%$BI=Bj(B	$B4(NdI9>2%b%G%k$NDj<02=(B
%
%$BMzNr(B	2001-07-09 $BB<EDBYMN(B
%

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

\Dtitle{$B4(NdI9>2%b%G%k$NDj<02=(B}
\Ddate{2001 $BG/(B 07 $B7n(B 11 $BF|(B}
\Dauthor{$BB<EDBYMN(B}
\Dchargrad

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

\begin{document}
% $B%?%$%H%k(B-------------------------------------
\maketitle
%-----------------------------------------------

%$BL\<!(B -----------------------------------------
\tableofcontents
%-----------------------------------------------
\Dparskip
\section{$B$O$8$a$K(B}

$B;d$OI9>230It$N5$8u>r7o$K$h$C$FI9>2$,$I$N$h$&$K?6Iq$&$+(B, 
$B$H$$$&$3$H$K6=L#$r;}$C$F$$$k(B.	

$B6qBNE*$JOC$H$7$F$ONc$($P(B, $BJ?6Q5$29$NJQ2=$,(B
$BI9>2$N<ANL<};Y$K$I$N$h$&$K1F6A$rM?$($F$$$k$+(B
$B$J$I$,$=$l$KEv$?$k(B.
$B$3$N$h$&$J6qBNE*$JLdBj$KBP$7$FCO5e>e$KB8:_$9$k(B
$B3F!9$NI92O!&I9>2$G$N1~Ez<B83$O9T$o$l$F$$$k(B. 
$B$7$+$78=<B$KB8:_$9$kI9>2$N@-<A$rM}2r$9$k$?$a$K$O(B
$B4JC1$JOHAH$G9M$($k$3$H$,M-8z$G$"$k(B($B$+$b$7$l$J$$(B).
$B$=$3$G2>A[E*$JI9>2$H5$8u>r7o$r@_Dj$7$F(B
$B$=$3$G$NI9>2$N4pK\E*$J?6Iq$$(B($BNc$($P(B, $BI98|!&29EY!&N.B.$J$I(B)
$B$K$D$$$FM}2r$9$k(B.

$B:#2s$N(B M $B%<%_$G$OI9>2N.F0%b%G%k$NCf$G:G$b4pK\E*$J%b%G%k$G$"$k(B
$B4(NdI9>2$K$D$$$FCmL\$7(B, $B$3$NI9>2$NDj<02=$r9T$C$?(B.

$B:#8e$N7W2h$H$7$F$O(B

\begin{enumerate}
\item $BDj<02=$K$h$C$FF@$i$l$kJ}Dx<07O$rN%;62=$7$F?tCM7W;;%3!<%I$r:n$k(B
\item $BMQ$$$?%3!<%I$N7k2L$H(B EISMINT $B$,9T$C$?%Y%s%A%^!<%/<B83$HHf3S$9$k(B
\item $B$5$^$6$^$J9=@.J}Dx<0(B($BNc$($P(B
	$B0[J}@-$N$"$kI9$N9=@.J}Dx<0$d(B CO$_2$$B$N9=@.J}Dx<0(B)
	$B$rMQ$$$?$NI9>2$N?6Iq$$$K$D$$$F7W;;$9$k(B.
\end{enumerate}

$B$N=g$K9T$C$F$$$/(B.

\section{$B4(NdI9>2%b%G%k$NDj<02=(B}

\subsection{$B9M$($k7O(B}

$B?^(B \ref{fig:1}$B$N$h$&$J?eJ?J}8~$KL58B$K9-$,$k6u4V$K(B
$BG/4V^>M\NL(B $M_s$ $B$G@.D9$9$kI9BN(B(ice body)$B$r9M$($k(B.
$B$3$NI9BN$OHs%K%e!<%H%s!&Hs05=L!&G.EAF3N.BN(B(Hutter 1983)$B$H$7$F07$&(B.
$BI9BN$N29EY$O==J,$KDc$$$b$N$H$7$FM;2r$O5/$3$i$J$$$H$9$k(B.
$BI9BN$N?eJ?8|$5(B $L$ $B$OI98|(B $H$ $B$KHf$Y$F==J,Bg$-$$$b$N$H$9$k(B.
$B$^$?I9BN$N=E$_$K$h$k4pHWCO7A$NJQ7A$O$J$$$H9M$($k(B.
$B?eJ?J}8~$N:BI8$r(B $x, y$, $B1tD>J}8~$r(B $z$ $B$H$9$k(B.

\begin{figure}[h]
\begin{center}
%\includegraphics[clip,keepaspectratio,height=4cm]{image/ice_sheet.eps}
\Depsf[12cm][]{image/ice_sheet.eps}
\caption{$B7O$N@_Dj(B}
\label{fig:1}
\end{center}
\end{figure}

\subsection{$BI9>2NO3X$N4pACJ}Dx<07O(B}

$B4pACJ}Dx<07O$O<!$N<ANL(B, $B1?F0NL(B, $B%(%M%k%.!<$NJ]B8B'$+$i$J$k(B.
\begin{eqnarray}
	\Ddiv \Dvect{v} &=& 0 		% $BO"B3$N<0(B
			\label{continity:01}\\ 
	0		 &=& 
		\Ddiv \Dvect{\sigma} + \rho \Dvect{g} % $B1?F0J}Dx<0(B 
			\label{moment:01}\\
	\rho \DD{c_p T}{t} &=& 
		- \Ddiv (k \Dgrad T)
		+ \mathrm{tr}(\Dvect{\sigma} \cdot \Dvect{D}) 
		% $B%(%M%k%.!<J]B8B'(B
			\label{enegy:01}
\end{eqnarray}
$BI9BN$NN.F0$,Hs>o$K$f$C$/$j$J$N$G1?F0NLJ]B8B'1&JU$O(B 0 $B$H$9$k(B.
%$BI9$+$i?e$KAjJQ2=$9$k$H$-$K@8$8$k@xG.(B $q$ $B$O4JC1$N$?$a(B, $B$3$3$G$O(B 
%$q = 0$ $B$H$9$k(B.
$B5-9f$N0UL#$O$=$l$>$l<!$NDL$j$G$"$k(B.
\begin{center}
		\begin{tabular}{cl}
	$\Dvect{v}$	 & $BB.EY%Y%/%H%k(B$=(u,v,w)$\\
	$\rho$		 & $BI9$NL)EY(B		 \\
	$\Dvect{\sigma}$ & $B1~NO%F%s%=%k(B		 \\
	$\Dvect{g}$ 	 & $B=ENO2CB.EY(B 		 \\
	$T$		 & $BI929(B			 \\
	$\Dvect{D}$	 & $BOD$_B.EY%F%s%=%k(B 	 \\ 
	$k$	 	 & $BG.EAF3EY(B 		 \\
	$c_p$  		 & $BI9$NHfG.(B 		 \\
		\end{tabular}
\end{center}

$B9=@.J}Dx<0$O0lHLE*$K(B
\begin{equation}
	\Dvect{D} = A(T) \tau^{n-1} \Dvect{\sigma'}
	\label{glen}
\end{equation}
$B$H$7$FI=$9(B(Glen $B$NN.F0B'(B).

$\tau$ $B$OJP:91~NO$N(B 2 $B<!$NITJQNL$GJP:91~NO%F%s%=%k(B 
$\sigma'$$B$rMQ$$$F<!$N$h$&$KI=$9(B.
\begin{equation}
	\tau^2 	= \frac{1}{2} \mathrm{tr}(\Dvect{\sigma'}^2)\\
\end{equation}

$BJP:91~NO%F%s%=%k(B $\Dvect{\sigma'}$$B$H05NO$O<!$N$h$&$KI=$9(B.
\begin{eqnarray}
	\Dvect{\sigma'} &=& p\Dvect{I} + \Dvect{\sigma} \\
	p &=& - \frac{\mathrm{tr}\Dvect{\sigma}}{3}
\end{eqnarray}
$\Dvect{I}$ $B$OC10L9TNs$G$"$k(B.

$B<0(B(\ref{glen})$B$K$*$1$k(B n $B$NCM$O1~NO<B83$d4QB,$GDj$a$i$l$kCM$G$"$j(B, 
$BIT3NDj$NMWAG$b$"$k$,(B, $BJ?6QCM$H$7$FDj?t(B 3 $B$rMQ$$$k(B(Paterson 1994).
$B%Q%i%a!<%?(B $A$ $B$O29EY$N4X?t$G%"%l%K%&%9$N<0(B,
\begin{equation}
	A(T) 	= A_0\exp \left( - \frac{Q}{R T} \right)
\end{equation}
$B$rMQ$$$k(B.
$A_0$ $B$NCM$O9=@.$9$kI9$N0[J}@-$dIT=cJ*$K0MB8$9$k(B. 
$B$3$3$G$O(B $A_0$ $B$O0lDj$H$9$k(B. 
$Q$ $B$O%/%j!<%W3h@-2=%(%M%k%.!<(B, R $B$O5$BNDj?t$G$"$k(B. 

%
% $B%Q%i%a!<%?(B$A(T)$ $B$O<!$N$h$&$KCJ3,E*$KCM$r;}$?$;$k(B.
% 1. A(T) = $B0lDj$H$7$F9M$($k(B
% 2. A(T) T = 263.15 K $B$GJQ2=$5$;$k(B
%   T < 263.15 K
%	Q = 60 kJ A_0 = 	
%%	A $B$NCM$O(B?
%   T > 263.15 K
%	Q =139 kJ
%

$BOD$_B.EY%F%s%=%k(B $\Dvect{D}$ $B$r0J2<$N$h$&$KDj5A$9$k(B.
\begin{equation}
	\Dvect{D} = 
		\frac{1}{2} 
		\left(\Dgrad \Dvect{v} + (\Dgrad \Dvect{v})^T \right)
\end{equation}



% $B6-3&>r7o(B ------------------------------------------------------
\subsection{$B6-3&>r7o(B}
\subsubsection{$B1?F03XE*6-3&>r7o(B}	
$BI9BN$NDlLL(B($z = 0$)$B1?F03XE*$J6-3&>r7o(B($BN.BNN3;R$,JI$r1[$($J$$(B)$B$O(B
$BCO7AE*$JJQ2=$,$J$/M;2r$b5/$-$J$$$N$GI9$NN.=P$O$J$$$N$G(B
\begin{equation}
	w(0) =	0
		%\DP{b}{t} + \Dvect{v} \cdot \Dgrad_H b + \frac{M_b}{\rho}
			\label{bound-k:01}\\
\end{equation}
$B$G$"$k(B. 

$BI=LL(B($z = h$)$B$G$O<!$N$h$&$KI=$9(B.
\begin{equation}
	w(h) 	= 
		\DP{h}{t} + \Dvect{v} \cdot \Dgrad_H h + \frac{M_s}{\rho}
			\label{bound-k:02}\\
\end{equation}
$B$3$3$G(B $\Dgrad_H $ $B$O(B
\begin{equation}
	\Dgrad_H = \left(\DP{}{x}, \DP{}{y}, 0 \right) 
\end{equation}
$B$G$"$k(B. $ M_s $ $B$OI=LL$N@5L#$N<ANL<};Y$G$"$k(B.

\subsubsection{$BNO3XE*6-3&>r7o(B}

$BI=LL$G$OI=LL$K1h$C$?1~NO$OF/$+$J$$$N$G(B
$BI=LL$KBP$9$kK!@~%Y%/%H%k$r(B $\Dvect{n} = n_i$ $B$H$9$k$H<!$N$h$&$KI=$9(B.
\begin{equation}
\left. (\sigma_{ij}n_j  - n_i \sigma_{ij} n_j) \right|_{z = h}  = 0
\end{equation}

% $B2r@b(B----------------------------------------------------------------------
% $B>e<0$O(B $B!XI=LL$KBP$9$k1~NO%Y%/%H%k!Y(B - $B!XI=LL$KBP$9$k1~NO$NK!@~J}8~%Y%/%H%k!Y(B
% $B$rI=$7$F$$$k(B.
% $B$7$+$7:8JUBh(B 2 $B9`$O@uI96a;w$K$h$C$FL5;k$9$k$N$G6a;wE,MQ8e$O(B
% \left. \sigma_{ij} \right|_{z = h} = 0 $B$H9M$($k(B.
% --------------------------------------------------------------------------

$BI9BNA40h$G05NOM;2rE@0J2<$N29EY$J$N$G(B
$BDlLL$G$OI9BN$O3j$Y$i$:8GDj$5$l$F$$$k$H$9$k(B.
\begin{equation}
	u(z=0) =0, \qquad v(z = 0) = 0
\end{equation}

% $B2r@b(B --------------------------------------------------------
% $BI9$,M;2r$7$F$$$k$HGv$$?eKl$,$G$-$k(B.
% $B$=$N$?$a7P83B'$K$h$j(B
% $BDlLL$G$N?eJ?B.EY$,DlLL$G$NQrCG1~NO$N4X?t$H$7$FI=8=$5$l$k(B.
% -------------------------------------------------------------


\subsubsection{$BG.3XE*6-3&>r7o(B}

$BDlLL$G$OCO3LG.N.NL(B $\gamma$ $B$rM?$($k(B.
\begin{equation}
	k \left. \DP{T}{z} \right|_{z=0} 
			= 
	\gamma 
\end{equation}

% $B0J2<$N>r7o$r;H$&$N$OBg>fIW$+!)(B $B$H$$$&M}M3$+$iA40h$G4(NdI9$H@_Dj$7$?(B
% 
% $B05NOM;2r29EY$r1[$($k>l9g$O$=$N29EY$G8GDj$7$F(B
% $B$=$l0J>e$N29EY>e>:$,5/$3$i$J$$$H$9$k(B.

$BI=LL29EY$O4X?t(B $T_s = T(\Dvect{x}, t)$$B$H$7$FE,59M?$($k(B
%\footnote{
%$BBeI=E*$JNc$H$7$FI98|$N4X?t$H$7$F<!$N$h$&$KM?$($k$3$H$b$"$k(B.
%\begin{equation}
%T_s = T_0 - \beta h
%\end{equation}
%$T_0$$B$O(B $z = 0$ $B$G$N4p=`29EY(B,  $\beta$ $BCGG.5$298:N($G$"$k(B.}
.

\subsection{$BD>@~D>9T:BI8$G$NI=<((B}
$BD>@~D>9T:BI8$G$NJ}Dx<07O$O0J2<$N$h$&$KI=8=$5$l$k(B. 

\begin{eqnarray}
&&	\DP{u}{x}  + \DP{v}{y} + \DP{w}{z} 
			= 0 \label{continity:02}\\
&&	\DP{\sigma_{xx}}{x} + \DP{\sigma_{xy}}{y} + \DP{\sigma_{xz}}{z} 
			= 0 \label{moment:02-1}\\
&&	\DP{\sigma_{yx}}{x} + \DP{\sigma_{yy}}{y} + \DP{\sigma_{yz}}{z} 
			= 0 \label{moment:02-2}\\
&&	\DP{\sigma_{zx}}{x} + \DP{\sigma_{zx}}{y} + \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}{ccc}
D_{xx}  & D_{xy} & D_{xz}\\
	& D_{yy} & D_{yz}\\
	&  	 & D_{zz}\\
		\end{array}
	\right) 
			= A(T)\tau^{n-1}
	\left(
		\begin{array}{ccc}
\sigma'_{xx}		& \sigma'_{xy} 		 & \sigma'_{xz}\\
		     	& \sigma'_{yy} 		 & \sigma'_{yz}\\
			&		  	 & \sigma'_{zz}\\
		\end{array}
	\right) 
					\label{flow-law:02}\\
&&	\tau^2 		= \frac{1}{2}
			  ({\sigma'_{xx}}^2 + {\sigma'_{yy}}^2 
				+{\sigma'_{zz}}^2) + ({\sigma'_{xy}}^2 
				+ {\sigma'_{yz}}^2 + {\sigma'_{zx}}^2)\\
&&	\sigma'_{ij}	= \sigma_{ij} + p \delta_{ij} \label{stress:01}\\
&&	p 		= - \frac{\sigma_{xx} + \sigma_{yy} + \sigma_{zz}}{3} 
			\label{unchange:01}\\
&&	\left(
		\begin{array}{ccc}
D_{xx}	& D_{xy} & D_{xz}\\
	& D_{yy} & D_{yz}\\
	&  	 & D_{zz}\\
		\end{array}
	\right) = 
	\left(
		\begin{array}{ccc}
\DP{u}{x} 		& \frac{1}{2}\left(\DP{u}{y}+\DP{v}{x}\right)
						&\frac{1}{2}\left(\DP{u}{z}+\DP{w}{x}\right)\\
		     	&\DP{v}{y} 		&\frac{1}{2}\left(\DP{v}{z}+ \DP{w}{y}\right)\\
			&		  	&\DP{w}{z}\\
		\end{array}
	\right)
				\label{stretch:01}
\end{eqnarray}

\subsection{$B@uI96a;w(B}
$BI9>2$N?eJ?J}8~$H1tD>J}8~$NFCD'E*D9$5$rHf3S$9$k$H(B
$BHs>o$KGv$$(B
\footnote{$B<B:]$KB8:_$9$kI9>2$N>l9g(B, $H/L \sim O(10^{-3})$}
$B$N$G@uI96a;w(B(shallow ice approximation)$B$r;\$9$3$H$G(B
$BJ}Dx<07O$O4JN,2=$5$l$k(B(Hutter 1983).

\subsubsection{$B1?F0NLJ]B8B'(B}

$B$^$:(B, $O((H/L)^2)$ $B$N9`$rL5;k$9$k6a;w$r;\$9(B(Blatter 1995).
$B<0(B(\ref{moment:02-3})$B$G?eJ?J}8~$N1~NO$N8{G[$,(B
$B1tD>J}8~$N8{G[$KHf$Y$FL5;k$G$-$k$N$G(B
\begin{equation}
	\left| \DP{\sigma_{zx}}{x} \right|, 
	\left| \DP{\sigma_{zy}}{y} \right|
	\ll 
	\left| \DP{\sigma_{zz}}{z} \right|
\end{equation}
$B$G$"$k(B.
$B=>$C$F(B, $B1tD>J}8~$N1?F0J}Dx<0$O<!$N$h$&$KI=$5$l$k(B.
\begin{equation}
	\DP{\sigma_{zz}}{z} = \rho  g
\end{equation}

$BBg5$05$r(B 0 $B$H$9$l$P(B, $BI=LL(B(z = h)$B$+$i(B z $B$^$G@QJ,$7$F(B, 
\begin{equation}
	\sigma_{zz} = - \rho  g (h - z)
\end{equation}
$B$,F@$i$l$k(B. 
$BJP:91~NO$NDj5A$+$i(B,
\begin{equation}
	\sigma'_{xx} + \sigma'_{yy} + \sigma'_{zz} = 0
\end{equation}
$B$G$"$k(B. 
$B=>$C$F(B, $\sigma_{xx}, \sigma_{yy}$ $B$O(B
\begin{eqnarray}
	\sigma_{xx} = 2 \sigma'_{xx} + \sigma'_{yy} + \sigma_{zz} 
	\label{sigma_xx}\\
	\sigma_{yy} = 2 \sigma'_{yy} + \sigma'_{xx} + \sigma_{zz}
	\label{sigma_yy}
\end{eqnarray}
$B$H$J$k(B. $B<0(B(\ref{sigma_xx}), (\ref{sigma_yy})
$B$r1?F0J}Dx<0$KBeF~$9$k$3$H$K$h$j(B
\begin{eqnarray}
	\DP{\sigma'_{xy}}{y} 
	+ \left(2 \DP{\sigma'_{xx}}{x}
	+ \DP{\sigma'_{yy}}{x} \right) 
	+ \DP{\sigma'_{xz}}{z} 
	- \rho  g  \DP{h}{x} 
	= 0  \label{moment:03-1}\\
	\DP{\sigma'_{xy}}{x} 
	+ \left(2 \DP{\sigma'_{yy}}{y} 
	+ \DP{\sigma'_{xx}}{y} \right) 
	+ \DP{\sigma'_{yz}}{z} 
	- \rho  g \DP{h}{y} 
	= 0 \label{moment:03-2}
\end{eqnarray}
$B$,F3$+$l$k(B. 

$B<!$K(B $O(H/L)$ $B$N9`$rL5;k$9$k6a;w$r9T$&(B(Blatter 1995).
\begin{eqnarray}
	\left|\DP{\sigma'_{ii}}{x_j}\right| 
		\ll 
	\left| \DP{\sigma_{iz}}{z} \right| 
			\qquad \mathrm{i,j =1, 2} \label{}\\
	\left|\DP{\sigma'_{xy}}{x_i}\right| 
		\ll 
	\left| \DP{\sigma_{iz}}{z} \right| 
			\qquad \mathrm{i =1, 2}\\
	|\sigma'_{ii}|, |\sigma'_{ij}| 
		\ll 
	|\sigma_{iz}| 	\qquad \mathrm{i,j =1,2} \label{shallow:01}
\end{eqnarray}
$B$3$l$K$h$j(B(\ref{moment:03-1}),(\ref{moment:03-2})$B$O(B 
\begin{eqnarray}
	\DP{\sigma'_{xz}}{z} = \rho  g \DP{h}{x} \\
	\DP{\sigma'_{yz}}{z} = \rho  g \DP{h}{y}
\end{eqnarray}
$B$H$J$k(B.

\subsubsection{$B%(%M%k%.!<J]B8B'(B}

$B%(%M%k%.!<J]B8B'$N3H;69`$*$h$SG4@-JQ7A9`$O<!$N$h$&$K6a;w$5$l$k(B.

\begin{eqnarray}
	k \Dlapla T 
		&\simeq& 
	k \DP[2]{T}{z}\\
	\sum_{i,j}D_{ij} \sigma_{ij} 
		&\simeq&
	2(D_{xz} \sigma'_{xz} 
	+ D_{yz} \sigma'_{yz})
\end{eqnarray}

$B$3$l$K$h$j%(%M%k%.!<J}Dx<0$O<!$N$h$&$KI=$5$l$k(B.
\begin{equation}
	\DP{T}{t}	= - \left[u \DP{T}{x} 
				+ v \DP{T}{y} 
				+ w \DP{T}{z} 
			    \right] 
		    	  + \frac{k }{\rho C_p}\DP[2]{T}{z} 
			  + 2(D_{xz}\sigma'_{xz} 
				+ D_{yz} \sigma'_{yz}) 
			 % - \frac{q}{\rho C_p}
\end{equation}

\subsubsection{$B9=@.J}Dx<0(B}

$B9=@.J}Dx<0$K4X78$9$k(B(\ref{unchange:01}), $B<0(B(\ref{stretch:01})$B$O(B
$B@uI96a;w$K$h$j(B
\begin{eqnarray}
	\tau^2  \simeq {\sigma'_{xz}}^2 + {\sigma'_{yz}}^2 \\
	D_{xz}  \simeq  \frac{1}{2}\DP{u}{z},  
		\qquad 
	D_{yz}  \simeq \frac{1}{2}\DP{v}{z}
	%
\end{eqnarray}
$B$H$J$k(B.

\subsubsection{$B6-3&>r7o(B}
$BNO3XE*6-3&>r7o$G$OI=LL$K1h$C$?1~NO$OF/$+$J$$$H$7$?(B.
$B@uI96a;w$N85$G$O(B $B<0(B(\ref{shallow:01})$B$J$N$G(B
$BI=LL$K$*$1$k1~NO$O(B
\begin{equation}
\sigma'_{xz} = 0,  \qquad  \sigma'_{yz} = 0
\end{equation}
$B$H$$$&>r7o$,M?$($i$l$k(B.

\section*{$B;29MJ88%(B}
%\begin{Dreference}
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\item Paerson, W.S.B. 1994. The Physics of glaciers(3rd ed.). Pergamon, Oxford


\end{itemize}
%\end{Dreference}


\end{document}



