<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%#LPRdvips main.dvi; lpr main.ps
%#!platex main.tex

\section{åŸºç¤Žæ–¹ç¨‹å¼ç³»}

ã“ã“ã§ã¯åŸºç¤Žæ–¹ç¨‹å¼ç³»ã®å®šå¼åŒ–ã«ã¤ã„ã¦è¨˜è¿°ã™ã‚‹.
æµä½“ã¯ãƒ–ã‚·ãƒã‚¹ã‚¯è¿‘ä¼¼ã‚’é©å¿œã—ãŸç£æ°—æµä½“ã§ã‚ã‚‹ã¨ã—,
é‡åŠ›åŠ&nbsp;é€Ÿåº¦ã¯å¯†åº¦ãŒä¸€æ§˜ãªç‰©ä½“ã®è‡ªå·±é‡åŠ›å&nbsp;´,
åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒã¯å†…éƒ¨ä¸€æ§˜ç†±æºã«ã‚ˆã‚‹ç†±ä¼å°Žè§£ã§ã‚ã‚‹ã¨ä»®å®šã™ã‚‹.
é€Ÿåº¦å&nbsp;´ã¨ç£å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«--ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¸åˆ†è§£ã—,
ãã®æ™‚é–“ç™ºå±•æ–¹ç¨‹å¼ã¨å¢ƒç•Œæ¡ä»¶ã‚’å®šå¼åŒ–ã™ã‚‹
(Zhang and Busse,1987, Chandrasekhar, 1961).

%--------------------------------------------------------------------------
 \subsection{ç³»ã®è¨­å®šã¨æ”¯é…æ–¹ç¨‹å¼}
 
 è§’é€Ÿåº¦ $\Omega$ ã§å›žè»¢ã™ã‚‹åŠå¾„ $r_{i},r_{o}$ ã®çƒã«æŒŸã¾ã‚ŒãŸ
 çƒæ®»å†…ã«ã‚ã‚‹å°Žé›»æ€§æµä½“ã«ã¤ã„ã¦è€ƒãˆã‚‹.
 çƒæ®»å†…ã®æµä½“ã¯ãƒ–ã‚·ãƒã‚¹ã‚¯æµä½“ã‹ã¤ç£æ°—æµä½“ã§ã‚ã‚‹ã¨ã—,
 åº§æ¨™ç³»ã‚’çƒæ®»ã®ä¸­å¿ƒã‚’åŽŸç‚¹ã¨ã™ã‚‹ç·¯åº¦çµŒåº¦åº§æ¨™ç³»
 $(\lambda, \varphi, r)$ ã¨ã™ã‚‹.
 ã“ã‚Œã‚’ \Dfigref{ç³»ã®å›³} ã«ç¤ºã™.

 \begin{figure}[htbp!]
  \begin{center}
   \Depsf[13cm]{./img/cordinate.eps}
   \caption{è€ƒãˆã¦ã„ã‚‹ç³»ã®æ¨¡å¼å›³} \Dfiglab{ç³»ã®å›³}
  \end{center}
 \end{figure}
 
 \newpage
 æ”¯é…æ–¹ç¨‹å¼ç³»ã¯
 å›žè»¢ç³»ã«ãŠã‘ã‚‹ãƒ–ã‚·ãƒã‚¹ã‚¯æµä½“ã®æ–¹ç¨‹å¼ç³»ã¨
 ç£æ°—æµä½“ã®æ–¹ç¨‹å¼ç³»ã¨ã‚’
 çµ„ã¿åˆã‚ã›ã‚‹ã“ã¨ã§å¾—ã‚‰ã‚Œã‚‹. 
 ä»¥ä¸‹ã«ã“ã‚Œã‚‰ã‚’æ›¸ãä¸‹ã™:
 \begin{eqnarray} 
  &amp;&amp;\Deqlab{é€£ç¶šã®å¼}
   \Ddiv{\Dvect{u}}=0, \\
  &amp;&amp;\Deqlab{ç£å&nbsp;´ã®ã‚½ãƒ¬ãƒŽã‚¤ãƒ€ãƒ«æ¡ä»¶}
   \Ddiv{\Dvect{B}} = 0, \\
  &amp;&amp;\Deqlab{é‹å‹•æ–¹ç¨‹å¼}
   \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
   + 2\Omega \Dvect{k}\times\Dvect{u}
   = -\frac{1}{\rho_{0}}\Dgrad p + \frac{\rho}{\rho_{0}}\Dvect{g} 
   +\nu \Dlapla \Dvect{u} 
   + \frac{1}{\mu_{0} \rho_{0}}(\Drot\Dvect{B})\times\Dvect{B},\nonumber \\
  &amp;&amp; \\
  &amp;&amp;\Deqlab{ç†±ã®å¼}
   \DP{T}{t} + (\Dvect{u}\cdot\Dgrad)T = \kappa\Dlapla{T} + Q, \\
  &amp;&amp;\Deqlab{èª˜å°Žæ–¹ç¨‹å¼}
   \DP{\Dvect{B}}{t} = 
   \Drot(\Dvect{u}\times\Dvect{B}) + \eta\Dlapla{\Dvect{B}}.
 \end{eqnarray}

 å„ã€…ã®è¨˜å·ãŒä½•ã‚’è¡¨ã—ã¦ã„ã‚‹ã‹ã¯æ¬¡ã®é€šã‚Šã§ã‚ã‚‹:
 \begin{quote}
  \begin{tabular}{ll}
   $\Dvect{u}$  
                  &amp; æµé€Ÿãƒ™ã‚¯ãƒˆãƒ«. \\
                  &amp;  æˆåˆ†æ¯Žã«æ›¸ãå&nbsp;´åˆã«ã¯,$\Dvect{u}=(u,v,w)$ \\
   $\Dvect{B}$    
                  &amp; ç£æŸå¯†åº¦ãƒ™ã‚¯ãƒˆãƒ«. \\
                  &amp; æˆåˆ†æ¯Žã«æ›¸ãå&nbsp;´åˆã«ã¯, 
                    $\Dvect{B}=(B_{\lambda},B_{\varphi},B_{r})$ \\
   $T$
                  &amp; æµä½“ã®å®Ÿéš›ã®æ¸©åº¦. \\ 
                  &amp; åŸºæœ¬å&nbsp;´ã®æ¸©åº¦$T_{0}$, 
                    ãã“ã‹ã‚‰ã®ãšã‚Œ $\theta$ ã‚’ç”¨ã„ã¦è¡¨ã™ã¨,\\
                  &amp; $T=T_{0}+\theta$ ã§ã‚ã‚‹. \\
   $\rho, \rho_{0}$
                  &amp; æµä½“ã®å¯†åº¦, åŠã³ãã®åŸºæœ¬å&nbsp;´ã®å¯†åº¦. \\
                  &amp; ã“ã“ã§ã¯ãƒ–ã‚·ãƒã‚¹ã‚¯æµä½“ã§ã‚ã‚‹ã“ã¨ã‚’ä»®å®šã—ã¦ã„ã‚‹ã®ã§,\\
                  &amp; $\rho = \rho_0(1-\alpha\theta)$ ã§ã‚ã‚‹. \\
   $p$       
                  &amp; æµä½“ã®å…¨åœ§åŠ›.\\
                  &amp; åŸºæœ¬å&nbsp;´ã®åœ§åŠ› $p_{0}$, 
                    ãã“ã‹ã‚‰ã®ãšã‚Œ $\pi$ ã‚’ç”¨ã„ã¦è¡¨ã™ã¨ \\
                  &amp; $p=p_{0} +\pi$ ã§ã‚ã‚‹.\\
   $\Dvect{k}$  
                  &amp; çƒæ®»ã®å›žè»¢è»¸æ–¹å‘ã®å˜ä½ãƒ™ã‚¯ãƒˆãƒ«
                    ( \Dfigref{ç³»ã®å›³}ã§ã®$z$ è»¸æ–¹å‘). \\
   $\Dvect{g}$    &amp; é‡åŠ›åŠ&nbsp;é€Ÿåº¦ãƒ™ã‚¯ãƒˆãƒ«.\\
   $Q$            &amp; å†…éƒ¨ç†±æº(æ¸©åº¦ä¸Šæ˜‡çŽ‡). \\
  \end{tabular}
 \end{quote}  
  ã¾ãŸ, ä»¥ä¸‹ã®ç‰©æ€§å®šæ•°ã¯ä¸€å®šã§ã‚ã‚‹ã¨ã™ã‚‹.
 \begin{quote}
  \begin{tabular}{ll}
   $\nu$          &amp; å‹•ç²˜æ€§çŽ‡.     \\
   $\kappa$       &amp; ç†±æ‹¡æ•£çŽ‡.     \\
   $\alpha$       &amp; ä½“ç©è†¨å¼µçŽ‡.   \\
   $\eta$         &amp; ç£æ°—æ‹¡æ•£çŽ‡.   \\
   $\mu_{0}$          &amp; é€ç£çŽ‡. 
  \end{tabular}
  \end{quote}  
 
 é‡åŠ›åŠ&nbsp;é€Ÿåº¦ $\Dvect{g}$ ã¨ã—ã¦
 å¯†åº¦ãŒã»ã¼ä¸€æ§˜ãªç‰©ä½“ã®è‡ªå·±é‡åŠ›ã‚’è€ƒãˆã‚‹äº‹ã¨ã—,
 \begin{equation}
  \Dvect{g} = -\gamma \Dvect{r}
 \end{equation}
 ã¨ã™ã‚‹.
 ã“ã“ã§  $\gamma$ ã¯å®šæ•°, 
 $\Dvect{r}$ ã¯çƒæ®»ã®ä¸­å¿ƒã‚’åŽŸç‚¹ã¨ã™ã‚‹å‹•å¾„æ–¹å‘ã®ä½ç½®ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚ã‚‹
 \footnote{
 é‡åŠ›åŠ&nbsp;é€Ÿåº¦ã«ã¯
 åŸºæœ¬å&nbsp;´ã‹ã‚‰ã®å¯†åº¦å¤‰åŒ–ã¨
 å›žè»¢ã®åŠ¹æžœã«ã‚ˆã‚‹éžçƒå¯¾ç§°ãªæˆåˆ†ãŒå­˜åœ¨ã™ã‚‹ãŒ,
 ç°¡å˜ã®ãŸã‚ã“ã“ã§ã¯ç„¡è¦–ã™ã‚‹.
 }.
 
 åŸºæœ¬å&nbsp;´ã¯å†…éƒ¨ä¸€æ§˜ç†±æºåˆ†å¸ƒ ($Q=const$) ã®ä¸‹ã§ã®
 ç£å&nbsp;´ã®ç„¡ã„é™æ­¢çŠ¶æ…‹ã§ã‚ã‚‹ã¨ã—,
 åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒã¨ã—ã¦
 \begin{equation}
  \Deqlab{åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒ}
  T_{0}(r) = -\frac{1}{2}\beta r^{2} + C
 \end{equation}
 ã‚’ä»®å®šã™ã‚‹.
 ã&nbsp;ã&nbsp;ã— $\displaystyle{\beta \equiv \frac{1}{3}\frac{Q}{\kappa}}$ ã¨ã—ãŸ
 \footnote{
 ç†±ä¼å°Žæ–¹ç¨‹å¼ã‚’æº€ãŸã™æ¸©åº¦åˆ†å¸ƒã«ã¯ $C'/r$ ã®ä¸å®šæ€§ãŒæ®‹ã£ã¦ã„ã‚‹ãŒ,
 å¾Œã®æ•°å­¦çš„ç°¡ä¾¿ã•ã®ãŸã‚ $C'=0$ ã®åˆ†å¸ƒã‚’é¸ã¶ã“ã¨ã¨ã™ã‚‹. 
 }
 .

 ã“ã®åŸºæœ¬å&nbsp;´ã‚’æ”¯é…æ–¹ç¨‹å¼ç³»
 \Deqref{é€£ç¶šã®å¼}ã€œç†±ã®å¼\Deqref{ç†±ã®å¼}ã‹ã‚‰åˆ†é›¢ã™ã‚‹ã¨,
 \begin{eqnarray} 
  &amp;&amp;\Deqlab{é€£ç¶šã®å¼'}
   \Ddiv\Dvect{u} =0, \quad \Ddiv\Dvect{B}=0 \\
  &amp;&amp;\Deqlab{é‹å‹•æ–¹ç¨‹å¼'}
   \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
   + 2\Omega \Dvect{k}\times\Dvect{u}
   = -\frac{1}{\rho_{0}}\Dgrad \pi 
     + \alpha \theta \gamma \Dvect{r}
   +\nu \Dlapla \Dvect{u} 
   + \frac{1}{\mu_{0} \rho_{0}}(\Drot\Dvect{B})\times\Dvect{B},\nonumber\\
  &amp;&amp; \\
  &amp;&amp;\Deqlab{ç†±ã®å¼'}
   \DP{\theta}{t} 
   + (\Dvect{u}\cdot\Dgrad)\theta - \beta \Dvect{u}\cdot \Dvect{r}
   = \kappa\Dlapla{\theta}, \\
  &amp;&amp;\Deqlab{èª˜å°Žæ–¹ç¨‹å¼'}
   \DP{\Dvect{B}}{t} = 
   \Drot(\Dvect{u}\times\Dvect{B}) + \eta\Dlapla{\Dvect{B}}
 \end{eqnarray}
 ã¨ãªã‚‹
 \footnote{
 åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒã¨ã—ã¦ \Deqref{åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒ} ã‚’é¸ã‚“ã&nbsp;ã“ã¨ã§,
 åŸºæœ¬å&nbsp;´ã®ç§»æµãŒ 
 $\Dvect{u}\cdot\Dgrad T_{0} = -\beta \Dvect{u}\cdot\Dvect{r}$
 ã¨ãªã‚‹. 
 ã™ãªã‚ã¡, æœ¬ãƒ¢ãƒ‡ãƒ«ã§ã¯åŸºæœ¬å&nbsp;´ã®æ¸©åº¦åˆ†å¸ƒãŒãƒ¢ãƒ‡ãƒ«ã«çµ„ã¿è¾¼ã¾ã‚Œã¦ã„ã‚‹.
 }
.
 
 æ¬¡ã«ã“ã‚Œã‚‰ã®æ–¹ç¨‹å¼ç³»ã‚’ä»¥ä¸‹ã®ã‚¹ã‚±ãƒ¼ãƒ«ã‚’ç”¨ã„ã¦ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹:
  \begin{quote}
   \begin{tabular}{lll}
    é•·ã•ã‚¹ã‚±ãƒ¼ãƒ« &amp; $D = r_{o} - r_{i}$ &amp; (çƒæ®»ã®åŽšã•)    \\
    æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ« &amp; $D^{2}/\nu$ &amp; (ç²˜æ€§æ‹¡æ•£æ™‚é–“)  \\
    é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\nu/D$ &amp; (ç²˜æ€§é€Ÿåº¦)\\
    æ¸©åº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\beta D^{2}$ &amp;\\
    ç£å&nbsp;´ã®ã‚¹ã‚±ãƒ¼ãƒ« 
    &amp; $\sqrt{\mu_{0} \rho_{0}}\nu/D$ 
    &amp; %ã‚¢ãƒ«ãƒ•ãƒ´ã‚§ãƒ³æ³¢ã®é€Ÿåº¦ ? 
    \\
   åœ§åŠ›ã®ã‚¹ã‚±ãƒ¼ãƒ« &amp;$\rho_{0}\nu^{2}/D^{2}$  &amp;
   \end{tabular}
  \end{quote}
  ãã®çµæžœ, æ–¹ç¨‹å¼ç³»ã¯
 \begin{eqnarray}
  &amp;&amp; \Deqlab{ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}
   \Ddiv\Dvect{u} = 0,~~~\Ddiv\Dvect{B}=0, \\
  &amp;&amp;\Deqlab{ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}
   \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
   + \tau \Dvect{k}\times\Dvect{u}
   = - \Dgrad \pi 
     + P_{r}^{-1}R_{a}
     \theta\Dvect{r}
   + \Dlapla \Dvect{u} 
   + (\Drot\Dvect{B})\times\Dvect{B},\nonumber \\
  &amp;&amp; \\
  &amp;&amp;\Deqlab{ç„¡æ¬¡å…ƒç†±ã®å¼}
   \DP{\theta}{t} + (\Dvect{u}\cdot\Dgrad)\theta -\Dvect{u}\cdot\Dvect{r} 
   =    P_{r}^{-1}\Dlapla{\theta}, \\
  &amp;&amp;\Deqlab{ç„¡æ¬¡å…ƒèª˜å°Žæ–¹ç¨‹å¼}
   \DP{\Dvect{B}}{t} - \Drot(\Dvect{u}\times\Dvect{B}) 
   =    P_{m}^{-1}\Dlapla{\Dvect{B}}
 \end{eqnarray}
 ã¨ãªã‚‹.
 ã“ã“ã§ $\Dvect{u},\pi,\Dvect{B},\theta$ ã¯, 
 å„ã€…
 ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸé€Ÿåº¦, 
 åœ§åŠ›ã®ãšã‚Œ, 
 ç£å&nbsp;´, 
 æ¸©åº¦ã®ãšã‚Œã‚’è¡¨ã™ã¨å®šç¾©ã—ãªãŠã—ãŸ.

 ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã«ã‚ˆã£ã¦ç¾ã‚ŒãŸç„¡æ¬¡å…ƒæ•°ã¯å„ã€…ä»¥ä¸‹ã®é€šã‚Šã§ã‚ã‚‹.
 $R_{a}$ ã¯ãƒ¬ã‚¤ãƒªãƒ¼æ•°, $\tau$ ã¯ã‚¨ã‚¯ãƒžãƒ³æ•° $E$ ã®é€†æ•°,
 (ã‚‚ã—ãã¯ãƒ†ã‚¤ãƒ©ãƒ¼æ•°ã®å¹³æ–¹æ&nbsp;¹),
 $P_{r}$ ã¯ãƒ—ãƒ©ãƒ³ãƒ‰ãƒ«æ•°, $P_{m}$ ã¯ç£æ°—ãƒ—ãƒ©ãƒ³ãƒ‰ãƒ«æ•°ã§ã‚ã‚Š,
 ãã‚Œãžã‚Œæ¬¡å¼ã§å®šç¾©ã•ã‚Œã‚‹
 \footnote{
 æ¥­ç•Œæ¨™æº–ã®ç„¡æ¬¡å…ƒåŒ–ã¨ç¨®ã€…ã®è«–æ–‡ã§ã®ç„¡æ¬¡å…ƒåŒ–ã«ã¤ã„ã¦ã¯,
 appendix å‚ç…§(æœªæ•´å‚™).
 }
:
 \begin{eqnarray}
  R_{a} &amp;\equiv&amp; \frac{\alpha \gamma \beta D^{6}}{\kappa \nu},\\
  \tau^{-1}  &amp;=&amp; E \equiv \frac{\nu}{2\Omega D^{2}}, \\
  P_{r} &amp;\equiv&amp; \frac{\nu}{\kappa}, \\
  P_{m} &amp;\equiv&amp; \frac{\nu}{\eta}.  
 \end{eqnarray}
 ã“ã‚Œã«åŠ&nbsp;ãˆã¦ç³»ã«ç¾ã‚Œã‚‹é‡è¦ãªãƒ‘ãƒ©ãƒ¡ãƒ¼ã‚¿ã¨ã—ã¦
 å†…å¾„å¤–å¾„æ¯” $\zeta \equiv r_{i}/r_{o}$ ãŒå­˜åœ¨ã™ã‚‹.
 
%--------------------------------------------------------------------------
\newpage
 \subsection{ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«ã‚ˆã‚‹æ–¹ç¨‹å¼ç³»ã®è¡¨ç¾} 
 
 é€£ç¶šã®å¼ãŠã‚ˆã³ç£å&nbsp;´ã®ã‚½ãƒ¬ãƒŽã‚¤ãƒ€ãƒ«æ¡ä»¶\Deqref{ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}
 ã‚ˆã‚Š, é€Ÿåº¦å&nbsp;´ã¨ç£å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«, ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’ç”¨ã„ã¦è¡¨ã™
ã“ã¨ãŒã§ãã‚‹:
 \begin{eqnarray}
  \Deqlab{é€Ÿåº¦åˆ†è§£}
  \Dvect{u} &amp;=&amp; \Drot(\psi \Dvect{r}) + \Drot\Drot(\phi\Dvect{r}), \\
  \Deqlab{ç£å&nbsp;´åˆ†è§£}
  \Dvect{B} &amp;=&amp; \Drot(g \Dvect{r}) + \Drot\Drot(h \Dvect{r}). 
 \end{eqnarray}
 ã“ã‚Œã‚‰ã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ« $\psi, \phi, g, h$ ã‚’ç”¨ã„ã¦
 æ–¹ç¨‹å¼ç³»ã‚’è¡¨ã™. 
 é‹å‹•æ–¹ç¨‹å¼ \Deqref{ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}ã«
 $\Dvect{r}\cdot\Drot$, $\Dvect{r}\cdot\Drot\Drot$ ã‚’ä½œç”¨ã™ã‚‹ã“ã¨ã§,
 \begin{eqnarray}
  \Deqlab{ãƒˆãƒ­é‹å‹•}
   &amp;&amp;
   \left[ \left( \Dlapla - \DP[]{}{t} \right) L_2 
   + \tau  \DP{}{\lambda} \right] \psi 
   - \tau Q \phi  
   =  
     \Dvect{r}\cdot\Drot \{( \Dvect{u}\cdot\Dgrad)\Dvect{u} 
   - (\Dvect{B}\cdot\Dgrad)\Dvect{B})\} \\
  \Deqlab{ãƒãƒ­é‹å‹•}
  &amp;&amp;\left[ \left( \Dlapla - \DP[]{}{t} \right) L_2 
     + \tau \DP{}{\lambda} \right] \Dlapla \phi 
   + \tau Q \psi - P_{r}^{-1}R_{a} L_2 \theta \nonumber \\
  &amp;&amp; \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
    =
    - \Dvect{r}\cdot\Drot\Drot \{(\Dvect{u}\cdot\Dgrad)\Dvect{u} 
     - (\Dvect{B}\cdot\Dgrad)\Dvect{B})\} 
 \end{eqnarray}
 ãŒå¾—ã‚‰ã‚Œã‚‹.
 å¼ä¸­ã®æ¼”ç®—å­ã¯å„ã€…æ¬¡å¼ã§å®šç¾©ã—ãŸ:
 \begin{eqnarray}
  \mu &amp;\equiv&amp; \sin\varphi, \\
  L_{2} 
   &amp;\equiv&amp;  -r^{2}\Dlapla +\DP{}{r}r^{2}\DP{}{r}  \nonumber \\
  &amp;=&amp; -\biggl[
   \frac{1}{\cos^{2}\varphi}\DP[2]{}{\lambda} 
   +
   \frac{1}{\cos\varphi}\DP{}{\varphi}\biggl(\cos\varphi\biggr)\DP{}{\varphi}
   \biggr] \nonumber \\
  &amp;=&amp; -\biggl[
   \frac{1}{1-\mu^{2}}\DP[2]{}{\lambda}
   + 
   \DP{}{\mu}\biggl\{(1-\mu^{2})\DP{}{\mu}\biggr\}
   \biggr],     \\
  Q  &amp;\equiv&amp; \Dvect{k}\cdot\Dgrad 
   - \Dinv{2}(    L_2 \Dvect{k}\cdot\Dgrad 
   + \Dvect{k}\cdot\Dgrad L_2   ) \nonumber \\
  &amp;=&amp; 
   - r\sin\varphi\Dlapla 
   + \biggl(L_{2} + r\DP{}{r}\biggr)
   \times
   \biggl(\sin\varphi\DP{}{r} 
   - \frac{cos\varphi}{r}\DP{}{\varphi}\biggr) \nonumber \\
  &amp;=&amp;
   -r\mu\Dlapla 
   +\biggl(L_{2} +r\DP{}{r}\biggr)
   \times
   \biggl(\mu\DP{}{r}- \frac{1-\mu^{2}}{r}\DP{}{\mu}\biggr). 
 \end{eqnarray}
 ç†±ã®å¼ \Deqref{ç„¡æ¬¡å…ƒç†±ã®å¼}
 ã‚’ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ã™ã¨
 \begin{eqnarray}
  \Deqlab{ãƒãƒ­ç†±}
   \biggl(P_{r}^{-1}\Dlapla - \DP{}{t}\biggr)\theta
   + L_{2}\phi &amp;=&amp; \Dvect{u}\cdot\Dgrad\theta.
 \end{eqnarray}
 èª˜å°Žæ–¹ç¨‹å¼ã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§ã®è¡¨ç¾ã¯
 \Deqref{ç„¡æ¬¡å…ƒèª˜å°Žæ–¹ç¨‹å¼} ã¸
 $\Dvect{r}\cdot\Drot, \Dvect{r}\cdot$ ã‚’ä½œç”¨ã™ã‚‹äº‹ã§å¾—ã‚‰ã‚Œã‚‹:
 \begin{eqnarray}
  \Deqlab{ãƒˆãƒ­èª˜å°Ž}
   \biggl(P_{m}^{-1}\Dlapla -\DP{}{t}\biggr)L_{2}g
   &amp;=&amp;- \Dvect{r}\cdot\Drot\Drot(\Dvect{u}\times\Dvect{B}),\\
  \Deqlab{ãƒãƒ­èª˜å°Ž}
   \biggl(P_{m}^{-1}\Dlapla -\DP{}{t}\biggr)L_{2}h
   &amp;=&amp; - \Dvect{r}\cdot\Drot(\Dvect{u}\times\Dvect{B}).
 \end{eqnarray}

 $\Dvect{r}\cdot\Drot, \Dvect{r}\cdot\Drot\Drot$ ã®åº§æ¨™è¡¨ç¾ã¯
   \begin{eqnarray*}
     \Dvect{r}\cdot(\Drot\Dvect{F})
       &amp;=&amp; \frac{1}{\cos\varphi}\DP{F_\varphi}{\lambda}
         - \frac{1}{\cos\varphi}\DP{}{\varphi}(F_\lambda\cos\varphi),\\
     \Dvect{r}\cdot(\Drot\Drot\Dvect{F})
       &amp;=&amp; \frac{1}{\cos\varphi}\DP{(\Drot\Dvect{F})_\varphi}{\lambda}
         - \frac{1}{\cos\varphi}\DP{}{\varphi}
              [(\Drot\Dvect{F})_\lambda\cos\varphi]\\
       &amp;=&amp; \frac{1}{\cos\varphi}\DP{}{\lambda}
                 \left(  \frac{1}{r}\DP{}{r}(rF_\lambda)
                        -\frac{1}{r\cos\varphi}\DP{F_r}{\lambda} \right)\\
       &amp; &amp;  - \frac{1}{\cos\varphi}\DP{}{\varphi}
              \left[\cos\varphi 
                     \left( \frac{1}{r}\DP{F_r}{\varphi} 
                           - \frac{1}{r}\DP{}{r}(rF_\varphi)\right)\right]\\
       &amp;=&amp; \frac{1}{\cos\varphi}\DP{}{\lambda}
                  \frac{1}{r}\DP{}{r}(r F_\lambda)
            + \frac{1}{\cos\varphi}\DP{}{\varphi}
                  \frac{\cos\varphi}{r}\DP{}{r}(rF_\varphi)\\
       &amp; &amp;  - \frac{1}{r\cos^2\varphi}\DP[2]{F_r}{\lambda}
            - \frac{1}{r\cos\varphi}\DP{}{\varphi}
                  \left(\cos\varphi\DP{F_r}{\varphi}\right)\\
       &amp;=&amp; \frac{1}{r}\DP{}{r}\left[r \left(
                \frac{1}{\cos\varphi}\DP{F_\lambda}{\lambda}
               + \frac{1}{\cos\varphi}\DP{}{\varphi}(F_\varphi\cos\varphi)
               \right)\right]
             + \frac{L_2 F_r}{r}.
   \end{eqnarray*}

  \subsubsection{é€Ÿåº¦å&nbsp;´åŠã³ç£å&nbsp;´ã¨ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¨ã®é–¢ä¿‚å¼}
  
  ãƒˆãƒ­ã‚¤ãƒ€ãƒ«, ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‹ã‚‰
  é€Ÿåº¦å&nbsp;´ã‚’æ±‚ã‚ã‚‹å¼ã¯æ¬¡ã®é€šã‚Šã§ã‚ã‚‹:
  \begin{eqnarray}
   u  &amp; = &amp;   
    \DP[]{~\psi}{\varphi} 
    + \Dinv{r\cos\varphi} 
    \DP[]{}{r}\left( r \DP[]{~\phi}{\lambda} \right),\\
   v  &amp; = &amp; 
    - \Dinv{\cos\varphi}\DP[]{~\psi}{\lambda}
    + \Dinv{r} 
    \DP[]{}{r}\left( r \DP[]{~\phi}{\varphi} \right),\\
   w  &amp; = &amp; 
    \frac{L_2 ~\phi}{r}.
  \end{eqnarray}
  åŒæ§˜ã«ç£å&nbsp;´ã®å„æˆåˆ†ã«ã¤ã„ã¦:
  \begin{eqnarray}
   B_{\lambda}  &amp; = &amp;   
    \DP[]{~g}{\varphi} 
    + \Dinv{r\cos\varphi} 
    \DP[]{}{r}\left( r \DP[]{~h}{\lambda} \right),\\
   B_{\varphi}  &amp; = &amp; 
    - \Dinv{\cos\varphi}\DP[]{~g}{\lambda}
    + \Dinv{r} 
    \DP[]{}{r}\left( r \DP[]{~h}{\varphi} \right),\\
   B_{r}  &amp; = &amp; 
    \frac{L_2 ~h}{r}.
  \end{eqnarray}

  é€†ã«, é€Ÿåº¦å&nbsp;´ã®å„æˆåˆ†ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’æ±‚ã‚ã‚‹ã«ã¯, 
  \Deqref{é€Ÿåº¦åˆ†è§£}ã«
  $\Dvect{r}\cdot$, $\Dvect{r}\cdot\Drot$ ã‚’ä½œç”¨ã•ã›ã‚Œã°ã‚ˆã„. 
  %
  \begin{eqnarray*}
    &amp; &amp; \Dvect{r}\cdot\Dvect{u} 
        = \Dvect{r}\cdot[\Drot(\psi\Dvect{r}) + \Drot\Drot(\phi\Dvect{r}) ]
        = \Dvect{r}\cdot\Drot\Drot(\phi\Dvect{r}) 
        = L_2\phi,\\
    &amp; &amp; \Dvect{r}\cdot\Drot\Dvect{u} 
        = \Dvect{r}\cdot\Drot[\Drot(\psi\Dvect{r}) + \Drot\Drot(\phi\Dvect{r}) ]
        = \Dvect{r}\cdot\Drot\Drot(\psi\Dvect{r}) = L_2\psi,
  \end{eqnarray*}
  %
  ã—ãŸãŒã£ã¦, 
  \begin{equation}
        \phi = L_2^{-1}(rw), \quad
        \psi = L_2^{-1}[\Dvect{r}\cdot(\Drot\Dvect{u})]. 
  \end{equation}
  %
  ã“ã“ã§ $L_2^{-1}$ ã¯ $L_2$ ã®é€†æ¼”ç®—å­ã§ã‚ã‚‹. 
  åŒæ§˜ã«, \Deqref{ç£å&nbsp;´åˆ†è§£}ã«
  $\Dvect{r}\cdot$, $\Dvect{r}\cdot\Drot$ ã‚’ä½œç”¨ã•ã›ã‚‹ã“ã¨ã«ã‚ˆã‚Š
  ç£å&nbsp;´ã®å„æˆåˆ†ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’æ±‚ã‚ã‚‹å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
  \begin{equation}
        h = L_2^{-1}(rB_r), \quad
        g = L_2^{-1}[\Dvect{r}\cdot(\Drot\Dvect{B})]. 
  \end{equation}

%   ã¾ãŸ , å¾Œã®ä¾¿åˆ©ã®ãŸã‚ $U,V$ ã‚’
%   \begin{eqnarray}
%    U  
%     &amp; \equiv &amp; u\cos\varphi  \nonumber \\
%     &amp;=&amp; \cos\varphi\DP[]{~\psi}{\varphi} 
%         + \Dinv{r} \DP[]{}{r}\left( r \DP[]{~\phi}{\lambda} \right) \nonumber\\
%     &amp;=&amp; (1-\mu^{2})\DP[]{~\psi}{\varphi} 
%         + \Dinv{r} \DP[]{}{r}\left( r \DP[]{~\phi}{\lambda} \right), \\
%    V  &amp; \equiv &amp; v\cos\varphi \nonumber \\ 
%     &amp;=&amp; -\DP[]{~\psi}{\lambda}
%     + \frac{\cos\varphi}{r} 
%     \DP[]{}{r}\left( r \DP[]{~\phi}{\varphi} \right) \nonumber \\
%     &amp;=&amp; 
%      -\DP[]{~\psi}{\lambda}
%      + \frac{1-\mu^{2}}{r} 
%     \DP[]{}{r}\left( r \DP[]{~\phi}{\varphi} \right) 
%   \end{eqnarray}
%   ã¨å®šç¾©ã—ã¦ãŠã.
%   åŒæ§˜ã«ç£å&nbsp;´ã«ã¤ã„ã¦ã‚‚
%   \begin{eqnarray}
%    B_{\lambda }\cos\varphi  
%     &amp;=&amp; \cos\varphi\DP[]{~g}{\varphi} 
%         + \Dinv{r} \DP[]{}{r}\left( r \DP[]{~h}{\lambda} \right) \nonumber\\
%     &amp;=&amp; (1-\mu^{2})\DP[]{~g}{\varphi} 
%         + \Dinv{r} \DP[]{}{r}\left( r \DP[]{~h}{\lambda} \right), \\
%    B_{\varphi}\cos\varphi 
%     &amp;=&amp; -\DP[]{~g}{\lambda}
%     + \frac{\cos\varphi}{r} 
%     \DP[]{}{r}\left( r \DP[]{~h}{\varphi} \right) \nonumber \\
%     &amp;=&amp; 
%      -\DP[]{~g}{\lambda}
%      + \frac{1-\mu^{2}}{r} 
%     \DP[]{}{r}\left( r \DP[]{~h}{\varphi} \right). 
%   \end{eqnarray}


%--------------------------------------------------------------------------
\newpage   
 \subsection{å¢ƒç•Œæ¡ä»¶}

\subsubsection{é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶}   

é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å¢ƒç•Œã‚’æ¨ªåˆ‡ã‚‹æµã‚ŒãŒå­˜åœ¨ã—ãªã„, ã§ã‚ã‚‹.
ã™ãªã‚ã¡:
\begin{equation}
 w = 0, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}.
\end{equation}

ã“ã‚Œã‚’ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°,
\begin{equation}
 \phi = 0, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}. 
\end{equation}

\subsubsection{åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶}   

åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ã¨ã—ã¦ã¯,
å¿œåŠ›ç„¡ã—(stress-free), ã‚‚ã—ãã¯æ»‘ã¹ã‚Šç„¡ã—(rigid)ã‚’è€ƒãˆã‚‹.

å¿œåŠ›ç„¡ã—ã®å&nbsp;´åˆã«ã¯
\begin{equation}
 \DP{}{r}\biggl(\frac{u}{r}\biggr) = \DP{}{r}\biggl(\frac{v}{r}\biggr) 
  = 0, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}.
\end{equation}

ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°
\begin{equation}
 \DP[2]{\phi}{r}
  = \DP{}{r}\biggl(\frac{\psi}{r}\biggr)
  = 0, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}. 
\end{equation}

æ»‘ã¹ã‚Šç„¡ã—ã®å&nbsp;´åˆã«ã¯,
\begin{equation}
 u = v = 0,~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}. 
\end{equation}

ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°
\begin{equation}
 \DP{\phi}{r} = \psi = 0,~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}. 
\end{equation}

\subsubsection{ç†±çš„å¢ƒç•Œæ¡ä»¶}

ç†±çš„å¢ƒç•Œæ¡ä»¶ã¨ã—ã¦ã¯, 
å¢ƒç•Œã«ãŠã„ã¦æ¸©åº¦å›ºå®š, 
ã‚‚ã—ãã¯ç†±ãƒ•ãƒ©ãƒƒã‚¯ã‚¹å›ºå®šã‚’è€ƒãˆã‚‹.

æ¸©åº¦å›ºå®šæ¡ä»¶ã®å&nbsp;´åˆã«ã¯
\begin{equation}
 \theta = 0, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}.
\end{equation}

ç†±ãƒ•ãƒ©ãƒƒã‚¯ã‚¹å›ºå®šã®å&nbsp;´åˆã«ã¯
\footnote{
åŸºæœ¬å&nbsp;´ã®ç†±ãƒ•ãƒ©ãƒƒã‚¯ã‚¹ãŒå¢ƒç•Œã«ãŠã„ã¦ä¸Žãˆã‚‰ã‚Œã¦ã„ã‚‹äº‹ã«ãªã‚‹.
},
\begin{equation}
 \DP{\theta}{r} = 0,~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}.
\end{equation}

\subsubsection{ç£å&nbsp;´ã®å¢ƒç•Œæ¡ä»¶}

ç£å&nbsp;´ã®å¢ƒç•Œæ¡ä»¶ã¨ã—ã¦
çƒæ®»ã®å¤–éƒ¨ãŒä¸å°Žä½“ã§ã‚ã‚‹å&nbsp;´åˆã¨,
å¤–éƒ¨ã¨å†…éƒ¨ã®é›»æ°—ä¼å°Žåº¦ãŒç­‰ã—ã„å&nbsp;´åˆã‚’è€ƒãˆã‚‹.

çƒæ®»ã®å¤–éƒ¨ãŒä¸å°Žä½“ã®å&nbsp;´åˆã«ã¯
ç£å&nbsp;´ãŒå¢ƒç•Œã«ãŠã„ã¦å¤–éƒ¨ã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«å&nbsp;´ã«æŽ¥ç¶šã•ã‚Œã‚‹.
ã™ãªã‚ã¡, 
å¤–éƒ¨ã®ç£å&nbsp;´ã‚’ $\Dvect{B}_{e}$, 
ãã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’ $W$ ã¨ã—ã¦
\begin{equation}
 \Dvect{B} = \Dvect{B}_{e}
, ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}.
\end{equation}
ãŸã&nbsp;ã— $\Dvect{B}_{e} = \Dgrad W,~~~\Dlapla W=0$ ã§ã‚ã‚‹. 

çƒæ®»å¤–éƒ¨ã¨å†…éƒ¨ã®é›»æ°—ä¼å°Žåº¦ãŒç­‰ã—ã„å&nbsp;´åˆã«ã¯,
ç£å&nbsp;´ã&nbsp;ã‘ã§ã¯ãªãé›»å&nbsp;´ã®æ°´å¹³æˆåˆ†ã‚‚
å¢ƒç•Œã«ãŠã„ã¦é€£ç¶šã¨ãªã‚‹:
\begin{eqnarray}
 \Deqlab{ç£å&nbsp;´å¢ƒç•Œ}
 \Dvect{B} = \Dvect{B}_{e}
  , ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}, \\
 \Deqlab{é›»å&nbsp;´å¢ƒç•Œ}
 \Dvect{E}_{H} = \Dvect{E}_{eH}
  , ~~~\mbox{at}~~~ r=\frac{\xi}{1-\xi}, \quad \frac{1}{1-\xi}
\end{eqnarray}
ã“ã“ã§ æ·»å­— $_H$ ã¯æ°´å¹³æˆåˆ†ã‚’è¡¨ã—ã¦ã„ã‚‹. 
çƒæ®»å¤–éƒ¨ã®é›»ç£å&nbsp;´ã¯åˆ¥é€”ä¸Žãˆã‚‰ã‚Œãªã‘ã‚Œã°ãªã‚‰ãªã„
\footnote{
åœ°çƒãƒ€ã‚¤ãƒŠãƒ¢ã§å†…æ&nbsp;¸ã‚’è§£ãéš›ã«è‰¯ãç”¨ã„ã‚‰ã‚Œã‚‹æ–¹æ³•ã¯, 
å†…æ&nbsp;¸ãŒè§’é€Ÿåº¦ $\Dvect{\omega}_{i}=\omega_i\Dvect{k}$
ã§å‰›ä½“å›žè»¢ã—ã¦ã„ã‚‹ã¨ã—ã¦, 
\[
 \Dvect{v}_{i} = \Dvect{\omega}_{i}\times\Dvect{r}
\]
ã‚’èª˜å°Žæ–¹ç¨‹å¼\Deqref{ç„¡æ¬¡å…ƒèª˜å°Žæ–¹ç¨‹å¼}ã¸ä»£å…¥ã—$\Dvect{B}_{e}$
ã‚’æ±‚ã‚ã‚‹, ã§ã‚ã‚‹. 
å†…æ&nbsp;¸ã®è§’é€Ÿåº¦ã®æ™‚é–“å¤‰åŒ–ã¯åˆ¥é€”ãƒˆãƒ«ã‚¯ã®å¼ã‚’ç”¨ã„ã¦æ±‚ã‚ã‚‰ã‚Œã‚‹.
}.


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