial \vy}{\partial z} \color{black} - {\color{black} +\mu \frac{\partial \vz}{\partial y}} \color{black}\right) \color{black} - -\frac{\partial \pressure}{\partial y}=0\\ - \\ - \displaystyle - {\color{black} \frac{\partial}{\partial x} \left( \mu \frac{\partial \vx}{\partial z} +\mu \frac{\partial \vz}{\partial x} \right) - +\frac{\partial}{\partial y} \left( \mu \frac{\partial \vy}{\partial z} +\mu \frac{\partial \vz}{\partial y} \right)} + - \frac{\partial}{\partial z} \left( 2 \mu \frac{\partial \vz}{\partial z} \right) - -\frac{\partial \pressure}{\partial z}- \rho g=0 - \end{array} \right. \\ - \\ - \displaystyle \frac{\partial \vx}{\partial x}+\frac{\partial \vy}{\partial y}+\frac{\partial \vz}{\partial z}=0 - \end{array} - \end{align*} - \end{block} - - \begin{block}{Glen's flow law} - \begin{equation} - \mu = \frac{B}{2 \, \epseff^{\frac{n-1}{n}} } - \end{equation} - \begin{itemize} - \item $B$ ice hardness - \item $n$ Glen's law coefficient ($n=3$) - \item $\epseff$ effective strain rate (second invariant) - \end{itemize} - \end{block} - - $\rightarrow$ Treatment of non-linearity with fixed point - - \note{For all models, mu is non linear (depends on the velocity derivatives) and therefore the - previous equations (for all three models) are nonlinear. We need an iterative scheme to solve - them and we use a fixed-point to do so. } -\end{frame} %}}} -\begin{frame} \frametitle{Material non-linearity} %{{{ - Treatment of non-linearity with fixed point: - - \begin{center} - \includegraphics[height=0.68\textheight]{convergence} - \end{center} - - Vertical velocity computed with incompressibility for 2d shelfy-stream and 3d Blatter/Pattyn - modes. - \note{ - We saw that the behavior law is non linear so this is the schematic we use to solve this - non-linearity problem. We use a fix-point or Picard method. For FS, on the right, we start from - an estimated velocity, we calculate the associated viscosity and use it in the weak formulation. - We find a new velocity and calcule the corresponding viscosity, and iterate until it converges - and we have reach the convergence criterion. For BP or SSA, the viscosity depends on the - horizontal velocity only. We therefore use the same loop with horizontal velocity and compute the - vertical velocity afterwards, using the incompressibility equations, schematic on the left. - } -\end{frame} %}}} - -\subsection{Ice flow equation} -\begin{frame} \frametitle{Flow equation}%{{{1 - {\tt setflowequation} is used to generate the approximation used to compute the velocity - - \begin{itemize} - \item Arguments: - \begin{enumerate} - \item model - \item approximation names - \item approximation domains - \end{enumerate} - \item Domains can be Argus files or array of element flags - \item Approximation available - \begin{itemize} - \item stokes (Full-Stokes model) - \item pattyn (Higher-order model) - \item macayeal (Shallow Shelf Approximation) - \item hutter (Shallow Ice Approximation) - \end{itemize} - \item Possibility of coupling models - \end{itemize} -\end{frame} -%}}} -\begin{frame} \frametitle{Flow equation}%{{{1 - {\tt setflowequation} is used to generate the approximation used to compute the velocity - - \begin{itemize} - \item Examples - \end{itemize} - - \lstinputlisting[basicstyle=\lstbasicfont\tiny,firstnumber=1,firstline=15,lastline=18]{Code/ex1.m} - - \begin{itemize} - \item To diplay the type of approximation: - \end{itemize} - - \lstinputlisting[firstnumber=1,firstline=20,lastline=20]{Code/ex1.m} -\end{frame} -%}}} -\begin{frame} \frametitle{Flow equation}%{{{1 - \begin{itemize} - \item To diplay the type of approximation: - \end{itemize} - - \lstinputlisting[firstnumber=1,firstline=20,lastline=20]{Code/ex1.m} - - \begin{center} - \includegraphics[height=0.65\textheight]{macayealpattyn} - \end{center} -\end{frame} -%}}} -\begin{frame} \frametitle{Flow equation class}%{{{1 - \lstinputlisting[firstnumber=1,firstline=1,lastline=13,basicstyle=\lstbasicfont\tiny]{Code/ex1.m} -\end{frame} -%}}} -\subsection{Diagnostic parameters} -\begin{frame} \frametitle{Diagnostic class}%{{{1 - \lstinputlisting[firstnumber=1,firstline=22,lastline=53,basicstyle=\lstbasicfont\Tiny]{Code/ex1.m} -\end{frame} -%}}} -\subsection{Boundary conditions} -\begin{frame} \frametitle{Boundary conditions}%{{{1 - Boundary conditions created automatically or manually - \begin{itemize} - \item Automatically: - \lstinputlisting[firstnumber=1,firstline=55,lastline=57,basicstyle=\lstbasicfont\normalsize]{Code/ex1.m} - \item Manually: fields to change - \begin{itemize} - \item {\tt md.diagnostic.spcvx} - \item {\tt md.diagnostic.spcvy} - \item {\tt md.diagnostic.spcvz} - \item {\tt md.diagnostic.icefront} - \end{itemize} - \end{itemize} - - \begin{itemize} - \item To diplay the boundary conditions - \end{itemize} - - \lstinputlisting[firstnumber=1,firstline=59,lastline=59]{Code/ex1.m} -\end{frame} -%}}} - -\section{Combining models} -\subsection{Methods implemented in ISSM} -\begin{frame} \frametitle{Models description} %{{{ - - \emph{''Everything should be made as simple as possible, but no simpler.''} Albert Einstein - - \vspace{2em} - - \centering - \renewcommand\arraystretch{1.4} - \begin{tabular}{l c c c} - Model & Dim. & Unknowns & Reference\\ - \hline - Full-Stokes (FS) & 3d & 4 & \cite{Stokes1845}\\ - Blatter-Pattyn (BP) & 3d & 2 + 1 & \cite{Blatter1995,Pattyn2003}\\ - Shallow shelf (SSA) & 2d & 2 + 1 & \cite{MacAyeal1989}\\ - Shallow ice (SIA) & 2d & 2 + 1 & \cite{Hutter1983}\\ - \hline - \end{tabular} - - \vspace{2em} - \begin{center} - \renewcommand\arraystretch{1.4} - \begin{tabular}{c c c c} - \includegraphics[height=0.35\textheight]{Hutter} & - \includegraphics[height=0.35\textheight]{MacAyeal} & - \includegraphics[height=0.35\textheight]{Pattyn} & - \includegraphics[height=0.35\textheight]{Stokes} \\ - \end{tabular} - \end{center} - - \note{Everything should be made as simple as possible, but no simpler. - We remain with three ice flow models, two are 3D and one 2D, two decouple horizontal and - vertical velocities and one computes the four components of velocity and pressure simultaneously. - These three models are valid on different areas and we want to use the simplest possible model - according that the assumptions made are valid so we can have an accurate model than is - computationally manageable. The SSA is a 2D model with no variation of velocity in the vertical - direction, so it is valid for ice shelves than mainly stread and fast ice streams whose motion is - mainly due to sliding over the bedrock. It does not work in areas where the ice is frozen at the - bedrock as there is almost no sliding in these areas. In these regions we can use a HO model as - there is vertical variations of velocity. This model is valid for most areas except where - patterns are complicated such as grounding lines and rough bed areas. In these areas, we think - only stokes can be used. So instead of relying on only one or the other model, we can try a - different approach and combine them.} -\end{frame} %}}} -\subsection{Penalties} -\begin{frame} \frametitle{Penalty method} %{{{ - \begin{itemize} - \item Only to couple SSA and HO - \item Very stiff spring to penalize differences between degrees of freedom - \end{itemize} - \begin{center} - \begin{tabular}{c c} - \includegraphics[width=0.45\textwidth]{penalized_vel_3} & - \includegraphics[width=0.45\textwidth]{penalties1} - \end{tabular} - \end{center} - - \vspace{1em} - Using penalties to couple models: - \lstinputlisting[basicstyle=\lstbasicfont\tiny,firstnumber=1,firstline=69,lastline=69]{Code/ex1.m} -\end{frame} %}}} -\subsection{Tiling method} -\begin{frame} \frametitle{Domain Decomposition} %{{{ - - \begin{itemize} - \item $\Omega=\Omb \cup \Omr$ - \item $\Omega_S=\Omb \cap \Omr \neq \varnothing $ - \item $ \VEC{u} = \rest{\VEC{u_1}}{\Omb}+ \rest{\VEC{u_2}}{\Omr} - \in \tilde{V} \left(\Omega\right) = \left( \Vb\left(\Omb\right) + \Vr \left(\Omr\right)\right)$ - \end{itemize} - - \begin{center} - \includegraphics[width=0.9\textwidth]{example11} - \vspace{-1em} - \end{center} - - \begin{equation*} - \text{Find } \VEC{u}=\rest{\VEC{u_1}}{\Omb}+ \rest{\VEC{u_2}}{\Omr} \in \tilde{V}, - \end{equation*} - \begin{equation*} - \forall \left(\VEC{v_1},\VEC{v_2}\right) \in \tilde{V} - \hspace{2em} - a\left(\VEC{u_1}+\VEC{u_2},\VEC{v_1}+\VEC{v_2}\right) = l \left(\VEC{v_1}+\VEC{v_2} \right) - \end{equation*} - - \begin{itemize} - \item[$\rightarrow$] Infinite number of solutions for the continuous problem - \end{itemize} - \note{ - Let suppose that $\Omega_1$ and $\Omega_2$ are two subdomains of the domain $\Omega$. These two - subdomains cover the entire domain $\Omega$. We impose that they overlap into a superposition zone - $\Omega_t$. The second domain can be strictly contained into the first one or contain part of the - corder of the global boundary. Any configuration is good as long as there is a transition zone - everywhere between the two models. $\Gamma_1$ and $\Gamma_2$ are the boundaries of the superposition - zone. THe solution u is now sum u1+u2 solutions corresponding to the - two subdomains. Similarly v=v1+v2. The problem is now a(u1+u2,v1+v2)=l(v1+v2). However, there is - a redundancy problem as there is no unique solution to this continuous problem. There is actually - an infinite number of solutions because of the superposition zone. Any couple u1+u2 is possible - as long as the sum remains the same. - } -\end{frame} %}}} -\begin{frame} \frametitle{Discretization} %{{{ - - \begin{center} - \includegraphics[height=0.3\textheight]{1d} - \end{center} - We take advantage of the discretization to avoid the redundancy: - \begin{itemize} - \item[$\rightarrow$] Create one layer of elements in the superposition zone - \end{itemize} - - \vspace{1em} - \begin{center} - \includegraphics[width=0.3\textwidth]{SquareTiling} - \end{center} - \note{ - To overcome this redundancy problem, we now take advantage of the discretization of the problem. - We limit the superposition zone to only one layer of elements. So the first domain is in blue, - the second one in red and they overlap into the yellow superposition zone that contains only one - layer of elements. Each grid is on the boundary $\Gamma_1$ or $\Gamma_2$ (no grids are strictly - inside the superposition zone). Additionnally, we impose homogeneous Dirichlet boundary - conditions to u1 and u2 on $\Gamma_1$ and $\Gamma_2$ respectively to have a continuous solution. With - these constraints, the solution is now unique for the discretized problem and the two - formulations are equivalent. - } -\end{frame} %}}} -\begin{frame} \frametitle{Multi-model formulation} %{{{ - Two different models: $a_1$, $a_2$ and $l_1$, $l_2$ - - \vspace{1em} - Find $\VEC{u}=\rest{\VEC{u_1}}{\Omb}+\rest{\VEC{u_2}}{\Omr} \in \left(\Vbt + \Vrt\right),$ - such that: - \begin{equation*} - \forall \VEC{v}=\rest{\VEC{v_1}}{\Omb}+\rest{\VEC{v_2}}{\Omr}\; \in \left(\Vbt + \Vrt\right) - \end{equation*} - \begin{multline*} - \underbrace{a_1\left(\rest{\VEC{u_1}}{\Omb},\rest{\VEC{v_1}}{\Omb}\right)}_{\color{blue}\text{model 1}} + - \underbrace{a_2\left(\rest{\VEC{u_2}}{\Omr},\rest{\VEC{v_2}}{\Omr}\right)}_{\color{red}\text{model 2}} +\\ - \underbrace{a_2\left(\rest{\VEC{u_1}}{\Omb},\rest{\VEC{v_2}}{\Omr}\right) - + a_1\left(\rest{\VEC{u_2}}{\Omr},\rest{\VEC{v_1}}{\Omb}\right)}_{\text{model coupling}}\\ - = \underbrace{l_1 \left(\rest{\VEC{v_1}}{\Omb}\right)}_{\color{blue}\text{model 1}} + - \underbrace{l_2 \left(\rest{\VEC{v_2}}{\Omr}\right)}_{\color{red}\text{model 2}} - \end{multline*} - - \begin{itemize} \setlength{\itemsep}{5pt} - \item Coupling different mechanical models - \item Easy to implement (local modification of stiffness matrices) - \end{itemize} - - \note{ - We can take advantage of this new formulation to introduce two different mechanical models on the - two subdomains.Here is the weak formulation of the problem. We are now working on a set V that is a union - of two spaces: V1 for the model 1 and V2 for the model 2.\\ - This gives the following weak formulation. We recognize the terms used for the mono-model a11, - l1, a22 and l2. We also have additional terms of coupling that combine the solution u1 with the - test functions v2 and vice versa. These terms exist only on the superposition zone where both - models are used. It is therefore easy to implement this method in an existing code, as the main - modification is to add the coupling terms in the stiffness matrix and only for the superposition - zone.} -\end{frame} %}}} -\subsection{Utilization} -\begin{frame} \frametitle{Flow equation}%{{{1 - {\tt setflowequation} is used to generate the approximation used to compute the velocity - - \begin{itemize} - \item Examples - \end{itemize} - - \lstinputlisting[basicstyle=\lstbasicfont\tiny,firstnumber=1,firstline=61,lastline=63]{Code/ex1.m} - - \begin{itemize} - \item Use {\tt exptool} to create EXP contours - \end{itemize} - - \lstinputlisting[firstnumber=1,firstline=65,lastline=65]{Code/ex1.m} -\end{frame} -%}}} -\begin{frame} \frametitle{Flow equation}%{{{1 - \begin{itemize} - \item To diplay the type of approximation: - \end{itemize} - - \lstinputlisting[firstnumber=1,firstline=67,lastline=67]{Code/ex1.m} - - \begin{center} - \includegraphics[height=0.65\textheight]{macayealpattyn2} - \end{center} -\end{frame} -%}}} - -\begin{frame}[allowframebreaks]{Bibliography}%{{{1 - \bibliographystyle{apalike} - \bibliography{references} - \end{frame}%}}} -%Thanks slide{{{ -\usebackgroundtemplate{\includegraphics[width=\paperwidth,height=\paperheight]{Thanksbg}} -\begin{frame}[plain] - \ghostframe - \begin{center} - \vspace*{-0.1\paperheight} - \hspace*{-0.15\paperwidth}\Huge{Thanks!} - \end{center} - \note{Thank you very much for your attention} -\end{frame} -\usebackgroundtemplate{} -%}}} -\end{document} Index: sm/workshop/2012/Talks/07_Ice_flow_models/Makefile =================================================================== --- /issm/workshop/2012/Talks/07_Ice_flow_models/Makefile (revision 14034) +++ (revision ) @@ -1,37 +1,0 @@ -#Makefile for beamer -TARGET=11_Ice_flow_models - -#export TEXINPUTS=$(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena//: - -all: one - -one: - pdflatex -halt-on-error -file-line-error $(TARGET).tex - open $(TARGET).pdf - -complete: - pdflatex -halt-on-error -file-line-error -draftmode $(TARGET).tex - bibtex $(TARGET) - pdflatex -halt-on-error -file-line-error -draftmode $(TARGET).tex - pdflatex -halt-on-error -file-line-error $(TARGET).tex - open $(TARGET).pdf - -bib: - bibtex $(TARGET) - -links: - ln -s $(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena/*.sty . - ln -s $(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena/general/*.sty . - ln -s $(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena/subtheme/*.sty . - ln -s $(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena/ML/*.sty . - ln -s $(ISSM_DIR)/publications/templates/beamer/Theme/Pasadena/graphics/JPL/*.pdf . - ln -s $(ISSM_DIR)/publications/bibtex/references.bib - -unlinks: - find . -maxdepth 1 -type l -print | xargs rm - -open: - open -a /Applications/Adobe\ Reader\ 9/Adobe\ Reader.app/ $(TARGET).pdf - -clean: - rm -f ${TARGET}.{dvi,ps,pdf,toc,log,aux,out,nav,snm,bbl,blg,vrb} Index: sm/workshop/2012/Talks/07_Ice_flow_models/mcode.sty =================================================================== --- /issm/workshop/2012/Talks/07_Ice_flow_models/mcode.sty (revision 14034) +++ (revision ) @@ -1,280 +1,0 @@ -%% -%% This is file `mcode.sty' -%% -%% It is supposed to help you easily include MATLAB source code -%% into LaTeX document, but have it nicely highlighted, using -%% the great listings package. -%% -%% PLEASE NOTE that this package does nothing but save you from -%% figuring out some configurations in setting up the LISTINGS -%% package. ALL the work is done by that package! Thus, please -%% refer your questions to the listings package documentation. -%% -%% Usage: You have three ways of including your MATLAB code. As -%% environment, as inline object and directly from an external -%% file. -%% -%% 1) Environment: -%% -%% \begin{lstlisting} -%% YOUR CODE HERE -%% \end{lstlisting} -%% -%% -%% 2) Inline object: -%% -%% Bla bla \mcode{CODEFRAGMENT} bla bla. -%% -%% -%% 3) Include external file (in environment form) -%% -%% \lstinputlisting{YOUR-FILE.m} -%% -%% -%% For your convenience this package has the following options: -%% -%% - bw if you intend to print the document (highlighting done -%% via text formatting (bold, italic) and shades of gray) -%% -%% - numbered if you want line numbers -%% -%% - autolinebreaks if you want the package to automatically -%% wrap your code. 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For ex. §text and some math: $x^2$§, which is -%% especially useful in comments for putting nicely typeset -%% equations etc. To get the same colour/style as in the rest -%% of the comment use \mcommentfont, i.e. §\mcommentfont $x^2$§ -%% -%% To change the font used, edit the first line in the "custo- -%% mise below" section. And feel free to edit other things as -%% well. Refer to the documentation of the listings package to -%% see what else you could do. If an extra small font is re- -%% quired, use {\fontfamily{pcr}\fontsize{3}{4.6}\selectfont} -%% in the definition of \lstbasicfont. -%% -%% Author: -%% Florian Knorn | florian@knorn.org | www.florian-knorn.com -%% -%% Version history: -%% 2.2 -- Bugfix (thanks Willi Gerbig!) -%% 2.1 -- Finally automatic detection between end and end -%% 2.0 -- New options for line breaking and literate prog. -%% 1.8 -- Fixed typo in documentation regarding §...§ -%% 1.7 -- Added MATLAB block comment syntax %{ ...... %} -%% 1.6 -- Added some infos, dealing with keyword ``end'' -%% 1.5 -- Tweaked check to see wether textcomp is loaded -%% 1.4 -- Fixed misconfig (mathescape now set to false) -%% 1.3 -- Purely cosmetic (tabs replaced by spaces) -%% 1.2 -- Added \lstset{showstringspaces=false} -%% 1.1 -- Added \mcode command and [final] option -%% 1.0 -- Release - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% D O N ' T T O U C H T H I S % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\def\fileversion{2.2} -\def\filedate{2011/08/26} - -\typeout{-- Package: `mcode' \fileversion\space <\filedate> --} -\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{mcode}[\filedate\space\fileversion] - -% for bw-option -\newif\ifbw -\DeclareOption{bw}{\bwtrue} - -% numbered option -\newif\ifnumbered -\DeclareOption{numbered}{\numberedtrue} - -% final option -\newif\iffinal -\DeclareOption{final}{\finaltrue} - -% autolinebreaks option -\newif\ifautolinebreaks -\DeclareOption{autolinebreaks}{\autolinebreakstrue} - -% literate programming (replace certain characters/relations -\newif\ifuseliterate -\DeclareOption{useliterate}{\useliteratetrue} - -% framed option -\newif\ifframed -\DeclareOption{framed}{\framedtrue} - -\DeclareOption*{% default - \PackageWarning{mcode}{Unknown option `\CurrentOption' !}% -} -\ProcessOptions - -\ifbw\typeout{ - settings optimized for printing (bw formating)} -\else\typeout{ - settings optimized for display (colour formating)}\fi -\ifnumbered\typeout{ - line numbering enabled}\else\fi -\ifuseliterate\typeout{ - literate programming (character replacements) enabled}\else\fi -\ifautolinebreaks\typeout{ - automatic line breaking enabled (careful, buggy!)}\else\fi -\ifframed\typeout{ - framed listings}\else\fi - -% This command allows you to typeset syntax highlighted Matlab -% code ``inline''. 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Index: sm/workshop/2012/Talks/07_Ice_flow_models/todo =================================================================== --- /issm/workshop/2012/Talks/07_Ice_flow_models/todo (revision 14034) +++ (revision ) @@ -1,24 +1,0 @@ -Ice flow models - 2d: SSA - SIA - 3d: SIA - Higher-order - Full-Stokes - SSA collapsed - Coupling models (SSA, HO, FS) - md.flowequation - -Boundary conditions: - Routines SetIceSheetBC, ... - Manually md.diagnostic.spcvx, ... - -Parameters of diagnostic: md.diagnostic - -Thermal model (only 3d) - md.thermal - -Prognostic - md.prognostic - -Transient model - md.transient