incompressible Hall MHD and its linear modes (Coriolis exists)
basic equations for incompressible HMHD media
$ \partial_t ( \bm U + \bm A/\alpha ) = \bm U \times (\nabla \times ( \bm U + \bm A/\alpha )) + \overbrace{ \eta \triangle ( {\rm{Pr_{m}}}\, \bm U + \bm A/\alpha ) }^{\nu\triangle\bm U -(\eta/\alpha)\bm J} - \nabla \Pi …①,
$ \partial_t ( -\bm A/\alpha ) = ( \bm U - \alpha\bm J ) \times ( \nabla \times ( -\bm A/\alpha ) ) + \nabla\phi/\alpha + \underbrace{ \eta \triangle ( -\bm A/\alpha ) }_{(\eta/\alpha)\bm J} …②
where $ \bm U is velocity field (of ion), $ \bm A ,$ \phi are electromagnetic potentials,$ \alpha is Hall term parameter,$ {\rm{Pr_{m}}} = \nu/\eta
①＋② derives$ \partial_t\bm U = \bm U \times \bm \Omega + \bm J \times \bm B - \nabla (P+\frac12U^2) + \nu \triangle\bm U
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] + \eta \triangle ( {\rm{Pr_{m}}}\,\alpha\bm \Omega + \bm B ) …①,
$ \partial_t ( -\bm B ) = \nabla \times \big[ ( \bm U - \alpha\bm J ) \times ( -\bm B ) \big] - \eta\triangle\bm B …②
where $ \alpha is Hall parameter,$ \Omega = \nabla \times U , $ J = \nabla \times B ,$ {\rm{Pr_{m}}} = \nu/\eta
dissipationless equations
$ \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[ ( 2\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} (2\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 :
$ - i \omega \underbrace{ \left[\begin{array}{cc} \alpha\sigma k + \frac{1}{\alpha\sigma k} & -\frac{1}{\alpha\sigma k} \\ -\frac{1}{\alpha\sigma k} & \frac{1}{\alpha\sigma k} \end{array}\right] }_{det = 1} \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right] = i k_{\parallel} \left[\begin{array}{cc} 2\alpha\Omega_{0} + B_{0} & 0 \\ 0 & -B_{0} \end{array}\right] \left[\begin{array}{c} \bm U \\ \bm V \end{array}\right]
$ \newcommand{\f}{\displaystyle} -\omega \underbrace{ \left[\begin{array}{cc} -B_{0} & \phantom{\bigg|\!\!} 0 \\ \phantom{\bigg|\!\!} 0 & 2\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] }_{=:\hat W, \ {\rm{where}}\ |\hat W| = -B_{0}(2\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}(2\alpha\Omega_{0} + B_{0}) }_{= \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{2\alpha\Omega_{0}}{\alpha \sigma k} $ \Longrightarrow $ \widetilde\Lambda + \widetilde\Lambda^{\dag} = - \alpha \sigma k + \frac{2\widetilde\Omega}{\sigma k} ,$ \widetilde\Lambda \widetilde\Lambda^{\dag} = - (2\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{2 \widetilde\Omega}{\sigma k} \Big)^2 + 4 ( 2\alpha \widetilde\Omega + 1 ) $ \left. = \Big( \alpha k + \frac{2\widetilde\Omega}{ k} \Big)^2 + 4 \right._{\phantom q}
$ \widetilde\Lambda = -\alpha\sigma k + \sigma \Big( \frac{\alpha k}{2} + \frac{\widetilde\Omega}{ k} \Big) \pm \sqrt{ \Big( \frac{\alpha k}{2} + \frac{\widetilde\Omega}{ k} \Big)^2 + 1 }^{\phantom l} $ = - \sigma \alpha k + \sigma \mu + s \sqrt{\mu^2+1}
$ = - \sigma \alpha k + s (\sqrt{\mu^2+1} + s \sigma \mu) $ = - \alpha \sigma k + s \gamma^{\sigma s}
$ \widetilde\Lambda^{\dag} = - \sigma \alpha k - s \gamma^{-\sigma s} where$ \mu := \frac{\alpha k}{2} + \frac{\widetilde\Omega}{ k} ,$ \gamma := \sqrt{\mu^2+1}+\mu