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\documentclass[12pt,a4j,notitlepage]{jarticle}
\usepackage{Dennou6}
\usepackage{psfrag}
\usepackage{ascmac}
\usepackage{times}
\Dtitle[2 �ż����ΰ�簵������ή]
       {2 ���� 2 �ż��������ΰ�Ǥν簵ή��\\��������ή����}
\Dauthor[�ݹ� ����]{�ݹ� ����, SPMODEL ��ȯ���롼��}
\Ddate{ʿ�� 17 ǯ 10 �� 21 ��} 
\Dparskip

\pagestyle{Dheadings}

\begin{document}

\maketitle

\section{�Ϥ����}

  ����ʸ�ϤǤ�, 2 ���� 2 �ż��������ΰ�ǤΦ��̽簵ή�Υ�ǥ��
  �������������ñ�ˤޤȤ�, 
  ��¸§���꼰���Ȥ��Υ��ڥ��ȥ�ɽ����Ƴ�Ф�, 
  �Ǹ�˸�����ή�¸�����������ˤĤ��ƽҤ٤�. 

\section{������������}

  \subsection{����������}

    $x$, $y$ �����˳ơ� $L_x$, $L_y$ �γȤ�������
    ����ΰ�Ǥ� 2 �����󰵽�ή�Τ�ͤ���. 
    �����������ϱ����������Ǥ���
    \footnote{Ƴ�Фʤɾܤ�����
      ��2 ���� 2 �ż��������ΰ�Ǥν簵ή�Υ�ǥ���꼰���פ�
      ���ȤΤ���}.
    %
    \begin{eqnarray}
      \Deqlab{ή���ؿ�����������}
        & & \DP{\zeta}{t} + J(\psi,\zeta) +\beta(y)\DP{\psi}{x}
              = (-1)^{p+1}\nu_{2p}\Dlapla[2p] \zeta, \\
        & &  \zeta(x,y,t) = \Dlapla\psi(x,y,t),
    \end{eqnarray}
    %
    ������ $\zeta(x,y,t)$ �����б���, $\psi(x,y,t)$ ��ή���ؿ��Ǥ���, 
    ®�پ� $(u,v)$ ��
    \begin{equation}
      \Deqlab{®�٤�ή���ؿ�}
      u = -\DP{\psi}{y}, \quad v = -\DP{\psi}{x}
    \end{equation}
    %
    �ʤ�ط�����. 

    ��������
    %
    \begin{eqnarray}
      \Deqlab{����ή���ؿ��������}
        & & \zeta(0,y) = \zeta(L_x,y), \quad \zeta(x,0) = \zeta(x,L_y), \\
        & & \psi(0,y) = \psi(L_x,y), \quad \psi(x,0) = \psi(x,L_y). 
    \end{eqnarray}
    %
    ������, Ϳ����ʪ���ѥ�᥿�� $\beta(y)$ ��
    %
    \begin{equation}
      \Deqlab{�١����ѥ�᥿���������}
      \beta(0) = \beta(L_y),
    \end{equation}
    %
    ���������Ƥ��ʤ���Фʤ�ʤ�. 
    $\beta$ ������ξ�礬������ $\beta$ �̥�ǥ�ȸƤФ���ΤǤ���. 

  \subsection{��ư���ͥ륮����¸§}

  \subsection{���󥹥ȥ��ե�����¸§}


\section{2 �ťա��ꥨ���Ÿ���ˤ��ɽ��}

  \subsection{2 �ťա��ꥨŸ��}

    ���ڥ��ȥ�ˡ�ˤ����ͷ׻���Ԥ������, 
    ��ʪ���̤򶭳�����������ȡ���ϤǤ��� 2 �ťա��ꥨ���Ÿ������. 
    ���ʤ��, 
    %
    \begin{eqnarray}
      \psi(x,y,t)
         &=& \sum_{k=-K}^{K} \sum_{l=-L}^{L}
               \tilde{\psi}_{kl}(t)
                 \exp\left(i\frac{2\pi x}{L_x}k\right)
                 \exp\left(i\frac{2\pi y}{L_y}l\right),\\
      \tilde{\psi}_{kl}(t)
         &=& \Dinv{L_x L_y}\int_0^{L_x}\int_0^{L_y}\psi(x,y,t)
                 \exp\left(-i\frac{2\pi k}{L_x}x\right)
                 \exp\left(-i\frac{2\pi l}{L_y}y\right) dx \ dy,
    \end{eqnarray}
    %
    $\tilde{\psi}_{kl}(t)$ ��ή���ؿ��Υ��ڥ��ȥ���ȿ� $k,l$ ��ʬ, 
    $K,L$ �Ϥ��줾�� $x,y$ �����������ȿ��Ǥ���. 
    ���� $\zeta(x,y,t)$ �Υ��ڥ��ȥ��Ʊ�ͤ� 
    $\tilde{\zeta}_{kl}(t)$ ���������. 

  \subsection{��ư���ͥ륮�����ڥ��ȥ�}

    ����ư���ͥ륮��
    $\Ddsty E_k(t)=\int_0^{L_x}\int_0^{L_y} \Dinv{2}\left[
        \left(\DP{\psi}{x}\right)^2+\left(\DP{\psi}{y}\right)^2 \right]
      d x d y $
    �򥹥ڥ��ȥ���ʬ��ɽ��. 
    $\tilde{k}=(2\pi k/L_x)$, $\tilde{l}=(2\pi l/L_y)$ ���֤�����, 
    �� 1 ���ܤ������������ѷ������, 
    %
    \begin{eqnarray*}
      \left(\DP{\psi}{x}\right)^2 
         &=& \left(\sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  i\tilde{k}\tilde{\psi}_{kl}(t)e^{i\tilde{k}x+i\tilde{l}y}
             \right)^2 \\
         &=& \left(
               \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                 i\tilde{k}\tilde{\psi}_{kl}(t)e^{i\tilde{k}x+i\tilde{l}y}
             \right)\cdot
             \left(
               \sum_{k'=-K}^{K} \sum_{l'=-L}^{L}
                 i\tilde{k'}\tilde{\psi}_{k'l'}(t)e^{i\tilde{k'}x+i\tilde{l}y}
             \right).
    \end{eqnarray*}
    %
    ���ΰ���ʬ $\Ddsty\int_0^{L_x}\int_0^{L_y} d x d y $ ����Ѥ������
    �ȿ��� 0 �Ǥʤ��Ȥ߹�碌���Ѥ������ʬ�� 0 �Ȥʤ�Τ�
    $k'=-k, l'=-l$ �δ�Ϳ�Τߤ��Τ���. �������ä�,
    %
    \begin{eqnarray*}
      \int_0^{L_x}\int_0^{L_y} \left(\DP{\psi}{x}\right)^2 d x d y
         &=& \int_0^{L_x}\int_0^{L_y}
                \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  i\tilde{k}\tilde{\psi}_{kl}(t)e^{i\tilde{k}x+i\tilde{l}y}
                  (-i\tilde{k})\tilde{\psi}_{-k-l}(t)
                       e^{-i\tilde{k}x-i\tilde{l}y}
             d x d y \\
         &=& \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  \tilde{k}^2\tilde{\psi}_{kl}\tilde{\psi}_{-k-l}
             \int_0^{L_x}\int_0^{L_y} d x d y \\
         &=& L_x L_y
              \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  \tilde{k}^2|\tilde{\psi}_{kl}|^2
    \end{eqnarray*}
    %
    ������ $\psi(x,y,t)$ ���¿��Ǥ����狼��������ط���
    %
    \begin{displaymath}
      \tilde{\psi}_{-k-l} = \tilde{\psi}_{kl}^*
    \end{displaymath}
    %
    ���Ѥ���\footnote{
      $\psi$ ���¿��Ǥ��뤫�� $\psi =\psi^*$ ���
      \begin{eqnarray}
      \psi(x,y,t)^*
         &=& \sum_{k=-K}^{K} \sum_{l=-L}^{L}
               \tilde{\psi}_{kl}^*
                 \exp\left(-i\frac{2\pi x}{L_x}k\right)
                 \exp\left(-i\frac{2\pi y}{L_y}l\right),\\
         &=& \sum_{k=-K}^{K} \sum_{l=-L}^{L}
               \tilde{\psi}_{-k-l}^*
                 \exp\left(i\frac{2\pi x}{L_x}k\right)
                 \exp\left(i\frac{2\pi y}{L_y}l\right),
      \end{eqnarray}
      �ƥա��ꥨ��ʬ����Ӥ���� $\tilde{\psi}_{-k-l}^*=\tilde{\psi}_{kl}$,
      ���ʤ��, $\tilde{\psi}_{-k-l}=\tilde{\psi}_{kl}^*$ ��������.}.
    ���դ��Υ������ꥹ����ʣ�Ƕ����ɽ���Ƥ���. 

    �� 2 ���ܤ�Ʊ�ͤ˷׻������
    %
    \begin{displaymath}
      \int_0^{L_x}\int_0^{L_y} \left(\DP{\psi}{y}\right)^2 d x d y
         = L_x L_y
              \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  \tilde{l}^2|\tilde{\psi}_{kl}|^2.
    \end{displaymath}
    %
    �������ä����ΰ���ʬ������ư���ͥ륮����
    \begin{eqnarray}
      E_k(t)&=&\int_0^{L_x}\int_0^{L_y} \Dinv{2}\left[
        \left(\DP{\psi}{x}\right)^2+\left(\DP{\psi}{y}\right)^2 \right]
         d x d y\nonumber\\
      &=&  L_x L_y\sum_{k=-K}^{K} \sum_{l=-L}^{L}
             \Dinv{2}(\tilde{k}^2+\tilde{l}^2)|\tilde{\psi}_{kl}|^2 \nonumber\\
      &=&  L_x L_y\sum_{k=-K}^{K} \sum_{l=-L}^{L}
          \Dinv{2}
          \left[ \left(\frac{2\pi k}{L_x}\right)^2
                +\left(\frac{2\pi l}{L_y}\right)^2 \right]
                   |\tilde{\psi}_{kl}|^2.\nonumber\\
    \end{eqnarray}
    %
    ���μ�����, ���ȿ��ǤΥ��ͥ륮��̩��, ���ʤ�����ͥ륮�����ڥ��ȥ��
    ���Τ褦������Ǥ���. 
    \begin{equation}
      {\cal E}(k,l,t) \equiv
          \Dinv{2}
           (\tilde{k}^2+\tilde{l}^2)|\tilde{\psi}_{kl}|^2
          = \Dinv{2}\left[ \left(\frac{2\pi k}{L_x}\right)^2
                +\left(\frac{2\pi l}{L_y}\right)^2 \right]
                   |\tilde{\psi}_{kl}|^2.
    \end{equation}
    %
    ���ΤȤ�����ư���ͥ륮����
    %
    \begin{equation}
      E_k(t) = L_x L_y\sum_{k=-K}^{K} \sum_{l=-L}^{L}{\cal E}(k,l,t).
    \end{equation}

  \subsection{���󥹥ȥ��ե������ڥ��ȥ�}

    �����󥹥ȥ��ե���
    $\Ddsty Q(t)=\Dinv{2}\int_0^{L_x}\int_0^{L_y} \zeta(x,y,t)^2 d x d y 
         =\Dinv{2}\int_0^{L_x}\int_0^{L_y} [\Dlapla\psi(x,y,t)]^2 d x d y$
    �򥹥ڥ��ȥ���ʬ��ɽ��. 
    \begin{eqnarray*}
      [\Dlapla\psi(x,y,t)]^2 
         &=& \left(\sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  (\tilde{k}^2+\tilde{l}^2)\tilde{\psi}_{kl}
                     e^{i\tilde{k}x+i\tilde{l}y}
             \right)^2 \\
         &=& \left(
               \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  (\tilde{k}^2+\tilde{l}^2)\tilde{\psi}_{kl}
                     e^{i\tilde{k}x+i\tilde{l}y}
             \right)\cdot
             \left(
               \sum_{k'=-K}^{K} \sum_{l'=-L}^{L}
                  (\tilde{k'}^2+\tilde{l}^2)\tilde{\psi}_{k'l'}
                     e^{i\tilde{k'}x+i\tilde{l'}y}
             \right).
    \end{eqnarray*}
    ���ΰ���ʬ $\Ddsty\int_0^{L_x}\int_0^{L_y} d x d y $ ����Ѥ������
    �ȿ��� 0 �Ǥʤ��Ȥ߹�碌���Ѥ������ʬ�� 0 �Ȥʤ�Τ�
    $k'=-k, l'=-l$ �δ�Ϳ�Τߤ��Τ���. �������ä�,
    %
    \begin{eqnarray*}
      Q(t)&=& \Dinv{2}\int_0^{L_x}\int_0^{L_y}[\Dlapla\psi(x,y,t)]^2  d x d y\\
         &=& \Dinv{2}\int_0^{L_x}\int_0^{L_y}
             \left(
               \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                  (\tilde{k}^2+\tilde{l}^2)\tilde{\psi}_{kl}
                     e^{i\tilde{k}x+i\tilde{l}y}
                  (\tilde{k}^2+\tilde{l}^2)\tilde{\psi}_{-k-l}
                     e^{-i\tilde{k}x-i\tilde{l}y}
             \right) d x d y \\
         &=& \Dinv{2}\sum_{k=-K}^{K} \sum_{l=-L}^{L}
                 (\tilde{k}^2+\tilde{l}^2)^2
                     \tilde{\psi}_{kl}\tilde{\psi}_{-k-l}
              \int_0^{L_x}\int_0^{L_y} d x d y \\
         &=& L_x L_y \sum_{k=-K}^{K} \sum_{l=-L}^{L}
                 \Dinv{2}(\tilde{k}^2+\tilde{l}^2)^2 |\tilde{\psi}_{kl}|^2\\
         &=&  L_x L_y \sum_{k=-K}^{K} \sum_{l=-L}^{L}
              \Dinv{2}\left[ \left(\frac{2\pi k}{L_x}\right)^2
                     +\left(\frac{2\pi l}{L_y}\right)^2 \right]^2
                   |\tilde{\psi}_{kl}|^2.
    \end{eqnarray*}
    %
    �������ä��ȿ����ȤΥ��󥹥ȥ��ե���̩�� ${\cal Q}(k,l,t)$������Ǥ���,
    %
    \begin{equation}
      {\cal Q}(k,l,t) 
         =  \Dinv{2}(\tilde{k}^2+\tilde{l}^2)^2 |\tilde{\psi}_{kl}|^2
         =  \Dinv{2}\left[ \left(\frac{2\pi k}{L_x}\right)^2
                   +\left(\frac{2\pi l}{L_y}\right)^2 \right]^2
                   |\tilde{\psi}_{kl}|^2.
    \end{equation}
    %
    ���ΤȤ������󥹥ȥ��ե�����
    \begin{equation}
      Q(t) =  L_x L_y \sum_{k=-K}^{K} \sum_{l=-L}^{L} {\cal Q}(k,l,t).
    \end{equation}

\section{�¸�����}



\end{document}


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