HMHD decomposition (Coriolis absent)

線形波の関数形をモード展開の基礎に置く

Linear wave: $ \footnotesize \left\{ \begin{array}{l} \partial_t \boldsymbol u = \boldsymbol B_0\cdot \nabla \boldsymbol b \\ \partial_t \boldsymbol b = \boldsymbol B_0\cdot \nabla (\boldsymbol u - \alpha \boldsymbol j ) \end{array} \right. $ \footnotesize \newcommand{\b}{\boldsymbol} \stackrel{Fourier}{\Longrightarrow} - i \omega \left[ \begin{array}{c} \b u \\ \b b \end{array} \right] = i \b B_0\cdot\vec k \left[ \begin{array}{cc} O & I \\ I & - \alpha \nabla \times \end{array} \right] \left[ \begin{array}{c} \b u \\ \b b \end{array} \right] = i B_0k_{||} \lambda \left[ \begin{array}{c} \b u \\ \b b \end{array} \right]

Eigenvalue problem: $ \footnotesize \lambda \left[ \begin{array}{c} \boldsymbol u \\ \boldsymbol b \end{array} \right] = \left[ \begin{array}{cc} O & I \\ I & - \alpha \nabla \times \end{array} \right] \left[ \begin{array}{c} \boldsymbol u \\ \boldsymbol b \end{array} \right] \Longleftrightarrow \left\{ \begin{array}{l} \lambda \boldsymbol u = \boldsymbol b \\ \lambda \boldsymbol b = \boldsymbol u - \alpha \boldsymbol j \end{array} \right.

Eigenfunction of curl operator (complex helical waves):$ \footnotesize \nabla \times \boldsymbol\phi_\sigma (\vec k) = \sigma K \boldsymbol\phi_\sigma (\vec k) where$ \footnotesize \sigma=\pm1 ,$ \footnotesize K = |\vec k| and postulate$ \footnotesize \int|\boldsymbol\phi_\sigma (\vec k)|^2 = 1

Eigenequation:$ \footnotesize \left| \begin{array}{cc} - \lambda & 1 \\ 1 & - \lambda - \alpha \sigma K \end{array} \right| = \lambda^2 + \alpha \sigma K \lambda -1 = 0 $ \footnotesize \Longrightarrow \lambda = - \frac{\alpha \sigma K}{2} + t \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} , where$ \footnotesize t = \pm 1

Eigenvector:$ \footnotesize \left[ \begin{array}{cc} O & 1 \\ 1 & - \alpha \sigma K \end{array} \right] \left[ \begin{array}{c} \bm\phi_\sigma (\vec k) \\ \lambda \bm\phi_\sigma (\vec k) \end{array} \right] = \left[ \begin{array}{c} \lambda \bm\phi_\sigma (\vec k) \\ (1 - \alpha \sigma K \lambda) \bm\phi_\sigma (\vec k) \end{array} \right] = \left[ \begin{array}{c} \lambda \bm\phi_\sigma (\vec k) \\ \lambda^2 \bm\phi_\sigma (\vec k) \end{array} \right] = \lambda \left[ \begin{array}{r} \bm\phi_\sigma (\vec k) \\ \lambda \bm\phi_\sigma (\vec k) \end{array} \right]

when $ \footnotesize \boldsymbol u = \boldsymbol\phi_\sigma (\vec k) ,$ \footnotesize \boldsymbol b = \lambda \boldsymbol\phi_\sigma (\vec k) ,$ \footnotesize \boldsymbol u - \alpha \boldsymbol j = \lambda^2 \boldsymbol\phi_\sigma (\vec k)

Each eigenvalue and its eigenfunction:

$ \footnotesize \sigma = -1 , \ t = + 1 : $ \footnotesize \lambda = \frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} =: \mu , whistler$ \footnotesize \frac1{\sqrt{\mu^2+1}}\left[ \begin{array}{r} \bm\phi_- (\vec k) \\ \mu \bm\phi_- (\vec k) \end{array} \right]

$ \footnotesize \sigma = -1 , \ t = - 1 : $ \footnotesize \lambda = \frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = - \frac 1 \mu , ion cyclotron$ \footnotesize \frac1{\sqrt{\mu^2+1}} \left[ \begin{array}{r} \mu \boldsymbol\phi_- (\vec k) \\ - \boldsymbol\phi_- (\vec k) \end{array} \right]

$ \footnotesize \sigma = + 1 , \ t = + 1 : $ \footnotesize \lambda = - \frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = \frac 1 \mu , ion cyclotron$ \footnotesize \frac1{\sqrt{\mu^2+1}} \left[ \begin{array}{r} \mu \boldsymbol\phi_+ (\vec k) \\ \boldsymbol\phi_+ (\vec k) \end{array} \right]

$ \footnotesize \sigma = + 1 , \ t = - 1 : $ \footnotesize \lambda = - \frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = - \mu , whistler$ \footnotesize \frac1{\sqrt{\mu^2+1}} \left[ \begin{array}{r} \bm\phi_+ (\vec k) \\ - \mu \bm\phi_+ (\vec k) \end{array} \right]

where$ \footnotesize \lambda = \lambda(\vec k,\sigma,s) = - \sigma s \left[ \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} + s \frac{\alpha K}{2} \right] = - \sigma s \mu^s ,$ \footnotesize \mu = \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} + \frac{\alpha K}{2}

Relation between $ \footnotesize (U,B) and$ \footnotesize (Z^{IC},Z^{Wh})

$ \footnotesize \left[ \begin{array}{r} \widehat u_- (\vec k) \\ \widehat b_- (\vec k) \end{array} \right] = \frac{\widehat Z_-^{Wh}}{\sqrt{\mu^2+1}} \left[ \begin{array}{r} 1 \\ \mu \end{array} \right] + \frac{\widehat Z_-^{IC}}{\sqrt{\mu^2+1}} \left[ \begin{array}{r} \mu \\ -1 \end{array} \right] = \frac{1}{\sqrt{\mu^2+1}} \left[ \begin{array}{rr} \mu & 1 \\ -1 & \mu \end{array} \right]\left[ \begin{array}{r} \widehat Z_-^{IC} \\ \widehat Z_-^{Wh} \end{array} \right]

$ \footnotesize \Longrightarrow \left[ \begin{array}{r} \widehat Z_-^{IC} \\ \widehat Z_-^{Wh} \end{array} \right] = \frac{1}{\sqrt{\mu^2+1}} \left[ \begin{array}{rr} \mu & -1 \\ 1 & \mu \end{array} \right] \left[ \begin{array}{r} \widehat u_- (\vec k) \\ \widehat b_- (\vec k) \end{array} \right] $ \footnotesize \overset{\alpha\to0}{\longrightarrow} \frac{1}{\sqrt{2}} \left[ \begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array} \right] \left[ \begin{array}{r} \widehat u_- (\vec k) \\ \widehat b_- (\vec k) \end{array} \right]

$ \footnotesize \left[ \begin{array}{r} \widehat u_+ (\vec k) \\ \widehat b_+ (\vec k) \end{array} \right] = \frac{\widehat Z_+^{Wh}}{\sqrt{\mu^2+1}} \left[ \begin{array}{r} 1 \\ - \mu \end{array} \right] + \frac{\widehat Z_+^{IC}}{\sqrt{\mu^2+1}} \left[ \begin{array}{r} \mu \\ 1 \end{array} \right] = \frac{1}{\sqrt{\mu^2+1}} \left[ \begin{array}{rr} \mu & 1 \\ 1 & -\mu \end{array} \right]\left[ \begin{array}{r} \widehat Z_+^{IC} \\ \widehat Z_+^{Wh} \end{array} \right]

$ \footnotesize \Longrightarrow \left[ \begin{array}{r} \widehat Z_+^{IC} \\ \widehat Z_+^{Wh} \end{array} \right] = \frac{1}{\sqrt{\mu^2+1}} \left[ \begin{array}{rr} \mu & 1 \\ 1 & - \mu \end{array} \right] \left[ \begin{array}{r} \widehat u_+ (\vec k) \\ \widehat b_+ (\vec k) \end{array} \right] $ \overset{\alpha\to0}{\longrightarrow} \frac{1}{\sqrt{2}} \left[ \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{r} \widehat u_+ (\vec k) \\ \widehat b_+ (\vec k) \end{array} \right]

Unitary relations

$ |\widehat u_x(\vec k)|^2 + |\widehat u_y(\vec k)|^2 + |\widehat u_z(\vec k)|^2 = |\widehat u_+(\vec k)|^2 + |\widehat u_-(\vec k)|^2

$ |\widehat b_x(\vec k)|^2 + |\widehat b_y(\vec k)|^2 + |\widehat b_z(\vec k)|^2 = |\widehat b_+(\vec k)|^2 + |\widehat b_-(\vec k)|^2

$ |\widehat u_+(\vec k)|^2 + |\widehat b_+(\vec k)|^2 = |\widehat Z_+^{Wh}(\vec k)|^2 + |\widehat Z_+^{IC}(\vec k)|^2

$ |\widehat u_-(\vec k)|^2 + |\widehat b_-(\vec k)|^2 = |\widehat Z_-^{Wh}(\vec k)|^2 + |\widehat Z_-^{IC}(\vec k)|^2

一般化渦度演算子の固有値問題

$ \newcommand{\f}{\small} \newcommand{\b}{\boldsymbol} \f \underbrace{ \left[ \begin{array}{r} \alpha\b\omega + \b b \\ \b b \end{array} \right] }_{\vec{\b\Omega}} = \left[ \begin{array}{r} \alpha\nabla\times\b u + \frac{\f\b u}{\f\alpha\nabla\times} - \frac{\f\b u - \alpha\b j}{\f\alpha\nabla\times} \\ \frac{\f\b u}{\f\alpha\nabla\times} - \frac{\f\b u - \alpha\b j}{\f\alpha\nabla\times} \end{array} \right] = \left[ \begin{array}{rr} \alpha\nabla\times + \frac{\f 1}{\f\alpha\nabla\times} & -\frac{\f 1}{\f\alpha\nabla\times} \\ \frac{\f 1}{\f\alpha\nabla\times} & -\frac{\f 1}{\f\alpha\nabla\times} \end{array} \right] \underbrace{ \left[ \begin{array}{r} \b u \\ \b u - \alpha\b j \end{array} \right] }_{\overrightarrow{\b V}}

$ \small \Lambda^2 - \alpha \sigma K \Lambda - 1 = 0 ,$ \small \Lambda + \Lambda^* = \alpha \sigma K ,$ \small \Lambda \Lambda^* = - 1 ,$ \small \Lambda = \frac12\left[ \alpha \sigma K + t \sqrt{(\alpha K)^2 + 4} \right]

$ \newcommand{\f}{\footnotesize} \f \left[ \begin{array}{rr} \alpha\sigma K + \frac{\f 1}{\f\alpha\sigma K} & -\frac{\f 1}{\f\alpha\sigma K} \\ \frac{\f 1}{\f\alpha\sigma K} & -\frac{\f 1}{\f\alpha\sigma K} \end{array} \right] \left[ \begin{array}{c} \Lambda \\ -\Lambda^* \end{array} \right] = \left[ \begin{array}{r} \alpha\sigma K \Lambda + \frac{\f \Lambda}{\f\alpha\sigma K} - \frac{\f -\Lambda^*}{\f\alpha\sigma K} \\ \frac{\f \Lambda}{\f\alpha\sigma K} - \frac{\f -\Lambda^*}{\f\alpha\sigma K} \end{array} \right]

$ \footnotesize = \left[ \begin{array}{r} \alpha\sigma K \Lambda + 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} \Lambda^2 \\ - \Lambda \Lambda^* \end{array} \right] = \Lambda \left[ \begin{array}{c} \Lambda \\ - \Lambda^* \end{array} \right] = \Lambda \left[ \begin{array}{c} \Lambda \\ 1/\Lambda \end{array} \right]

$ \footnotesize \sigma = -1 , \ t = + 1 : $ \footnotesize \Lambda = - \frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = \frac{1}{\mu} ,$ \footnotesize - \Lambda^* = \frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = \mu , $ \footnotesize \left[ \begin{array}{r} \frac{1}{\mu} \boldsymbol\phi_- (\vec k) \\ \mu \boldsymbol\phi_- (\vec k) \end{array} \right] Wh-

$ \footnotesize \sigma = -1 , \ t = - 1 : $ \footnotesize \Lambda = - \frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = -{\mu} ,$ \footnotesize - \Lambda^* = \frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = -\frac1\mu , $ \footnotesize \left[ \begin{array}{r} -{\mu} \boldsymbol\phi_- (\vec k) \\ -\frac1\mu \boldsymbol\phi_- (\vec k) \end{array} \right] IC-

$ \footnotesize \sigma = +1 , \ t = + 1 : $ \footnotesize \Lambda = \frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = {\mu} ,$ \footnotesize - \Lambda^* = -\frac{\alpha K}{2} + \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = \frac1\mu , $ \footnotesize \left[ \begin{array}{r} {\mu} \boldsymbol\phi_+ (\vec k) \\ \frac{1}\mu \boldsymbol\phi_+ (\vec k) \end{array} \right] IC+

$ \footnotesize \sigma = +1 , \ t = - 1 : $ \footnotesize \Lambda = \frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = -\frac1{\mu} ,$ \footnotesize - \Lambda^* = -\frac{\alpha K}{2} - \sqrt{\Big(\frac{\alpha K}{2}\Big)^2 + 1} = -\mu , $ \footnotesize \left[ \begin{array}{r} -\frac1{\mu} \boldsymbol\phi_+ (\vec k) \\ -\mu \boldsymbol\phi_+ (\vec k) \end{array} \right] Wh+

$ \footnotesize \newcommand{\b}{\boldsymbol} \b u = \Lambda \b\phi_\sigma ,$ \footnotesize \newcommand{\b}{\boldsymbol} \b u - \alpha \b j= \frac1\Lambda \b\phi_\sigma ,$ \footnotesize \newcommand{\b}{\boldsymbol} \alpha \b j = ( \Lambda - \frac1\Lambda ) \b\phi_\sigma = \frac{\Lambda^2 - 1}\Lambda \b\phi_\sigma = \frac{\alpha\sigma K\Lambda}\Lambda \b\phi_\sigma = \alpha \sigma K \b\phi_\sigma ,$ \footnotesize \newcommand{\b}{\boldsymbol} \b b = \b\phi_\sigma

$ \footnotesize \newcommand{\b}{\boldsymbol} - \b a / \alpha = - \b\phi_\sigma / \alpha \sigma K ,$ \footnotesize \newcommand{\b}{\boldsymbol} \b u + \b a / \alpha = \Lambda \b\phi_\sigma + \b\phi_\sigma / \alpha \sigma K = (\alpha\sigma K\Lambda + 1) \b\phi_\sigma/\alpha\sigma K = \Lambda^2 \b\phi_\sigma/\alpha\sigma K