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\section{é‰›ç›´é›¢æ•£åŒ–}

 Glatzmaier(1984) ã«å¾“ã„, æ–¹ç¨‹å¼ã‚’é‰›ç›´æ–¹å‘ã«é›¢æ•£åŒ–ã™ã‚‹. 
 é‰›ç›´å¾®åˆ†ã¯ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã‚’ç”¨ã„ã¦è©•ä¾¡ã™ã‚‹.

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 \subsection{é‰›ç›´é›¢æ•£åŒ–}
 
 å‹•å¾„ã‚’å·®åˆ†åŒ–ã—, ä¸Šã®å±¤ã‹ã‚‰ä¸‹ã®å±¤ã¸ã¨ç•ªå·ã‚’ã¤ã‘ã‚‹.

 æ&nbsp;¼å­ç‚¹ã‚’ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•ãƒã‚¤ãƒ³ãƒˆ(ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°ãŒæ¥µåœ°ã‚’ã¨ã‚‹ç‚¹)ã«ã¨ã‚‹ãŸã‚, 
 å‹•å¾„ã‚’æ¬¡å¼ã®ã‚ˆã†ã«åº§æ¨™å¤‰æ›ã™ã‚‹:
 \begin{eqnarray}
  r   &amp;=&amp; \frac{r_{o} + r_{i}}{2} + \frac{r_{o}-r_{i}}{2}\xi,\\
  \xi &amp;=&amp; \frac{2r - (r_{o}+r_{i})}{r_{o}-r_{i}}.
 \end{eqnarray}
 ã“ã®å¤‰æ›ã«ã‚ˆã‚Š, å‹•å¾„ã¯ $\xi$ ã‚’ç”¨ã„ã¦åŒºé–“ $[-1,1]$ ã§è¡¨ç¾ã•ã‚Œã‚‹
 (ä¸Šéƒ¨å¢ƒç•ŒãŒ $1$, ä¸‹éƒ¨å¢ƒç•ŒãŒ$-1$).

 åº§æ¨™ $\xi$ ä¸Šã§ $K+1$ å€‹ã®æ&nbsp;¼å­ç‚¹ã‚’æ¬¡ã®ã‚ˆã†ã«å®šç¾©ã™ã‚‹:
 \begin{equation}
  \xi_{k} = \cos\biggl(\frac{\pi k}{K} \biggr),\quad k = 0,1,\cdots K.
 \end{equation}
 å„æ&nbsp;¼å­ç‚¹ã§ã®ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°ã®å€¤ã¯
 \begin{equation}
  T_{l}(\xi_{k}) = \cos\biggl(\frac{l k \pi}{K} \biggr).
 \end{equation}

 æ&nbsp;¼å­ç‚¹ã§ã®ç‰©ç†é‡ã‚’ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°ã§å±•é–‹ã™ã‚‹.     
 ä¾‹ãˆã° $\tilde{\psi}(r,t)_{n}^{m}$ ã«ã¤ã„ã¦,
 \begin{equation}
  \psi_{nl}^{m}(t) 
   = \biggl(\frac{2}{K}\biggr)
   \sum_{k=0}^{K}{}''
   \tilde{\psi}_{n}^{m}(\xi_{k},t) T_{l}(\xi_{k}), 
   \quad l = 0,1,\cdots K.
 \end{equation}
 ãŸã&nbsp;ã— $\sum{}''$ ã¯
 æœ€åˆã¨æœ€å¾Œã®é&nbsp;…ã‚’ 2 åˆ†ã® 1 å€ã—ã¦å’Œã‚’ã¨ã‚‹ã“ã¨ã‚’è¡¨ã—ã¦ã„ã‚‹. 
 åŒæ§˜ã«ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•å±•é–‹ä¿‚æ•°ã‹ã‚‰æ&nbsp;¼å­ç‚¹ã§ã®å€¤ã‚’æ±‚ã‚ã‚‹ã«ã¯
 \begin{equation}
  \tilde{\psi}_{n}^{m}(\xi_{k},t) 
   = \sum_{l=0}^{K}{}''\psi_{jn}^{m}(t) T_{j}(\xi_{k}).
 \end{equation}
 ä»¥ä¸Šã®çµæžœã¯, 
 $\tilde{\phi}(r,t)_{n}^{m},
  \tilde{g}(r,t)_{n}^{m}, \tilde{h}(r,t)_{n}^{m}
  \tilde{\theta}(r,t)_{n}^{m}$
 ã«ã¤ã„ã¦ã‚‚ãã®ã¾ã¾æˆç«‹ã™ã‚‹.

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 \subsection{é‰›ç›´å¢ƒç•Œæ¡ä»¶}

 å†…éƒ¨é&nbsp;˜åŸŸã®æ&nbsp;¼å­ç‚¹ã«ãŠã„ã¦,
 å·®åˆ†åŒ–ã—ãŸæ–¹ç¨‹å¼ã‚’æ™‚é–“ç©åˆ†ã—ãŸå¾Œã«,
 å¢ƒç•Œæ¡ä»¶ã‚’æº€ãŸã™ã‚ˆã†ã«å¢ƒç•Œã«ãŠã‘ã‚‹æ&nbsp;¼å­ç‚¹ä¸Šã®é–¢æ•°å€¤ã‚’å®šã‚ã‚‹. 

 \subsubsection{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}
  
 ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«é–¢ã—ã¦ã¯ 
 $r_k$ ($ k=1,\cdots K-1$ )ã®ç‚¹ã§æ™‚é–“ç©åˆ†ã‚’å®Ÿè¡Œã™ã‚‹.

 å¿œåŠ›ç„¡ã—æ¡ä»¶ã®å&nbsp;´åˆ, å¢ƒç•Œå€¤ã¯å†…éƒ¨é&nbsp;˜åŸŸã‚’è¨ˆç®—ã—ãŸå¾Œ
 \begin{eqnarray}
   \tilde{\psi}_{n}^{m}(r_{0},t) 
    = \frac{r_{0}}{r_1}\tilde{\psi}_{n}^{m}(r_1,t) , \quad
   \tilde{\psi}_{n}^{m}(r_{K},t)
    = \frac{r_{K}}{r_{K-1}}\tilde{\psi}_{n}^{m}(r_{K-1},t)
 \end{eqnarray}
 ã§å®šã‚ã‚‹. 

 æ»‘ã¹ã‚Šç„¡ã—æ¡ä»¶ã®å&nbsp;´åˆã¯
 \begin{eqnarray}
   \Deqlab{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«å¢ƒç•Œ:rigid}
    \tilde{\psi}_{n}^{m}(r_{0},t) = 0, \quad   
    \tilde{\psi}_{n}^{m}(r_{K},t) = 0.
 \end{eqnarray}
 ã¨ãªã‚‹.

 \subsubsection{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

 ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¯
 $r_{k},~(k=2,\cdots K-2)$ ã®ç‚¹ã§ 
 $(D_{n}\tilde{\phi}_{n}^{m})$ ã«ã¤ã„ã¦æ™‚é–“ç©åˆ†ã‚’å®Ÿè¡Œã—ãŸå¾Œ,
 $D_{n}$ ã®é€†æ¼”ç®—(ç©åˆ†)ã‚’ãŠã“ãªã£ã¦
 $\tilde{\phi}_{n}^{m}$ ã‚’å®šã‚ã‚‹.
 ãã®ãŸã‚ã«ã¯å¢ƒç•Œå€¤å•é¡Œ
 \begin{eqnarray}
   \Deqlab{å¢ƒç•Œå€¤å•é¡Œ}
   &amp;&amp; 
   \biggl(\DD[2]{}{r} + \frac{2}{r}\DD[]{}{r} - \frac{n(n+1)}{r^{2}}\biggr) 
   \tilde{\phi}_{n}^{m} 
   = (D_{n}\tilde{\phi}_{n}^{m}),  \\
   \Deqlab{å¢ƒç•Œå€¤:å¿œåŠ›ç„¡ã—}
   &amp;&amp; 
    \tilde{\phi}_n^m = \DP[2]{\tilde{\phi}_n^m}{r} = 0
          \quad \mbox{at} \quad  r = r_{i}, r_{o}. 
 \end{eqnarray}
 ã‚’è§£ã‹ã­ã°ãªã‚‰ãªã„(ä¸Šå¼ã¯å¿œåŠ›ç„¡ã—æ¡ä»¶ã®å&nbsp;´åˆ).
 æ»‘ã¹ã‚Šç„¡ã—æ¡ä»¶ã®å&nbsp;´åˆã«ã¯ã€\Deqref{å¢ƒç•Œå€¤:å¿œåŠ›ç„¡ã—}ã‚’
 \begin{equation}
    \tilde{\phi}_n^m = \DP{\tilde{\phi}_n^m}{r} = 0
          \quad \mbox{at} \quad  r = r_{i}, r_{o}. 
 \end{equation}
 ã¸ç½®ãæ›ãˆã‚Œã°ã‚ˆã„.
    
 \Deqref{å¢ƒç•Œå€¤å•é¡Œ}, \Deqref{å¢ƒç•Œå€¤:å¿œåŠ›ç„¡ã—}
 ã‚’ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã§è¡¨ç¾ã™ã‚‹ã¨
 \begin{eqnarray}
     &amp;&amp;\sum_{l=0}^{K}{}''
          \biggl(    
               \DD[2]{T_j(r_k)}{r} 
             + \frac{2}{r_k} \DD[]{T_j(r_k)}{r}
             - \frac{n(n+1)}{r_k^2} T_j(r_k)
         \biggr) \phi_{jn}^m 
      =  D_n \tilde{\phi}_n^m(r_k), \nonumber \\
     &amp;&amp; {}{}{}k = 2, \cdots K-2, \\
     &amp;&amp; \Deqlab {é‹å‹•æ¡ä»¶}
      \sum_{l=0}^{K}{}''T_j(r_0) \phi_{jn}^m = 0, 
      \quad 
      \sum_{l=0}^{K}{}''T_j(r_K) \phi_{jn}^m = 0, \\
     &amp;&amp; \Deqlab{ãƒãƒ­ã‚¤ãƒ€ãƒ«:free}
      \sum_{l=0}^{K}{}''\DD[2]{T_j(r_0)}{r} \phi_{jn}^m = 0, 
      \quad
      \sum_{l=0}^{K}{}''\DD[2]{T_j(r_K)}{r} \phi_{jn}^m = 0.
 \end{eqnarray}
 ã¨, $\phi_{jn}^m$ ã«é–¢ã™ã‚‹é€£ç«‹ 1 æ¬¡æ–¹ç¨‹å¼ã¨ãªã‚‹. ã“ã‚Œã‚’ LU åˆ†è§£ã§è§£ã,
 $\phi_{jn}^m $ ã‚’å®šã‚ã‚‹. 
 
 æ»‘ã¹ã‚Šç„¡ã—æ¡ä»¶ã®å&nbsp;´åˆã¯ \Deqref{ãƒãƒ­ã‚¤ãƒ€ãƒ«:free}ãŒ
 \begin{eqnarray}
     \Deqlab{ãƒãƒ­ã‚¤ãƒ€ãƒ«:rigid}
      \sum_{l=0}^{K}{}''\DD[]{T_j(r_0)}{r} \phi_{jn}^m = 0, 
      \quad 
      \sum_{l=0}^{K}{}''\DD[]{T_j(r_K)}{r} \phi_{jn}^m = 0,
 \end{eqnarray}
 ã¨ãªã‚‹.

 \subsubsection{æ¸©åº¦}

 æ¸©åº¦ã«é–¢ã—ã¦ã¯ $r_k, k=1,\cdots K-1$ ã®ç‚¹ã§æ™‚é–“ç©åˆ†ã‚’å®Ÿè¡Œã™ã‚‹. 
 å¢ƒç•Œå€¤ã¯å†…éƒ¨é&nbsp;˜åŸŸã‚’è¨ˆç®—ã—ãŸå¾Œ(æ¸©åº¦å›ºå®šã®å&nbsp;´åˆ)
 \begin{equation}
  \Deqlab{æ¸©åº¦å›ºå®š}
   \tilde{\Theta}_n^m(r_0,t) = 0, \ \ \tilde{\Theta}_n^m(r_K,t) = 0,
 \end{equation}
 ã§å®šã‚ã‚‹. ç†±ãƒ•ãƒ©ãƒƒã‚¯ã‚¹å›ºå®šã®å&nbsp;´åˆã¯ \Deqref{æ¸©åº¦å›ºå®š} ã‚’
 \begin{equation}
  \tilde{\Theta}_n^m(r_0,t) = \tilde{\Theta}_n^m(r_1,t), \ \ 
   \tilde{\Theta}_n^m(r_K,t) = \tilde{\Theta}_n^m(r_{K-1},t),
 \end{equation}
 ã«ç½®ãæ›ãˆã‚‹. 

 \subsubsection{ç£å&nbsp;´}

 å…ˆãšå¢ƒç•Œå¤–éƒ¨ãŒä¸å°Žä½“ã®å&nbsp;´åˆã‚’è€ƒãˆã‚‹.
 æŽ¥ç¶šã™ã¹ããƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«å&nbsp;´ $W$ ã‚’çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ã™ã‚‹ã¨, 
 \begin{eqnarray}
      W &amp; = &amp; \sum_{n,m} W_{n}^{m} r^{-n-1} Y_{n}^{m} \quad (r &gt; r_{o}), \\
        &amp; = &amp; \sum_{n,m} W_{n}^{m} r^{n} Y_{n}^{m} \quad  (r &lt; r_{i}).
 \end{eqnarray}
 å¾“ã£ã¦å„æˆåˆ†ã¯
 \begin{eqnarray}
  B_\lambda \cos\theta
   &amp; = &amp;  \sum_{n,m} im W_{n}^{m} r^{-n-2} Y_{n}^{m} \quad  (r &gt; r_{o}), \\
   &amp; = &amp;  \sum_{n,m} im W_{n}^{m} r^{n-1} Y_{n}^{m} \quad  (r &lt; r_{i}),\\
  B_\mu \cos\theta
   &amp; = &amp;  \sum_{n,m} W_{n}^{m} r^{-n-2}(1-\mu^2)\DP{Y_{n}^{m}}{\mu} 
   \quad  ( r&gt;r_o ), \\
   &amp; = &amp;  \sum_{n,m} W_{n}^{m} r^{n-1}(1-\mu^2)\DP{Y_{n}^{m}}{\mu} 
   \quad  ( r&lt;r_i ),\\
  B_r 
   &amp; = &amp; -\sum_{n,m} (n+1)W_{n}^{m} r^{-n-2} Y_{n}^{m} \quad  (r &gt; r_{o}), \\
   &amp; = &amp;  \sum_{n,m} n W_{n}^{m} r^{n-1} Y_{n}^{m} \quad  (r &lt; r_{i}).
 \end{eqnarray}
 ã“ã‚Œã«å¯¾ã—ã¦, ç£å&nbsp;´ã®å„æˆåˆ†ãŒé€£ç¶šã§ãªã‘ã‚Œã°ãªã‚‰ãªã„. 
 ä¸€æ–¹, ç£å&nbsp;´ã®é‰›ç›´æˆåˆ† \Deqref{ç£å&nbsp;´é‰›ç›´} ã¨æ¯”è¼ƒã™ã‚‹ã“ã¨ã«ã‚ˆã‚Š, 
 \begin{eqnarray}
  \frac{n(n+1)h_n^m}{r}= -(n+1)W_{n}^{m} r^{-n-2} 
   \quad  (r =r_{o}), \\
  \frac{n(n+1)h_n^m}{r}= n W_{n}^{m} r^{n-1} 
   \quad  (r = r_{i}).
 \end{eqnarray}
 ã•ã‚‰ã«æ°´å¹³æˆåˆ†ã‚’ \Deqref{ç£å&nbsp;´çµŒåº¦}, \Deqref{ç£å&nbsp;´ç·¯åº¦}ã¨æ¯”è¼ƒã—ã¦
 \begin{eqnarray}
  g_n^m =0, \quad
   \Dinv{r}\DP{(r h_n^m)}{r} = W_{n}^{m} r^{-n-2}, \quad (r=r_{o}),\\
  g_n^m =0, \quad 
   \Dinv{r}\DP{(r h_n^m)}{r} = W_{n}^{m} r^{n-1}, \quad  (r=r_{i}),
 \end{eqnarray}
 $W_{n}^{m}$ ã‚’ä»£å…¥ã—ã¦, $h_n^m$ ã®å¢ƒç•Œæ¡ä»¶ãŒå¾—ã‚‰ã‚Œã‚‹: 
 \begin{eqnarray}
  \DP{h_n^m}{r} + \frac{(n+1)h_{n}^{m}}{r} = 0, \quad  (r=r_{o}),\\
  \DP{h_n^m}{r} - \frac{nh_{n}^{m}}{r} = 0, \quad  (r=r_{i}),
 \end{eqnarray}
 å¾“ã£ã¦ã¾ã¨ã‚ã‚‹ã¨, 
 ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ç£å&nbsp;´ã«é–¢ã—ã¦ã¯ $r_k, k=1,\cdots K-1$ ã®ç‚¹ã§
 æ™‚é–“ç©åˆ†ã‚’å®Ÿè¡Œã™ã‚‹. å¢ƒç•Œå€¤ã¯å†…éƒ¨é&nbsp;˜åŸŸã‚’è¨ˆç®—ã—ãŸå¾Œ, 
 \begin{equation}
  \tilde{g}_n^m(r_{0},t) = 0, \quad \tilde{g}_n^m(r_{K},t) = 0, 
 \end{equation}
 ã§å®šã‚ã‚‹. 
 ãƒãƒ­ã‚¤ãƒ€ãƒ«ç£å&nbsp;´ã‚‚ $r_k, k=1,\cdots K-1$ ã®ç‚¹ã§æ™‚é–“ç©åˆ†ã‚’å®Ÿè¡Œã™ã‚‹. 
 ãã®å¾Œå¢ƒç•Œæ¡ä»¶ã«ã‚ˆã‚Šå¢ƒç•Œã®å€¤ã‚’å®šã‚ã‚‹:
 \begin{eqnarray}
  &amp;&amp; \DP{h_n^m}{r} + \frac{(n+1)h_n^m}{r} = 0, \quad  
         \mbox{at} \quad  r=r_{0}, \\
  &amp;&amp; \DP{h_n^m}{r} - \frac{nh_n^m}{r} = 0, \quad  
         \mbox{at} \quad  r=r_{K}. 
 \end{eqnarray}
 ã¾ãŸ, ã“ã®æ™‚ã®å¤–éƒ¨ç£å&nbsp;´ã¯æ¬¡å¼ã§è¡¨ã•ã‚Œã‚‹:
 \begin{equation}
  W_{n}^{m} = -n (r^{n+1} h_n^m)_{r=r_0} \ \ (r &gt; r_{o}), \quad 
  W_{n}^{m} = \biggl[\frac{(n+1)h_n^m}{r^{n}}\biggr]_{r=r_K},\ \ (r &lt; r_{i}).
 \end{equation}
 
 æ¬¡ã«çƒæ®»å†…ã®æµä½“ã¨çƒæ®»ã®èŠ¯ã¨ãªã‚‹é&nbsp;˜åŸŸ(ä»¥ä¸‹, å†…æ&nbsp;¸)ã¨ã®
 é›»æ°—ä¼å°Žåº¦ãŒç­‰ã—ã„å&nbsp;´åˆã‚’, 
 ç‰¹ã«å†…æ&nbsp;¸ãŒç³»ã«å¯¾ã—ã¦è§’é€Ÿåº¦ $\Dvect{\omega}^{i}=\omega^{i}\Dvect{k}$
 ã§å‰›ä½“å›žè»¢ã—ã¦ã„ã‚‹å&nbsp;´åˆã«ã¤ã„ã¦è€ƒãˆã‚‹(ä»¥ä¸‹, æ·»å­—$^{i}$ã¯å†…æ&nbsp;¸ã‚’è¡¨ã™).
 
 å†…æ&nbsp;¸ã®ç£å&nbsp;´ $\Dvect{B}^{i}$ åŠã³è§’é€Ÿåº¦ $\omega_{i}$ ã¯,
 å‰›ä½“å›žè»¢ã™ã‚‹å°Žä½“ã®èª˜å°Žæ–¹ç¨‹å¼ã¨
 è‡ªè»¢è»¸å‘¨ã‚Šã®ãƒˆãƒ«ã‚¯ã®é‡£åˆã‹ã‚‰, 
 \begin{eqnarray}
  \Deqlab{å†…æ&nbsp;¸èª˜å°Ž}
  \DP{\Dvect{B}^{i}}{t} 
   &amp;=&amp; \omega^{i}\DP{\Dvect{B}^{i}}{\lambda} + P_{m}^{-1}\Dlapla\Dvect{B}^{i}\\
  \Deqlab{ãƒˆãƒ«ã‚¯}
  I\DP{\omega^{i}}{t} &amp;=&amp; \Gamma_{M}+\Gamma_{\nu}
 \end{eqnarray}
 ã«ã‚ˆã£ã¦æ±‚ã‚ã‚‰ã‚Œã‚‹.
 ã“ã“ã§ $I$ ã¯å†…æ&nbsp;¸ã®è‡ªè»¢è»¸å‘¨ã‚Šã®æ…£æ€§ãƒ¢ãƒ¼ãƒ¡ãƒ³ãƒˆã§ã‚ã‚Š, 
 $\Gamma_{M}$  ã¯ãƒ­ãƒ¼ãƒ¬ãƒ³ãƒ„åŠ›ã«ã‚ˆã‚‹ãƒˆãƒ«ã‚¯ã‚’, 
 $\Gamma_{\nu}$ ã¯ç²˜æ€§å¿œåŠ›ã«ã‚ˆã‚‹ãƒˆãƒ«ã‚¯ã‚’è¡¨ã™.

 \begin{itemize}
  \item å›žè»¢å¯¾ç§°æ€§ã‚’ç”¨ã„ã‚‹:\\
	$\Dvect{B}^{i}(\lambda,\varphi,-r) 
	 = \Dvect{B}^{i}(\lambda \pm \pi, \pi - \varphi,r)$
  \item $h_{n}^{mi}(-r) = (-1)^{l+1}h_{n}^{mi}(r)$
  \item ãƒã‚§ãƒœã‚·ã‚§ãƒ•ã§è¨ˆç®—ã™ã‚‹ã®ã¯, $r=[0,r]$ ã¾ã§.
  \item å¢ƒç•Œæ¡ä»¶ã¯ rigid â†’ ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«, ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å‹•å¾„å¾®åˆ†ãŒé€£ç¶š
  \item ãƒˆãƒ«ã‚¯ã¯â€¦ã‚€ãƒ¼ã‚“.
 \end{itemize}

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