%#! platex master_thesis-M2.tex
\chapter{$B?tCM7W;;$N7k2L(B} \label{chap:result}

$BK\>O$G$O(B, $BK\8&5f$G9T$J$C$??tCM7W;;$N7k2L$r$^$H$a$k(B. TO03 $B$N7k2L$H$NHf3S(B
$B$r9T$J$$(B, $B:G8e$K$^$H$a$H9M;!$r=R$Y$k(B.

\section{$B7W;;@_Dj(B}

\begin{table}[htb]
  \begin{center}
    \caption{$B7W;;@_Dj(B}
    \begin{tabular}{l||c|c|c|c} \bhline{1.5pt}
      & case1 & case2 & case3 & case4\\ \hline\hline
     $\Delta\nu$ $B$NCM(B & 0 & $-5.280\times10^{-17}$
             & $5.280\times10^{-17}$ & $1.560\times10^{-16}$ \\\hline
     $B2rA|EY(B & $1024^2$ & \multicolumn{3}{c}{$512^2$} \\ \hline
     $B@ZCGGH?t(B& $341$ & \multicolumn{3}{c}{$170$}  \\ \hline
     $B2r@O;~4V(B & $t=1800 \sim 2000$ & \multicolumn{3}{c}{$t=1300 \sim 1500$} \\
    \bhline{1.5pt}
    \end{tabular}
   \label{tab:setting}
  \end{center}
\end{table}

$BI=(B \ref{tab:setting} $B$K7W;;@_Dj$r<($9(B. case1 $B$G$O(B, Larichev and Held
(1995) $B$d(B Tung and Orland (2003) $B$HF1$8$/(B, $B>eAX$H2<AX$GG4@-78?t$,F1$8>l(B
$B9g$N7W;;@_Dj$G$"$k(B. case2 $B$G$O(B, $B2<AX$NG4@-78?t$,%<%m$H$J$k$h$&$K(B
$B@_Dj$7$?(B. case3 $B$H(B case4 $B$G$O(B, $B2<AX$NG4@-78?t$,>eAX$NG4@-78?t$KHf$Y(B
$B$F$=$l$>$l(B 2 $BG\(B, 3 $BG\$K$J$kCM$K@_Dj$7$?(B. $B$3$l$i$O(B, Gkioulekas (2014)\cite{G14} $B$G2<(B
$BAX$NG4@-78?t$,>eAX$NG4@-78?t$h$j$bBg$-$$>l9g(B, (\ref{eq:futousiki}) $B$,@.(B
$BN)$9$k2DG=@-$,$"$k$H=R$Y$i$l$F$*$j(B, $B$=$l$r8!>Z$9$k$?$a$N@_Dj$G$"$k(B.

case1 $B$O(B, $BBh(B \ref{chap:cal_summary} $B>O$G=R$Y$?=i4|CM$+$i%9%?!<%H$7(B,
$t \in [0,1000]$ $B$N;~4V$O2rA|EY(B $256^2$ $B$G(B, $B%9%T%s%"%C%W$r$7(B,
$t \in (1000,1500]$ $B$N;~4V$O2rA|EY(B $512^2$, $t \in (1500,2000]$ $B$N;~4V$O2rA|EY(B
$1024^2$ $B$G7W;;$r9T$J$$(B, $t \in [1800,2000]$ $B$N7W;;7k2L$r2r@O$KMQ$$$?(B. case2,
case3, case4 $B$K$D$$$F(B $B$O(B, case1 $B$N(B $t=1000$ $B$NCM$r=i4|CM$H$7$FMQ$$(B,
$t=1500$ $B$^$G2rA|EY(B $512^2$ $B$G;~4VH/E8$5$;$?(B. $B2r@O$K$O(B $t \in
[1300,1500]$ $B$N7W;;7k2L$rMQ$$$?(B.   

\section{$B=i4|$N%(%M%k%.!<$NH/E8(B}

\begin{figure}[htb]
 \centering 
 \includegraphics[bb=0 0 783 594,scale=.35]{./results/result1/ektot_initial.png}
 \caption[$B=i4|$N%(%M%k%.!<H/E8$NMM;R(B. ]{$B=i4|$N%(%M%k%.!<H/E8(B. $B=D<4$O%(%M(B
 $B%k%.!<$NBg$-$5(B($\log$$B%9%1!<%k(B), $B2#<4$O;~9o(B. $BE@@~$N79$-$O(B, $B7905IT0BDj$N(B
 $B@~7?O@$+$iF3=P$5$l$k(B, $B:GB.@.D9N((B $\alpha_{\mathrm{max}}$ $B$N79$-$r(B, $B0lE@(B
 $B:?@~$N79$-$O(B, $t=30$ $B$K$*$1$k%(%M%k%.!<%9%Z%/%H%k$N%T!<%/$NGH?t(B
 ($k = \sqrt{2}$)$B$K$*$1$k@.D9N((B$\alpha_{k=\sqrt{2}}$ $B$r<($9(B.}  
 \label{fig:ektot_initial}
\end{figure}

\begin{figure}[htb]
 \centering 
 \includegraphics[bb=0 0 620 654,scale=.35]{./results/result1/ek-production.png}
 \caption[$B@~7?H/E8$N;~4V$K$*$1$k@5L#$N%(%M%k%.!<CmF~N((B $k_x
 Uk_d^2\hat{\psi}(\bvec{k}) \hat{\tau}^{\ast}(\bvec{k})$ $B$N%9%Z%/%H%k9=(B
 $BB$(B.]{$B@~7?H/E8$N;~4V$K$*$1$k@5L#$N%(%M%k%.!<CmF~N((B $k_x
 Uk_d^2\hat{\psi}(\bvec{k}) \hat{\tau}^{\ast}(\bvec{k})$ $B$N%9%Z%/%H%k9=(B
 $BB$(B. $B2#<4$,?eJ?J}8~$NGH?t(B $k$, $B=D<4$,;~9o(B $t$ $B$rI=$9(B.}  
 \label{fig:develop_initial}
\end{figure}

$B$3$N%b%G%k$G$O(B, $BGH?t6u4V$GHs>o$K>.$5$JGH?t$K0M$i$J$$?6I}$H%i%s%@%`$J0LAj$r(B
$B=i4|>r7o$H$7$FM?$($F$$$k$N$G(B, $BH/E8$O7905IT0BDj$N@~7?@.D9$K=>$&$H9M$((B
$B$i$l$k(B. $B?^(B \ref{fig:ektot_initial} $B$K(B $t \in [0,250]$ $B$N%(%M%k%.!<$NH/E8(B
$B$r<($9(B. $B%(%M%k%.!<$N;X?tH!?tE*@.D9$O(B $t\lesssim 50$ $B$G8+$i$l$k(B. $B@~7?O@$K(B
$B$h$k$H(B, $B:GB.@.D9N((B $\alpha_\mathrm{max}$ $B$O(B $k_x=0.64 k_d\sim 6.4,
k_y=0$ $B$N$H$-(B, $\alpha_{\mathrm{max}}\sim 0.205$ $B$G$"$k$+$i(B, $B%(%M%k%.!<(B
$B$,(B e $BG\$K$J$k;~4V$O(B $1/\alpha_{\mathrm{max}}\sim 4.89$ $B$G$"$k(B. $B$7$+$7(B,
$B$3$N;~4V(B($t\lesssim 50$)$B$K$*$1$k(B 2 $B<!85%(%M%k%.!<%9%Z%/%H%k$N%T!<%/$O(B
$(k_x,k_y)=(1,1)$ $B$K$"$k$N$G(B, $B%T!<%/$NA4GH?t$O(B $k=\sqrt{2}$ $B$G$"$k(B. $B$3$NGH(B
$B?t$r;}$C$?>qMp$N@.D9N($O(B $\alpha_{k=\sqrt{2}} \sim 0.049$ $B$G$"$k(B
\footnote{$B@.D9N($NF3=P$O(B, $BIUO?;2>H(B.}. $B?^(B \ref{fig:ektot_initial} $B$+$i(B,
$B=i4|$NH/E8$O@.D9N((B $\alpha_{k=\sqrt{2}}$ $B$GH/E8$7$F$$$k$3$H$,$o$+$k(B. $B$D$^$j(B,
$t\lesssim 50$ $B$K$*$1$k7O$NH/E8$O7905IT0BDj$K$h$C$F5/$-$F$$$k(B.

$B@5L#$N%(%M%k%.!<CmF~9`(B $U k_d^2 (\partial\psi/\partial x)$ $B$NGH?t%9%Z%/(B
$B%H%k$r?^(B \ref{fig:develop_initial} $B$K<($9(B. $B7905IT0BDj$N@~7?O@$GFCD'$E$1(B
$B$i$l$kH/E8$r<($9(B $t\lesssim 50$ $B$G$O(B, $B$3$N%9%Z%/%H%k$O(B $k\approx k_d=10$
$B6aK5$h$j$bDcGH?tNN0h$GM-0U$JCM$r;}$C$F$$$k(B. $B$7$+$7(B, $t\gtrsim 50$ $B$G$O(B,
$B%9%Z%/%H%k$N%T!<%/$O$h$jDcGH?tB&$K%7%U%H$7$F$$$/(B. $B$3$l$O(B, $B=g05%(%M%k%.!<(B
$B$N5U%+%9%1!<%I$K$h$C$F=g05%(%M%k%.!<$,DcGH?tB&$K=8@Q$9$k$?$a$K5/$3$k$b$N(B
$B$G$"$k$H9M$($i$l$k(B. $B$7$?$,$C$F(B, $t\gtrsim 50$ $B$OHs@~7?9`$,Bn1[$7$?H/E8CJ(B
$B3,$H9M$($i$l(B, $B<B:]$KHs@~7?9`$,@~7?9`$rBn1[$7$F$$$k(B($B?^>JN,(B). 

\section{case1 $B$N7k2L(B}

\subsection{$B%(%M%k%.!<$H%(%s%9%H%m%U%#!<$NH/E8(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1440 810,scale=.2]{./results/result1/ektot.png}
 \caption[case1 $B$N%(%M%k%.!<$NH/E8(B. ]{case1 $B$N%(%M%k%.!<$NH/E8(B. $B@D?'$,A4(B
 $B%(%M%k%.!<(B, $B\t?'$,=g05%b!<%I$N%(%M%k%.!<(B, $BNP?'$,7905%b!<%I$N%(%M%k%.!<(B
 $B$r<($9(B.}  
 \label{fig:ektot_result1}
\end{figure}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1440 810,scale=.2]{./results/result1/enstot.png}
 \caption[case1 $B$N%(%s%9%H%m%U%#!<$NH/E8(B.]{case1 $B$N%(%s%9%H%m%U%#!<$NH/(B
 $BE8(B. $B@D?'$,A4%(%s%9%H%m%U%#!<(B, $B\t?'$,=g05%b!<%I$N%(%s%9%H%m%U%#!<(B, $BNP?'(B
 $B$,7905%b!<%I$N%(%s%9%H%m%U%#!<$r<($9(B.}  
 \label{fig:enstot_result1}
\end{figure}

$B?^(B \ref{fig:ektot_result1} $B$H?^(B \ref{fig:enstot_result1} $B$K(B case1 $B$N%(%M(B
$B%k%.!<$H%(%s%9%H%m%U%#!<$N;~4VJQ2=$r$>$l$>$l<($9(B. $B@~7?H/E8$,=*$o$C$?8e$O(B,
$B%(%M%k%.!<$O=g05%b!<%I$,(B, $B%(%s%9%H%m%U%#!<$O7905%b!<%I$,Bn1[$7$F$$$k(B. $B%9(B
$B%T%s%"%C%W$N8e$K$I$A$i$b$"$kCM$N$"$?$j$G?6F0$7$F$*$j(B, $BE}7WE*J?9U>uBV$KC#(B
$B$7$F$$$k$H9M$($i$l$k(B. 

\subsection{$B120L>l$NMM;R(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 798
 944,scale=.3]{./results/result1/vor.png}
 \caption[$t=1935$ $B$N;~9o$K$*$1$k120L>l$NMM;R(B.]{$t=1935$ $B$N;~9o$K$*$1$k(B
 $B120L>l$NMM;R(B. $B:8>e$,=g05%b!<%I(B, $B1&>e$,7905%b!<%I(B, $B:82<$,>eAX(B, $B1&2<$,2<(B
 $BAX$N120L$r<($9(B.}  
 \label{fig:vor_result1}
\end{figure}

case1 $B$N(B $t=1935$ $B$K$*$1$k120L>l$NMM;R$r?^(B \ref{fig:vor_result1} $B$K<($9(B.
$B=g05%b!<%I$N120L>l$NMM;R$O(B 2 $B<!850lMMEyJ}@-MpN.$N120L>l$NMM;R$H$h$/;w$F(B
$B$*$j(B, $B6u4V$N=j!9$K<4BP>N$J12$,E@:_$9$k(B. $B0lJ}(B, $B7905%b!<%I$N120L>l$O120L$N(B
$B6/$$$H$3$m$,KON.$7$N$h$&$K8=$l$k(B. $B$3$N7k2L$O(B, Larichev and Held
(1995) $B$H@09gE*$J7k2L$G$"$k(B. $B>eAX$H2<AX$N120L$O6&$K7905%b!<%I(B
$B$N120L>l$K;w$F$*$j(B, $B35$M>eAX$H2<AX$O8_$$$K5UId9f$G$"$k(B. 

\subsection{$BGH?t6u4V$K$*$1$k2r@O7k2L(B}

\subsubsection{$B%(%M%k%.!<%9%Z%/%H%k(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1440 703,scale=.26]{./results/result1/eksp.png}
 \caption[case1 $B$N;~4VJ?6Q$7$?%(%M%k%.!<%9%Z%/%H(B
 $B%k$N7k2L$H(B TO03 $B$N%(%M%k%.!<%9%Z%/%H%k$NHf3S(B. ]{case1 $B$N(B $t \in
 [1800,2000]$ $B$G;~4VJ?6Q$7$?%(%M%k%.!<%9%Z%/%H%k$N7k2L(B($B:8(B)$B$H(B TO03 $B$N%(%M(B
 $B%k%.!<%9%Z%/%H%k(B($B1&(B)$B$NHf3S(B. $B:8?^$N@D?'$OA4%(%M%k%.!<%9%Z%/%H%k(B, $B\t?'$O(B
 $B=g05%b!<%I$N%(%M%k%.!<%9%Z%/%H%k(B, $BNP?'$O7905%b!<%I$N%(%M%k%.!<%9%Z%/%H(B
 $B%k$r<($9(B. $B1&?^$NB@$$@~$O$=$l$>$l;~4VJ?6Q$7$?A4%(%M%k%.!<%9%Z%/%H%k$r<((B
 $B$7(B, $B<B@~$,;~9o(B 231 $B$+$i(B 347, $BGK@~$,;~9o(B 347 $B$+$i(B 463 $B$N;~4VJ?6Q$H$J$C$F(B
 $B$$$k(B. $B:Y$$GK@~$O(B $k^{-3}$ $B$N79$-(B, $B:Y$$E@@~$O(B $k^{-5/3}$ $B$N79$-$r$=$l$>$l(B
 $B<($9(B. }  
 \label{fig:eksp}
\end{figure}

$B?^(B \ref{fig:eksp} $B$K(B case1 $B$N>l9g$H(B TO03 $B$N>l9g$N%(%M%k%.!<%9%Z%/(B
$B%H%k$NHf3S$r<($9(B. $B%(%M%k%.!<CmF~GH?t$O@~7?$N7905IT0BDjO@$K$h$k$H(B $k_d$
$B6aK5$G$"$k$,(B, $B$3$N7O$G$O?^(B \ref{fig:develop_initial} $B$G<($5$l$F$$$k$h$&(B
$B$K:GDcGH?t$G$"$k$3$H$rCm0U$7$F$*$/(B. case1 $B$G$OJQ7AH>7B$KBP1~$9$kGH?t(B
$k_d = 10$ $B$h$j$bDcGH?tB&$G$O(B, $B%9%Z%/%H%k$N79$-$,(B $k^{-5/3}$, $B9bGH?tB&$G(B
$B$O(B $k^{-3}$ $B$K6a$/$J$C$F$$$k(B\footnote{$B$3$l$i$O$=$l$>$l%(%M%k%.!<47@-NN0h(B
$B$H%(%s%9%H%m%U%#!<47@-NN0h$K$h$k(B $B$b$N$G$"$k$H9M$($i$l$k$,(B, $B8e$N@a$G;60o(B
$B$H6/@)$N%9%Z%/%H%k$d%(%M%k%.!<%U%i%C%/%9$N7k2L$r<($7$F$+$i7kO@IU$1$k$3$H(B
$B$K$9$k(B.
}. $B$^$?(B, case1 $B$G$O%(%M%k%.!<;60oGH?t$,(B $k^{\mathcal{E}}_{\mathrm{diss}}
\simeq 232$, $B%(%s%9%H%m%U%#!<;60oGH?t$,(B $k^{\mathcal{Q}}_{\mathrm{diss}}
\simeq 234$ $B$H$J$C$?(B  
\footnote{
$B%(%M%k%.!<;60oGH?t$H%(%s%9%H%m%U%#!<;60oGH?t$O<!852r@O$+$iF3=P$9$k$3$H$,(B
$B=PMh$k(B($BIUO?(B\ref{chap:dim_analysis}). 
}. $B?^(B
\ref{fig:eksp} $B$N:8B&$N7k2L$r8+$k$H(B, $k\approx 230$ $BIU6a$G%(%M%k(B
$B%.!<%9%Z%/%H%k$,5^9_2<$7$F$*$j(B, $B$3$N7W;;7k2L$O;60oNN0h$^$G2rA|$G$-$F$$$k(B
$B$3$H$,$o$+$k(B. $B0lJ}(B, TO03 $B$G$O(B, $BDcGH?tB&$G$O(B $\beta$ $B9`$,8z$$$F(B
$B$*$j(B, $B%(%M%k%.!<%9%Z%/%H%k$O(B $E(k) \propto k^{-5/3}$ $B$K$J$C$F$$$J$$(B. $B9b(B
$BGH?tB&$G$O(B, $E(k) \propto k^{-3}$ $B$NNN0h$,$"$k$,Hs>o$K69$/(B, $B$`$7$m(B,
$E(k)\propto k^{-5/3}$ $B$,82Cx$G$"$k(B. TO03 $B$G$O(B, $B6/@)GH?t$h$j$b9bGH?tB&$N(B
$E(k)\propto k^{-3}$ $B$NNN0h$O%(%s%9%H%m%U%#!<47@-NN0h$G(B, $E(k)\propto
k^{-5/3}$ $B$NNN0h$O%(%M%k%.!<%+%9%1!<%I$,%(%s%9%H%m%U%#!<%+%9%1!<%I$rBn1[(B
$B$9$k%(%M%k%.!<47@-NN0h$G$"$k$H$7$F$$$k$,(B, $B%(%M%k%.!<$d%(%s%9%H%m%U%#!<$N(B
$B6/@)$d;60o$N%9%Z%/%H%k$O<($5$l$F$$$J$$(B. $B$^$?(B, $B%(%M%k%.!<$d%(%s%9%H%m%U%#!<(B
$B$N;60oGH?t$,$$$/$i$J$N$+$bL@<($5$l$F$$$J$$(B. $B%(%M%k%.!<%9%Z%/%H%k$N7W;;7k(B
$B2L$O@ZCGGH?t6aK5$G5^=T$J79$-$r;}$C$F$$$J$$$?$a(B, $B;60oNN0h$O2rA|$5$l$F$$$J(B
$B$$$3$H$,?dB,$G$-$k(B.  

\subsubsection{$B%(%M%k%.!<$N<};Y2r@O(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1004
 554,scale=.3]{./results/result1/ekflux.png}
 \caption[case1 $B$N%(%M%k%.!<%U%i%C%/%9(B. ]{case1 $B$N%(%M%k%.!<%U%i%C%/%9(B.
 $B:8$OA4GH?tNN0h(B, $B1&$O9bGH?tNN0h$N?^(B.}   
 \label{fig:ek-flux_result1}
\end{figure}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1404
 604,scale=.25]{./results/result1/ek-tendency.png}
 \caption[case1 $B$N%(%M%k%.!<$K$*$1$k6/@)$H;60o$N%9%Z%/%H%k(B.]{case1 $B$N%((B
 $B%M%k%.!<$K$*$1$k6/@)$H;60o$N%9%Z%/%H%k(B. $B@V?'$,6/@)(B, $B@D?'$,%(%/%^%s%@%s(B
 $B%T%s%0$K$h$k;60o(B, $BNP?'$,G4@-$K$h$k;60o$r<($9(B. $B:8$OA4GH?tNN0h(B, $B1&$OJQ7AH>(B
 $B7B$h$j$b9bGH?tB&$NNN0h$r<($9(B. }   
 \label{fig:ek-tendency_result1}
\end{figure}

$B%(%M%k%.!<%U%i%C%/%9$r?^(B \ref{fig:ek-flux_result1} $B$K<($9(B. $B$3$l$K$h$k$H(B,
$k_d$ $B$h$j$bDcGH?tB&$O%U%i%C%/%9$,0lDj$K$J$C$F$$$J$$$,(B, $B@5$NCM$G$"$k$3(B
$B$H$+$i(B, $BDcGH?tB&$+$i9bGH?tB&$K%(%M%k%.!<$,%+%9%1!<%I$7$F$$$k$3$H$,$o$+$k(B.
$BDL>o$N(B 2 $B<!850lMMEyJ}@-MpN.$K$*$1$k%(%M%k%.!<47@-NN0h$G9bGH?tB&$+$iDcGH?tB&$X$N5U%+%9%1!<%I$,$"$j(B, $B%(%M%k%.!<%U%i%C%/%9$O0lDj$G(B, $B$=$3$G(B $k^{-5/3}$
$B$N%9%Z%/%H%k$,7A@.$5$l$k(B.$B%(%M%k%.!<47@-NN0h$H$J$C$F$$$k(B. $B$7$+$7(B, $B>e5-$N(B
$B7k2L$+$i(B, case1 $B$N$3$NNN0h$O%(%M%k%.!<47@-NN0h$K$J$C$F$$$k$H$O8@$($J$$(B.
$B$^$?(B, $B6/@)$H;60o$N%9%Z%/%H%k$r<($7$??^(B \ref{fig:ek-tendency_result1} $B$+(B
$B$i(B, case1 $B$G$O:GDcGH?t$K$*$$$F6/@)9`(B $U\,k_d^2 \partial\psi/\partial x$
$B$K$h$k6/@)$,$"$j(B, $B$3$l$K$h$C$F9bGH?tB&$K%(%M%k%.!<$,%+%9%1!<%I$7$F$$$k$H(B
$B7kO@IU$1$i$l$k(B. $B$h$C$F(B, $B?^(B \ref{fig:eksp-hikaku} $B$G<($5$l$?(B case1 $B$N%(%M(B
$B%k%.!<%9%Z%/%H%k$NDcGH?tB&$N79$-(B $k^{-5/3}$ $B$O$3$NNN0h$,%(%M%k%.!<47@-NN(B
$B0h$G$"$k$?$a$G$O$J$/(B, $B6vA3(B $k^{-5/3}$ $B$K6a$$79$-$K$J$C$?$@$1$G$"$k$H9M$((B
$B$i$l$k(B. $B0lJ}(B, $B9bGH?tB&$G$O(B, $B%(%M%k%.!<;60oN($NCM(B($9.70\times10^{-5}$)$B$H(B
$B%(%M%k%.!<%U%i%C%/%9$NCM(B($BLs(B $1.00\times10^{-4}$)$B$rHf3S$9$k$H(B, $B$o$:$+$J$,$i(B
$B%(%M%k%.!<$,9bGH?tB&$K%+%9%1!<%I$7$F$$$k$3$H$,J,$+$k(B. 
 
\subsubsection{$B%(%s%9%H%m%U%#!<$N<};Y2r@O(B}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1004
 554,scale=.3]{./results/result1/ensflux.png}
 \caption[case1 $B$N%(%s%9%H%m%U%#!<%U%i%C%/%9(B. ]{case1 $B$N%(%s%9%H%m%U%#!<(B
 $B%U%i%C%/%9(B. $B:8$OA4GH?tNN0h(B, $B1&$O9bGH?tNN0h$N?^(B.}   
 \label{fig:ens-flux_result1}
\end{figure}

\begin{figure}[h]
 \centering 
 \includegraphics[bb=0 0 1404
 604,scale=.25]{./results/result1/ens-tendency.png}
 \caption[$B6/@)$H;60o$N%(%s%9%H%m%U%#!<%9%Z%/%H%k(B. ]{case1 $B$N6/@)$H;60o$N(B
 $B%(%s%9%H%m%U%#!<%9%Z%/%H%k(B. $B@V?'$,6/@)(B, $B@D?'$,%(%/%^%s%@%s%T%s%0$K$h$k(B
 $B;60o(B, $BNP?'$,G4@-$K$h$k;60o$r<($9(B. $B:8$OA4GH?t(B, $B1&$O(B $k_d$ $B$h$j$b9bGH?t(B
 $BB&$N%9%Z%/%H%k(B.}   
 \label{fig:ens-tendency_result1}
\end{figure}

$B%(%s%9%H%m%U%#!<%U%i%C%/%9$r?^(B \ref{fig:ens-flux_result1} $B$K<($9(B. $B$3$l$K(B
$B$h$k$H(B, $BA4GH?t$KEO$C$F%(%s%9%H%m%U%#!<$O9bGH?tB&$K%+%9%1!<%I$7$F$*$j(B, $BFC(B
$B$K(B, $k_d$ $B$h$j$b9bGH?tB&$G$O%(%s%9%H%m%U%#!<%U%i%C%/%9$,$[$\0lDj$H$J$C(B
$B$F$$$k(B\footnote{
$B%(%s%9%H%m%U%#!<;60oN($,(B $6.22$ $B$G%(%s%9%H%m%U%#!<%U%i%C%/%9$NCM$OLs(B
$6.2$ $B$J$N$G(B, $B$[$\D`$j9g$C$F$$$k(B.
}. $B$^$?(B, $B%(%s%9%H%m%U%#!<$N6/@)$H;60o$N%9%Z%/%H%k(B($B?^(B
\ref{fig:ens-tendency_result1}) $B$K$h$k$H(B, $B$3$NNN0h$G$O6O$+$J$,$i%(%/%^%s(B
$B%@%s%T%s%0$K$h$k;60o$,$"$k$b$N$N(B, $B6/@)$d;60o$O$[$\L5$$$b$N$H$_$J$;$k(B. $B?^(B
\ref{fig:ens-tendency_result1} $B$N(B $k_d$ $B$h$j$bBg$-$JGH?tNN0h$K$*$1$k%(%s(B
$B%9%H%m%U%#!<%U%i%C%/%9$N0lDjCM$+$i$N%:%l$O(B, $B$3$N%(%/%^%s;60o$N8z2L$K$h$k(B
$B$b$N$G$"$k(B. $B$h$C$F(B, $B$3$NNN0h$O%(%s%9%H%m%U%#!<47@-NN0h$H$_$J$9$3$H$,$G$-(B,$B%9(B
$B%Z%/%H%k$N79$-$,(B $k^{-3}$ $B$H$J$C$?7k2L$H@09gE*$G$"$k(B. $B$^$?(B, $B%(%s%9%H%m(B
$B%U%#!<;60oGH?t(B $k^{\mathcal{Q}}_{\mathrm{diss}}\approx 234$ $B$"$?$j$G$OG4(B
$B@-$K$h$k;60o$,$h$/Mx$$$F$$$k$3$H$b$o$+$k(B. 

\subsection{(\ref{eq:futousiki}) $B$N8!>Z(B}

\begin{figure}[h]
 \centering
 \includegraphics[bb=0 0 668
 572,scale=.3]{./results/result1/check_sign.png}
 \caption{case1 $B$G$N(B(\ref{eq:futousiki}) $B$N8!>Z(B. $B@V?'$,(B
 $k^2\Pi_{\mathrm{E}}-\Pi_{\mathrm{Q}}$, $B@D?'$,(B $k^2\Pi_{\mathrm{E}}$,
 $BNP?'$,(B $\Pi_{\mathrm{Q}}$ $B$NGH?tJ,I[$rI=$9(B.}  
 \label{fig:check-sign_result1}
\end{figure}

case1 $B$G(B (\ref{eq:futousiki}) $B$,@.N)$9$k$+$r8!>Z$7$?$N$,?^(B
\ref{fig:check-sign_result1} $B$G$"$k(B. $k_d$ $B$h$j$bBg$-$JGH?tNN0h$K$*$$$F(B,
$B9bGH?tB&$K9T$/$KO"$l$F(B $k^2$ $B$r3]$1$?%(%M%k%.!<%U%i%C%/%9$O3N$+$KBg$-$/(B
$B$J$C$F$$$k$,(B, $k^2\Pi_{\mathrm{E}}$ $B$,(B $\Pi_{\mathrm{Q}}$ $B$h$j$bBg$-$/$J(B
$B$kA0$K(B, $B%(%s%9%H%m%U%#!<;60oGH?t$KE~C#$7$F$$$k(B. $B$h$C$F(B, $B$3$N@_Dj$G$O(B,
TO03 $B$G<gD%$5$l$F$$$?(B (\ref{eq:futousiki}) $B$,@.N)$9$kNN0h$O8=$l$J$+$C$?(B.  

\section{$B>eAX$H2<AX$GG4@-78?t$rJQ$($?>l9g(B}

$B<!$K(B, case2, case3, case4 $B$N7W;;7k2L$r<($9(B. case2 $B$G$O?^(B
\ref{fig:ektot_case2} $B$G%(%M%k%.!<$N;~4VJQ2=$r<($7$?$h$&$K(B, $B@5DjCM$G$"$k(B
$B$Y$-%(%M%k%.!<$,Ii$K$J$j(B, $B?tCM7W;;$,GKC>$7$F$7$^$C$?(B. case3, case4 $B$N%((B
$B%M%k%.!<$d%(%s%9%H%m%U%#!<$N;~4VJQ2=$d120L>l$NMM;R$O(B case1 $B$HBg$-$/JQ$o(B
$B$i$J$+$C$?(B.

\begin{figure}[htb]
 \centering
 \includegraphics[bb=0 0 904 654,scale=.30]{./results/ektot_case2.png}
 \caption[case2 $B$N%(%M%k%.!<$N;~4VJQ2=(B.]{case2 $B$N%(%M%k%.!<$N;~4VJQ2=(B.
 $BESCf$+$iHs@~7?:nMQ$N?6I}$,Bg$-$/$J$j(B, $B%(%M%k%.!<$,Ii$K$J$k$J$I(B, $B?tCM7W(B
 $B;;$,GKC>$7$F$7$^$C$F$$$k(B.} 
 \label{fig:ektot_case2}
\end{figure}

\begin{figure}[htb]
 \centering
 \includegraphics[bb=0 0 807 904,scale=.40]{./results/eksp-hikaku.png}
 \caption[case1-4 $B$N%(%M%k%.!<%9%Z%/%H%k$NHf3S(B. ]{$B%(%M%k%.!<%9%Z%/%H%k$N(B
 $BHf3S(B. $B:8>e$,(B case1, $B1&>e$,(Bcase2, $B:82<$,(Bcase3, $B1&2<$,(B case4 $B$N>l9g$r<($7(B
 $B$F$$$k(B. $B@D?'$OA4%(%M%k%.!<%9%Z%/%H%k(B, $B\t?'$O=g05%b!<%I$N%(%M%k%.!<%9%Z(B
 $B%/%H%k(B, $BNP?'$O7905%b!<%I$N%(%M%k%.!<%9%Z%/%H%k$r<($9(B. $BE@@~$O(B
 $k^{-5/3}$, $B0lE@:?@~$O(B $k^{-3}$ $B$N79$-$r$=$l$>$lI=$9(B.} 
 \label{fig:eksp-hikaku}
\end{figure}

case1$\sim$4 $B$N>l9g$N%(%M%k%.!<%9%Z%/%H%k$r?^(B \ref{fig:eksp-hikaku} $B$K<((B
$B$9(B. case3, case4 $B$K$D$$$F$O(B, case1 $B$H;w$?7k2L$K$J$C$?(B. $B0lJ}(B, case2 $B$K$D(B
$B$$$F$O(B, $B9bGH?tB&$G%9%Z%/%H%k$N79$-$,(B $k^{-3}$ $B$H$J$C$F$$$kNN0h$O69$/(B, $B@Z(B
$BCGGH?t$K6a$$NN0h$G$O%9%Z%/%H%k$N79$-$,4K$d$+$K$J$C$F$*$j(B, $k^{-5/3}$ $B$K(B
$B6a$$79$-$K$J$C$F$$$k(B.$B$3$l$O(B, $B?^(B \ref{fig:eksp} $B$N1&$K<($7$?(B TO03 $B$N%(%M(B
$B%k%.!<%9%Z%/%H%k$N7k2L$HHs>o$K;w$F$$$k(B. $B$7$+$7$J$,$i(B, case2 $B$N7W;;$O>e$G(B
$B=R$Y$?$h$&$KHsJ*M}E*$J?6$kIq$$$r<($7(B, $BDj>o@-$O<B8=$7$F$$$J$$$3$H$+$i(B,
$k^{-5/3}$ $B$N7k2L$b?.Mj$9$k$KB-$i$J$$(B.

\begin{figure}[htb]
 \centering
 \includegraphics[bb=0 0 800 906,scale=.40]{./results/check-sign.png}
 \caption[case1-4 $B$N(B(\ref{eq:futousiki}) $B$N8!>Z7k2L$NHf(B
 $B3S(B.]{(\ref{eq:futousiki}) $B$N8!>Z7k2L$NHf3S(B. $B:8>e$,(B case1, $B1&>e$,(B case2,
 $B:82<$,(B case3, $B1&2<$,(B case4 $B$N>l9g$r<($7$F$$$k(B. $B$$$:$l$N?^$K$*$$$F$b(B, $B\t(B
 $B?'$,(B $k^2\Pi_{\mathrm{E}}-\Pi_{\mathrm{Q}}$, $B@D?'$,(B $k^2\Pi_{\mathrm{E}}$,
 $BNP?'$,(B $\Pi_{\mathrm{Q}}$ $B$NGH?tJ,I[$r<($9(B.} 
 \label{fig:futousiki-hikaku}
\end{figure}

$B?^(B \ref{fig:futousiki-hikaku} $B$K(B (\ref{eq:futousiki}) $B$N8!>Z7k2L$NHf3S$r(B
$B<($9(B. $B$$$:$l$N>l9g$G$b(B, (\ref{eq:futousiki}) $B$O@.N)$7$J$+$C$?(B. 

\section{$B$^$H$a$H9M;!(B}

\begin{table}[htb]
 {\footnotesize 
  \begin{center}
    \caption{$B%(%M%k%.!<;60oN((B}
    \begin{tabular}{l|cccc} \bhline{1.5pt}
     & case1  & case2 & case3 & case4 \\\hline\hline
     $\psi\cdot J(\psi,\bnabla^2\psi)$
       & $1.62\times10^{-17}$ & $1.87\times10^{-16}$
               & $-5.97\times10^{-18}$ & $-2.97\times10^{-17}$\\ 
     $\psi\cdot J(\tau,\bnabla^2\tau)$
       & $7.17\times10^{-2}$ & $8.20\times10^{-2}$
               & $2.86\times10^{-2}$ & $5.30\times10^{-2}$\\ 
     $\psi\cdot U\partial\bnabla^2\tau/\partial x$
       & $1.53\times10^{-3}$ & $1.44\times10^{-3}$
               & $1.09\times10^{-3}$ & $1.35\times10^{-3}$ \\ 
     $\psi\cdot \kappa\bnabla^2(\psi-\tau)/2$
       & $-7.31\times10^{-2}$ & $-8.09\times10^{-2}$
               & $-3.33\times10^{-2}$ & $-5.44\times10^{-2}$\\ 
     $\psi\cdot [\nu\bnabla^8(\bnabla^2\psi)(\bnabla^2-k_d^2)\tau\}]$
     & \multirow{2}{*}{$-4.85\times10^{-5}$} &
             \multirow{2}{*}{$-1.29\times10^{-4}$} 
               & \multirow{2}{*}{$-9.43\times10^{-5}$} &
                     \multirow{2}{*}{$-1.86\times10^{-4}$}\\ 
     \hspace{2pt}$+\frac{1}{2}\Delta\nu\bnabla^8\{\bnabla^2\psi
     (\bnabla^2-k_d^2)\tau\}]$ \\ \hline
     $\tau\cdot J(\tau,\bnabla^2\psi)$
       & $4.42\times10^{-18} $ & $9.00\times10^{-19}$
               & $-3.75\times10^{-20}$ & $-4.75\times10^{-19}$\\
     $\tau\cdot J(\psi,\bnabla^2\tau)$
       & $-7.17\times10^{-2} $ & $-8.20\times10^{-2}$
               & $-2.86\times10^{-2}$ & $-5.31\times10^{-2}$\\
     $\tau\cdot J(\psi,-k_d^2\tau)$
       & $-1.46\times10^{-17}$ & $8.10\times10^{-18}$
               & $1.80\times10^{-18}$ & $4.69\times10^{-18}$\\ 
     $\tau\cdot U\partial\bnabla^2\psi/\partial x$
       & $-1.53\times10^{-3}$ & $-1.44\times10^{-3}$
               & $-1.09\times10^{-2}$ & $-1.35\times10^{-3}$\\
     $\tau\cdot Uk_d^2\partial\psi/\partial x$
       & $7.24\times10^{-2}$ & $8.20\times10^{-2}$ 
               & $2.98\times10^{-2}$ & $5.46\times10^{-2}$ \\
     $\tau\cdot \kappa\bnabla^2(\tau-\psi)/2$
       & $1.07\times10^{-3}$ & $-2.20\times10^{-4}$
               & $2.01\times10^{-4}$ & $2.23\times10^{-4}$\\
     $\tau\cdot[\nu\bnabla^8(\bnabla^2-k_d^2)\tau$
         & \multirow{2}{*}{$-4.85\times10^{-5}$} &
             \multirow{2}{*}{$-1.29\times10^{-4}$} 
             & \multirow{2}{*}{$-9.30\times10^{-5}$} &
                     \multirow{2}{*}{$-1.85\times10^{-4}$} \\
     \hspace{2pt}$+\frac{1}{2}\Delta\nu\bnabla^8\{(\bnabla^2-k_d^2)\tau
     - \bnabla^2\psi\}]$ 
     \\ \bhline{1.5pt}
    \end{tabular}
   \label{tab:ek-dissipation-rate}
  \end{center}
 }
 \end{table}

\begin{table}[htb]
 {\footnotesize
  \begin{center}
    \caption{$B%(%s%9%H%m%U%#!<;60oN((B}
    \begin{tabular}{l|cccc} \bhline{1.5pt}
     & case1  & case2 & case3 & case4 \\\hline\hline
     $\bnabla^2\psi\cdot J(\psi,\bnabla^2\psi)$
       & $4.85\times10^{-16}$ & $-1.21\times10^{-15}$
               & $-1.57\times10^{-16}$ & $3.09\times10^{-16}$\\ 
     $\bnabla^2\psi\cdot J(\tau,\bnabla^2\tau)$
       & $3.89\times10^{0}$ & $4.26\times10^{0}$
               & $1.47\times10^{0}$ & $2.76\times10^{0}$\\ 
     $\bnabla^2\psi\cdot U\partial\bnabla^2\tau/\partial x$
       & $1.22\times10^{-1}$ & $7.74\times10^{-2}$
               & $1.08\times10^{-1}$ & $1.17\times10^{-1}$\\ 
     $\bnabla^2\psi\cdot \kappa\bnabla^2(\psi-\tau)/2$
       & $-7.62\times10^{-1}$ & $-2.10\times10^{0}$
               & $-3.28\times10^{-1}$ & $-5.00\times10^{-1}$\\ 
     $\bnabla^2\psi\cdot[\nu\bnabla^8(\bnabla^2\psi)$
       & \multirow{2}{*}{$-3.11\times10^{0}$} &
             \multirow{2}{*}{$-2.08\times10^{0}$} 
               & \multirow{2}{*}{$-1.26\times10^{0}$} &
                     \multirow{2}{*}{$-2.38\times10^{0}$} \\ 
     \hspace{2pt}
     $+\frac{1}{2}\Delta\nu\bnabla^8\{\bnabla^2\psi(\bnabla^2-k_d^2)\tau\}]$& 
         \\ \hline 
     $(\bnabla^2-k_d^2)\tau\cdot J(\tau,\bnabla^2\psi)$
       & $-3.89\times10^{0}$ & $-4.26\times10^{0}$
               & $-1.47\times10^{0}$ & $-2.76\times10^{0}$\\
     $(\bnabla^2-k_d^2)\tau\cdot J(\psi,\bnabla^2\tau)$
       & $-7.17\times10^{0}$ & $-8.20\times10^{0}$
               & $-2.86\times10^{0}$ & $-5.31\times10^{0}$\\
     $(\bnabla^2-k_d^2)\tau\cdot J(\psi,-k_d^2\tau)$
       & $7.17\times10^{0}$ & $8.20\times10^{0}$
               & $2.86\times10^{0}$ & $5.31\times10^{0}$\\ 
     $(\bnabla^2-k_d^2)\tau\cdot U\partial\bnabla^2\psi/\partial x$
       & $-2.76\times10^{-1}$ & $-2.22\times10^{-1}$
               & $-2.16\times10^{-1}$ & $-2.52\times10^{-1}$\\
     $(\bnabla^2-k_d^2)\tau\cdot Uk_d^2\partial\psi/\partial x$
       & $7.39\times10^{0}$ & $8.34\times10^{0}$
               & $3.08\times10^{0}$ & $5.59\times10^{0}$\\
     $(\bnabla^2-k_d^2)\tau\cdot \kappa\bnabla^2(\tau-\psi)/2$
       & $-2.04\times10^{-1}$ & $-1.70\times10^{0}$
               & $-1.03\times10^{-1}$ & $-1.70\times10^{-1}$\\
     $(\bnabla^2-k_d^2)\tau\cdot[\nu\bnabla^8(\bnabla^2-k_d^2)\tau$
         & \multirow{2}{*}{$-3.11\times10^{0}$} &
             \multirow{2}{*}{$-2.08\times10^{0}$} 
                 & \multirow{2}{*}{$-1.25\times10^{-0}$} &
                     \multirow{2}{*}{$-2.39\times10^{0}$} \\
     \hspace{2pt}$+\frac{1}{2}\Delta\nu\bnabla^8\{(\bnabla^2-k_d^2)\tau
     - \bnabla^2\psi \}]$&
     \\ \bhline{1.5pt}
    \end{tabular}
   \label{tab:ens-dissipation-rate}
  \end{center}
 }
\end{table}

$BI=(B \ref{tab:ek-dissipation-rate} $B$K%(%M%k%.!<$NJQ2=N($KBP$9$k(B
(\ref{eq:tro_ekeq-ws}) $B$H(B (\ref{eq:cli_ekeq-ws}) $B$N3F9`$N4sM?$r(B, $BI=(B
\ref{tab:ens-dissipation-rate} $B$K%(%s%9%H%m%U%#!<$NJQ2=N($KBP(B
$B$9$k(B (\ref{eq:tro_enseq-ws}) $B$H(B (\ref{eq:cli_enseq-ws}) $B$N3F9`$N4sM?$r$=(B
$B$l$>$l<($9(B. $B$$$:$l$N>l9g$b(B, $B%(%M%k%.!<J]B8B'$H%(%s%9%H%m%U%#!<J]B8B'$,@.(B
$BN)$7$F$$$k(B. 

TO03 $B$G$O>eAX$H2<AX$NG4@-78?t$N0c$$$K$D$$$F$O8@5Z$5$l$F$*$i$:(B, $B$b$7H`$i(B
$B$N<gD%$,@5$7$1$l$P(B case1 $B$N@_Dj$G(B (\ref{eq:futousiki}) $B$,@.N)$7$F$$$kNN(B
$B0h$G(B N-G $B%9%Z%/%H%k$,7A@.$5$l$k$O$:$G$"$k$,(B, $B7k2L$H$7$F$OA4GH?tNN0h$G(B
(\ref{eq:futousiki}) $B$O@.N)$7$J$+$C$?(B. Gkioulekas (2014) $B$G$O>eAX$H2<AX(B
$B$NG4@-78?t$rJQ$($k$3$H$K$h$j(B (\ref{eq:futousiki}) $B$,@.N)$9$k2DG=@-$,<(:6(B
$B$5$l$F$$$?$,(B, $B>eAX$H2<AX$GG4@-78?t$rJQ$($?>l9g(B case2, case3, case4 $B$G$b(B
$BA4GH?tNN0h$G(B (\ref{eq:futousiki}) $B$,@.N)$7$J$+$C$?(B. $B$7$+$7(B, $BG4@-$,>eAX$h(B
$B$j$b2<AX$NJ}$,<e$$(B case2 $B$G$O(B, TO03 $B$GF@$i$l$?%(%M%k%.!<%9%Z%/%H%k$K6a$$(B
$B7A$H$J$C$F$*$j(B, $B9bGH?tNN0h$G(B$k^{-5/3}$ $B$N%9%Z%/%H%k$,7A@.$5$l$F$$$k$h$&(B
$B$K8+$($k(B. $B$3$l$O(B, $B9bGH?tNN0h$G$NG4@-$K$h$k;60o$,<e$$$?$a$K(B, $B%(%M%k%.!<$,(B
$B7O$N30$K;60o$7$-$l$F$$$J$$$?$a(B, $B%9%Z%/%H%k$,$a$/$j>e$,$C$F$$$k$?$a$@$H9M(B
$B$($i$l$k(B. $B<B:](B, case2 $B$N2<AX$NG4@-78?t$O(B $\nu+\Delta\nu = 0$ $B$H$J$C$F$*(B
$B$j(B, $B;60oGH?t$,Dj5A$G$-$:(B, $B;60oNN0h$r2rA|$9$k$3$H$,$G$-$J$$(B.

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