(U,B)-representation of incompressible HMHD
非圧縮HMHDの$ \big<V_{\tilde k}\big|\hat M\big|[V_{\tilde p},V_{\tilde q}]\big> は$ V=(U,-\alpha J) ,$ V=(U,V) ならばsemi-direct, direct product group上の力学系として記述できる。これを$ (U,B) で書き下したいときに必要となる数学的構造は何か？
homomorphic(?) mapping of the right hand side of generalized vorticity equations
$ \small \newcommand{\U}{\underbrace} q^* ( \U { \hat D_D \hat\iota_{V_D} \hat D_D }_{\hat D_D L_{V_D} = L_{V_D} \hat D_D}\, \U{ \hat M_D V_D }_{\underline{M}_{D}}) = $ \small \newcommand{\U}{\underbrace} \overbrace{ \U{ q^* \hat D_D q_*^{-1} }_{\hat D_S}\, \U{ q_* \hat\iota_{V_D} (q^*)^{-1} }_{\hat\iota_{V_S}}\, \U{ q^* \hat D_D q_*^{-1} }_{\hat D_S} }^{D_SL_{V_S}\underline M=\ L_{V_S}D_S\underline M}\, \overbrace{ \U{ q_* \hat M_D (q^*)^{-1} }_{\hat M_Z}\, \U{ q^* V_D }_{V_S} }^{\underline{M}_Z}
$ \newcommand{\ds}{\displaystyle} \newcommand{\bg}{\phantom{\Big|\!\!}} \footnotesize V_Z := q^*V_D = \left\{\begin{array}{cc} I & O \bg \\ \ds \frac{1}{\alpha\nabla\times} & \ds -\frac{1}{\alpha\nabla\times} \end{array}\right\} \left[\begin{array}{c} U \bg \\ V \bg \end{array}\right]_D = \left\{\begin{array}{c} U \bg \\ \ds\frac{U-V}{\alpha\nabla\times} \end{array}\right\} = \left\{\begin{array}{c} U \\ B \end{array}\right\} …これは$ q^* を決める式
$ \footnotesize \underline M_Z := q_* M_D= \left\{\begin{array}{cc} I & I \\ 0 & -\alpha\nabla\times \end{array}\right\} \left[\begin{array}{r} U+A/\alpha \\ -A/\alpha \end{array}\right]_D = \left\{\begin{array}{c} U \\ B \end{array}\right\} …これは$ q_* を決める式
$ \bm qの逆行列の関係
$ \footnotesize {}^tq_* q^* = \left\{\begin{array}{cc} \phantom{\Big|\!\!} I & O \\ I & -\alpha\nabla\times \phantom{\Big|\!\!} \end{array}\right\} \left\{\begin{array}{cc} I & O \phantom{\Big|\!\!} \\ \displaystyle \frac{1}{\alpha\nabla\times} & \displaystyle -\frac{1}{\alpha\nabla\times} \end{array}\right\} = \left[\begin{array}{cc} I & O \\ O & I \end{array}\right] therfore$ {}^tq_*=(q^*)^{-1} ,
$ \footnotesize {}^t({}^tq_*q^*) = {}^tq^* q_* = \left[\begin{array}{cc} I & O \\ O & I \end{array}\right] therefore$ (q_*)^{-1} = {}^tq^*
$ \footnotesize {}^tV_Z \underline M_Z = {}^t(q^* V_D)(q_* \underline M_D) = {}^tV_D {}^tq^* q_* \underline M_D = {}^tV_D \underline M_D
$ \footnotesize q_*M_D = q_* \hat M_D V_D = q_* \hat M_D {}^tq_* q^* V_D
目標地点
$ \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
①＋②…$ \partial_t\bm U = \bm U \times \bm \Omega + \bm J \times \bm B - \nabla (\Pi-\phi/\alpha)
$ -\alpha\nabla\times ②…$ \partial_t \bm B = \nabla\times( \bm U - \alpha\bm J ) \times \bm B
運動量演算子$ \footnotesize q_* \hat M_D (q^*)^{-1} = \left\{\begin{array}{cc} I & I \\ 0 & -\alpha\nabla\times \end{array}\right\} \left[\begin{array}{rr} I + \frac{1}{(\alpha\nabla\times)^2} & - \frac{1}{(\alpha\nabla\times)^2} \\ - \frac{1}{(\alpha\nabla\times)^2} & \frac{1}{(\alpha\nabla\times)^2} \end{array}\right] \left\{\begin{array}{cc} I & O \\ I & -\alpha\nabla\times \end{array}\right\}
$ \footnotesize = \left\{\begin{array}{cc} I & O \\ \frac{1}{\alpha\nabla\times} & -\frac{1}{\alpha\nabla\times} \end{array}\right] \left\{\begin{array}{cc} I & O \\ I & -\alpha\nabla\times \end{array}\right\} = \left\{\begin{array}{cc} I & O \\ O & I \end{array}\right\} ,
渦度$ \footnotesize W_Z := q^* W_D = q^* \hat D_D (q_*)^{-1} q_* M_D = q^* \hat D_D {}^tq^* q_* \hat M_D (q^*)^{-1} q^* V_D
外微分$ \footnotesize q^* \hat D_D (q_*)^{-1}= \left\{\begin{array}{cc} I & O \\ \frac{1}{\alpha\nabla\times} & -\frac{1}{\alpha\nabla\times} \end{array}\right\} \left[\begin{array}{cc} C_i\alpha\nabla\times & O \\ O & C_e\alpha\nabla\times \end{array}\right] \left\{\begin{array}{cc} I & \frac{1}{\alpha\nabla\times} \\ O & -\frac{1}{\alpha\nabla\times} \end{array}\right\} = \left[\begin{array}{cc} C_i\alpha\nabla\times & O \\ C_i & -C_e \end{array}\right] \left\{\begin{array}{cc} I & \frac{1}{\alpha\nabla\times} \\ O & -\frac{1}{\alpha\nabla\times} \end{array}\right\}
$ \footnotesize = \left[\begin{array}{cc} C_i\alpha\nabla\times & C_i \\ C_i & \frac{C_i+C_e}{\alpha\nabla\times} \end{array}\right]
渦度$ \footnotesize W_Z = \hat D_Z \hat M_Z V_Z = \left[\begin{array}{cc} C_i\alpha\nabla\times & C_i \\ C_i & \frac{C_i+C_e}{\alpha\nabla\times} \end{array}\right] \left[\begin{array}{cc} I & O \\ O & I \end{array}\right] \left[\begin{array}{c} U \\ B \end{array}\right] = \left[\begin{array}{cc} C_i(\alpha\Omega + B) \\ C_i U + \frac{C_i+C_e}{\alpha} A \end{array}\right]
ヘリシティ密度$ \footnotesize H = C_i(\alpha\Omega + B) U + (C_i U + \frac{C_i+C_e}{\alpha} A) B = C_i(\alpha\Omega + B)( U + \frac{A}{\alpha}) + \frac{C_e}{\alpha} AB
内部積$ \footnotesize \hat\iota_{V_Z} = q_*(-\widehat\iota_{V_D}) {}^tq_*= \left\{\begin{array}{cc} I & I \\ 0 & -\alpha\nabla\times \end{array}\right\} \left[\begin{array}{cc} \frac{1}{C_i\alpha}U\times & O \\ O & \frac{1}{C_e\alpha}V\times \end{array}\right] \left\{\begin{array}{cc} I & O \\ I & -\alpha\nabla\times \end{array}\right\} = \left[\begin{array}{cc} \frac{1}{C_i\alpha}U\times & \frac{1}{C_e\alpha}V\times \\ O & -\frac{1}{C_e}\nabla\times V\times \end{array}\right] \left\{\begin{array}{cc} I & O \\ I & -\alpha\nabla\times \end{array}\right\}
$ \footnotesize = \left[\begin{array}{cc} \frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*) & -\frac{1}{C_e}V\times(\nabla\times*) \\ -\frac{1}{C_e}\nabla\times(V\times*) & \frac{\alpha}{C_e} \nabla\times( V\times(\nabla\times*) ) \end{array}\right] ,
$ \bm{\partial_t\underline M_Z = \hat\iota_{V_Z} \hat D_Z \underline M_Z} の右辺の作用素$ \footnotesize \hat\iota_{V_Z} \hat D_Z = \left[\begin{array}{cc} \frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*) & -\frac{1}{C_e}V\times(\nabla\times*) \\ -\frac{1}{C_e}\nabla\times(V\times*) & \frac{\alpha}{C_e}\nabla\times(V\times(\nabla\times*)) \end{array}\right] \left[\begin{array}{cc} C_i\alpha\nabla\times & C_i \\ C_i & \frac{C_i+C_e}{\alpha\nabla\times} \end{array}\right] = \left[\begin{array}{cc} U\times(\nabla\times*) & J \times * \\ O & \nabla\times((U - \alpha J)\times*) \end{array}\right]
$ \tiny (\frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*)) C_i\alpha(\nabla\times*) + (-\frac{1}{C_e}V\times(\nabla\times*)) C_i = U\times(\nabla\times*)
$ \tiny (\frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*)) C_i + (-\frac{1}{C_e}V\times(\nabla\times*)) \frac{C_i+C_e}{\alpha\nabla\times} = \frac{1}{\alpha}(U\times*) - \frac{1}{\alpha}(V\times*) = J \times *
$ \tiny (-\frac{1}{C_e}\nabla\times(V\times*)) (C_i\alpha\nabla\times) + (\frac{\alpha}{C_e}\nabla\times(V\times(\nabla\times*))) C_i = O
$ \tiny (-\frac{1}{C_e}\nabla\times(V\times*))C_i + (\frac{\alpha}{C_e}\nabla\times( V\times(\nabla\times*)))\frac{C_i+C_e}{\alpha\nabla\times} = \nabla\times(V\times*) = \nabla\times((U - \alpha J)\times*)
変分原理を巻き戻しLie括弧の表式を掘り起こす
$ \footnotesize 0 = \delta S = \int dV\Big[ \bm u \odot (\dot{\bm\xi} + [\bm\xi,\bm U]) \Big] = \int dV\Big[ -\dot{\bm u} \cdot \bm\xi + \underbrace{ \bm u \cdot [\bm\xi,\bm U]} \Big]
$ \footnotesize \int dV\Big[\xi \cdot [U\times(\nabla\times u) + J \times b] + \eta \cdot [ \nabla\times((U - \alpha J)\times b)] \Big]
$ \footnotesize = \int dV\Big[ u \cdot (\nabla\times(\xi\times U)) + b\cdot ( \xi \times J ) + b \cdot ((\nabla\times\eta) \times (U - \alpha J))\Big]
$ \footnotesize = \int dV\Big[ u \cdot (\nabla\times(\xi\times U)) + b \cdot \big[ \xi \times (\nabla\times B) + (\nabla\times\eta) \times U - \alpha (\nabla\times\eta) \times (\nabla\times B)\big] \Big]
$ \footnotesize \big[ (\xi,\eta) , (U,B) \big] = \big( \nabla\times(\xi\times U) , \overbrace{ \xi \times (\nabla\times B) + (\nabla\times\eta) \times U - \alpha (\nabla\times\eta) \times (\nabla\times B) }^{not\ solenoidal} \!\!\!\!\!\!\!\! \underbrace{ \ \ - \nabla \Pi \ \ }_{should\ be\ necessarry} \!\!\!\!\!\!\!\! \big)
$ \upsilon \nu
$ \footnotesize 0 = \delta S = \int dV\Big[ \bm u \odot (\dot{\bm\nu} + [\bm\nu,\bm U]) \Big] = \int dV\Big[ -\dot{\bm u} \cdot \bm\nu + \underbrace{ \bm u \cdot [\bm\nu,\bm U]} \Big]
$ \footnotesize \int dV\Big[\nu \cdot [U\times(\nabla\times u) + J \times b] + \beta \cdot [ \nabla\times((U - \alpha J)\times b)] \Big]
$ \footnotesize = \int dV\Big[ u \cdot (\nabla\times(\nu\times U)) + b\cdot ( \nu \times J ) + b \cdot ((\nabla\times\beta) \times (U - \alpha J))\Big]
$ \footnotesize = \int dV\Big[ u \cdot (\nabla\times(\nu\times U)) + b \cdot \big[ \nu \times (\nabla\times B) + (\nabla\times\beta) \times U - \alpha (\nabla\times\beta) \times (\nabla\times B)\big] \Big]
$ \footnotesize \big[ (\nu,\beta) , (U,B) \big] = \big( \nabla\times(\nu\times U) , \overbrace{ \nu \times (\nabla\times B) + (\nabla\times\beta) \times U - \alpha (\nabla\times\beta) \times (\nabla\times B) }^{not\ solenoidal} \!\!\!\!\!\!\!\! \underbrace{ \ \ - \nabla \Pi \ \ }_{should\ be\ necessarry} \!\!\!\!\!\!\!\! \big)
$ \footnotesize \big[ (\xi,\eta) , (U,B) \big] の第2成分の式変形
$ \footnotesize \underline{\xi \times (\nabla\times B)} + \underbrace{(\nabla\times\eta) \times U - \alpha (\nabla\times\eta) \times (\nabla\times B)}
$ \footnotesize = \frac1\alpha\xi \times U - \underline{\frac1\alpha \xi\times} (U - \underline{\alpha\nabla\times B}) + \underbrace{(\nabla\times\eta) \times V} = \boxed{ \frac1\alpha \xi \times U - \frac1\alpha \zeta \times V } ,where$ \footnotesize \zeta = \xi - \alpha \nabla\times \eta ,$ \footnotesize V = U - \alpha \nabla\times B
$ \footnotesize \Big[ \big[ (\xi,\eta) , (U,B) \big] , (u,b) \Big] の第2成分
$ \footnotesize \frac1\alpha \big\{\nabla\times(\xi\times U)\big\} \times u - \frac1\alpha \Big\{[\nabla\times(\xi\times U)]-\alpha\nabla\times\big[\frac1\alpha \xi \times U - \frac1\alpha \zeta \times V\big]\Big\} \times v
$ \footnotesize = \frac1\alpha \big\{\nabla\times(\xi\times U)\big\} \times u - \frac1\alpha \big\{\nabla\times(\zeta \times V) \big\} \times v
渦度方程式からLie括弧の表式を検算する
$ \footnotesize \dot\Omega + [\Omega,U] = 0 \Longleftrightarrow \dot\Omega = [U,\Omega]
$ \footnotesize [U,*] = \hat D_Z \hat\iota_{V_Z} = \left[\begin{array}{cc} C_i\alpha\nabla\times & C_i \\ C_i & \frac{C_i+C_e}{\alpha\nabla\times} \end{array}\right] \left[\begin{array}{cc} \frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*) & -\frac{1}{C_e}V\times(\nabla\times*) \\ -\frac{1}{C_e}\nabla\times(V\times*) & \frac{\alpha}{C_e}\nabla\times(V\times(\nabla\times*)) \end{array}\right]
$ \footnotesize = \left[\begin{array}{cc} \nabla\times(U\times*) & O \\ J\times* & (U - \alpha J ) \times (\nabla\times*) \end{array}\right]
$ \tiny C_i\alpha\nabla\times(\frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*)) + C_i(-\frac{1}{C_e}\nabla\times(V\times*)) = \nabla\times(U\times*)
$ \tiny C_i\alpha\nabla\times(-\frac{1}{C_e}V\times(\nabla\times*)) + C_i(\frac{\alpha}{C_e}\nabla\times(V\times(\nabla\times*))) = O
$ \tiny C_i(\frac{1}{C_i\alpha}(U\times*) + \frac{1}{C_e\alpha}(V\times*)) + \frac{C_i+C_e}{\alpha\nabla\times}(-\frac{1}{C_e}\nabla\times(V\times*)) = \frac{1}{\alpha}(U\times*) - \frac{1}{\alpha}(V\times*) = J \times *
$ \tiny C_i(-\frac{1}{C_e}V\times(\nabla\times*)) + \frac{C_i+C_e}{\alpha\nabla\times}(\frac{\alpha}{C_e}\nabla\times(V\times(\nabla\times*))) = V\times(\nabla\times*) = (U - \alpha J)\times(\nabla\times*)
渦度作用素の固有値問題$ \footnotesize \hat W_Z = \left[\begin{array}{cc} C_i\alpha\sigma k & C_i \\ C_i & \displaystyle \frac{C_i+C_e}{\alpha\sigma k} \end{array}\right] ；$ \footnotesize \Lambda + \underline\Lambda = \displaystyle C_i \alpha\sigma k + \frac{C_i+C_e}{\alpha\sigma k} ；$ \footnotesize \Lambda \underline\Lambda = C_i C_e ；
$ \footnotesize (\Lambda - \underline\Lambda) = (\Lambda + \underline\Lambda)^2 -4 \Lambda \underline\Lambda = \displaystyle ( C_i \alpha\sigma k + \frac{C_i+C_e}{\alpha\sigma k} )^2 - 4 C_i C_e = C_i^2 \left[ ( \alpha\sigma k + \frac{1+C}{\alpha\sigma k} )^2 - 4 C - 4 + 4 \right] = C_i^2 \left[ ( \alpha\sigma k - \frac{1+C}{\alpha\sigma k} )^2 + 4 \right]
$ \footnotesize \therefore \Lambda_\sigma^s(k) = \frac{C_i}{2}(\alpha\sigma k + \frac{1+C}{\alpha\sigma k}) + s \frac{C_i}{2}\sqrt{( \alpha\sigma k - \frac{1+C}{\alpha\sigma k} )^2 + 4} = C_i \bigg[ \alpha\sigma k - \sigma (\frac{\alpha k}{2} - \frac{1+C}{2\alpha k}) + s \sqrt{( \frac{\alpha k}{2} - \frac{1+C}{2\alpha k} )^2 + 1}\ \bigg]
$ \footnotesize = C_i \Bigg\{ \sigma \alpha k + s \bigg[ \sqrt{( \frac{\alpha k}{2} - \frac{1+C}{2\alpha k} )^2 + 1} - s \sigma (\frac{\alpha k}{2} - \frac{1+C}{2\alpha k}) \bigg] \Bigg\}
$ \footnotesize = C_i ( \sigma \alpha k + s \gamma^{- \sigma s} )
where$ \footnotesize \gamma = \gamma(k) = \gamma(k;\alpha,C) := \sqrt{( \frac{\alpha k}{2} - \frac{1+C}{2\alpha k} )^2 + 1} + (\frac{\alpha k}{2} - \frac{1+C}{2\alpha k})
$ \footnotesize \Lambda = C_i \sigma \alpha k + C_i s \gamma^{- \sigma s} \Longleftrightarrow C_i \sigma \alpha k -\Lambda = - C_i s \gamma^{- \sigma s} ,$ \footnotesize \underline\Lambda = C_i \sigma \alpha k - C_i s \gamma^{\sigma s} \Longleftrightarrow C_i \sigma \alpha k - \underline\Lambda = C_i s \gamma^{\sigma s}
Since$ \footnotesize \hat W_Z - \Lambda I = \left[\begin{array}{cc} C_i\alpha\sigma k - \Lambda & C_i \\ C_i & \underline\Lambda - C_i \alpha\sigma k \end{array}\right] = \left[\begin{array}{cc} -C_is\gamma^{-\sigma s} & C_i \\ C_i & - C_i s \gamma^{\sigma s} \end{array}\right] , $ \footnotesize V_Z \propto \left[\begin{array}{r} s\gamma^{\sigma s} \bm\phi_\sigma(\vec k) \\ \bm\phi_\sigma(\vec k) \end{array}\right]
$ \footnotesize B = \bm\phi_\sigma(\vec k) ,$ \footnotesize J = \sigma k \bm\phi_\sigma(\vec k) ,$ \footnotesize U = s\gamma^{\sigma s} \bm\phi_\sigma(\vec k) ,$ \footnotesize V = U - \alpha J = (s \gamma^{\sigma s} - \alpha \sigma k) \bm\phi_\sigma(\vec k) = -\frac{\underline\Lambda}{C_i} \bm\phi_\sigma(\vec k)
$ \footnotesize \alpha\nabla\times U + B = \alpha\sigma k\ s\gamma^{\sigma s} + 1 = s\gamma^{\sigma s} (\alpha\sigma k + s \gamma^{-\sigma s} ) = s\gamma^{\sigma s} \frac{\Lambda}{C_i}
If $ 1+C=0 ,$ \footnotesize \Lambda_\sigma^s(k) = C_i \left( \frac{\alpha\sigma k}{2} + s \sqrt{( \frac{\alpha k}{2} )^2 + 1}\ \right) = C_i s \left( \sqrt{( \frac{\alpha k}{2} )^2 + 1} + \sigma s \frac{\alpha k}{2} \right) = C_i s \gamma_0^{\sigma s} where$ \footnotesize \gamma_0 = \sqrt{( \frac{\alpha k}{2} )^2 + 1} + \frac{\alpha k}{2}
固有値、固有ベクトルも(U,V), (U,J)と無矛盾な結果が出ているので、定式化としては大丈夫だろう。
(U,B)記述の利点は一般化速度、一般化運動量の変数が同じになることである。