%#! platex master_thesis-M2.tex
\chapter{$B$O$8$a$K(B}

\section{$B8&5f$NGX7J(B}

\subsection{$BBg5$$N%9%Z%/%H%k$N7A@.(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1305 3047,scale=.1]{./intro/b_and_s_1983.png}
 \caption[FGGE $B%G!<%?%;%C%H$N%9%Z%/%H%k2r@O$N7k2L(B. [Boar and Shepherd
 (1983) Fig.1(a)\rbrack ]{FGGE $B%G!<%?%;%C%H$N%9%Z%/%H%k2r@O$N7k2L(B. $B2#<4$OGH?t(B,
 $B=D<4$O%(%M%k%.!<%9%Z%/%H%k$r<($9(B. $B0lE@:?@~$ODj>o@.J,$r<($7(B, $BDcGH?tB&$G(B
 $BBn1[$7$F$$$k(B. $BE@@~$OHsDj>o@.J,$r<($7(B, $B9bGH?tB&$GBn1[$7$F$$$k(B. $B<B@~$O$=(B
 $B$l$i$N9g7W$N%9%Z%/%H%k$r<($7$F$$$k(B. [Boar and Shepherd (1983)
 Fig.1(a)]}  
 \label{fig:b_and_s_1983}
\end{figure}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1339 1005,scale=.15]{./intro/2d-spectra.png}
 \caption[2 $B<!850lMMEyJ}@-MpN.$K$*$1$k%(%M%k%.!<%9%Z%/%H%k$N357A$H%(%M%k(B
 $B%.!<$d%(%s%9%H%m%U%#!<$NN.$l$N35G0?^(B. [Vallis (2006) Fig.8.7\rbrack ]{2
 $B<!850lMMEyJ}@-MpN.$K$*$1$k%(%M%k%.!<%9%Z%/%H%k$N357A$H%(%M%k%.!<$d%(%s(B
 $B%9%H%m%U%#!<$NN.$l$N35G0?^(B. $B%(%M%k%.!<$d%(%s%9%H%m%U%#!<$r$"$k69$$BS0h(B
 $B$KCmF~$7$?$H$-(B, $B$=$l$h$j$bDcGH?tB&$K$O%(%M%k%.!<$,%+%9%1!<%I$7(B, $B%(%M%k(B
 $B%.!<47@-NN0h$,7A@.$5$l(B, $B%9%Z%/%H%k$N79$-$O(B $k^{-5/3}$ $B$H$J$k(B. $B0lJ}(B, $B9b(B
 $BGH?tB&$K$O(B, $B%(%s%9%H%m%U%#!<$,%+%9%1!<%I$7(B, $B%(%s%9%H%m%U%#!<47@-NN0h$,(B
 $B7A@.$5$l(B, $B%9%Z%/%H%k$N79$-$O(B $k^{-3}$ $B$H$J$k(B\cite{Vallis}}.  
 \label{fig:2d-spectra}
\end{figure}

$BCO5e$K$*$1$kBg5$$d3$MN$NBg5,LO1?F0$O(B, $B$7$P$7$P(B 2 $B<!85N.BNE*$J@-<A$r;}$D(B
$B$3$H$,CN$i$l$F$$$k(B. $B$=$NM}M3$H$7$F(B, $B4v2?3XE*$KGv$$$3$H(B, $B2sE>7O$G$"$k$3$H(B,
$BL)EY@.AX$r$7$F$$$k$3$H$J$I$,5s$2$i$l$k(B. $B<B:]$K(B, $B$$$/$D$+$N4QB,;v(B
$B<B$+$i(B, $BCO5eBg5$$NBg5,LO1?F0$N(B 2 $B<!85N.BNE*$J@-<A$,3NG'$5$l$F$$$k(B. $B$=$N(B
$B0lNc$H$7$F(B, Boar and Shepherd (1983) ($B0J2<(B, BS83 $B$HN,5-$9$k(B)$B$K$h$C$F9T$o(B
$B$l$?(B FGGE $B%G!<%?%;%C%H$N2r@O7k2L$r?^(B
\ref{fig:b_and_s_1983} $B$K<($9(B. $B$3$l$K$h$k$H(B, $BEl@>GH?t(B $k$ $B$,(B 14 $B$+$i(B 25
$B$N4V$G$O(B,$B%9%Z%/%H%k$N79$-$,GH?t(B $k$ $B$N(B $-3$ $B>h$K$J$C$F$$$k$3$H$,$o$+$k(B.
$B@~7?$N7905IT0BDjO@$K$h$k$H(B, $BJQ7AH>7B6aK5$NGH?t$r;}$D>qMp$,:GB.@.D9>qMp$G$"$k$?$a(B, $BJQ(B
$B7AH>7B$"$?$j$GJ?6QN.$+$i%(%M%k%.!<$,CmF~$5$l$k(B. $B7O$N6/@)(B
$B$OGH?t$,(B 8 $B$+$i(B 10 $B$NBS0h$G$J$5$l$k$H9M$($i$l$k$?$a(B, $k^{-3}$ $B$N%9%Z%/%H(B
$B%k$,7A@.$5$l$F$$$k$N$O(B, $B6/@)$5$l$F$$$kBS0h$h$j$b9bGH?tB&$G$"$k(B. $B0lJ}(B, 2
$B<!850lMMEyJ}@-MpN.$N6/@)GH?t$h$j$b9bGH?tNN0h$K7A@.$5$l$k%(%s%9%H%m%U%#!<(B
$B47@-NN0h$G$O(B, $B%(%M%k%.!<%9%Z%/%H%k$N79$-$O(B $k^{-3}$ $B$H$J$k$3$H$,CN$i$l$F(B
$B$$$k(B\cite{K67}($B35G0?^$r?^(B \ref{fig:2d-spectra} $B$K<($9(B). $B$=$3$G(B, $BBg5$$K$*(B
$B$1$k(B $k^{-3}$ $B$N%9%Z%/%H%k$O(B 2 $B<!850lMMEyJ}@-MpN.$K$*$1$k%(%s%9%H%m%U%#!<(B
$B47@-NN0h$N%9%Z%/%H%k$HN`;w$N$b$N$G$"$k$H9M$($i$l$F$$$k(B.  

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1440 1080,scale=.20]{./intro/n_and_g_1985.png}
 \caption[GASP $B$N9R6u5!4QB,$K$h$k7w3&LLIU6a$N%G!<%?$N%9%Z%/%H%k2r@O$N7k2L(B.
 [Nastrom and Gage (1985) Fig.3 $B$r0lIt2C9)(B\rbrack]{GASP $B$N9R6u5!4QB,$K$h(B
 $B$k7w3&LLIU6a$N%G!<%?$N%9%Z%/%H%k2r@O$N7k2L(B. $B2#<4$OGH?t(B,$B=D<4$O%(%M%k%.!<(B
 $B%9%Z%/%H%k$r<($9(B. $B:8$+$i(B, $BEl@>Iw(B, $BFnKLIw(B, $B290L$N%9%Z%/%H%k$r<($7(B, $BFnKL(B
 $BIw$H290L$N%9%Z%/%H%k$OJX59E*$K$=$l$>$l(B 1 $B7e$:$D9bGH?tB&$K$:$i$7$F$+$+$l(B
 $B$F$$$k(B. $BD>@~$O(B, $k^{-3}$ $B$H(B $k^{-5/3}$ $B$N79$-$r<($7$F$$$k(B. $B?'$,EI$i$l$?(B
 $BBS0h$O(B, $BCO5eBg5$$K$*$1$k%(%M%k%.!<CmF~GH?tBS0h$r<($9(B.[Nastrom and Gage
 (1985) Fig.3 $B$r0lIt2C9)(B]}  
 \label{fig:n_and_g_1985}
\end{figure}

$B0lJ}(B, 2 $B<!850lMMEyJ}@-MpN.$NM}O@$G$O@bL@$G$-$J$$Bg5$$N4QB,7k2L$b$"$k(B. $B?^(B
\ref{fig:n_and_g_1985} $B$O(B Nastrom and Gage (1985) $B$K$h$C$F<($5$l$?(B, GASP
$B$N9R6u5!4QB,$K$h$k7w3&LLIU6a$N%G!<%?$N%9%Z%/%H%k2r@O$N7k2L$G$"$k(B. $BAm4Q5,(B
$BLO%9%1!<%k(B($10^{3}\sim 10^{4} \si{km}$)$B$K$*$$$F$O(B, BS83 $B$HF1(B 
$BMM$K(B $k^{-3}$ $B$N%9%Z%/%H%k$K$J$C$F$$$k$3$H$,J,$+$k(B. $B$7$+$7(B, $BAm4Q5,LO$h$j(B
$B$b>.$5$J%9%1!<%k$G$O(B, $k^{-5/3}$ $B$N%9%Z%/%H%k$r<($7$F$$$k(B. $BCO5eBg5$$NJQ7AH>7B$O$*$h(B
$B$=(B $10^{4}\si{km}$ $B$N%9%1!<%k$G$"$k$?$a(B, $k^{-5/3}$ $B$N%9%Z%/%H%k$,7A@.$5(B
$B$l$F$$$k$N$O(B, $B%(%M%k%.!<CmF~GH?t$h$j$b9bGH?tB&$G$"$k(B. 2 $B<!850lMMEyJ}@-Mp(B
$BN.$NM}O@$G$O(B, $B%(%M%k%.!<CmF~GH?t$h$j$bDcGH?tB&$G%(%M%k%.!<47@-NN0h$N%9%Z(B
$B%/%H%k$,(B $k^{-5/3}$ $B$H$J$k$3$H$,CN$i$l$F$$$k$?$a(B, $B$3$N%9%Z%/%H%k$N7A@.$N%a(B
$B%+%K%:%`$O(B, 2 $B<!850lMMEyJ}@-MpN.$NM}O@$G$O@bL@$9$k$3$H$,$G$-$J$$(B. 

$k^{-3}$ $B$H(B $B9bGH?tB&$N(B $k^{-5/3}$ $B$N%9%Z%/%H%k$O$=$N8e$NBg5$Bg=[4D%b%G%k(B
$B$d5R4Q2r@O%G!<%?$rMQ$$$?8&5f$K$*$$$F$bIQHK$KF@$i$l$F$$$k(B
\cite{HTO08}\cite{TTZ11}\cite{BES13}. $B$=$3$G(B, $B$3$l$OBg5$$NBg(B
$B5,LO1?F0$NIaJWE*$JFCD'$G$"$k$H$5$l(B, $B$7$P$7$P(B Nastrom-Gage $B%9%Z%/%H%k(B($B0J(B
$B2<(B, N-G $B%9%Z%/%H%k$HN,5-$9$k(B)$B$H8F$P$l$k(B. N-G $B%9%Z%/%H%k$,7A@.$5$l$k%a%+(B
$B%K%:%`$O(B, 2$B<!850lMMEyJ}@-MpN.$NM}O@$H$OJL$N4QE@$+$i@bL@$9$kI,MW$,$"$k(B,
$B$H9M$($i$l$F$-$?(B. $B$3$N@bL@$KBP$9$k%"%W%m!<%A$O0lHL$K(B 3 $B$D$KJ,N`$G$-$k(B
\cite{FK15} :  
\begin{enumerate}
 \item $BBPN.$K$h$C$F@8@.$5$l$?%(%M%k%.!<$NBg$-$J%9%1!<%k$X$N5U%+%9%1!<%I(B
       $B$K$h$C$F7A@.(B(Lilly 1983).
 \item $BH/;6N.$K$h$C$F7A@.(B($BNc$($P(B, Koshyk and Hamilton 2001).
 \item $BBg$-$J%9%1!<%k$+$i$N%(%M%k%.!<$ND>@\E*$J%+%9%1!<%I(B(Tung and
       Orland 2003). 
\end{enumerate}
1. $B$N%?%$%W$N@bL@$O8=:_$G$OH]Dj$5$l$F$$$k(B. 2. $B$N%?%$%W$N@bL@$O8=:_$b$C$H(B
$B$b$i$7$$@bL@$H$5$l$F$$$k(B. 3. $B$N%?%$%W$N@bL@$K$D$$$F$O(B, $BIaJWE*$+H]$+$O:#(B
$B$N$H$3$m7hCe$7$F$$$J$$(B. $BK\8&5f$G$O$3$N(B 3. $B$N%?%$%W$N@bL@$K$D$$$F>\$7$/8&(B
$B5f$9$k(B. 

\subsection{2 $BAX%b%G%k$K$h$k:F8=(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 716 492,scale=.3]{./intro/t-o.png}
 \caption{Tung and Orland (2003) $B$G<gD%$5$l$kCO9UN.MpN.$N%(%M%k%.!<%9%Z(B
 $B%/%H%k$N7A@.%a%+%K%:%`$N35G0?^(B. [Tung and Orland (2003) \cite{TO03} Fig.2(c)] }
 \label{fig:t_and_o_2}
\end{figure}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1440 1080,scale=.20]{./intro/flux_hikaku.png}
 \caption[Tung and Orland (2003) $B$G<($5$l$?%(%M%k%.!<%U%i%C%/%9$H%(%s%9(B
 $B%H%m%U%#!<%U%i%C%/%9$N7k2L(B. [Tung and Orland (2003) Fig.7 $B$r0lIt2C(B
 $B9)(B\rbrack ]{$B>e$O%(%M%k%.!<%U%i%C%/%9(B $\Pi_{\mathrm{E}}(k)$ $B$NGH?tJ,I[$r(B
 $B<($7(B, $B2<$O%(%M%k%.!<%U%i%C%/%9$KGH?t$N(B 2 $B>h$r$+$1$?$b$N(B
 $k^2\Pi_{\mathrm{E}}(k)$ $B$H%(%s%9%H%m%U%#!<%U%i%C%/%9(B
 $\Pi_{\mathrm{Q}}(k)$$B$NBg$-$5$NHf3S$r<($9(B. $B2<$N?^$+$i(B, $B%(%M%k%.!<%U%i%C(B
 $B%/%9$,%(%s%9%H%m%U%#!<%U%i%C%/%9$h$j$bBn1[$9$kNN0h$,8=$l$k$3$H$,$o$+$j(B,
 $B$=$NGH?t$h$j$b9bGH?tB&$K$*$$$F%(%M%k%.!<%9%Z%/%H%k$N79$-$,(B $k^{-5/3}$
 $B$K$J$k$HH`$i$O<gD%$7$F$$$k(B. [Tung and Orland (2003) \cite{TO03} Fig.7
 $B$r0lIt2C9)(B]} \label{fig:t_and_o_7} 
\end{figure}

3. $B$N%?%$%W$N@bL@$O(B Tung and Orland (2003)($B0J2<(B, TO03 $B$HN,5-(B
$B$9$k(B) $B$K$h$C$FDs>'$5$l$?(B. $BH`$i$O(B, $BI8=`E*$J=`CO9U(B 2 $BAX%b%G%k(B($\beta$ $B%A%c(B
$B%M%k$N=`CO9U(B 2 $BAX%b%G%k(B)$B$rMQ$$$F(B N-G $B%9%Z(B
$B%/%H%k$r:F8=$7(B, $B=`CO9U(B 2 $BAX%b%G%k$K$*$1$k%(%M%k%.!<$H%(%s%9%H%m%U%#!<$N(B
$BN.$l$K$D$$$F0J2<$N$h$&$K<gD%$7$F$$$k(B:  
\begin{enumerate}
 \renewcommand{\labelenumi}{$B<gD%(B(\arabic{enumi})}
 \item $B=`CO9U(B 2 $BAX%b%G%k$G$O(B, $B%(%s%9%H%m%U%#!<47@-NN0h$K$*$$$F!V1#$l$?%((B
       $B%M%k%.!<%+%9%1!<%I(B(hidden energy cascade)$B!W$,$"$k(B($B?^(B \ref{fig:t_and_o_2}).
 \item $B6/@)GH?t$h$j$b9bGH?tB&$NNN0h$K$*$$$F(B, $B%(%M%k%.!<%+%9%1!<%I$,%(%s%9%H(B
       $B%m%U%#!<%+%9%1!<%I$h$jBn1[$9$kGH?tBS0h$,=P8=$9$k(B($B?^(B
       \ref{fig:t_and_o_7}). $B$=$NBS0h$K$*$$$F(B, 
       $B0J2<$NITEy<0$,@.N)$9$k(B:
       \begin{equation}
        k^2\Pi_{\mathrm{E}}(k) - \Pi_{\mathrm{Q}}(k) > 0. \label{eq:futousiki}
       \end{equation}
       $B$3$3$G(B, $\Pi_{\mathrm{E}}(k)$ $B$O%(%M%k%.!<%U%i%C%/%9(B,
       $\Pi_{\mathrm{Q}}(k)$ $B$O%(%s%9%H%m%U%#!<%U%i%C%/%9$rI=$9(B.
       (\ref{eq:futousiki}) $B$rK~$?$9NN0h$K$*$$$F(B, $B%(%M%k%.!<47@-NN0h%9%Z(B
       $B%/%H%k(B $k^{-5/3}$ $B$,7A@.$5$l$k(B.
 \end{enumerate}
$B2a5n$N8&5f$G(B, 2 $B<!85(B Navier-Stokes $BJ}Dx<07O$K$*$$$F(B, $B%(%M%k%.!<$r(B
$BCmF~$7$F$$$kBS0h0J30$NA4$F$NGH?t$G(B $k^2\Pi_{\mathrm{E}}(k) -
\Pi_{\mathrm{Q}}(k) < 0$ $B$H$J$k$3$H$,<($5$l$F$$$k(B(Gkioulekas and Tung
(2007)\cite{GT07}). $B=`CO9U(B 2 $BAX%b%G%k$K$*$$$F$O(B, Gkioulekas
(2014)\cite{G14} $B$K$*$$$F(B, (\ref{eq:futousiki}) $B$,@.N)$9$kGH?tBS$,B8:_$9(B
$B$k2DG=@-$,$"$k$3$H$,M}O@E*$K<(:6$5$l$F$$$k$N$_$G(B, $BIaJWE*$K(B
(\ref{eq:futousiki}) $B$,@.N)$9$k$3$H$O>ZL@$5$l$F$$$J$$(B.

$B$^$?(B, TO03 $B$G$O=`CO9U(B 2 $BAX%b%G%k$G$N?tCM<B83$N7k2L$,<($5$l$F$$$k$,(B, $B%G!<(B
$B%?2r@O$,IT==J,$G$"$j(B, $B>e5-$N<gD%$,K\Ev$K@5$7$$$N$+5?Ld$,;D$kE@$,$$$/$D(B
$B$+$"$k(B.

\section{$B8&5f$NL\E*(B}

$B0J>e$NGX7J$rF'$^$(K\8&5f$G$O(B, TO03 $B$GDs>'$5$l$?=`CO9U(B 2 $BAX%b%G%k$K$*$1(B
$B$k(B N-G $B%9%Z%/%H%k$N7A@.%a%+%K%:%`$,IaJWE*$G$"$k$+H]$+$K$D$$$F5DO@$9$k(B.
$B$=$N$?$a$K(B, $B=`CO9U(B 2 $BAX%b%G%k$rMQ$$$??tCM%7%_%e%l!<%7%g%s$r9T$$(B, N-G $B%9(B
$B%Z%/%H%k$,7A@.$5$l$k$+$I$&$+$r3N$+$a$?(B. $B$^$?(B, TO03 $B$G$O>\:Y$K<($5$l$F$$(B
$B$J$+$C$?GH?t6u4VFb$K$*$1$k%(%M%k%.!<$d%(%s%9%H%m%U%#!<$NN.$l$K4X$9$k2r@O(B
$B$H9M;!$r9T$$(B, (\ref{eq:futousiki}) $B$,@.N)$9$k$+H]$+$r3NG'$7$?(B. 

$BK\O@J8$N9=@.$O0J2<$NDL$j$G$"$k(B: $BBh(B 2 $B>O$G?tCM7W;;$KMQ$$$?%b%G%k$K$D$$$F(B
$B=R$Y$k(B. $B;YG[J}Dx<0$NF3=P$K$D$$$F$O(B, $BIUO?(B\ref{chap:governing_equation}$B$r(B
$B;2>H$5$l$?$$(B. $BBh(B 3 $B>O$G$OGH?t6u4V$K$*$1$k;YG[J}Dx<0$N=EMW$J@-<A(B, $BFC$KJ](B
$BB8B'$K$D$$$F3NG'$9$k(B. $B$3$A$i$b(B, $B>ZL@$K$D$$$F$OIUO?(B\ref{chap:wave-eq}$B$r;2(B
$B>H$5$l$?$$(B. $BBh(B 4 $B>O$G$O(B,  (\ref{eq:futousiki}) $B$N@.N)2DH]$K$D$$$F?t3XE*(B
$B$J>ZL@$r;n$_$k(B. $BBh(B 5 $B>O$G$O(B, $BK\8&5f$G9T$C$??tCM<B83$H2r@O$N7k2L$r<($9(B.
$BBh(B 6 $B>O$G$O(B(\ref{eq:futousiki}) $B$N@.N)$K$D$$$F$NJL$N4QE@$+$i$N5DO@$r9T(B
$B$$(B, $B7kO@$r=R$Y$k(B.  

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