Hall magnetohydrodynamics (incompressiblity assumed)

disspationless HMHD (to derive linear wave equation)

$ \partial_t \boldsymbol u + ( \boldsymbol u \cdot \nabla ) \boldsymbol u = - \nabla P + \boldsymbol j \times \boldsymbol b

$ \partial_t \boldsymbol b = \nabla \times ( ( \boldsymbol u -\alpha \boldsymbol j ) \times \boldsymbol b )

Linear wave equation (under an ambient uniform magnetic field $ \boldsymbol b_0 )

$ \left\{ \begin{array}{l} \partial_t \boldsymbol u = - \nabla P + \boldsymbol j \times \boldsymbol b_0 \\ \partial_t \boldsymbol b = \nabla \times ( ( \boldsymbol u -\alpha \boldsymbol j ) \times \boldsymbol b_0 ) \end{array} \right. \Longrightarrow \left\{ \begin{array}{l} \partial_t \boldsymbol u = (\boldsymbol b_0 \cdot \nabla ) \boldsymbol b - \nabla P \\ \partial_t \boldsymbol b = ( \boldsymbol b_0 \cdot \nabla ) ( \boldsymbol u -\alpha \boldsymbol j ) \end{array} \right.

$ \Longleftrightarrow \frac{\partial}{\partial t} \left[ \begin{array}{c} \boldsymbol u \\ \boldsymbol b \end{array} \right] = \left[ \begin{array}{cc} O & \boldsymbol b_0 \cdot \nabla \\ \boldsymbol b_0 \cdot \nabla & - \alpha (\boldsymbol b_0 \cdot \nabla)\nabla \times \end{array} \right] \left[ \begin{array}{c} \boldsymbol u \\ \boldsymbol b \end{array} \right] = \boldsymbol b_0 \cdot \nabla \overbrace{ \left[ \begin{array}{cc} O & I \\ I & \boxed{- \alpha \nabla \times} \end{array} \right] }^{principal\ part} \left[ \begin{array}{c} \boldsymbol u \\ \boldsymbol b \end{array} \right]

The boxed block contains the curl operator so that the eigenmode of HMHD should be described by the eigenmode of the curl operator (in other words, the the velocity and magnetic fields of an eigenmode have three dimensionally coupled components).

dispersion relation (assuming that $ \boldsymbol u, \boldsymbol b \propto \exp( i ( \vec{k} \cdot \vec{x} - \omega t )) )

formally $ \left| \begin{array}{cc} i \omega & i \boldsymbol b_0 \cdot \vec k \\ i \boldsymbol b_0 \cdot \vec k & i \omega - \alpha i (\boldsymbol b_0 \cdot \vec k) (i \vec k \times) \end{array} \right| = 0 .

using non-zero eigenvalues of curl operator:

$ \left| \begin{array}{cc} i \omega & i b_0 k_{\|} \\ i b_0 k_{\|} & i \omega - \alpha i b_0 k_{\|} (\sigma K) \end{array} \right| = -\omega(\omega - \alpha b_0 k_{\|} (\sigma K)) + (b_0 k_{\|})^2 = 0 ,

where $ b_0 k_{\|} = \boldsymbol b_0 \cdot \vec k,\ \ \ \sigma=\pm1,\ \ \ K = \sqrt{ k_x^2 + k_y^2 + k_z^2 } .

Solutions for the phase velocity: $ \omega = \frac{ 1 }{2} b_0 k_{\|} \sigma \alpha K \pm \frac12 \sqrt{ (b_0 k_{\|})^2 ( \alpha^2 K^2 + 4 ) } = \frac{ b_0 k_{\|} }{ 2 } \sigma s \Big[ \sqrt{ \alpha^2 K^2 + 4 } + s \alpha K \Big] , where $ s=+1 (whistler wave), $ s=-1 (ion cyclotron wave). Note: $ b_0 k_{\|} may be negative for some $ \vec k 's.

$ \lambda := \sqrt{ \left(\frac{\alpha K}{2}\right)^2 + 1 } + \frac{\alpha K}{2} \approx 1 + \frac{\alpha K}{2} + \frac12\left(\frac{\alpha K}{2}\right)^2

Note: eigenvalues of curl operator on$ \mathbb{T}^3 (or$ \mathbb{R}^3 )

The curl operator has three real-valued, non-degenerated eigenvalues; that is,$ \lambda = K,\ -K,\ 0 where$ K= \sqrt{ k_x^2 + k_y^2 + k_z^2 } .

Note that$ - K^2 is the eigenvalue of Laplace operator $ \partial_x^2 + \partial_y^2 + \partial_z^2 .

Note: spherical base vector functions on wavenumber space to construct the curl operator eigenmodes

reference axis vector: $ \hat a , wave number vector: $ \vec k = ( k_x, k_y, k_z ) , $ \phi -direction vector: $ \widehat{\boldsymbol{\phi}}(\vec k) := \hat a \times \vec k , $ \theta -direction vector: $ \widehat{\boldsymbol{\theta}}(\vec k) := \widehat{\boldsymbol{\phi}}(\vec k) \times \vec k ;

When $ \hat a := (1,0,0) , $ \phi -direction vector: $ \widehat{\boldsymbol{\phi}}(\vec k) = (0,-k_z,k_y) , $ \theta -direction vector: $ \widehat{\boldsymbol{\theta}}(\vec k) = (- k_y^2 - k_z^2 , k_x k_y , k_x k_z ) ;

Helical vector: $ \widehat{\boldsymbol{\theta}} + i\sigma K\widehat{\boldsymbol{\phi}} ;

its curl: $ i\vec k \times ( \widehat{\boldsymbol{\theta}} + i\sigma K\widehat{\boldsymbol{\phi}} ) = i K^2 \widehat{\boldsymbol{\phi}} + \sigma K \widehat{\boldsymbol{\theta}} = \sigma K ( \widehat{\boldsymbol{\theta}} + i \sigma K \widehat{\boldsymbol{\phi}} ) , because $ \vec k \times \widehat{\boldsymbol{\theta}} = \vec k \times ( \widehat{\boldsymbol{\phi}} \times \vec k ) = K^2 \widehat{\boldsymbol{\phi}} - (\widehat{\boldsymbol{\phi}} \cdot \vec k) \vec k = K^2 \widehat{\boldsymbol{\phi}} and $ \vec k \times \widehat{\boldsymbol{\phi}} = -\widehat{\boldsymbol{\theta}} ;

norm of vectors: $ | \widehat{\boldsymbol{\theta}}(\vec k) | = K \, K_\perp ,$ | \widehat{\boldsymbol{\phi}}(\vec k) | = K_\perp where$ K_\perp := |\vec k - (\vec k \cdot \hat a) \hat a|

Eigenvector of (pseudo-differential) curl operator

$ \widehat{\boldsymbol{\theta}}(\vec k) + i \sigma K \widehat{\boldsymbol{\phi}}(\vec k) for $ \hat a=(0,-1,0) ;←mathematicaで無造作に計算したら、これに相当する答えがでた

eigenvector for $ \lambda = K :

$ \left[ \begin{array}{ccc} 0 & -i k_z & i k_y \\ i k_z & 0 & - i k_x \\ -i k_y & i k_x & 0 \end{array} \right] \left[ \begin{array}{c} - k_x k_y - i K k_z \\ k_z^2 + k_x^2 \\ - k_y k_z + i K k_x \end{array} \right] = \left[ \begin{array}{c} -ik_z(k_z^2 + k_x^2)+ik_y(- k_y k_z + i K k_x) \\ ik_z(- k_x k_y - i K k_z)-ik_x(- k_y k_z + i K k_x) \\ -ik_y(- k_x k_y - i K k_z)+ik_x(k_z^2 + k_x^2) \end{array} \right] = K \left[ \begin{array}{c} - k_x k_y - i K k_z \\ k_z^2 + k_x^2 \\ - k_y k_z + i K k_x \end{array} \right]

eigenvector for $ \lambda = -K :

$ \left[ \begin{array}{ccc} 0 & -i k_z & i k_y \\ i k_z & 0 & - i k_x \\ -i k_y & i k_x & 0 \end{array} \right] \left[ \begin{array}{c} k_x k_y - i K k_z \\ - k_z^2 - k_x^2 \\ k_y k_z + i K k_x \end{array} \right] = \left[ \begin{array}{c} -ik_z(- k_z^2 - k_x^2)+ik_y(k_y k_z + i K k_x) \\ ik_z(k_x k_y - i K k_z)-ik_x(k_y k_z + i K k_x) \\ -ik_y(k_x k_y - i K k_z)+ik_x(- k_z^2 - k_x^2) \end{array} \right] = - K \left[ \begin{array}{c} k_x k_y - i K k_z \\ - k_z^2 - k_x^2 \\ k_y k_z + i K k_x \end{array} \right]

norm of these vectors

$ {\rm Re}:(k_x k_y)^2+(- k_z^2 - k_x^2)^2+(k_y k_z)^2=(k_x^2+k_z^2)(k_x^2+k_y^2+k_z^2)\ \ \ {\rm Im}:(- K k_z)^2+(K k_x)^2=K^2(k_z^2+k_x^2)

orthogonal triplet

$ (k_x,k_y,k_z),\ \ \ (- K k_z,0,K k_x),\ \ \ \pm(-k_x k_y,k_x^2+k_z^2,-k_y k_z)

eigenvector for $ \lambda = 0 :

$ \left[ \begin{array}{ccc} 0 & -i k_z & i k_y \\ i k_z & 0 & - i k_x \\ -i k_y & i k_x & 0 \end{array} \right] \left[ \begin{array}{c} k_x \\ k_y \\ k_z \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right]

(real-valued) eigenfunction of curl operator on $ \mathbb{T}^3

$ \widehat{\boldsymbol{\theta}}(\vec k) \cos\phi - \sigma K \widehat{\boldsymbol{\phi}}(\vec k) \sin\phi for $ \hat a=(0,-1,0) where $ \phi = \vec{k} \cdot \vec{x} - \omega t

eigenfunction for $ \lambda = K :

$ \left[ \begin{array}{ccc} 0 & -\partial_z & \partial_y \\ \partial_z & 0 & - \partial_x \\ -\partial_y & \partial_x & 0 \end{array} \right] \left[ \begin{array}{c} - k_x k_y \cos\phi + K k_z \sin\phi \\ (k_z^2 + k_x^2) \cos\phi \\ - k_y k_z \cos\phi - K k_x \sin\phi \end{array} \right]

$ = \left[ \begin{array}{c} -\partial_z((k_z^2 + k_x^2) \cos\phi)+\partial_y(- k_y k_z \cos\phi - K k_x \sin\phi) \\ \partial_z(- k_x k_y \cos\phi + K k_z \sin\phi)-\partial_x(- k_y k_z \cos\phi - K k_x \sin\phi) \\ -\partial_y(- k_x k_y \cos\phi + K k_z \sin\phi)+\partial_x((k_z^2 + k_x^2) \cos\phi) \end{array} \right] = \left[ \begin{array}{c} k_z K^2 \sin\phi - k_y K k_x \cos\phi \\ K (k_z^2 + k_x^2) \cos\phi \\ - k_x K^2 \sin\phi - k_y K k_z \cos\phi \end{array} \right]

eigenfunction for $ \lambda = -K :

$ \left[ \begin{array}{ccc} 0 & -\partial_z & \partial_y \\ \partial_z & 0 & - \partial_x \\ -\partial_y & \partial_x & 0 \end{array} \right] \left[ \begin{array}{c} k_x k_y \cos\phi + K k_z \sin\phi \\ - (k_z^2 + k_x^2) \cos\phi \\ k_y k_z \cos\phi - K k_x \sin\phi \end{array} \right]

$ = \left[ \begin{array}{c} -\partial_z(-(k_z^2 + k_x^2) \cos\phi)+\partial_y(k_y k_z \cos\phi - K k_x \sin\phi) \\ \partial_z(k_x k_y \cos\phi + K k_z \sin\phi)-\partial_x(k_y k_z \cos\phi - K k_x \sin\phi) \\ -\partial_y(k_x k_y \cos\phi + K k_z \sin\phi)+\partial_x(-(k_z^2 + k_x^2) \cos\phi) \end{array} \right] = \left[ \begin{array}{c} - k_z K^2 \sin\phi - k_y K k_x \cos\phi \\ K (k_z^2 + k_x^2) \cos\phi \\ k_x K^2 \sin\phi - k_y K k_z \cos\phi \end{array} \right]