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%	2001-07-25	    ����ξϤ��Խ�
%	2001-10-10	    ��ɹ����������������������˲���
%       2001-11-08          ����ǽ�Ϸ�ǡ�Τ��ṽ����ľ��
%                              ������ǥ�(�꼰��) �� �ǽ����֤�Ǻ�
%			       z ��ɸ�Ϥ�ɽ��
%                              Υ����ǥ롦�¸�������̤Υɥ�����Ȥ�    
%			       
%       2002-01-08           ����ξϤδ���
%                            ��Ω�Ƥθ�ľ��

\documentclass[a4j,12pt]{jarticle}
\usepackage{Dennou6}

%\usepackage[dvips]{graphicx}
%
\Dtitle{����ɹ����ǥ���꼰��}
\Ddate{2001 ǯ 11 �� 09 ��}
\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��(ice body)��ͤ���.
����ɹ�Τ���˥塼�ȥ��󰵽̡�Ǯ��Ƴή��(Hutter 1983)�Ȥ��ư���.
ɹ�Τβ��٤Ͻ�ʬ���㤤��ΤȤ���ͻ��ϵ�����ʤ��Ȥ���.
ɹ�Το�ʿ���� $L$ ��ɹ�� $H$ ����٤ƽ�ʬ�礭����ΤȤ���.
�ޤ�ɹ�ΤνŤߤˤ������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{����������}

�����������ϼ��μ�����¸§, ��ư����¸§, Ǯ�ϳؤμ�����ӹ�������������ʤ�.
\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}
ɹ�Τ�ήư�����ˤ�ä���ʤΤǱ�ư����¸§���դ� 0 �Ȥ���.
%ɹ���������Ѳ�����Ȥ�����������Ǯ $q$ �ϴ�ñ�Τ���, �����Ǥ� 
%$q = 0$ �Ȥ���.
����ΰ�̣�Ϥ��줾�켡���̤�Ǥ���.
\begin{center}
		\begin{tabular}{cl}
	$\Dvect{v}$	 & ®�٥٥��ȥ�$=(u,v,w)$\\
	$\rho$		 & 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(T) \tau^{n-1} \Dvect{\sigma'}
	\label{glen}
\end{equation}
�Ȥ���ɽ��(Glen ��ήư§).

$\tau$ ���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󥽥� $\Dvect{\sigma'}$�Ȱ��Ϥϼ��Τ褦��ɽ��.
\begin{eqnarray}
	\Dvect{\sigma'} &=& p\Dvect{I} + \Dvect{\sigma} \\
	p &=& - \frac{\mathrm{tr}\Dvect{\sigma}}{3}
\end{eqnarray}
$\Dvect{I}$ ��ñ�̹���Ǥ���.
 ��(\ref{glen})�ˤ����� n ���ͤϱ��ϼ¸����¬���������ͤǤ���.
% �Գ�������Ǥ⤢�뤬, ʿ���ͤȤ������ 3 ���Ѥ���(Paterson 1994).
�ѥ�᡼�� $A$ �ϲ��٤δؿ��ǥ���˥����μ�,
\begin{equation}
	A(T) 	= A_0\exp \left( - \frac{Q}{R T} \right)
\end{equation}
���Ѥ���.
$A_0$ ���ͤϹ�������ɹ�ΰ��������Խ�ʪ�˰�¸����. 
�����Ǥ� $A_0$ �ϰ���Ȥ���. 
$Q$ �ϥ��꡼�׳��������ͥ륮��, R �ϵ�������Ǥ���. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% �ѥ�᡼��$A(T)$ �ϼ��Τ褦���ʳ�Ū���ͤ��������.
% 1. A(T) = ����Ȥ��ƹͤ���
% 2. A(T) T = 263.15 K ���Ѳ�������
%   T < 263.15 K
%	Q = 60 kJ A_0 = 	
%	A ���ͤ�?
%   T > 263.15 K
%	Q =139 kJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�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}
   & &	\DP{u}{x} + \DP{w}{z} 
		= 0 
		\label{continity:02}\\
   & &	\DP{\sigma_{xx}}{x} + \DP{\sigma_{xz}}{z} 
		= 0 
		\label{moment:02-1}\\
   & &	\DP{\sigma_{zx}}{x} + \DP{\sigma_{zz}}{z} 
		= \rho g 
		\label{moment:02-3}\\
   & & 	\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_{zz}\\
	      \end{array}\right) 
		= A(T)\tau^{n-1}
	\left(\begin{array}{cc}
	\sigma'_{xx}	& \sigma'_{xz}\\
			& \sigma'_{zz}\\
	      \end{array}\right) 
		\label{flow-law:02}\\
   & &	\tau^2 	
		= \frac{1}{2}
		  ({\sigma'_{xx}}^2 + {\sigma'_{zz}}^2) 
		+ {\sigma'_{zx}}^2\\
   & &	\sigma'_{ij}	= \sigma_{ij} + p \delta_{ij} 
		\label{stress:01}\\
   & &	p	= - \frac{\sigma_{xx} + \sigma_{zz}}{2} 
		\label{pressure:01}\\
   & &	\left(\begin{array}{cc}
	D_{xx}	& D_{xz}\\
		& D_{zz}\\
	      \end{array}\right) 
		= 
	      \left(\begin{array}{cc}
	\DP{u}{x} 	&\frac{1}{2}\left(\DP{u}{z}+\DP{w}{x}\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��.


% ������� ------------------------------------------------------
\subsection{�������}
\subsubsection{��ư��Ū�������}	
ɹ�Τ�����($z = 0$)�Ǥ�ɹ��ή�ФϹͤ��ʤ��Τ�
\begin{equation}
	w(0) =	0
		%\DP{b}{t} + \Dvect{v} \cdot \Dgrad_H b + \frac{M_b}{\rho}
			\label{bound-k:01}\\
\end{equation}
�Ǥ���.
�ޤ�ɽ��($z = h$)�Ǥ�, ��̣�μ��̼��� $M_s$�ˤ���Ϳ�ˤ��
   \begin{equation}
	w(h)	= \DP{h}{t} + u \DP{h}{x} - \frac{M_s}{\rho}
		\label{bound-k:02}
   \end{equation}
�Ȥʤ�.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%��3 ����ɽ���ξ���
%\begin{equation}
%	w(h) 	= 
%		\DP{h}{t} + \Dvect{v} \cdot \Dgrad_H h + \frac{M_s}{\rho}
%			\label{bound-k:02}\\
%\end{equation}
%������ $\Dgrad_H $ ��
%\begin{equation}
%	\Dgrad_H = \left(\DP{}{x}, \DP{}{y}, 0 \right) 
%\end{equation}
%�Ǥ���. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
������$ M_s $ ��$M_s = M_s(x, h)$�Ȥ���.

\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
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ����
% �弰�� 
%��ɽ�̤��Ф�����ϥ٥��ȥ�� - ��ɽ�̤��Ф�����Ϥ�ˡ�������٥��ȥ��
% ��ɽ���Ƥ���.
% ������������ 2 �����ɹ����ˤ�ä�̵�뤹��ΤǶ��Ŭ�Ѹ��
% \left. \sigma_{ij} \right|_{z = h} = 0 �ȹͤ���.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
���̤Ǥϳ�٤餺���ꤵ��Ƥ���Ȥ���Τ�
\begin{equation}
	u(z=0) =0 %, \qquad v(z = 0) = 0
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ɹ��ͻ�򤷤Ƥ�����������줬�Ǥ���.
% ���Τ���и�§�ˤ��
% ���̤Ǥο�ʿ®�٤����̤Ǥ����Ǳ��Ϥδؿ��Ȥ���ɽ�������.
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Ǯ��Ū�������}
���̤Ǥ��ϳ�Ǯή�� $\gamma$ ��Ϳ����.
\begin{equation}
	k \left. \DP{T}{z} \right|_{z=0} 
			= 
	- \gamma
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% �ʲ��ξ���Ȥ��Τ�����פ��� �Ȥ�����ͳ��������Ǵ���ɹ�����ꤷ��
% 
% ����ͻ���٤�ۤ�����Ϥ��β��٤Ǹ��ꤷ��
% ����ʾ�β��پ徺��������ʤ��Ȥ���.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
ɽ�̲���$T_s$��$T_s=T(x, h)$
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\footnote{
%��ɽŪ����Ȥ���ɹ���δؿ��Ȥ��Ƽ��Τ褦��Ϳ���뤳�Ȥ⤢��.
%\begin{equation}
%T_s = T_0 - \beta h
%\end{equation}
%$T_0$�� $z = 0$ �Ǥδ�ಹ��,  $\beta$ ��Ǯ������Ψ�Ǥ���.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{��ɹ���}

\subsection{����������}
ɹ���ο�ʿ�������� $L$ �α�ľ�������� $H$ ��٤ƽ�ʬ������
\footnote{�����ǤϽĲ���� $H/L \sim O(10^{-3})$���٤ȹͤ��Ƥ���.}
�Τ���ɹ���(shallow ice approximation)��ܤ����Ȥ�
��ά�������(Hutter 1983). �Ĳ���� $\varepsilon \equiv H/L$�Ȥ���.

\subsubsection{������¸§}

������¸§(\ref{continity:02})��̵����������.
��ʿ®�٥�������, ����ӱ�ľ®�٥�������� $U$, $W$ �Ȥ����
\begin{equation}
 \frac{U}{L}\DP{u}{x} + \frac{W}{H}\DP{w}{z} = 0
\end{equation}

�Ǥ���. 
�ƹ�Υ�������򤽤줾��$U/L, W/H$�Ȥ����
��ư����¸§��

\begin{equation}
 \DP{u}{x} + \DP{w}{z} = 0
\end{equation}

�Ȥʤ�. $W$ �Υ�������� $W = \varepsilon U$ �Ƚ񤱤�. 


\subsubsection{��ư����¸§}

�к����� $\sigma'_{xx}, \sigma'_{xz}, \sigma'_{zz}$�Υ��������
���줾�� $\Sigma'_{xx}, \Sigma'_{xz}, \Sigma'_{zz}$�Ȥ����
����������(\ref{flow-law:02})���Ĥ�®�٤������(\ref{stretch:01})����

\begin{eqnarray}
\frac{A(T) \tau^{n-1}\sigma'_{xx}}
 {A(T) \tau^{n-1}\sigma'_{xz}} 
 &=&
 \frac{D_{xx}}{D_{xz}}  \nonumber\\
%
 \frac{\Sigma'_{xx}}{\Sigma'_{xz}} 
 &\sim& 
\frac{\frac{W}{H}}
          {\frac{1}{2}\left(\frac{U}{H} 
		       + \frac{W}{L}\right)}\nonumber \\
 &\sim&
 \frac{\varepsilon}{1 + \varepsilon^2} \nonumber
\end{eqnarray}
�Ȥʤ�. $\Sigma'_{zz}/\Sigma'_{xz}$ ��Ʊ�ͤ˵�ޤ�.
�ʾ��� $\Sigma'_{xx}, \Sigma'_{zz}$ �� $\Sigma'_{xz}$ 
��Ȥäưʲ��Τ褦��ɽ�����.

\begin{eqnarray}
\Sigma'_{xx} \sim \varepsilon \Sigma'_{xz} \\
\Sigma'_{zz} \sim \varepsilon \Sigma'_{xz} 
\end{eqnarray}

��ư������(\ref{moment:02-1}), (\ref{moment:02-3})
��嵭�ǻȤ�줿���������$\sigma_{zz}$�Υ��������
$\Sigma_{zz}$ �Ȥ���̵����������.

��������(\ref{moment:02-1})�α����� 1 ��� $\sigma_{xx}$��
��(\ref{pressure:01})����
\[
 \sigma_{xx} = 2 \sigma'_{xx} + \sigma_{zz}
\]
��Ȥä�
\[
 2 \DP{\sigma'_{xx}}{x} + \DP{\sigma_{zz}}{x} + \DP{\sigma_{xz}}{z} 
		= 0 
\]
�Ȥ���. 

\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  \nonumber\\
 \varepsilon \Sigma'_{xz} \DP{\sigma_{zx}}{x} 
  + \Sigma_{zz} \DP{\sigma_{zz}}{z} = \rho_0 g_0 H \rho g \nonumber
\end{eqnarray}

�Ȥʤ�. $\rho_0$, $g_0$ �� $\rho$, $g$�Υ�������Ǥ���.
$\Sigma'_{xz}$, $\Sigma_{zz}$�Υ�������Ϥ��줾��
\begin{eqnarray}
 \Sigma'_{xz} = \rho_0 g_0 H \varepsilon \nonumber\\
 \Sigma_{zz}  = \rho_0 g_0 H  \nonumber
\end{eqnarray}
�Ǥ���Ȥ���ȱ�ư��������
\begin{eqnarray}
 2 \varepsilon^2 \DP{\sigma'_{xx}}{x}
  + \DP{\sigma_{zz}}{x}
  + \DP{\sigma'_{xz}}{z} = 0 \\
 \varepsilon^2 \DP{\sigma_{zx}}{x} 
  + \DP{\sigma_{zz}}{z} = \rho g
\end{eqnarray}
�Ȥ�����. $\varepsilon^2$ �ι��̵�뤹��ȱ�ư��������
\begin{eqnarray}
\DP{\sigma_{zz}}{x}
  + \DP{\sigma'_{xz}}{z} = 0 \\
\DP{\sigma_{zz}}{z} = \rho g
\end{eqnarray}
�Ȥʤ�. �� 2 ���ˤ�����ɽ��($z = h$)�ˤ�����
 $\sigma_{zz}$ �������(�絤��) $P_s$ �Ȥ���� 
$\sigma_{zz} = P_s - \rho g(h - z)$
�ʤΤ��� 1 ������������

\begin{equation}
 0  =  - \rho g \DP{h}{x} + \DP{\sigma_{xz}}{z}
\end{equation}

�Ȥʤ�.


\subsubsection{Ǯ�ϳؤμ�}

���� $t$ �Υ�������� $t_0$, ���� $T$ �Υ�������� $T_0$ 
�Ȥ���Ǯ�ϳؤμ�(4.18)��̵����������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} \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}

�ƹ�ˤ����� $O(\varepsilon)$, $O(\varepsilon^2)$ �Υ������ι��
̵�뤹��. ����ˤ�ä�Ϳ������Ǯ�ϳؤμ���
\begin{eqnarray*}
\DP{T}{t} &=&  
 		      - \left(u \DP{T}{x} + w \DP{T}{z} \right)
		      + \frac{k}{\rho c_p}\DP[2]{T}{z} 
		      + \frac{1}{\rho c_p} \DP{u}{z} \sigma_{zx} \\
\end{eqnarray*}
�Ǥ���.

\subsubsection{����������}

�к����Ϥ� 2 ���������� $\tau$ �Υ������� $\tau_0$ ��
5.1.2 ����Ѥ����к����ϤΥ��������Ƴ�������
\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}}  \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��̵�뤹���$\tau_0 = \Sigma'_{xz}$.
��ä� 2 ���������̤�
\[
 \tau = \sigma'_{xz}
\]
�Ǥ���.
��äƹ�����������
\begin{eqnarray}
	\left(\begin{array}{cc}
	D_{xx}  & D_{xz}\\
		& D_{zz}\\
	      \end{array}\right) 
		= A(T){\sigma'_{xz}}^{n-1}
	\left(\begin{array}{cc}
	\sigma'_{xx}	& \sigma'_{xz}\\
			& \sigma'_{zz}\\
	      \end{array}\right) 
\end{eqnarray}
�Ȥʤ�. 



\subsubsection{��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	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��ɹ����ˤ�äƻ������꼰�ϰʲ��Τ褦�ˤʤ�.
\begin{eqnarray}
 &&   \DP{u}{x} + \DP{w}{z} = 0 \label{shallow-continity} \\
% \DP{h}{t} + \DP{uh}{x} + \frac{M_s}{\rho} = 0 \\
 &&   0  =  - \rho g \DP{h}{x} + \DP{\sigma_{xz}}{z}
	\label{shallow-moment}  \\
 &&   \DP{T}{t} = 
  - \left(u \DP{T}{x} + w \DP{T}{z} \right)
  + \frac{k}{\rho g}\DP[2]{T}{z} 
  + \frac{1}{\rho g}\DP{u}{z} \sigma_{zx}  \label{shallow-temp}\\
 && 	\left(\begin{array}{cc}
	\DP{u}{x}  & \frac{1}{2}\left(\DP{u}{z} \right)\\
		& \DP{w}{z}\\
	      \end{array}\right) 
		= A(T){\sigma'_{xz}}^{n-1}
	\left(\begin{array}{cc}
	\sigma'_{xx}	& \sigma'_{xz}\\
			& \sigma'_{zz}\\
	      \end{array}\right)
	\label{shallow-glen}
\end{eqnarray}
\pagebreak

\subsection{��ɹ������Ѥ����������}

ɽ�̤˱�ä����Ϥ���ɹ����ˤ����� $z$ ������ʬ��
$x$ ������ʬ����٤ƽ�ʬ�������ȹͤ����

\begin{equation}
\sigma_{xz}|_{z=h} = 0
\end{equation}
��ɽ���Ǥ���.
����ˤ�ä���ɹ������Ǥζ�������
\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)\\
u(z = 0) &=& 0 \\
\left. \sigma_{xz}\right|_{z = h} &=& 0 \\
\left. k \DP{T}{z} \right| &=& - \gamma \\
T_s &=& T_s(x, h)
\end{eqnarray}
�Ȥ���.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{����������������֤�����}

��ɹ������Ǥ� 2 ����ɹ���β򤯤٤������������ȶ�������Ϳ����.
��������ɹ����®�پ�$u, w$, ɹ�� $h$����뤿���
���˷����֤���������.

\subsection{®�پ�}

®�پ� $u$, $w$ ��ɹ���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�. 
�������������(\ref{shallow-glen})����������. 
�Ĥ�®�٤��������$D = \frac{1}{2}\DP{u}{z} $���ä��Τ�
$u$ �ϰʲ��Τ褦��ɽ�����.

\begin{equation}
u =  - 2(\rho g)^n \left|\DP{h}{x}\right|^{n-1}\DP{h}{x} 
	\int^{z}_{0}A(T)(h - z')^n \Dd z' 
	\label{u}
\end{equation}

$w$ �� $u$ ��Ϣ³�μ������ޤ�.

\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}

������ $F$ �ϼ�(\ref{u})
\begin{eqnarray}
F &=&  \int^{h}_{0} u \Dd z \\
  &=& 2 (\rho g)^n \left( \DP{h}{x} \right)^{n} 
	\int^{h}_{0} \Dd z \int^{z}_{0} A(T)(h - z)^{n} \Dd z \label{flux}
\end{eqnarray}
�Ȥʤ�.

\pagebreak

\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(T)\sigma^{n+1}_{zx} \\
  & & F = - 2 (\rho g)^n \left|\DP{h}{x}\right|^{n-1}
          \DP{h}{x}
	  \int^{h}_{0} \Dd z 
	  \int^{z}_{0} A(T)(h - z)^{n} \Dd z \\
  & & u(z) =  - 2(\rho g)^n \left|\DP{h}{x}\right|^{n-1}
          \DP{h}{x}
	  \int^{z}_{0}A(T)(h - z')^n \Dd z'  \\
  & & w(z) = - \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}



\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. 
			Journal of Glaciology 41(138), 333-344
\item Hutter, K. 1983. Theoretical glaciolog: 
		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. 
		Annals og Glaciology 23, 1--12
\item Paerson, W.S.B. 1994. The Physics of glaciers(3rd ed.). Pergamon, Oxford
\end{itemize}
%\end{Dreference}


\end{document}




