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通常のモードと連続モードの
共鳴による不安定
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九州大学 応用力学研究所

伊賀 啓太

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iga@riam.kyushu-u.ac.jp


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(2003年夏 GFDセミナー, 2003/09/09 奈井江)          

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共鳴を起こす中立モードの対がない場合(1)

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いくつかの不安定では\
不安定モードが2つの中立波動の共鳴でうまく理解できた。

しかし、このような対がうまく見つからないことがある。

一般的に、不安定モードの位相速度の実部は\
基本流の流速の最大値と最小値の間にあることが多い。
→基本場の流速と一致することが多い。
→Lin (1945)の定理よりそこに中立波は存在できないから、\
不安定モードにつながる中立波が見つからない。

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臨界層不安定

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臨界層不安定はそのような不安定の一つ。
←本来、中立波だった波が、基本場が存在して\
臨界点を持つことになったために不安定になったもの。

不安定モードにつながる中立波が一つしかない。
→これは中立波の共鳴で説明できるか?

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基本流の流速の範囲内には連続モードがある。
→連続モードが関わった共鳴と見ることはできないであろうか?

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臨界層不安定の理論的考察

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状況設定 (Iga 1999a)

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(Iga, 1999: Fluid, Dyn. Res., 25, 63, Fig.1)

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臨界層不安定の理論的考察

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摂動展開よってこのモードの位相速度の虚部は

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%% 以下, eps ファイルを作る TeX ファイル
%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-ci-theory eqn"
%
\begin{displaymath}
     |c_i^{(1)}|
     = \frac{\pi H_1^2(y_c) Q_1'(y_c) (c_r^{(0)}-U_2)
          |v_1^{(1)}(y_c)|^2}
          {\displaystyle k^2\left|U_1'(y_c)\right|
           \int \left[ H_2 |u_2^{(0)}|^2 + H_2 |v_2^{(0)}|^2
                         + |p_2^{(0)}|^2/(g\rho\Delta\rho) \right] dy}
\end{displaymath}
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%image "./eqn/5-ci-theory.eps" 0 350 350 1

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	Q_1'(y_c) (c_r-U_2) > 0の時:\
固有値の虚部が正負の2つのモード
	Q_1'(y_c) = 0の時:\
固有値の虚部が0のモードが1つ
	Q_1'(y_c) (c_r-U_2) < 0の時:\
正則なモードはない

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臨界層不安定の理論的考察

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不安定が起きる時、下層の構造はどのように表されるか?
←下層のみで解いた時の解を用いてどのように表されるか?
→下層の1層問題の固有関数で展開してみる。

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連続モードの直交性(浅水系):

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%% 以下, eps ファイルを作る TeX ファイル
%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-orthogonality eqn"
%
\begin{eqnarray*}
     && \int
          \left( u_c(y;y_{ca}) h_c^*(y;y_{cb}) + h_c(y;y_{ca}) u_c^*(y;y_{cb})
               - \frac{H_1^2 q_c(y;y_{ca}) q_c^*(y;y_{cb})}{Q_1'(y)} \right) dy
\\
     && \hspace{20mm} = -\frac{\pi^2H_1^2(y_{ca})Q_1'(y_{ca})}
                              {{U_1'}^2(y_{ca})}
                 \left( C_{{\rm I\! I}r}^2(y_{ca})
                      + C_{{\rm I\! I}i}^2(y_{ca}) \right)
          \delta(y_{ca}-y_{cb})
\end{eqnarray*}
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%image "./eqn/5-orthogonality.eps" 0 300 300 1

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臨界層不安定の理論的考察

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連続モードの係数C(y_c)は
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%filter "./bin/latex2eps-with-ext-dir.sh 5-coefficients-basic eqn"
%
\begin{eqnarray*}
     && C(y_c) =
          -\frac{{U_1'}^2(y_c)}
                {\pi^2H_1^2(y_c)Q_1'(y_c)
                        \left( C_{{\rm I\! I}r}^2(y_c)
                             + C_{{\rm I\! I}i}^2(y_c) \right)}
\\
     &&\hspace{15mm} \times \int
          \left( u_1(y) h_c^*(y;y_c) + h_1(y) u_c^*(y;y_c)
               - \frac{H_1^2(y) q_1(y) q_c^*(y;y_c)}{Q_1'(y)} \right) dy
\end{eqnarray*}
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%image "./eqn/5-coefficients-basic.eps" 0 300 300 1

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不安定モードと展開する連続モード渦位はそれぞれ
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%% 以下, eps ファイルを作る TeX ファイル
%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-q-v-unstable-continuous eqn"
%
\begin{eqnarray*}
     &&
     q_1(y) = - \frac{Q_1'(y) v_1(y)}{-ik \left( c - U_1(y) \right)}
\\
     &&
     q_c(y;y_c) = - {\cal P} \frac{Q_1'(y) v_c(y;y_c)}
                                  {-ik \left( c_c(y_c) - U_1(y) \right)}
          + C_{{\rm I\! I}i}(y_c) \frac{\pi Q_1'(y_c)}{U_1'(y_c)}
            \delta (y-y_c)
\end{eqnarray*}
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%image "./eqn/5-q-v-unstable-continuous.eps" 0 350 350 1

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臨界層不安定の理論的考察

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展開係数は次のように求められる
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%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-coefficient-complete eqn"
%
\begin{displaymath}
     C(y_c)
     \simeq - \frac{U_1'(y_c) v_1(y_c)}
                 {\pi k
                 [ \mbox{sgn}(c_i U_1'(y_c)) C_{{\rm I\! I}r}(y_c)
                                         - i C_{{\rm I\! I}i}(y_c) ]}
     \cdot \frac{1}{(c - c_c(y_c))}
\end{displaymath}
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c_c 〜 c_r付近でのふるまいは
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%% 以下, eps ファイルを作る TeX ファイル
%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-coefficient-absolute-argument eqn"
%
\begin{eqnarray*}
     &&
     |C(y_c)| \sim
     \left[ \frac{c_i^2}{(c_c(y_c) - c_r)^2 + c_i^2} \right]
               ^{\frac{1}{2}}
\\
     &&
     \mbox{arg}(C(y_c)) \sim
          \arctan \left( \frac{c_c(y_c) - c_r}{c_i} \right)
\end{eqnarray*}
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	c_c=c_rにピーク
	ピークの値の1/√2倍になるところの幅の半分は|c_i|
	この間に位相はπ/4ずれる
→下層の構造は、\
位相速度がc_r-|c_i| < c_c < c_r+|c_i|\
の連続モードの重ね合わせでほぼ表される。

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臨界層不安定の理論的考察

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連続モードの擬運動量は
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%%
%filter "./bin/latex2eps-with-ext-dir.sh 5-momentum-q eqn"
%
\begin{displaymath}
     M \equiv \int \left( \overline{hu}
          - \frac{H^2 \overline{q^2}}{2 Q'} \right) dy
     = \frac{1}{4} \int \left( hu^* + h^*u
          - \frac{H^2 |q|^2}{Q'} \right) dy .
\end{displaymath}
%endfilter
%image "./eqn/5-momentum-q.eps" 0 350 350 1
%%

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連続モードの渦位は臨界点で無限大
→
	連続モードの擬運動量の絶対値は無限大
	連続モードの擬運動量の符合はQ'(y_c)と逆符号

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臨界層不安定の数値計算

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状況設定 (Iga 1999a)

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(Iga, 1999: Fluid, Dyn. Res., 25, 63, Fig.5)

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臨界層不安定の数値計算

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臨界層付近での振舞い (Iga 1999a)

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(Iga, 1999: Fluid, Dyn. Res., 25, 63, Fig.6)

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臨界層不安定の数値計算

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不安定を作る連続モードの割合 (Iga 1999a)
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(Iga, 1999: Fluid, Dyn. Res., 25, 63, Fig.7)

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臨界層不安定の数値計算

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擬運動量の流れ (Iga 1999a)

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(Iga, 1999: Fluid, Dyn. Res., 25, 63, Fig.8)

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