<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{æ°´å¹³é›¢æ•£åŒ–}

 æ°´å¹³æ–¹å‘ã®é›¢æ•£åŒ–ã«ã¯æ“¬ã‚¹ãƒšã‚¯ãƒˆãƒ«å¤‰æ›ã‚’ç”¨ã„ã‚‹(Glatzmaier,1984).
 ç·šå½¢é&nbsp;…ã«é–¢ã™ã‚‹æ°´å¹³å¾®åˆ†ã«ã¤ã„ã¦ã¯
 çƒé¢èª¿å’Œé–¢æ•°ã‚’åŸºåº•ã¨ã™ã‚‹ç›´è¡Œé–¢æ•°å±•é–‹ã«ã‚ˆã£ã¦è©•ä¾¡ã—,
 éžç·šå½¢é&nbsp;…ã¯æ&nbsp;¼å­ç‚¹ä¸Šã§å€¤ã‚’è¨ˆç®—ã™ã‚‹.
 
%------------------------------------------------------------------------
 \subsection{ã‚¹ãƒšã‚¯ãƒˆãƒ«å±•é–‹}

 æ°´å¹³æ–¹å‘ã®å±•é–‹é–¢æ•°ç³»ã¨ã—ã¦, 
 æ¼”ç®—å­ $L_{2}$ ã®å›ºæœ‰é–¢æ•°ã§ã‚ã‚‹
 çƒé¢èª¿å’Œé–¢æ•° $Y^{m}_{n}(\lambda,\mu)$ ã‚’ç”¨ã„ã‚‹.
 $Y^{m}_{n}(\lambda,\mu)$ ã¯ãƒ«ã‚¸ãƒ£ãƒ³ãƒ‰ãƒ«é™ªé–¢æ•° $P^{m}_{n}(\mu)$ ã‚’ç”¨ã„ã¦
 \begin{equation}
  Y^{m}_{n}(\lambda,\mu) \equiv P^{m}_{n}(\mu) e^{im\lambda}
 \end{equation}
 ã§å®šç¾©ã•ã‚Œ, æ¼”ç®—å­ $L_{2}$ ã«ã¤ã„ã¦
 \begin{equation}
  L_{2} Y^{m}_{n}(\lambda,\mu) = n(n+1)Y^{m}_{n}(\lambda,\mu)
 \end{equation}
 ã‚’æº€ãŸã™. ãŸã&nbsp;ã— $n \geq |m|$ ã§ã‚ã‚‹.

 ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ« $\psi$ ã® $Y^{m}_{n}(\lambda,\mu)$ ã«ã‚ˆã‚‹
 é›¢æ•£çš„ãªå±•é–‹ $\psi_{ij}$ ã¯
 \begin{equation}
  \psi_{ij} \equiv \psi(\lambda_{i},\mu_{j}) 
   = \mathcal{R}_{e} \sum^{N}_{m=-N}\sum^{N}_{n=|m|}
     \tilde{\psi}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j}),
 \end{equation}
 ãã®é€†å¤‰æ› $\tilde{\psi}^{m}_{n}$ ã¯
 \begin{eqnarray}
  \tilde{\psi}^{m}_{n}
   &amp;=&amp; \frac{1}{4\pi}\int^{1}_{-1}\Dd \mu\int^{\pi}_{0}\Dd \lambda
       \psi(\lambda,\mu)Y^{m*}_{n}(\lambda,\mu) \nonumber \\
   &amp;=&amp; \frac{1}{I}\sum^{I}_{i=1}\sum^{J}_{j=1}
       \psi(\lambda_{i},\mu_{j})Y^{m*}_{n}(\lambda_{i},\mu_{j})\omega_{j}
 \end{eqnarray}
 ã§ã‚ã‚‹
 \footnote{
 ã“ã“ã§ã¯æ³¢æ•°åˆ‡æ–­ã¨ã—ã¦ä¸‰è§’å½¢åˆ‡æ–­ã‚’ç”¨ã„ã¦ã„ã‚‹.
 }.
 é€†å¤‰æ›ã«ãŠã„ã¦ç©åˆ†ã‚’å’Œã«ç½®ãæ›ãˆã¦è©•ä¾¡ã™ã‚‹éš›ã¯,
 $\lambda$ ç©åˆ†ã«ã¤ã„ã¦ Gauss ã®å°å½¢å…¬å¼ã‚’,
 $\mu$ ç©åˆ†ã«ã¤ã„ã¦ã¯ Gauss--Legendre ã®ç©åˆ†å…¬å¼ã‚’ç”¨ã„ãŸ.
 $\mu_{j}$ ã¯ Gauss ç·¯åº¦, $\omega_{j}$ ã¯ Gauss è·é‡,
 $\lambda_{i}$ ã¯ç­‰é–“éš”ã®æ&nbsp;¼å­ã§ã‚ã‚‹.

 ä»¥ä¸‹ã§ã¯ç°¡å˜ã®ãŸã‚, å’Œã®è¨˜å·ã‚’ 
 $\displaystyle{\sum^{N}_{m=-N}\sum^{N}_{n=|m|}=\sum_{m,n},\quad \sum^{I}_{i=1}\sum^{J}_{j=1}=\sum_{i,j}}$
 ã¨è¡¨ã™äº‹ã¨ã™ã‚‹.

 æ°´å¹³å¾®åˆ†ã‚’å«ã‚€é&nbsp;…ã®æ&nbsp;¼å­ç‚¹ã«ãŠã‘ã‚‹å€¤ã¯,
 ã‚¹ãƒšã‚¯ãƒˆãƒ«å±•é–‹ã«ã‚ˆã‚Šå„ã€…æ¬¡å¼ã§æ±‚ã‚ã‚‰ã‚Œã‚‹:
 \begin{eqnarray}
  \biggl(\DP{\psi}{\lambda}\biggr)_{ij} 
   &amp;=&amp; 
     \mathcal{R}_{e}\sum_{m,n}\sum_{i,j}
     im\tilde{\psi}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j}), \\
  %
  \biggl( (1-\mu^{2}) \DP{\psi}{\mu} \biggr)_{ij} 
   &amp;=&amp;
     \mathcal{R}_{e}\sum_{m,n}\sum_{i,j}
     \tilde{\psi}^{m}_{n}(1-\mu^{2})\DP{}{\mu}Y^{m}_{n}(\lambda_{i},\mu_{j}). 
 \end{eqnarray}

 ã¾ãŸ, éžç·šå½¢é&nbsp;…ã‚’æ&nbsp;¼å­ç‚¹ä¸Šã§è©•ä¾¡ã™ã‚‹éš›ã«ç¾ã‚Œã‚‹æ°´å¹³å¾®åˆ†ã‚’è¨ˆç®—ã—ã¦ãŠã:
 \begin{eqnarray}
  \biggl(\widetilde{\frac{1}{1-\mu^2}\DP{\psi}{\lambda}}\biggr)^{m}_{n}   
   &amp;=&amp; 
     \frac{1}{4\pi}\int^{1}_{-1}\Dd \mu\int^{\pi}_{0}\Dd \lambda
     \biggl(\frac{1}{1-\mu^{2}}\DP{\psi}{\lambda}\biggr)Y^{m*}_{n}(\lambda,\mu) 
     \nonumber \\
   &amp;=&amp; 
     \frac{1}{4\pi}\int^{1}_{-1}\Dd \mu\int^{\pi}_{0}\Dd \lambda
     ~im\psi(\lambda,\mu)\frac{1}{1-\mu^{2}}Y^{m*}_{n}(\lambda,\mu) 
     \nonumber \\
   &amp;=&amp; 
     \frac{1}{I}\sum_{i,j}
     im\psi_{ij}Y^{m*}_{n}(\lambda_{i},\mu_{j}) \frac{\omega_{j}}{1-\mu^{2}} \\
  \biggl(\widetilde{\DP{\psi}{\mu}}\biggr)^{m}_{n} 
   &amp;=&amp;
     \frac{1}{4\pi}\int^{1}_{-1}\Dd \mu \int^{\pi}_{0}\Dd \lambda
     \biggl(\DP{\psi}{\mu}\biggr)Y^{m*}_{n}(\lambda,\mu) \nonumber \\
   &amp;=&amp;
    -\frac{1}{4\pi}\int^{1}_{-1}\Dd \mu \int^{\pi}_{0}\Dd \lambda
     ~\psi\DP{}{\mu}Y^{m*}_{n}(\lambda,\mu) \nonumber \\
   &amp;=&amp;
     \frac{1}{I}\sum_{ij}
     ~\psi_{ij}(1-\mu_{j}^{2})
      \DP{}{\mu}Y^{m*}_{n}(\lambda_{i},\mu_{j})\frac{\omega_{j}}{1-\mu^{2}}.
 \end{eqnarray}
 ã“ã“ã§ $U\psi$ ãŒæ¥µ $\displaystyle{\mu=\pm1(\varphi=\pm\frac{\pi}{2})}$
 ã«ãŠã„ã¦ $0$ ã¨ãªã‚‹äº‹ã‚’ç”¨ã„ã¦ã„ã‚‹.
 
 ã•ã‚‰ã« $L_{2}$ ã‚’å«ã‚€é&nbsp;…ã®è©•ä¾¡ã«ã¯,
 \begin{equation}
  (\widetilde{L_{2}\psi})^{m}_{n}= n(n+1)\tilde{\psi}^{m}_{n}
 \end{equation}
 ã‚’ç”¨ã„ã‚‹.
 
 ä»¥ä¸Šã®è¨ˆç®—çµæžœã¯, 
 $\psi$ ã&nbsp;ã‘ã§ã¯ãªã $\phi,g,h,\theta$ ã«ã¤ã„ã¦ã‚‚ãã®ã¾ã¾æˆç«‹ã™ã‚‹.

\iffalse
%--------------------------------------------------------------------------
\newpage 
 \subsection{éžç·šå½¢é&nbsp;…ã®è©•ä¾¡}
 
 é‹å‹•æ–¹ç¨‹å¼, ç†±ã®å¼, èª˜å°Žæ–¹ç¨‹å¼ã®éžç·šå½¢é&nbsp;…ã‚’è¨ˆç®—ã—,
 ãã‚Œã‚‰ã‚’å±•é–‹ã™ã‚‹. 
 é€Ÿåº¦ãƒ™ã‚¯ãƒˆãƒ« $\Dvect{u}$ ã®å„æˆåˆ†ã‚’ $U,V$ ã§è¡¨ã™ã¨
 \begin{equation}
  \Dvect{u} = (u,v,w) 
            = \biggl(\frac{U}{\cos\varphi},\frac{V}{\cos\varphi},w\biggr) 
 \end{equation}
 ã¨ãªã‚‹. 
 ã“ã‚Œã‚’ç”¨ã„ã¦éžç·šå½¢é&nbsp;…ã‚’è¡¨ã™($f$ ã¯ä»»æ„ã®ã‚¹ã‚«ãƒ©ãƒ¼é‡ã¨ã™ã‚‹): 
 \begin{eqnarray}
  \Dvect{u}\cdot\Dgrad f
   &amp;=&amp; 
     \frac{u}{r\cos\varphi}\DP{f}{\lambda}
   + \frac{v}{r}\DP{f}{\varphi} 
   + w \DP{f}{r}
   \nonumber\\         
   &amp;=&amp; 
     \frac{U}{r\cos^{2}\varphi}\DP{f}{\lambda}
   + \frac{V}{r\cos\varphi}\DP{f}{\varphi} 
   + w \DP{f}{r}
   \nonumber\\         
   &amp;=&amp; 
     \frac{U}{r(1-\mu^{2})}\DP{f}{\lambda}
   + \frac{V}{r}\DP{f}{\mu} 
   + w \DP{f}{r}
   \nonumber\\         
  %
   \Ddiv\Dvect{u} 
   &amp;=&amp; 
      \frac{1}{r\cos\varphi}\DP{u}{\lambda}
    + \frac{1}{r\cos\varphi}\DP{}{\varphi} ( v\cos\varphi)
    + \frac{1}{r^2}\DP{}{r}(r^2 w) \nonumber\\
   &amp;=&amp;  
      \frac{1}{r\cos^2\varphi}\DP{U}{\lambda} 
    + \frac{1}{r\cos\varphi}\DP{V}{\varphi}
    + \frac{1}{r^2}\DP{}{r}(r^2 w) \nonumber\\
   &amp;=&amp;  
      \frac{1}{r(1-\mu^2)}\DP{U}{\lambda}
    + \DP{V}{\mu}
    + \frac{1}{r^2}\DP{}{r}(r^2 w), \\ 
    %
  \Dvect{r}\cdot(\Drot\Dvect{u})
   &amp;=&amp;  
     \frac{1}{\cos\varphi}
      \left\{
       \DP{ v }{\lambda} - \DP{}{\varphi}( u\cos\varphi )
      \right\}  \nonumber\\
   &amp;=&amp;   
      \frac{1}{\cos^{2}\varphi}\DP{V}{\lambda} 
    - \frac{1}{\cos\varphi}\DP{U}{\varphi} \nonumber\\
   &amp;=&amp;   
      \frac{1}{1-\mu^{2}}\DP{V}{\lambda} - \DP{U}{\mu}, \\
    %
  \Dvect{r}\cdot (\Drot\Drot\Dvect{u})
   &amp;=&amp;  
      \frac{1}{\cos\varphi}
       \left\{
	   \DP{}{\lambda} (\Drot\Dvect{u})_{\varphi}  
	 - \DP{}{\varphi} (\Drot\Dvect{u})_{\lambda}\cos\varphi
       \right\} \nonumber\\
   &amp;=&amp;  
      \frac{1}{\cos\varphi}
       \biggl[
	   \DP{}{\lambda}\frac{1}{r}
	    \left\{ 
	     \DP{}{r}( r u )  -  \frac{1}{\cos\varphi}\DP{w}{\lambda} 
	    \right\} \nonumber \\
   &amp;&amp;{}
    - \DP{}{\varphi}\frac{1}{r}
	    \left\{
	     \DP{w}{\varphi} - \DP{}{r}(r v) 
            \right\} 
	    \cos\varphi
       \biggr]  \nonumber\\
   &amp;=&amp;  
   \frac{1}{r}\DP{}{r}
      \biggl\{
        \frac{1}{\cos^{2}\varphi}\DP{}{\lambda}(ru\cos\varphi)
      \biggr\}
    +
    \frac{1}{r}\DP{}{r}
      \biggl\{
        \frac{1}{\cos\varphi}\DP{}{\varphi}(rv\cos\varphi)
      \biggr\}
    \nonumber\\
   &amp;&amp; 
    -\frac{1}{r\cos^{2}\varphi}\DP[2]{w}{\lambda}
    -\frac{1}{r\cos\varphi}\DP{}{\phi}\biggl(\cos\varphi\DP{w}{\varphi}\biggr)
    \nonumber\\
   &amp;=&amp;  
    \frac{1}{r}\DP{}{r}
      \biggl\{
        r\biggl(\frac{1}{1-\mu^{2}}\DP{U}{\lambda}\biggr)
      \biggr\}
    +
    \frac{1}{r}\DP{}{r}
      \biggl(r\DP{V}{\mu}\biggr)
    +
    \frac{L_{2}w}{r}.
 \end{eqnarray}
 ã‚ˆã£ã¦ã‚¹ãƒšã‚¯ãƒˆãƒ«ç©ºé–“ã§ã¯
 \begin{eqnarray}
  (\Ddiv\Dvect{u})^{m}_{n} 
   &amp;=&amp; 
      \frac{1}{r}\biggl(\frac{1}{1-\mu^2}\DP{U}{\lambda}\biggr)^{m}_{n} 
    + \frac{1}{r}\biggl(\DP{V}{\mu}\biggr)^{m}_{n} 
    + \frac{1}{r^2}\DP{}{r}r^2 (w)^{m}_{n} , \\ 
  (\Dvect{r}\cdot\Drot\Dvect{u})^{m}_{n} 
   &amp;=&amp; 
      \biggl(\frac{1}{1-\mu^{2}}\DP{V}{\lambda} \biggr)^{m}_{n} 
      - \biggl(\DP{U}{\mu}\biggr)^{m}_{n} ,\\
  (\Dvect{r}\cdot\Drot\Drot\Dvect{u})^{m}_{n}
   &amp;=&amp; 
    \frac{1}{r}\DP{}{r}
      \biggl\{
        r\biggl(\frac{1}{1-\mu^{2}}\DP{U}{\lambda}\biggr)^{m}_{n}
      \biggr\}
    +
    \frac{1}{r}\DP{}{r}\biggl\{
    r\biggl(\DP{V}{\mu}\biggr)^{m}_{n}
    \biggr\}
    +
    \frac{n(n+1)}{r}(w)^{m}_{n}. \nonumber \\
 \end{eqnarray}
\fi
%---------------------------------------------------------------------------
\newpage 
 \subsection{æ–¹ç¨‹å¼ã®ã‚¹ãƒšã‚¯ãƒˆãƒ«è¡¨ç¾}
 
 ã‚¹ãƒšã‚¯ãƒˆãƒ«å±•é–‹ã•ã‚ŒãŸãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§æ–¹ç¨‹å¼ç³»ã‚’æ›¸ãè¡¨ã™.
 %ã“ã“ã§, $n=0$ ã®å±•é–‹ä¿‚æ•°ã¯æ&nbsp;¼å­ç‚¹ã§ã®å€¤ã‚’æ±‚ã‚ã‚‹ãŸã‚ã«ã¯ä¸ç”¨ã§ã‚ã‚‹ã‹ã‚‰
 %ä»¥ä¸‹ã§ã¯

 \subsubsection{é‹å‹•æ–¹ç¨‹å¼}

 ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«:
 \begin{eqnarray}
  \DP{}{t}\tilde{\psi}^{m}_{n}
   &amp;=&amp;
   G^{m}_{n}\tilde{\psi}^{m}_{n} -\tau\frac{1}{n(n+1)}(\tilde{Q\phi})^{m}_{n}
   \nonumber \\
   &amp;&amp;{} - \frac{1}{n(n+1)}
    [ \Dvect{r}\cdot\Drot(\Dvect{u}\cdot\Dgrad\Dvect{u}-\Dvect{B}\cdot\Dgrad\Dvect{B})]^{m}_{n}.\quad(n\neq 0)
 \end{eqnarray}
 ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«:
 \begin{eqnarray}
  \DP{}{t}(D_{n}\tilde{\phi}^{m}_{n})
   &amp;=&amp;
   G^{m}_{n}(D_{n}\tilde{\phi}^{m}_{n}) 
   + \tau\frac{1}{n(n+1)}(\tilde{Q\psi})^{m}_{n} 
   - P_{r}^{-1}R_{a}\tilde{\theta}^{m}_{n}
   \nonumber \\
   &amp;&amp;{} + \frac{1}{n(n+1)}
    [ \Dvect{r}\cdot\Drot\Drot(\Dvect{u}\cdot\Dgrad\Dvect{u}-\Dvect{B}\cdot\Dgrad\Dvect{B})]^{m}_{n}.\quad(n\neq 0)
 \end{eqnarray}
 ã“ã“ã§ $G^{m}_{n}, D_{n}$ ã¯æ¬¡å¼ã§å®šç¾©ã•ã‚Œã‚‹:
 \begin{eqnarray}
  D_{n} \equiv \DP[]{}{r}+\frac{2}{r}\DP{}{r} -\frac{n(n+1)}{r^{2}}, \quad 
  G^{m}_{n} \equiv \frac{im}{n(n+1)} + D_{n}.
 \end{eqnarray}
 
 \subsubsection{ç†±ã®å¼}
 \begin{equation}
  \DP{}{t}\tilde{\theta}^{m}_{n}
   =   D_{n}\tilde{\theta}^{m}_{n} + n(n+1)\tilde{\theta}^{m}_{n} 
     - P_{r}^{-1}(\Dvect{u}\cdot\Dgrad\theta)^{m}_{n}
 \end{equation}

 \subsubsection{èª˜å°Žæ–¹ç¨‹å¼}
 ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«
 \begin{eqnarray}
  \DP{}{t}\tilde{g}^{m}_{n}
   &amp;=&amp;
     P_{m}^{-1}D_{n}\tilde{g}^{m}_{n} 
   + \frac{1}{n(n+1)}
    [ \Dvect{r}\cdot\Drot(\Dvect{u}\times\Dvect{B})]^{m}_{n}.\quad(n\neq 0)
 \end{eqnarray} 
 ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«
 \begin{eqnarray}
  \DP{}{t}\tilde{h}^{m}_{n}
   &amp;=&amp;
     P_{m}^{-1}D_{n}\tilde{h}^{m}_{n} 
   - \frac{1}{n(n+1)}
    [ \Dvect{r}\cdot\Drot\Drot(\Dvect{u}\times\Dvect{B})]^{m}_{n}.\quad(n\neq 0)
 \end{eqnarray} 
 
 \subsubsection{é€Ÿåº¦å&nbsp;´ã®è¡¨ç¾}

 \begin{eqnarray}
  U_{ij}(r,t) 
   &amp;=&amp; 
   \sum_{m,n}
     \biggl\{
         \tilde{\psi}^{m}_{n}(1-\mu^{2})\DP{}{\mu}Y^{m}_{n}(\lambda_{i},\mu_{j})
       + im\frac{1}{r}\DP{}{r}(r\tilde{\phi}^{m}_{n})Y^{m}_{n}(\lambda_{i},\mu_{j})
     \biggr\} \nonumber \\ \\
  V_{ij}(r,t) 
   &amp;=&amp; 
   \sum_{m,n}
     \biggl\{
       - im\tilde{\psi}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j})
       + \frac{1}{r}\DP{}{r}(r\tilde{\phi}^{m}_{n})(1-\mu^{2})\DP{}{\mu}Y^{m}_{n}(\lambda_{i},\mu_{j})
     \biggr\} \nonumber \\ \\
  w_{ij}(r,t) 
   &amp;=&amp; 
   \sum_{m,n} \frac{n(n+1)}{r}\tilde{\phi}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j})
 \end{eqnarray}

\subsubsection{ç£å&nbsp;´ã®è¡¨ç¾}

 \begin{eqnarray}
  \Deqlab{ç£å&nbsp;´çµŒåº¦}
   B_{\lambda_{ij}}(r,t)\cos\varphi 
   &amp;=&amp; 
   \sum_{m,n}
     \biggl\{
         \tilde{g}^{m}_{n}(1-\mu^{2})\DP{}{\mu}Y^{m}_{n}(\lambda_{i},\mu_{j})
       + im\frac{1}{r}\DP{}{r}(r\tilde{h}^{m}_{n})Y^{m}_{n}(\lambda_{i},\mu_{j})
     \biggr\} \nonumber \\ \\
  \Deqlab{ç£å&nbsp;´ç·¯åº¦}
  B_{\mu_{ij}}(r,t)\cos\varphi 
   &amp;=&amp; 
   \sum_{m,n}
     \biggl\{
       - im\tilde{g}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j})
       + \frac{1}{r}\DP{}{r}(r\tilde{h}^{m}_{n})(1-\mu^{2})\DP{}{\mu}Y^{m}_{n}(\lambda_{i},\mu_{j})
     \biggr\} \nonumber \\ \\
  \Deqlab{ç£å&nbsp;´é‰›ç›´}
  B_{r_{ij}}(r,t) 
   &amp;=&amp; 
   \sum_{m,n} \frac{n(n+1)}{r}\tilde{h}^{m}_{n}Y^{m}_{n}(\lambda_{i},\mu_{j})
 \end{eqnarray}


</pre></body></html>