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%       2002-01-08           ����ξϤδ���
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\documentclass[a4j,12pt]{jarticle}
\usepackage{Dennou6}

%\usepackage[dvips]{graphicx}
%
\Dtitle{����ɹ����ǥ���꼰��}
\Ddate{2002 ǯ 07 �� 20 ��}
\Dauthor{¼�� ����}
\Dchargrad


\def\thefigure{\thesection.\arabic{figure}}
\def\theequation{\thesection.\arabic{equation}}
\def\thetable{\thesection.\arabic{table}}

\begin{document}
%\renewcommand�ˤ������
%\renewcommand{\thefigure}{\thesection.\arabic{figure}}
%�Ȥ����ͤ˺��������

% �����ȥ�-------------------------------------
\maketitle
%-----------------------------------------------

%�ܼ� -----------------------------------------
\tableofcontents
%-----------------------------------------------

%----------------------------------------------


% ����ɹ����ǥ�------------------------------
\Dparskip
\section{�ͤ����}

  �� \ref{fig:1}�Τ褦�ʿ�ʿ������̵�¤˹�������֤�
  ǯ��޾���� $M_s$ ����Ĺ����ɹ����ͤ���.
  ������ɹ���β��٤Ͻ�ʬ���㤤��ΤȤ���ͻ��Ϲͤ��ʤ�.
  ���Τ褦�����ؤˤ�����ͻ���ʲ���ɹ����
  {\bfseries ����ɹ��}({\itshape cold ice sheet})
  �ȸƤ�.
  ����ɹ���� Hutter 1983 �˽��ä�
  �󰵽���˥塼�ȥ�ή�ΤȤ��ư���.
  ɹ���ο�ʿ�������� $L$ �ϱ�ľ�������� $H$ ��
  ��٤ƽ�ʬ�礭����ΤȤ���.
  �ޤ�ɹ���νŤߤˤ������Ϸ����ѷ��Ϥʤ��ȹͤ���.
  ��ʿ������ $x$, ��ľ������ $z$ �� 2 ����ľ��ľ����ɸ���Ѥ���.

  \begin{figure}[h]
   \begin{center}
    %\includegraphics[clip,keepaspectratio,height=4cm]{image/ice_sheet.eps}
    \Depsf[12cm][]{image/2Dice_sheet.eps}
    \caption{�Ϥ�����}
    \label{fig:1}
   \end{center}
  \end{figure}

% ����
  \pagebreak


\section{ɹ���ϳؤδ�����������}
\subsection{����������}

  ����ɹ���δ����������Ϥ��줾��
  \begin{eqnarray}
	\Ddiv \Dvect{v} &=& 0 		% Ϣ³�μ�
			\label{continity:01}\\ 
	0		 &=& 
		\Ddiv \Dvect{\sigma} + \rho \Dvect{g} % ��ư������ 
			\label{moment:01}\\
	\rho \DD{c_p T}{t} &=& 
		- \Ddiv (k \Dgrad T)
		+ \mathrm{tr}(\Dvect{\sigma} \cdot \Dvect{D}) 
		% Ǯ�ϳ�������
			\label{enegy:01}
  \end{eqnarray}
  �Ǥ���.
% ��������󥰤��̤Ǥ����ȹԤ��٤���
  ɹ����ήư�����ˤ�ä���ʤΤǲ�®�ٹ��̵�뤹��(Hutter 1983).
  ����ΰ�̣�Ϥ��줾�켡���̤�Ǥ���.
  \begin{center}
   \begin{tabular}{cl}
    $\Dvect{v}$	 & ®�٥٥��ȥ�$=(u,v)$  \\
    $\rho$		 & ɹ��̩��		 \\
    $\Dvect{\sigma}$ & ���ϥƥ󥽥�		 \\
    $\Dvect{g}$ 	 & ���ϲ�®�� 		 \\
    $T$		 & ɹ��			 \\
    $\Dvect{D}$	 & �Ĥ�®�٥ƥ󥽥� 	 \\ 
    $k$	 	 & Ǯ��Ƴ�� 		 \\
    $c_p$		 & ɹ����Ǯ		 \\
   \end{tabular}
  \end{center}

  �����������ϰ���Ū��
  %
  \begin{equation}
   \Dvect{D} = A \left(\frac{\tau}{\eta_{0}}\right)^{n-1} \Dvect{\sigma'}
    \label{glen}
  \end{equation}
  %
  �Ȥ���ɽ��(Glen 1955
  \footnote{
  ����θ���Ǥ�ɹ�ι����������� $\eta_{0}$ ��ޤޤʤ�ɽ���򤷤Ƥ���
  . ������, ������Ȥ٤���� $n$ ���ͤ��Ѥ����٤�Ǵ��Ψ
  �μ������Ѥ��ʤ���Фʤ�ʤ��ΤǾ�Ҥ�ɽ�����Ѥ��뤳�Ȥˤ���.
  }).

  $\eta_{0} $ ��ɹ�ι���Ψ�Ǥ���.
  $\tau$ ���к����Ϥ� 2 ���������̤��к����ϥƥ󥽥��
  $\sigma'$���Ѥ��Ƽ��Τ褦��ɽ��.
  \begin{equation}
   \tau^2 	= \frac{1}{2} \mathrm{tr}(\Dvect{\sigma'}^2)\\
  \end{equation}

  �к����ϥƥ󥽥� $\Dvect{\sigma'}$�ϼ��Τ褦��ɽ��.
  \begin{eqnarray}
       \Dvect{\sigma'} = p\Dvect{I} + \Dvect{\sigma} 
  \end{eqnarray}
   $\Dvect{I}$ ��ñ�̹���Ǥ���, $p$ �ϰ��ϤǤ���.
   ���� $p$ ���󰵽̤Ǥ��뤳�Ȥ���ʿ�ѿ�ľ���Ϥ�ɽ����
   \footnote{
   Malvern 1969 �ˤ���
   �˥塼�ȥ�Ǵ��ή�Τˤ����ƥ��ȡ������ξ���������, �⤷����
   �󰵽�ή�ΤǤ�����ϰ��Ϥ�ʿ�ѿ�ľ���ϤǤ���ȸ�����. ������
   ��˥塼�ȥ�ή�Τξ���
   ɬ������ʿ�ѿ�ľ���Ϥ�ɽ����Ȥϸ¤�ʤ�.}.
   \begin{equation}
       p = - \frac{\mathrm{tr}\Dvect{\sigma}}{3}
   \end{equation}
  ��(\ref{glen})�ˤ����� n ���ͤϱ��ϼ¸����¬���������ͤǤ���.
  �ѥ�᡼�� $A$ �ϲ��٤δؿ��ǥ���˥����μ�,
  \begin{equation}
	A 	= A'\exp \left( - \frac{Q}{R T} \right)
  \end{equation}
  ���Ѥ���(Glen 1955).
  $A'$ ���ͤϹ�������ɹ�ΰ��������Խ�ʪ�˰�¸����.
  �����Ǥ� $A'$ �ϰ���Ȥ���. 
  $Q$ �ϥ��꡼�׳��������ͥ륮��, R �ϵ�������Ǥ���. 

  �Ĥ�®�٥ƥ󥽥� $\Dvect{D}$ ��ʲ��Τ褦���������.
  \begin{equation}
	\Dvect{D} = 
		\frac{1}{2} 
		\left(\Dgrad \Dvect{v} + (\Dgrad \Dvect{v})^T \right)
  \end{equation}

\subsubsection*{ľ��ľ���ɸ�Ǥ�ɽ��}
  ľ��ľ���ɸ�Ǥν�ɽ���Ȱʲ��Τ褦�ˤʤ�.

  \begin{eqnarray}
   % Ϣ³�μ�
   & &	\DP{u}{x} + \DP{w}{z} 
		= 0 
		\label{continity:02}\\ 
   % ��ư������(x ��ɸ)
   & &	\DP{\sigma_{xx}}{x} + \DP{\sigma_{xz}}{z} 
		= 0 
		\label{moment:02-1}\\
   % ��ư������(z ��ɸ)
   & &	\DP{\sigma_{zx}}{x} + \DP{\sigma_{zz}}{z} 
		= \rho g 
		\label{moment:02-3}\\
   % Ǯ�ϳؤμ�
   & & 	\rho  c_p \DP{T}{t} 
		= - \rho  c_p(\Dvect{v} \cdot \Dgrad)T 
		  + k \Dlapla T 
		  + \sum_{i,j} D_{ij}\sigma_{ij} 
		%  + q 
		\label{enegy:02}\\
   % ����������
   & &	\left(\begin{array}{cc}
	D_{xx}  & D_{xz}\\
	D_{zx}  & D_{zz}\\
	      \end{array}\right) 
		= A\left(\frac{\tau}{\eta}\right)^{n-1}
	\left(\begin{array}{cc}
	\sigma'_{xx}	& \sigma'_{xz}\\
	\sigma'_{zx}	& \sigma'_{zz}\\
	      \end{array}\right) 
		\label{flow-law:02}\\
   % �к����Ϥ� 2 ����������
   & &	\tau^2 	
		= \frac{1}{2}
		  ({\sigma'_{xx}}^2 + {\sigma'_{zz}}^2) 
		+ {\sigma'_{zx}}^2\\
   % �к����ϥƥ󥽥�������
   & &	\sigma'_{ij}	= \sigma_{ij} + p \delta_{ij} 
		\label{stress:01}\\
   % ���Ϥ������
   & &	p	= - \frac{\sigma_{xx} + \sigma_{zz}}{2} 
		\label{pressure:01}\\
   % ���ٰ�¸�ؿ�
   & & A        = A'\exp \left( - \frac{Q}{R T} \right)\\
   % �ѷ�®�٥ƥ󥽥�������
   & &	\left(\begin{array}{cc}
	D_{xx}	& D_{xz}\\
	D_{zx}	& D_{zz}\\
	      \end{array}\right) 
		= 
	      \left(\begin{array}{cc}
	\DP{u}{x} 	&\frac{1}{2}\left(\DP{u}{z}+\DP{w}{x}\right)\\
	\frac{1}{2}\left(\DP{w}{x}+\DP{u}{z}\right)		&\DP{w}{z}\\
		\end{array}\right)
		\label{stretch:01}
  \end{eqnarray}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ϣ³�μ�(\ref{continity:02})��
% ��(\ref{flow-law:02}), (\ref{stress:01}), (\ref{pressure:01})
% ���鼫��������Ω�ĤΤǷ��, �ѿ�$u, w, T, 
% \sigma_{xx}, \sigma_{xz}, \sigma_{zx}, \sigma_{zz},  
% D_{xx}, D_{xz}, D_{zx}, D_{zz}, 
% \sigma'_{xx}, \sigma'_{xz}, \sigma'_{zx}\sigma'_{zz}, \tau, p $
% 17 ���Ф��Ƽ��� 18 �Ǥ�����Ω��.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% ������� ------------------------------------------------------

\subsection{�������}
\subsubsection{��ư��Ū�������}	
  ɹ��������($z = 0$)�Ǥ�
  \begin{equation}
	w(0) =	0
	\label{besal-velocity-w}\\
  \end{equation}
  �Ǥ���.
  �ޤ�ɽ��($z = h$)�Ǥ�, ǯ��޾���� $M_s$�ˤ���Ϳ�ˤ��
   \begin{equation}
	w(h)	= \DP{h}{t} + u \DP{h}{x} - \frac{M_s}{\rho}
		\label{surface-velocity-w}
   \end{equation}
  �Ȥʤ�.
  ������$ M_s $ �Ͼ��Ȼ��֤˰�¸����Ȥ���
  $M_s = M_s(x, h)$�Ȥ�, ��ξ���ǯ�־����̤�ɽ��.

\subsubsection{�ϳ�Ū�������}
  ɽ�̤Ǥ�ɽ�̤˱�ä����Ϥ�Ư���ʤ��Τ�
  ɽ�̤��Ф���ˡ���٥��ȥ�� $\Dvect{n}= n_i$ �Ȥ���ȼ��Τ褦��ɽ��.
  \begin{equation}
   \left. (\sigma_{ij}n_j  -  
    \sigma_{kl} n_l n_k n_i) \right|_{z = h}  = 0
   \label{surface-stress}
  \end{equation}

  ���̤Ǥϳ�٤餺���ꤵ��Ƥ���Ȥ���Τ�
  \begin{equation}
   u(z=0) =0 \label{besal-velocity-u}
  \end{equation}
  �Ǥ���.


\subsubsection{Ǯ��Ū�������}
  ���̤Ǥ��ϳ�Ǯή�� $\gamma$ ��Ϳ����.
  \begin{equation}
	k \left. \DP{T}{z} \right|_{z=0}
			= 
	- \gamma . \label{besal-thermal-flux}
  \end{equation}
  ɽ�̲���$T_s$�Ͼ��Ȼ��֤δؿ��Ȥ���
  $T_s=T(x, h)$
  �Ȥ���.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\footnote{
% ���ɹ��, ���꡼�����ɹ���Τ褦�ʶ���Ū�ʾ���
% �и������Ѥ���. ���ΰ�Ĥ��ʲ��Ǥ���(Abe 1993).
%\begin{equation}
%  T_s = T_0 - \beta h
%\end{equation}
%$T_0$�� $z = 0$ �Ǥδ�ಹ��,  $\beta$ ��Ǯ������Ψ�Ǥ���.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{��ɹ���}
  ɹ���ο�ʿ�������� $L$ �ϱ�ľ�������� $H$ ����٤ƽ�ʬ������.
  \footnote{�����ǤϽĲ���� $H/L \sim O(10^{-3})$���٤ȹͤ��Ƥ���.}
  ��ä�{\bfseries ��ɹ���}
  ({\itshape shallow ice approximation})(Hutter 1983)
  ���Ѥ���,  �Ϥδ�ά����Ԥ�.
  % ��ɹ����β��⤬�ʤ�.
\subsection{����������}

  ź�� ${*}$ ���դ����ѿ���̵�����̤Ȥ���,
  ���ηϤ���ɽŪ�ʥ�������Ϥ��줾��ʲ��Τ褦��ɽ��.
  \begin{eqnarray*}
   &&(x, z) = (L x_{*}, H z_{*})\\
   && t = t_{0} t_{*}\\
   &&(u, w) = (U u_{*}, W w_{*})\\
   &&(\sigma_{xx}, \sigma_{xz}, \sigma_{zz}) =
    (\Sigma_{xx}{\sigma_{xx}}_{*} , 
    \Sigma_{xz}{\sigma_{xz}}_{*}, 
    \Sigma_{zz}{\sigma_{zz}}_{*}) \\
   &&(\sigma'_{xx}, \sigma'_{xz}, \sigma'_{zz}) =
    (\Sigma'_{xx}{\sigma'_{xx}}_{*} , 
    \Sigma'_{xz}{\sigma'_{xz}}_{*}, 
    \Sigma'_{zz}{\sigma'_{zz}}_{*}) \\
   && A = A_{0} A_{*}\\
   && \rho = \rho_{0} \rho_{*}\\
   && g = g_{0} g_{*}\\
   && T = T_{0} T_{*}\\
   && \tau = \tau_{0} \tau_{*} \\
    && \eta = \eta_{0} \eta_{*}
  \end{eqnarray*}
  �Ĳ���� $\varepsilon \equiv H/L$�Ȥ���.

\subsubsection{Ϣ³�μ�}
  ������¸§(\ref{continity:02})��̵����������.
  %
  \begin{equation}
   \frac{U}{L}\DP{u_{*}}{x_{*}} + \frac{W}{H}\DP{w_{*}}{z_{*}} = 0,
  \end{equation}
  %
  �Ǥ���. 
  �ƹ�Υ������� $U/L, W/H$��Ʊ���٤ʤΤ�
  \begin{eqnarray}
   W = \varepsilon U \label{W}
  \end{eqnarray}
  �Ȥ���.
  Ϣ³�μ���
  %
  \begin{equation}
   \DP{u_{*}}{x_{*}} + \DP{w_{*}}{z_{*}} = 0 \label{shallow-continity}
  \end{equation}
  %
  �Ȥʤ�.
  

\subsubsection{��ư������}
  $\sigma_{xx}$�ϼ�(\ref{pressure:01})����
  \[
  \sigma_{xx} = 2 \sigma'_{xx} + \sigma_{zz}
  \]
  �δط���ߤ����Τ�, ��ư��������
  �ʲ��Τ褦�˽񤭴�������.
  \begin{eqnarray*}
   2 \DP{\sigma'_{xx}}{x} + \DP{\sigma_{zz}}{x} + \DP{\sigma_{xz}}{z} 
		= 0 \\
   \DP{\sigma_{zx}}{x} + \DP{\sigma_{zz}}{z} 
		= \rho g 
  \end{eqnarray*}

  �����μ���̵�����������
  \begin{eqnarray}
   2 \varepsilon \Sigma'_{xx}\DP{{\sigma'_{xx}}_{*}}{x_{*}} 
    +\varepsilon \Sigma_{zz}\DP{{\sigma_{zz}}_{*}}{x_{*}} 
    +\Sigma_{xz}            \DP{{\sigma_{xz}}_{*}}{z_{*}} 
		= 0, \label{moment03-1}\\
     \varepsilon \Sigma_{zx}\DP{{\sigma_{zx}}_{*}}{x_{*}} 
    +\Sigma_{zz}\DP{{\sigma_{zz}}_{*}}{z_{*}} 
		= \rho_{*} g_{*} H \rho g \label{moment03-2} 
  \end{eqnarray}
  �Ȥʤ�. 

  $\Sigma'_{xx}$ �� $\Sigma_{xz}$ �����
  ����������(\ref{flow-law:02})���Ѥ����
  \begin{eqnarray}
   \frac{\Sigma'_{xx}}{\Sigma_{xz}}
    &=& O\left(\frac{D_{xx}}{D_{xz}}\right) \label{ratio_sigma}
  \end{eqnarray}
  �Ǥ���. 
  �����Ǥ����Ǳ��Ϥ� $\sigma_{xz} = \sigma'_{xz}$�δط����Ѥ���.
  �ޤ��Ĥ�®�٤������(\ref{stretch:01})�ȼ�(\ref{W})����
  \begin{eqnarray}
   \frac{D_{xx}}{D_{xz}} 
    &=& \frac{\DP{u}{x}}{\frac{1}{2}\left(\DP{u}{z}+\DP{w}{x}\right)} 
    \nonumber\\
    &=& 
     O\left(\frac{U}{L}\bigg/ \left(\frac{U}{H} + \frac{W}{L}\right)\right) 
     \nonumber\\
    &=& 
     O(\varepsilon) \label{ratio_D}
  \end{eqnarray}
  �Ǥ��뤳�Ȥ��狼��. 
  ��(\ref{ratio_sigma}), ��(\ref{ratio_D})����
  \begin{equation}
   \Sigma'_{xx} = \varepsilon \Sigma_{xz}
  \end{equation}
  �Ȥ���.
  ��äƼ�(\ref{moment03-1})��
  \begin{eqnarray}
      2 \varepsilon^2\Sigma_{xz}\DP{{\sigma'_{xx}}_{*}}{x_{*}} 
    +\varepsilon \Sigma_{zz}\DP{{\sigma_{zz}}_{*}}{x_{*}} 
    +\Sigma_{xz}            \DP{{\sigma_{xz}}_{*}}{z_{*}} 
		= 0  \label{moment03-1'}
  \end{eqnarray}
  �Ȥʤ�.
  �����Ǽ�(\ref{moment03-1'})�����Ĥꤢ������ˤ�
  $\Sigma_{xz}$, $\Sigma_{zz}$�Υ�������Ͼ��ʤ��Ȥ�
  \begin{equation}
   \frac{\Sigma_{xz}}{\Sigma_{zz}} = O(\varepsilon)
  \end{equation}
  �Ǥʤ���Фʤ�ʤ�. ��ä�
  \begin{equation}
   \Sigma_{xz} = \varepsilon \Sigma_{zz}
  \end{equation}
  �Ȥ���. �ʾ��걿ư��������
  \begin{eqnarray}
   2 \varepsilon^2 \DP{{\sigma'_{xx}}_{*}}{x_{*}}
    + \DP{{\sigma_{zz}}_{*}}{x_{*}}
    + \DP{{\sigma_{xz}}_{*}}{z_{*}} = 0 \\
   \varepsilon^2 \DP{{\sigma_{zx}}_{*}}{x_{*}} 
    + \Sigma_{zz}\DP{{\sigma_{zz}}_{*}}{z_{*}} =  \rho_0 g_0 H \rho_{*} g_{*}
  \end{eqnarray}
  �Ȥʤ�. 
  $\varepsilon^2$ �ι��̵�뤷, 
  $\Sigma_{zz}  = \rho_0 g_0 H$�Ȥ����
  ��ư��������
  \begin{eqnarray}
   \DP{{\sigma_{zz}}_{*}}{x_{*}}
    + \DP{{\sigma_{xz}}_{*}}{z_{*}} = 0 \label{moment03-3}\\
   \DP{{\sigma_{zz}}_{*}}{z_{*}} = \rho_{*} g_{*}\label{moment03-4}
  \end{eqnarray}
  �Ȥʤ�.
  %
  ��ư�������α�ľ��ʬ(\ref{moment03-4})����
  ɽ��($z_{*} = h_{*}$)�ˤ�����
  ${\sigma_{zz}}_{*}$ ������� $P_s$ �Ȥ���� 
  ${\sigma_{zz}}_{*} = P_s - \rho_{*} g_{*}(h_{*} - z_{*})$
  �Ǥ���. ������ʿ��ʬ�μ�(\ref{moment03-3})��
  ���������, ��ư��������
  %
  \begin{equation}
   0  =  - \rho_{*} g_{*} \DP{h_{*}}{x_{*}} + \DP{{\sigma_{xz}}_{*}}{z_{*}}
    \label{shallow-moment}
  \end{equation}
  %
  �Ȥʤ�.


\subsubsection{Ǯ�ϳؤμ�}

  Ǯ�ϳؤμ�(\ref{enegy:02})��̵����������Ȱʲ��Τ褦�ˤʤ�.

  \begin{eqnarray}
   \DP{T_{*}}{t_{*}} 
    &+& \frac{U t_0}{L} u_{*} \DP{T_{*}}{x_{*}} 
    + \frac{W t_0}{H} w_{*} \DP{T_{*}}{z_{*}} \nonumber \\
   &=&    \frac{k t_0}{c_p \rho_{0}} \frac{1}{H^2}
    \left(\varepsilon ^2 \DP[2]{T_{*}}{x_{*}} +
     \DP[2]{T_{*}}{z_{*}}    \right) \nonumber\\
   &+& \frac{g H}{c_p T_0} \frac{U t_0}{L}
    \left( \varepsilon^2 \left( \DP{u_{*}}{x_{*}} {\sigma_{xx}}_{*}
			  + \DP{w_{*}}{z_{*}}{\sigma_{zz}}_{*}\right) 
    + \left( \DP{u_{*}}{z_{*}} 
       + \varepsilon^2 \DP{w_{*}}{x_{*}} \right){\sigma_{xz}}_{*}
   \right) 
  \end{eqnarray}

  ������$
  t_0 = \frac{L}{U}
  = H^2\frac{c_p \rho_0}{k}
  = \frac{L}{U}\frac{g_0 H}{c_p T_0}$
  �Ȥ���\footnote{
  ɹ���ΰ�ή����, Ǯ�Ȼ�����, Ǵ�����֤Ϥ��줾��
  \begin{eqnarray*}
   \mathrm{��ή����} \qquad t_0 &=& \frac{L}{U}  \\
    &\sim& \frac{10^6 \mathrm{m}}{10 \mathrm{m/a}} \\
    &\sim& 10^{12} \mathrm{s}\\
   \mathrm{Ǯ�Ȼ�����} \qquad t_0 &=& H^2\frac{c_p \rho_0}{k}\\
    &\sim& (10^3 \mathrm{m})^2
      \frac{2000 \mathrm{J/kg K \cdot 10^3 \mathrm{kg/m^3}}}{2 \mathrm{W/m K}} \\
    &\sim& 10^{12} \mathrm{s}\\
   \mathrm{Ǵ������} \qquad t_0 &=& \frac{L}{U}\frac{g_0 H}{c_p T_0}  \\
    &\sim& \frac{10^6 \mathrm{m}}{10 \mathrm{m/a}} 
     \frac{10 \mathrm{kg m/s^2} \cdot 10^3 \mathrm{m}}
     {2000 \mathrm{J/kg K} \cdot 10 \mathrm{K}} \\
     &\sim& 10^{12} \mathrm{s}
  \end{eqnarray*}
  ��Ʊ���٤Ǥ���.
  }. 
  �ƹ�ˤ����� $O(\varepsilon^2)$ �ι������
  ̵�뤹��� 
  \begin{eqnarray*}
   \DP{T_{*}}{t_{*}} &=&
 		      - \left(u_{*} \DP{T_{*}}{x_{*}}
			 + w_{*} \DP{T_{*}}{z_{*}} \right)
		      + \frac{k}{\rho_{0} c_p}\DP[2]{T_{*}}{z_{*}} 
		      + \frac{1}{\rho_{0} c_p} \DP{u_{0}}{z_{*}} 
		      {\sigma_{xz}}_{*}  \label{shallow-temp}\\
  \end{eqnarray*}
  �Ȥʤ�.

\subsubsection{����������}

  �к����Ϥ� 2 ���������� $\tau$ ��̵�����������
  \begin{eqnarray}
   \tau^2 &=&  
    \frac{1}{2}\left({\sigma'_{xx}}^2 + {\sigma'_{zz}}^2 \right)
    +  {\sigma'_{xz}}^2\nonumber \\
   {\tau_{0}}^2 {\tau_{*}}
    &=& 
    \frac{1}{2}\left({\Sigma'_{xx}}^2 {{\sigma'_{xx}}_{*}}^2
		+ {\Sigma'_{zz}}^2{{\sigma'_{zz}}_{*}}^2 \right)
    + {\Sigma'_{xz}}^2 {{\sigma'_{xz}}_{*}}^{2}  \nonumber \\
   &=&   \frac{1}{2}\varepsilon^2 
    {\Sigma'_{xz}}^2 \left({{\sigma'_{xx}}_{*}}^2 + {{\sigma'_{zz}}_{*}}^2 
		    \right)
    + {\Sigma'_{xz}}^2 {{\sigma'_{xz}}_{*}}^2  \nonumber 
  \end{eqnarray}
  �Ȥʤ�.
  $\varepsilon$ �ι��̵�뤷�� $\tau_{0} = \Sigma'_{xz}$ �Ȥ���.
  ��ä� 2 ���������̤�
  \[
  \tau_{*} = {\sigma'_{xz}}_{*}
  \]
  �Ȥʤ�.
  
  ������������̵�����������
  \begin{eqnarray}
   \left(\begin{array}{cc}
    \DP{u_{*}}{x_{*}}  & 
     \frac{1}{2}\left(\DP{u_{*}}{{z}_{*}} 
		 + \varepsilon^2 \DP{w_{*}}{x_{*}}\right)\\
	  & \DP{w_{*}}{z_{*}}\\
	 \end{array}\right) 
   =  A_{0}A_{*}\left(\frac{\Sigma'_{xz} {\sigma'_{xz}}_{*}}
		 {\eta_{0}\eta_{*}}\right)^{n-1}
   \left(\begin{array}{cc}
    \frac{L}{U}\Sigma'_{xx} {\sigma'_{xx}}_{*}	& 
     \frac{H}{U} \Sigma'_{xz}{\sigma'_{xz}}_{*}\\   &
      \frac{H}{W}\Sigma'_{zz}{\sigma'_{zz}}_{*}\\
	 \end{array}\right) 
  \end{eqnarray}
  �Ȥʤ�. $\Sigma'_{zz}$, $\Sigma'_{xz}$�δط���
  ��(\ref{ratio_sigma}), ��(\ref{ratio_D})����
  $\Sigma'_{xx} = \varepsilon \Sigma'_{xz}$�Ȥ����Τ�Ʊ�ͤ�
  $\Sigma'_{zz} = \varepsilon \Sigma'_{xz}$�Ȥ���.
  �����Ƥ���� $\epsilon^2$ �ι��̵�뤹��ȹ�����������
  \begin{eqnarray}
   \left(\begin{array}{cc}
    \DP{u_{*}}{x_{*}}  & 
     \frac{1}{2}\DP{u_{*}}{{z}_{*}} \\
	  & \DP{w_{*}}{z_{*}}\\
	 \end{array}\right) 
   = A_{*}\left(\frac{{\sigma'_{xz}}_{*}}
		 {\eta_{*}}\right)^{n-1}
   \left(\begin{array}{cc}
     {\sigma'_{xx}}_{*}	& 
    {\sigma'_{xz}}_{*}\\   &
    {\sigma'_{zz}}_{*}\\
	 \end{array}\right) \label{shallow-glen}
  \end{eqnarray}
  �Ȥʤ�. ���ΤȤ� $A_{0}$ ��
  $ A_{0} =\left(\frac{\Sigma'_{xz}}{\eta_{0}}\right)^{1-n} \Sigma_{xz}
    \frac{U}{H}$ �Ȥ���.


\subsection{��ɹ������Ѥ����������}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ��ɹ����ˤ�äƶ�����줿���ؤ�ή��
%
% ������ �� \DP{\sigma_{zz}}{z} ������äƤ���
% ��\DP{\sigma_{zz}}{x}  �� \DP{\sigma_{xz}}{z} ������äƤ���
% 
% �ʾ�� 2 ���ˤĤ��Ƹ��ڤ��ʤ��ƤϤ����ʤ������...
% ���ޤ�����Ƚ񤫤�Ƥ��ʤ��ΤǤ��Ф餯�֤��Ƥ���.
% �ʲ�����ʸ���ɤळ�Ȥ򾩤��
% 
% ��Heinz Blatter. 1995.
%	 Velocity and stress fields in grounded glaciers: 
%	a simple algorithm for including deviatoric stress gradients
%   J.Glaciol 41(138), 333-344
% ��C.J.Van Der Veen. 1989.
%	Force Budget: I. Theory and Numerical Methods
%   J.glaciol 35(119), 53 -- 60	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsubsection{��ư��Ū�������}
  ��ɹ����ˤ�ä��Ѥ�餺
  \begin{eqnarray}
   w(z = 0) &=& 0 \label{kinem-01}\\
   w(z = h) &=& \DP{h}{t} + u \DP{h}{x} - \frac{M_s}{\rho} \label{kinem-02}\\
   M_s &=& M_s(x, h)
  \end{eqnarray}
  �Ǥ���.
  
  \subsubsection{�ϳ�Ū�������}
  
  ɽ�̤Ǥξ��Ǥ��뼰(\ref{surface-stress})��񤭲���.
  $z = h$ �ˤ�����
  \begin{eqnarray*}
   &&   (\sigma_{xx} n_x + \sigma_{xz} n_z) \Dvect{e_1}  
    + (\sigma_{zx} n_x + \sigma_{zz} n_z) \Dvect{e_3}\\
   &&~~~~~~~~~~~~~     -   [ (\sigma_{xx} n_x + \sigma_{xz} n_z) n_x 
    + (\sigma_{zx} n_x + \sigma_{zz} n_z) n_z ]
    (n_x \Dvect{e_1} + n_z \Dvect{e_3})
     =   0
  \end{eqnarray*}
  �Ǥ���. $\Dvect{e_1}, \Dvect{e_3}$ �Ϥ��줾�� $x, z$
  ��������ñ�̥٥��ȥ�Ȥ���. 
  ɽ�̤��Ф���ˡ���٥��ȥ�$\Dvect{n}$ �� $x, z$ ��������ʬ��
  ���줾��$n_x, n_z$�Ȥ���. 
  $n^2_x + n^2_z = 1$��ȤäƤޤȤ��ȼ��Τ褦�˽񤭴�������.
  \begin{eqnarray*}
   \left((\sigma_{xx} - \sigma_{zz})n_x {n_z}^2 
    + \sigma_{xz} n_z(- n^2_x + n^2_z)\right)\Dvect{e_1}  
   + \left((\sigma_{zz} - \sigma_{xx})n^2_x n_x
    + \sigma_{xz} n_x(n^2_x - n^2_z)\right)\Dvect{e_3}  
    =   0 
  \end{eqnarray*}
  ɽ�̤��Ф���ˡ���٥��ȥ����ɹ����θ��Ǥ�
  $n_x / n_z << 1$ �Ǥ���.
  $n_x/n_z = \delta$ �Ȥ����.
  \begin{eqnarray*}
   \left(\delta n^3_z(\sigma_{xx} - \sigma_{zz})
    + \sigma_{xz} n^3_z(- \delta^2 + 1)\right)\Dvect{e_1}  
   + \left(\delta^2 n^3_z(\sigma_{zz} - \sigma_{xx})
    + \sigma_{xz} n^3_z \delta (\delta^2 - 1)\right)\Dvect{e_3}  
    =   0 
  \end{eqnarray*}
  �Ȥʤ�. $\delta, \delta^2, \delta^3$ �ι��̵�뤷,
  $z = h$�ˤ������������Ϥξ���
  \begin{eqnarray}
   \sigma_{xz} \Dvect{e_1}= 0
  \end{eqnarray}
  �Ȥ���.

  ���̤Ǥ��Ѥ�餺
  \begin{eqnarray}
      u(z = 0) &=& 0 
  \end{eqnarray}
  �Ǥ���.

  \subsubsection{Ǯ�ϳ�Ū�������}
  ��ɹ����ˤ�ä��Ѥ�餺
  \begin{eqnarray}
   \left. k \DP{T}{z} \right| &=& - \gamma \\
   T_s &=& T_s(x, h)
  \end{eqnarray}
  �Ǥ���.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagebreak



 \section{����������������֤�����}

  ��ɹ������Ǥ� 2 ����ɹ���β򤯤٤������������ȶ�������Ϳ����.
  ɹ���δ�����������®�پ� $u, w$, ɹ�� $h$ �μ�, Ǯ�ϳؤμ���
  ɽ�����. �����Ǥ�®�پ�$u, w$�����ɹ�� $h$�μ���Ƴ�Ф���.

  \subsection{®�پ�}

  ®�پ� $u$, $w$ ��ɹ�� $h$ �δؿ��Ȥ��ƽ񤭴�������.
  ��ư������(\ref{shallow-moment})�� $h$ ���� $z$ �ޤ���ʬ�����
  $\sigma_{xz}$ ��
  \begin{equation}
   \sigma_{xz} = - \rho g \DP{h}{x}(h -z) \label{shear-stress}
  \end{equation}
  �Ȥʤ�. 
  ����ˤ������������(\ref{shallow-glen})�����г���ʬ��
  �������� $0$ ���� $z$ �ޤ���ʬ����� $u$ ����ޤ�.

  \begin{equation}
   u =  - 2\left(\frac{\rho g}{\eta}\right)^n \left|\DP{h}{x}\right|^{n-1}\DP{h}{x} 
    \int^{z}_{0}A(h - z')^n \Dd z' 
    \label{u}
  \end{equation}

  $w$ ��Ϣ³(\ref{shallow-continity})�μ��� $0$ ���� $z$ �ޤ���ʬ
  ����.

  \begin{equation}
   w(z) = - \int^{z}_{0} \DP{u}{x} \Dd z \label{w}
  \end{equation}



  \subsection{ɹ���μ�}
  ɹ���η� $h$ ����뤿���
  Ϣ³�μ�(\ref{shallow-continity})�� $0$ ����$h$ �ޤ���ʬ��,
  ��ư��Ū�������(\ref{kinem-01}), (\ref{kinem-02}) 
  ����ɹ���λ���ȯŸ�μ��ˤ���.
  
  \begin{equation}
   \DP{h}{t} = -\DP{F}{x} + \frac{M_s}{\rho}\label{thick}
  \end{equation}

  ������ $F$ �� $u$ �Υե�å�����
  \begin{eqnarray}
   F &=&  \int^{h}_{0} u \Dd z \\
   &=& 2 \left(\frac{\rho g}{\eta}\right)^n \left( \DP{h}{x} \right)^{n} 
    \int^{h}_{0} \Dd z \int^{z}_{0} A(h - z)^{n} \Dd z \label{flux}
  \end{eqnarray}
  �Ȥ�����.


  \subsection{�򤯤٤����ȶ������}

  �ʾ�ε������򤯤٤�����ɹ���μ�(\ref{thick})��
  ���٤μ�(\ref{shallow-temp})�Ǥ���, 
  ���μ���򤯤���� $f$, $u$, $w$, $\sigma_{xz}$ ��
  ��(\ref{flux}), (\ref{u}), (\ref{w}), (\ref{shear-stress}) 
  ��ɬ�פǤ���.
  �ޤȤ�Ƶ��Ҥ����

  \begin{eqnarray}
   & &\DP{h}{t} 
    = -\DP{F}{x} + \frac{M_s}{\rho}\\
   & &\DP{T}{t} 
    =  -  \left(u \DP{T}{x} + w \DP{T}{z} \right) 
    + \frac{k}{\rho c_p}\DP[2]{T}{z} 
    +\frac{2}{\rho c_p} A\sigma^{n+1}_{zx} \\
   & & F = \int^{h}_{0} u \Dd z \nonumber\\
   & & ~~   = - 2 (\rho g)^n \left|\DP{h}{x}\right|^{n-1}
    \DP{h}{x}
    \int^{h}_{0} \Dd z 
    \int^{z}_{0} A(h - z)^{n} \Dd z \\
   & & u =  - 2(\rho g)^n \left|\DP{h}{x}\right|^{n-1}
    \DP{h}{x}
    \int^{z}_{0}A(h - z')^n \Dd z'  \\
   & & w = - \int^{z}_{0} \DP{u}{x}\Dd z \\
   & & \sigma_{xz} = - \rho g \DP{h}{x}(h -z) 
  \end{eqnarray}
  
  �����ƶ�������
  \begin{eqnarray}
   M_s = M_s(x, h)\\
   \left. k \DP{T}{z} \right|_{z=0} = - \gamma \\
   T_s = T_s(x, h)
  \end{eqnarray}
  �Ǥ���.


\section*{����ʸ��}
%\begin{Dreference}
\begin{itemize}
\item Blatter, H. 1995. Velocity and stress fields in grounded glaciers:
			a simple algorithm for including deviatoric 
			stress gradients. 
      {\itshape Journal of Glaciology} {\bfseries 41}(138), 333-344
\item Glen, J.W. 1955. The creep of polycrystalline ice. {\itshape
      Proc.R.Soc. London}, Ser.A, {\bfseries 228}, 519-538
\item Hutter, K. 1983. {itshape Theoretical glaciology: 
      material science of ice and 
      the mechanics of glaciers and ice sheets.}
      D. Reidel Publishing Co./Tokyo, Terra Scintific
      Publishing Co.
\item Huybrechts,P., T.Payne, and The EISMINT Intercomparison group. 1996. 
      The EISMINT benchmarks for testing ice-sheet models. 
      {\itshape Annals of Glaciology} {\bfseries 23}, 1--12
\item Malvern, L.E. 1969. {\itshape Introduction to the Mechanics of a
      Continuous Medium}.  Perntice-Hall
\item Paerson, W.S.B. 1994. {\itshape The Physics of glaciers(3rd ed.).} Pergamon, Oxford
\end{itemize}
%\end{Dreference}


\end{document}

