linear waves of incompressible dissipationless Hall MHD
generalized vorticity equation for incompressible dissipationless HMHD media
$ \partial_t ( \bm U + \bm A/\alpha ) = \bm U \times (\nabla \times ( \bm U + \bm A/\alpha )) - \nabla \Pi …①,
$ \partial_t ( -\bm A/\alpha ) = ( \bm U - \alpha\bm J ) \times ( \nabla \times ( -\bm A/\alpha ) ) + \nabla\phi/\alpha …②
where $ \bm U is velocity field (of ion), $ \bm A ,$ \phi are electromagnetic potentials,$ \alpha is Hall term parameter,$ \Pi is pressure,$ \bm J is current density field.
①＋② derives$ \partial_t\bm U = \bm U \times \bm \Omega + \bm J \times \bm B - \nabla (\Pi-\phi/\alpha) where $ \bm B = \nabla\times\bm A
vorticity equations for incompressible HMHD media
$ \partial_t ( \alpha\bm \Omega + \bm B ) = \nabla \times \big[ \bm U \times ( \alpha\bm \Omega + \bm B ) \big] \cdots \alpha\nabla\times ① ,$ \partial_t ( -\bm B ) = \nabla \times \big[ ( \bm U - \alpha\bm J ) \times ( -\bm B ) \big] \cdots \alpha\nabla\times ②
where $ \bm\Omega = \nabla \times \bm U , $ \bm J = \nabla \times \bm B
SInce $ \nabla\times(\bm A\times\bm B) = (\bm B\cdot\nabla)\bm A - (\bm A\cdot\nabla)\bm B if $ \nabla\cdot\bm A = \nabla\cdot\bm B = 0 , these equations read
$ \partial_t ( \alpha\bm\Omega + \bm B ) = \big[ ( \alpha\bm\Omega + \bm B ) \cdot \nabla \big] \bm U - ( \bm U \cdot \nabla) ( \alpha\bm\Omega + \bm B ) ,
$ \partial_t ( -\bm B ) = ( -\bm B \cdot \nabla )( \bm U - \alpha\bm J ) - \big[ ( \bm U - \alpha\bm J ) \cdot \nabla \big] (-\bm B)
linearized equations when uniform ambient magnetic field and coriolis effect exist
$ \partial_t ( \alpha\bm\Omega + \bm B ) = \big[ ( \alpha\bm\Omega_0 + \bm B_0 ) \cdot \nabla \big] \bm U ,$ \partial_t ( -\bm B ) = ( -\bm B_0 \cdot \nabla )( \bm U - \alpha\bm J )
matrix operator form (where$ \bm V is the electron velocity:$ \bm V = \bm U - \alpha\bm J , integral operator$ \frac{1}{\nabla\times} is the inverse of $ \nabla\times )
$ \frac{\partial}{\partial t} \left[\begin{array}{cc} \alpha\nabla\times + \frac{1}{\alpha\nabla\times} & -\frac{1}{\alpha\nabla\times} \\ -\frac{1}{\alpha\nabla\times} & \frac{1}{\alpha\nabla\times} \end{array}\right] \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right] = \left[\begin{array}{cc} (\alpha\bm\Omega_0 + \bm B_0)\cdot\nabla & 0 \\ 0 & -\bm B_0\cdot\nabla \end{array}\right] \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right]
substituting $ U, \, V \propto \exp[i(\vec k\cdot\vec x - \omega t)] and$ \phi(\vec{k},\sigma;\vec{x}) , the eigenfunction of curl operator ($ \nabla\times\phi = \sigma k \phi ,$ k = |\vec k| , $ \sigma = \pm 1 ), and $ \bm\Omega_0 = \Omega_0\bm e_{\parallel} ,$ \bm B_0 = B_0\bm e_{\parallel} ,$ k_{\parallel} = \bm e_{\parallel} \cdot \vec k = k \cos\theta :
$ \newcommand{\ds}{\displaystyle} - i \omega \underbrace{ \left[\begin{array}{rr} \ds \alpha\sigma k + \frac{1}{\alpha\sigma k} & \ds -\frac{1}{\alpha\sigma k} \phantom{\bigg|\!\!} \\ \ds -\frac{1}{\alpha\sigma k} & \ds \frac{1}{\alpha\sigma k} \phantom{\bigg|\!\!} \end{array}\right] }_{det = 1} \left[\begin{array}{c} \bm U \phantom{\bigg|\!\!} \\ \bm V \phantom{\bigg|\!\!} \end{array}\right] = i k_{\parallel} \left[\begin{array}{cc} \alpha\Omega_{0} + B_{0} & 0 \phantom{\bigg|\!\!} \\ 0 & -B_{0} \phantom{\bigg|\!\!} \end{array}\right] \left[\begin{array}{c} \bm U \phantom{\bigg|\!\!} \\ \bm V \phantom{\bigg|\!\!} \end{array}\right]
$ \newcommand{\f}{\displaystyle} -\omega \underbrace{ \left[\begin{array}{cc} -B_{0} & \phantom{\bigg|\!\!} 0 \\ \phantom{\bigg|\!\!} 0 & \alpha\Omega_{0} + B_{0} \end{array}\right] \left[\begin{array}{rr} \f \alpha\sigma k + \frac{1}{\alpha\sigma k} & \f \phantom{\bigg|\!\!} -\frac{1}{\alpha\sigma k} \\ \f -\frac{1}{\alpha\sigma k} & \f \phantom{\bigg|\!\!} \frac{1}{\alpha\sigma k} \end{array}\right] }_{\normalsize =:\hat W, \ {\rm{where}}\ |\hat W| = -B_{0}(\alpha\Omega_{0} + B_{0}) = \Lambda\Lambda^{\dag}} \left[\begin{array}{c} \phantom{\bigg|\!\!} \bm U \\ \phantom{\bigg|\!\!} \bm V \end{array}\right] = \underbrace{ -B_{0}(\alpha\Omega_{0} + B_{0}) }_{\normalsize = \Lambda \Lambda^{\dag}} k_{\parallel} \left[\begin{array}{c} \phantom{\bigg|\!\!} \bm U \\ \phantom{\bigg|\!\!} \bm V \end{array}\right]
substituting the eigenfunction of $ \hat W ,
$ -\omega \Lambda \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right] = \Lambda \Lambda^{\dag} k_{\parallel} \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right] , i.e. phase velocity is $ \omega = - k_{\parallel} \Lambda^{\dag} = - B_{0}k_{\parallel} \widetilde\Lambda^{\dag}
eigenvalues are (introducing $ \widetilde\Lambda := \frac{\Lambda}{B_{0}} ,$ \widetilde\Omega := \frac{\Omega_{0}}{B_{0}} ):
$ \Lambda + \Lambda^{\dag} = -B_{0} \alpha \sigma k + \frac{\alpha\Omega_{0}}{\alpha \sigma k} $ \Longrightarrow $ \widetilde\Lambda + \widetilde\Lambda^{\dag} = - \alpha \sigma k + \frac{\widetilde\Omega}{\sigma k} ,$ \widetilde\Lambda \widetilde\Lambda^{\dag} = - (\alpha\widetilde\Omega + 1) ,
$ (\widetilde\Lambda - \widetilde\Lambda^{\dag})^2 = (\widetilde\Lambda + \widetilde\Lambda^{\dag})^2 -4 \widetilde\Lambda \widetilde\Lambda^{\dag} = \Big( -\alpha \sigma k + \frac{\widetilde\Omega}{\sigma k} \Big)^2 + 4 ( 2\alpha \widetilde\Omega + 1 ) $ \left. = \Big( \alpha k + \frac{\widetilde\Omega}{ k} \Big)^2 + 4 \right._{\phantom q}
$ \widetilde\Lambda = -\alpha\sigma k + \sigma \Big( \frac{\alpha k}{2} + \frac{\widetilde\Omega}{2 k} \Big) \pm \sqrt{ \Big( \frac{\alpha k}{2} + \frac{\widetilde\Omega}{2 k} \Big)^2 + 1 }^{\phantom l} $ = \sigma ( - \alpha k + \mu + s \sqrt{\mu^2+1} ) $ = \sigma ( - \alpha k + s \gamma^s )
$ \widetilde\Lambda^{\dag} = \sigma ( - \alpha k + \mu - s \sqrt{\mu^2+1} ) $ = \sigma ( - \alpha k - s \gamma^{-s} ) where$ \mu := \frac{\alpha k}{2} + \frac{\widetilde\Omega}{2 k} ,$ \gamma := \sqrt{\mu^2+1}+\mu
(U,B)で記述してみる
$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \frac{\partial}{\partial t} \A{r}{ \alpha\bm\Omega + \bm B \\ \bm B } = \A{rr}{ ( \alpha\bm\Omega_0 + \bm B_0 ) \cdot \nabla & O \\ O & \bm B_0 \cdot \nabla } \A{l}{ \bm U \\ \bm U - \alpha\bm J }
$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \frac{\partial}{\partial t} \A{cc}{ \alpha\nabla\times & I \\ O & I } \A{r}{ \bm U \\ \bm B } = \A{rr}{ ( \alpha \bm\Omega_0 + \bm B_0 ) \cdot \nabla & O \\ O & \bm B_0 \cdot \nabla } \A{cc}{ I & O \\ I & -\alpha \nabla \times } \A{l}{ \bm U \\ \bm B }
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \frac{\partial}{\partial t} \A{cc}{ \tilde\alpha & 1 \\ 0 & 1 } \A{r}{ \bm U \\ \bm B } = \A{rr}{ ( \alpha \bm\Omega_0 + \bm B_0 ) \cdot \nabla & O \\ O & \bm B_0 \cdot \nabla } \A{rr}{ 1 & 0 \\ 1 & -\tilde\alpha } \A{l}{ \bm U \\ \bm B } where$ \footnotesize \tilde\alpha = \alpha\sigma k
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \frac{\partial}{\partial t} \A{r}{ \bm U \\ \bm B } = \frac{1}{\tilde\alpha} \A{cc}{ 1 & -1 \\ 0 & \tilde\alpha } \A{rr}{ ( \alpha \bm\Omega_0 + \bm B_0 ) \cdot \nabla & O \\ O & \bm B_0 \cdot \nabla } \A{rr}{ 1 & 0 \\ 1 & -\tilde\alpha } \A{l}{ \bm U \\ \bm B }
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} = \frac{1}{\tilde\alpha} \A{cc}{ 1 & -1 \\ 0 & \tilde\alpha } \A{rr}{ ( \alpha \bm\Omega_0 + \bm B_0 ) \cdot \nabla & O \\ \bm B_0 \cdot \nabla & - \tilde\alpha \bm B_0 \cdot \nabla } \A{l}{ \bm U \\ \bm B }
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} = \frac{1}{\tilde\alpha} \A{cc}{ \alpha \bm\Omega_0 \cdot \nabla & \tilde\alpha \bm B_0 \cdot \nabla \\ \tilde\alpha \bm B_0 \cdot \nabla & - (\tilde\alpha)^2 \bm B_0 \cdot \nabla } \A{l}{ \bm U \\ \bm B }
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} = \A{cc}{ \displaystyle \frac{\alpha \bm\Omega_0 \cdot \nabla}{\tilde\alpha} & \bm B_0 \cdot \nabla \\ \bm B_0 \cdot \nabla & - \tilde\alpha \bm B_0 \cdot \nabla } \A{l}{ \bm U \\ \bm B }
$ \footnotesize \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\f}{\phantom{\Big|\!\!}} = B_0 k_\parallel \A{cc}{ \displaystyle {\tilde\Omega}/{\sigma k} & 1 \\ 1 & - \alpha\sigma k } \A{l}{ \bm U \\ \bm B } $ \footnotesize \lambda + \lambda^\dag = \frac{\tilde\Omega}{\sigma k} - \alpha\sigma k $ \footnotesize \lambda \lambda^\dag = - \alpha\tilde\Omega - 1 …結局、当然だが、固有値は同じ
$ \footnotesize (\lambda - \lambda^\dag)^2 = \Big( \frac{\tilde\Omega}{k} - \alpha k \Big)^2 + 4 \alpha\Omega + 4 = \Big( \frac{\tilde\Omega}{k} + \alpha k \Big)^2 + 4