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\documentclass[a4j,12pt]{jarticle}
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\usepackage{Dennou6}                      % dennou-sty-6 �����

\Dtitle[��ž�������嶯����ή]        % �إå��������ȥ�, ��ǥ�̾��Ʊ��
{��ž�������嶯����ή������꼰��}
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\Dauthor[�ݹ� ����]{�ݹ� ����}            % ����
\Ddate{2015 ǯ 7 �� 2 ��}                 % ����
\Dnoparindent                             % ����λ������򤷤ʤ�
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% ����Ū�ʾ��ṽ���ϰʲ����̤�. 
%
%   1. �Ϥ���� 
%   2. ������������
%   3. Υ����
%   4. ���ѥ⥸�塼��Ȥ���¾������
%   5. ���ͼ¸�
%   6. ����ʸ��
%   -. �ռ�
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% ɬ�פ˱����ƹ������ѹ����뤳��.
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\begin{document}

\maketitle                                % �����ȥ�κ���
\tableofcontents                          % �ܼ��κ���
\pagebreak                                

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\Dparskip
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\newpage                                  % ���ڡ���
\section{����}

����ʸ��Ǥ�, ��ž�������嶯����ή������꼰����
���ͷ׻��Τ����Υ�����ˤĤ��Ƶ��Ҥ���. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���
\section{������������}
�����Ǥϻ����������Ϥȶ������򵭤�. 

  \subsection{������������}
    �����������Ϥϲ�ž�ϤǤε��������������ϤǤ���. 



  \subsection{���١�ȯ�����Ѥ���������������}

    ���������������Ϥ��٤�ȯ�����Ѥ���ɽ�������ϤĤ��ΤȤ���Ǥ���. 
    \begin{eqnarray}
      \DP{\zeta }{t} &=& -\frac{1}{a(1-\mu ^{2})}
                          \DP{(f+\zeta)U}{\lambda}
                        - \frac{1}{a}\DP{(f+\zeta)V}{\mu} \nonumber \\
                      & & -d_\zeta + F(\lambda,\mu,t),
                        \Deqlab{����������} \\
      \DP{D}{t} &=& \frac{1}{a(1-\mu ^{2})}
                    \DP{(f+\zeta)V}{\lambda}
                    - \frac{1}{a}\DP{(f+\zeta)U}{\mu} 
                    - \Dlapla \left[g (h+h_{s}) + E\right] \nonumber \\
                & & -d_D,
                    \Deqlab{ȯ��������} \\
      \DP{h}{t} &=& - \frac{1}{a(1-\mu ^{2})}\DP{(hU)}{\lambda}
                    - \frac{1}{a}\DP{(hV)}{\mu}  
                    -d_h,
                    \Deqlab{������¸�μ�} 
    \end{eqnarray}
    %
    �Ƶ���������\Dtabref{�ѿ�, ʪ����������}�˼�����.
    ���� $\zeta $ ��ȯ�� $D$ , ñ�̼��̤�����α�ư���ͥ륮�� $E$, 
    �黻�� $\nabla ^{2}$ �ϰʲ��Τ褦����������.
    %
    \begin{eqnarray}
      \zeta &=& \frac{1}{a(1-\mu ^{2})}\DP{V}{\lambda} - \frac{1}{a}\DP{U}{\mu}, \\
      D     &=& \frac{1}{a(1-\mu ^{2})}\DP{U}{\lambda} + \frac{1}{a}\DP{V}{\mu}, \\
      E     &=& \frac{U^{2}+V^{2}}{2(1-\mu ^{2})}, \\ 
      \nabla^{2} A &=& \frac{1}{a^{2}(1-\mu ^{2})}\DP[2]{A}{\lambda}
                  + \frac{1}{a^{2}}\DP{}{\mu}\left[(1-\mu^{2})\DP{A}{\mu}\right].
    \end{eqnarray}
    %
    ������ $A$ ��Ǥ�դΥ����顼�Ǥ���. ή���ؿ� $\psi $ ��®�٥ݥƥ󥷥��
    $\chi$ ��Ƴ�������,
    %
    \begin{eqnarray}
      U &=& -\frac{1-\mu^ {2}}{a}\DP{\psi}{\mu}
            +\frac{1}{a}\DP{\chi}{\lambda}, \\
      V &=& \frac{1}{a}\DP{\psi}{\lambda}
            + \frac{1-\mu^{2}}{a}\DP{\chi}{\mu}, \\
      \zeta &=& \nabla ^{2} \psi, \\
      D     &=& \nabla ^{2} \chi
    \end{eqnarray}
    ��ɽ�����. 

    \Deqref{����������}���դ� $F(\lambda,\mu,t)$ �ϱ��ٶ�����Ǥ���. 
    ���ζ���Ū�ʷ��ϼ¸��������ǵ��Ҥ���. 
    $d_\zeta,d_D,d_h$ �ϳ��ѿ����Ф��뻶����ɽ���Ƥ���(��������). 

    \begin{table}[h]
     \begin{center}
      \begin{tabular}{c l l  } \hline
       ����    &\qquad\qquad&  �ѿ�/ʪ�����   \\ \hline 
       $\lambda$&& ����            \\
       $\phi$   && ����            \\
       $\mu $   && $\sin$ ���� ($\equiv \sin \phi$) \\
       $t$      && ����            \\ 
       $u$      && ��������ή®    \\ 
       $v$      && ��������ή®    \\ 
       $U$      && $u\cos \phi $  \\ 
       $V$      && $v\cos \phi $  \\ 
       $h$      && ή�Τ�������    \\ 
       $\zeta$  && ����            \\ 
       $D$      && ȯ��            \\
       $E$      && ��ư���ͥ륮�� \\
       $\psi$   && ή���ؿ�        \\
       $\chi$   && ®�٥ݥƥ󥷥�� \\
       $f$      && $2\Omega \sin \phi$ \\
       $\Omega$ && ��ž��®��      \\
       $g$      && ���ϲ�®��     \\
       $a$      && ����Ⱦ��       \\
       $F$      && ���ٶ���       \\
       $\nu_{m}$  && ��ʿǴ������   \\
       $\kappa_{h}$  && ��ʿ�Ȼ�����   \\
       $N_{m}$  && ĶǴ���μ���(��ʿ��ץ饷����γ���) \\
       $N_{h}$  && Ķ�Ȼ��μ���(��ʿ��ץ饷����γ���) \\ \hline
      \end{tabular} 
     \end{center}
     \caption{�ѿ�, ʪ����������}
     \Dtablab{�ѿ�, ʪ����������}
    \end{table}

    ��λ�����ʬ��������ι����Τ����, 
    $h$ ��ʿ����ʬ $\overline{h}$ �Ȥ��줫��ξ��� $h'$ ���Ѥ���
    $h=\overline{h}+h'(\lambda,\mu,t)$ ��ɽ���Ƥ�����, 
    %
    \begin{eqnarray}
      \DP{\zeta }{t} &=& -\frac{1}{a(1-\mu ^{2})}
                          \DP{(f+\zeta)U}{\lambda}
                        - \frac{1}{a}\DP{(f+\zeta)V}{\mu} \nonumber \\
                      & & -d_\zeta  + F(\lambda,\mu,t),
                        \Deqlab{������������} \\
      \DP{D}{t} &=& \frac{1}{a(1-\mu ^{2})}
                    \DP{(f+\zeta)V}{\lambda}
                    - \frac{1}{a}\DP{(f+\zeta)U}{\mu} 
                    - \Dlapla \left[g (h'+h_{s}) + E\right] \nonumber \\
                & & -d_D
                    \Deqlab{����ȯ��������} \\
      \DP{h}{t} &=& - \frac{1}{a(1-\mu ^{2})}\DP{(h'U)}{\lambda}
                    - \frac{1}{a}\DP{(h'V)}{\mu}  
                    - \bar{h}D
                    - d_h
                    \Deqlab{���������¸�μ�} 
    \end{eqnarray}

    ��������Ȥ��ƥ쥤�꡼�໤, �˥塼�ȥ���Ѥ����ĶǴ���Ȼ���
    ��θ����. ���ζ���Ū�ʷ���
    %
    \begin{eqnarray}
      d_\zeta &=& - \zeta/\tau_R + (-1)^{N_m-1}\nu_{N_m}(\Dlapla)^{N_m}\zeta,\\
      d_D &=& - D/\tau_R + (-1)^{N_m-1}\nu_{N_m}(\Dlapla)^{N_m}D,\\
      d_h &=& - h'/\tau_N + (-1)^{N_h-1}\kappa_{N_h}(\Dlapla)^{N_h}h',
    \end{eqnarray}
    %
    ������ $\tau_R,\tau_N$ �Ϥ��줾��쥤�꡼�໤����ӥ˥塼�ȥ���Ѥλ����, 
    $N_m$, $N_h$ ��ή��줪��ӹ��پ���Ф���ĶǴ�����Ȼ��μ���, 
    $\nu_{N_m}, \kappa_{N_m}$ ��ĶǴ�����Ȼ������Ǥ���. 

    \subsection{�������}

    �������ϰ���, ���������Ȥ�˼����������Ǥ���. 
    \begin{eqnarray}
      \zeta (\lambda + 2\pi, \phi)  &=& \zeta (\lambda, \phi), \\
      \zeta (\lambda , \phi + 2\pi) &=& \zeta (\lambda, \phi).
    \end{eqnarray}
    �ʤɤ�ɽ�����.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���
\section{��ʿΥ����}

  ������Ǥ��������Ϥζ���Υ�����ˤĤ�������, 
  �ץ��������Ǽºݤ��Ѥ����Ƥ����������򵭽Ҥ���.

  �ʻ�������������ˤ����ֳ�, ���������ˤϥ��������پ���֤�, 
  ʪ���� A ��ʻ�������ͤ����������Τ� $A_{i,j}$ ��ɽ�����Ȥˤ���. 
  ź�� $i,j$ �ϳʻ��� $(\lambda _{i}, \mu _{j})$ ����ͤǤ��뤳�Ȥ򼨤�.

  $A_{i,j}$ ���Ф���Υ�����ڥ��ȥ��Ѵ��� $\hat{A}_{m,n}$ �Ǥ���魯.
  ���ʤ��, 
  %
  \begin{eqnarray}
   A_{i,j} &=& \sum _{n=0}^{M}\sum _{m=-n}^{n}
                   P_{n}^{m}(\mu)e ^{im\lambda} \hat A_{m,n}, \\
   \hat A_{m,n} &=& \frac{1}{2I} \sum _{j=1}^{J} \sum _{i=1}^{I}
                    A _{i,j}e^{-i m \lambda _{i}}P_{n}^{m}(\mu_{j})w_{j},
  \end{eqnarray}
  %
  ����������\Deqref{������������},\Deqref{����ȯ��������},
  \Deqref{���������¸�μ�},
  �򥹥ڥ��ȥ��Ѵ������
  %
  \begin{eqnarray}
    \DP{\hat\zeta _{m,n}}{t} 
         &=& -\widehat{\left[\frac{1}{a(1-\mu^{2})}
                        \DP{(f+\zeta_)U}{\lambda}\right]}_{m,n}
             - \widehat{\left[\frac{1}{a}\DP{(f+\zeta)V}{\mu}\right]}_{m,n} 
             - \widehat{d_\zeta}_{n,m}
             + \hat{F}_{n,m}(t)
          \Deqlab{���ڥ��ȥ뱲��������} \\
    \DP{\hat{D}_{m,n}}{t} 
         &=&   \widehat{\left[\frac{1}{a(1-\mu^2)}
                            \DP{(f+\zeta)V}{\lambda}\right]}_{m,n}
             - \widehat{\left[
                  \frac{1}{a}\DP{(f+\zeta)U}{\mu} \right]}_{m,n}\nonumber \\
         & & -g\Dlapla\hat{h}'_{m,n}
             - \widehat{\left[\Dlapla\left\{ gh_s+E \right\}\right]}_{m,n}
             - \widehat{d_D}_{,n,m}, 
         \Deqlab{���ڥ��ȥ�ȯ��������} \\
    \DP{\hat{h}_{m,n}}{t} 
         &=& - \widehat{\left[\frac{1}{a(1-\mu^2)}
                              \DP{(h'U)}{\lambda}\right]}_{m,n}
             - \widehat{\left[\frac{1}{a}\DP{(h'V)}{\mu}\right]}_{m,n}
             -\bar{h}\hat{D}_{m,n}
             - \widehat{d_h}_{n,m}
         \Deqlab{���ڥ��ȥ������¸�μ�} 
  \end{eqnarray}
  %
  ��������Υ��ڥ��ȥ��Ѵ���, �ʻ�����Ǥ��ͤ��Ѥ������������ɾ����, 
  ����򥹥ڥ��ȥ��Ѵ�����(�Ѵ�ˡ)��Ŭ�Ѥ��뤳�Ȥˤ���. 

  ��������Υ��ڥ��ȥ�ɽ����, 
  % 
  \begin{eqnarray}
    & &
    \widehat{d_\zeta}_{m,n} = -\nu_{m,n}\hat{\zeta}_{m,n}, \
    \widehat{d_D}_{m,n} = -\nu_{m,n}\hat{D}_{m,n}, \
    \widehat{d_h}_{m,n} = -\kappa_{m,n}\hat{h'}_{m,n}, \\
    & &
    \nu_{m,n}    =  1/\tau_R + \nu_{N_m}[n(n+1)]^{N_m}, \quad
    \kappa_{m,n} =  1/\tau_N + \kappa_{N_m}[n(n+1)]^{N_m}.
  \end{eqnarray}
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���
\section{������ʬ}

  �����Ǥϻ�����ʬˡ�ˤĤ��Ƶ��Ҥ�, �ץ��������Ǽºݤ��Ѥ����Ƥ�������
  ���򵭽Ҥ���. �ʲ��Ǥ� $\Delta t$ ����ֳʻҴֳ�, ���� $t, t+\Delta t$ 
  �ˤ����� $\hat \zeta_{m,n}$ ���ͤ򤽤줾��
  $\hat \zeta_{m,n}^{\tau}, \hat \zeta_{m,n}^{\tau+1}$ ����ɽ��. 

  % \subsection{��󥲡����å� \& ����󥯡��˥��륽��ˡ}

    ������ʬ�ο��ͥ�������Ȥ���, 
    �����ˤϥ���󥯡��˥��륽��ˡ, ����ʳ��ι��
    4 ���Υ�󥲡����å�ˡ���Ȥ߹�碌���Ѥ�������꼰����Ԥʤ�. 

    �ޤ���������۲�ˡ(�����顼��������)�ˤ����ʬ����. 
    %
    \begin{equation}
      \DD{\hat\zeta_{m,n}}{t} = \hat{F}_{n,m}(t) 
    \end{equation}
    %
    ���˻����ʳ��ι�˴ؤ��ƥ�󥲡����å�ˡ����ʬ����. 
    \Deqref{���ڥ��ȥ뱲��������},
    \Deqref{���ڥ��ȥ�ȯ��������}, 
    \Deqref{���ڥ��ȥ������¸�μ�}����
    ������Ȼ����ʳ��λ����Ѳ���
    %
    \begin{eqnarray}
      & &\left(\DD{\hat\zeta_{m,n}}{t}\right)_{ND}
       [\hat{\zeta}_{m,n},\hat{D}_{m,n},\hat{h'}_{m,n}]
        = -\widehat{\left[\frac{1}{a(1-\mu^{2})}
                          \DP{(f+\zeta)U}{\lambda}\right]}_{m,n}
         - \widehat{\left[\frac{1}{a}\DP{(f+\zeta)V}{\mu}\right]}_{m,n} 
            \nonumber\\\Deqlab{���ڥ��ȥ뱲�ٻ����Ѳ�����ʤ�}\\
      & & \left(\DD{\hat{D}_{m,n}}{t}\right)_{ND}
       [\hat{\zeta}_{m,n},\hat{D}_{m,n},\hat{h'}_{m,n}] 
           = \widehat{\left[\frac{1}{a(1-\mu^2)}
                              \DP{(f+\zeta)V}{\lambda}\right]}_{m,n}
             - \widehat{\left[
                  \frac{1}{a}\DP{(f+\zeta)U}{\mu} \right]}_{m,n}
              \nonumber \\
           & & \hspace{60mm}
              -g\Dlapla\hat{h}^{'\tau}_{m,n}
               - \widehat{\left[\Dlapla\left\{ g h_s+E \right\}\right]}_{m,n}
            \Deqlab{���ڥ��ȥ�ȯ�������Ѳ�����ʤ�} \\
      & & \left(\DD{\hat{h}_{m,n}}{t}\right)_{ND}
       [\hat{\zeta}_{m,n},\hat{D}_{m,n},\hat{h'}_{m,n}]
            = - \widehat{\left[\frac{1}{a(1-\mu^2)}
                                \DP{(h'U)}{\lambda}\right]}_{m,n}
               - \widehat{\left[\frac{1}{a}\DP{(h'V)}{\mu}\right]}_{m,n}
               -\bar{h}\hat{D}_{m,n}
        \nonumber\Deqlab{���ڥ��ȥ���̻����Ѳ�����ʤ�}\\
    \end{eqnarray}
    %
    ���������. ����������������Ѥ����Ƥ���ʻ�����������줿�ѿ���
    �ƥ��ڥ��ȥ��ѿ�
    $\hat{\zeta}_{m,n},\hat{D}_{m,n},\hat{h'}_{m,n}$
    ���饹�ڥ��ȥ���Ѵ����Ƶ�����. 

    ��󥲡����å�ˡ�ˤ����ʬ�μ��ϤĤ��Τ褦�ˤʤ�. 
    ������������������˼��򵭤�. 
    �� 1 �ʤ�
    %
    \begin{eqnarray}
      \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k1}
       &=& \left(\DD{\hat\zeta_{m,n}}{t}\right)_{ND}
          [\hat{\zeta}^{\tau}_{m,n},\hat{D}^{\tau}_{m,n},
           \hat{h'}^{\tau}_{m,n}],\\ 
      \hat\zeta^{\tau+1/2}_{m,n} &=& \hat\zeta^{\tau}_{m,n} + 
        \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k1}\cdot \frac{(\Delta t)}{2},
    \end{eqnarray}
    %
    �� 2 �ʤ�
    \begin{eqnarray}
      \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k2}
       &=& \left(\DD{\hat\zeta_{m,n}}{t}\right)_{ND}
          [\hat{\zeta}^{\tau+1/2}_{m,n},\hat{D}^{\tau+1/2}_{m,n},
           \hat{h'}^{\tau+1/2}_{m,n}], \\
      \hat\zeta^{*\tau+1/2}_{m,n} &=& \hat\zeta^{\tau}_{m,n} + 
        \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k2}\cdot \frac{(\Delta t)}{2},
    \end{eqnarray}
    %
    �� 3 �ʤ�
    %
    \begin{eqnarray}
      \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k3}
       &=& \left(\DD{\hat\zeta_{m,n}}{t}\right)_{ND}
          [\hat{\zeta}^{*\tau+1/2}_{m,n},\hat{D}^{*\tau+1/2}_{m,n},
           \hat{h'}^{*\tau+1/2}_{m,n}], \\
      \hat\zeta^{*\tau+1}_{m,n} &=& \hat\zeta^{\tau}_{m,n} + 
        \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k3}\cdot (\Delta t),
    \end{eqnarray}
    %
    �� 4 �ʤ�
    %
    \begin{eqnarray}
      \left(\DD{\hat\zeta_{m,n}}{t}\right)^{k4}
       = \left(\DD{\hat\zeta_{m,n}}{t}\right)_{ND}
          [\hat{\zeta}^{*\tau+1}_{m,n},\hat{D}^{*\tau+1}_{m,n},
           \hat{h'}^{*\tau+1}_{m,n}], 
    \end{eqnarray}
    %
    �ǽ�Ū��
    %
    \begin{eqnarray}
      \hat\zeta^{\tau+1,ND}_{m,n} &=& \hat\zeta^{\tau}_{m,n} + 
        (\Delta t)\left[
            \Dinv{6}\left(\DD{\hat\zeta_{m,n}}{t}\right)^{k1}
          + \Dinv{3}\left(\DD{\hat\zeta_{m,n}}{t}\right)^{k2}
          + \Dinv{3}\left(\DD{\hat\zeta_{m,n}}{t}\right)^{k3}
          + \Dinv{6}\left(\DD{\hat\zeta_{m,n}}{t}\right)^{k4}
          \right], \nonumber\\
    \end{eqnarray}
    %
    �ʾ�μ���ȯ�� $\hat{D}_{m,n}$, �Ѱ̾��� $\hat{h'}_{m,n}$ ��
    �Ĥ��Ƥ�ʿ�Ԥ��Ʒ׻����Ƥ������Ȥ�, �󻶰�����ʬ����λ����. 

    ��󥲡����å�ˡ�ˤ����ʬ�ΤΤ�, ����फ��ν�����Ԥʤ�. 
    ���٤���˼���, 
    \begin{displaymath}
      \frac{\zeta^{\tau+1}_{m,n}-\zeta^{\tau+1,ND}_{m,n}}{\Delta t}
       = -\nu_{m,n}\frac{\zeta^{\tau+1}_{m,n}+\zeta^{\tau+1,ND}_{m,n}}{2}
    \end{displaymath}
    %
    �������ä�
    \begin{equation}
      \zeta^{\tau+1}_{m,n} = 
      \frac{1- \nu_{m_n}(\Delta t)/2}{1+ \nu_{m_n}(\Delta t)/2}
         \cdot\zeta^{\tau+1,ND}_{m,n},
    \end{equation}
    %
    �Ƚ��������. ȯ�� $\hat{D}_{m,n}$, �Ѱ̾��� $\hat{h'}_{m,n}$ ��
    �Ĥ��Ƥ�Ʊ�ͤ˷׻�����. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���
\section{���ѥ⥸�塼��Ȥ���¾������}

  ���ڥ��ȥ��Ѵ��ȵ��Ѵ�, ��ʬ�黻�� SPMODEL �饤�֥�� (spml) ��
  w\_module\_sjpack �˴ޤޤ��ؿ����Ѥ��ƹԤ�. 
  ���ڥ��ȥ��Ѵ��ȵ��Ѵ��κݤο�����ʬ������������Ѥ��ƹԤ�. 
  spml �����̤ǻ��Ѥ��� ISPACK �λ��ͤ���, 
  �ʻ����� $I$ �� $2^n$ �Ǥʤ���Фʤ�ʤ�. 
  ��������η׻��ˤ�ä������륨�ꥢ���󥰤��ɤ�����, 
  �ʻ����� $I,J$ �������ȿ� $M$ ���Ф� $I>3M+1,J>3M/2$ ���������褦
  ��Ϳ����.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���

\section{Scott and Polvani (2008) ����� Saito and Ishioka (2015) �μ¸�}

  ��ž�������嶯����ή���������Ȥ���, 
  Scott and Polvani (2008) ����� Saito and Ishioka (2015)�ˤ��Ԥ�줿
  �¸��򻲹ͤˤ������ꤹ��. 

  �������ɽ����
  %
  \begin{eqnarray}
    & &
    F(\lambda,\mu,t) 
    = 2\mbox{Re}\left[\sum_{n=n_f-\Delta n/2}^{n_f+\Delta n/2}
                      \sum_{m=1}^n \hat{f}_{m,n}(t) P_n^m(\mu)e^{im\lambda}
                \right], \\
    & &
    \hat{f}_{m,n}(t) 
       = \sqrt{\frac{2n(n+1)\varepsilon_0}{(2n+1)\Delta n}}\xi_{m,n}(t), \quad
    \xi_{m,n}(t) = \frac{e^{i\theta_{m,n}(t)}}{\sqrt{\Delta t}}.
  \end{eqnarray}
  %
  ������  $\varepsilon_0$ �ϥ��ͥ륮������Ψ���礭����ɽ���ѥ�᥿��,
  $P_n^m(\mu)$ �� $2$ �˵��ʲ����줿�른���ɥ���ȡ��, 
  $\theta_{m,n}(t)$ �϶�� $[0,2\pi]$ �Ǥΰ�������Ǥ���, 
  �ƻ��֥��ƥåפˤ����ƹ��������
  \footnote{
    w\_module\_sjpack �Υ��ڥ��ȥ��ѿ����Ѥ����
    l\_nm(n,m) (m>0) �����ڥ��ȥ뷸���μ¿���, 
    l\_nm(n,m) (m<0) �����ڥ��ȥ뷸���ε�������
    ��Ǽ���Ƥ���Τ�, $m \geq 1$ ���Ф���
    %
    \begin{displaymath}
      w\_F(l\_(n,m)) = \sqrt{\frac{2n(n+1)\varepsilon_0}
                            {(2n+1)\Delta n\Delta t}}
                       \cos\theta_{m,n}(t),\quad
      w\_F(l\_(n,-m)) = \sqrt{\frac{2n(n+1)\varepsilon_0}
                             {(2n+1)\Delta n\Delta t}}
                       \sin\theta_{m,n}(t).
    \end{displaymath}
  }. 

  Ĺ����������Ȼ��֥���������Ⱦ�� $a$ �ȼ�ž���� $2\pi/\Omega$ ��
  ���֤Τ���ñ�Ǥ���. 
  %
  ���פ��ϳإѥ�᥿����
  �ѷ�Ⱦ�� $\Ddsty \frac{L_D}{a}=\frac{\sqrt{g\bar{h}}}{2\Omega}$. 
  %
  \begin{table}[h]
    \begin{center}
      \begin{tabular}{|l|c|} \hline
        �¸��ѥ�᥿��                   & ������\\\hline
        �����ȿ� $N_T$                   & 170, (683) \\
        ���֥��ƥå� $\Delta t$          & 0.0125 \\
        ��Ⱦ�� $a$                       & 1\\
        ��ž��®�� $\Omega$              & $2\pi$ \\
        ���ϲ�®�� $g$                   & ? \\
        ʿ�ѿ���   $\bar{h}$             & ? \\
        �ѷ�Ⱦ��   $L_D$                 & $(1),0.1,(0.025)$ \\
        ���ͥ륮������Ψ $\varepsilon_0$ & $5.0\times 10^7$ \\
        �������濴�ȿ� $n_f$             & 42 \\
        �������ȿ��� $\Delta n$          & 4  \\
        �쥤�꡼�໤�λ���� $\tau_R$    & $\infty$ \\
        �˥塼�ȥ���Ѥλ���� $\tau_N$  & $0.25 L_D^{-2}$\\
        ĶǴ������ $N_m$                 & 4 ($\nabla^8$)\\
        Ķ�Ȼ����� $N_m$                 & 4 ($\nabla^8$)\\
        ĶǴ������ $\nu_4$               & $10/[N_T(N_T+1)]^4$\\
        Ķ�Ȼ����� $\nu_4$               & $10/[N_T(N_T+1)]^4$\\\hline
      \end{tabular}
      \caption{�¸��ѥ�᥿������}
      \Dtablab{�¸��ѥ�᥿��}
    \end{center}
  \end{table}

  \begin{table}[h]
    \begin{center}
      \begin{tabular}{|lc|cccc|} \hline
                  & OpenMP �� & 1    & 2 & 4 & 6 \\\hline
                  & real time & ...s &   &   &   \\ 
        Intel ??? & usr time  & ...s &   &   &   \\\hline
      \end{tabular}
      \caption{10 ���׻��ˤ����������ȿ� $T170$ �Ǥη׻����� (sec)}
      \Dtablab{�׻�����}
    \end{center}
  \end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage                                  % ���ڡ���

\section{����ʸ��}
\begin{description}
 % \item Asselin, R. A., 1972: 
 %       Frequency filter for time integrations.
 %       {\it Mon. Wea. Rev.}, {\bf 100}, 487--490.

 \item �в�����, 2004:
       ���ڥ��ȥ�ˡ�ˤ����ͷ׻�����, �����ؽ��ǲ�, 232pp.

 \item Krishnamurti, T. N., H. S. Bedi, and V. M. Hardiker, 1998:
       An introduction to global spectral modeling. 
       Ocford Univ. Press, 253 pp.

%  \item Satoh, M., 2004: Atmospheric circulation dynamics and circulation 
%        models.
%        Springer-Verlag, 643 pp.

 \item �ݹ� ����, ������ ��ʿ, �в� ����, ���� ����, �ⶶ ˧��, 
   ���� ���, ���� ����, �� �Ͳ�, spmodel ��ȯ���롼��, 2013:
   ����Ū�ϵ�ή���ϳإ��ڥ��ȥ��ǥ뽸 (SPMODEL),
   http://www.gfd-dennou.org/library/spmodel/, �ϵ�ή����Ǿ�����.

 \item Scott, R. K., Polvani, L. M., 2007: 
   Forced-dissipative shallow-water turbulence
   on the sphere and the atmospheric circulation of the giant planets. 
   {\it J. Atmos. Sci.}, {\bf 64}, 3158--3176.

 \item Scott, R. K., Polvani, L. M., 2008: 
   Equatorial superrotation in shallow water atmospheres. 
   {\it Geophys. Res. Lett.}, {\bf 35}, L24202.

 \item Saito, I., Ishioka, K., 2015:
   Mechanism for the Formation of Equatorial Superrotation
   in Forced Shallow-Water Turbulence with Newtonian Cooling.
   {\it J. Atmos. Sci.}, {\bf 72}, 1466--1483.

\end{description}

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\end{document}

