重写推三场方程如下
\begin{equation}
\label{eq:3field}
\begin{aligned}
\rho \dv{U}{t}
&=
B^2 \bm{b} \cdot \grad \frac{J_\parallel}{B}
+ 2 \bm{b} \cp \bm{\kappa} \cdot \grad P \\
\pdv{\psi}{t} &= – \frac{1}{B_0}\nabla_\parallel \phi \\
\pdv{t} P &= – \frac{\bm{b_0}}{B_0} \cp \grad \phi \cdot \grad P \\
U &= \frac{1}{B} \nabla_{\perp}^2 \phi \\
J_\parallel &= J_{\parallel 0} – \frac{1}{\mu_0} B_0 \nabla^2_\perp \psi \\
\end{aligned}
\end{equation}
涡量方程线性化如下：
\begin{equation*}
\begin{aligned}
\rho_0 \qty(\pdv{U_1}{t} + \frac{\bm{b_0}}{B_0} \cp \grad \phi_1 \cdot \grad U_1)
&=
B^2 \qty( \bm{b_0} \cdot \grad \frac{J_{\parallel 1}}{B_0} –
\bm{b_0} \cp \grad \psi_1 \cdot \grad \frac{J_{\parallel 0} + J_{\parallel 1}}{B_0})
+ 2 \bm{b_0} \cp \bm{\kappa_0} \cdot \grad P \\
\end{aligned}
\end{equation*}
势能方程线性化如下：
\begin{equation*}
\begin{aligned}
\pdv{\psi_1}{t} =& – \frac{1}{B_0}
\bm{b_0} \cdot \grad \phi_1
– \bm{b_0} \cdot \grad \psi_1 \cp \grad \phi_0
– \bm{b_0} \cdot \grad \psi_0 \cp \grad \phi_1 \\
\pdv{\psi_1}{t} =& – \frac{1}{B_0}
\bm{b_0} \cdot \grad \phi_1 \\
\end{aligned}
\end{equation*}
整理其他几个方程，可得到线性化之后的方程组如下：
\begin{equation}
\label{eq:3field_1}
\begin{aligned}
\rho_0 \pdv{U_1}{t}
=& – \frac{\rho_0}{B_0} \bm{b_0} \cp \grad \phi_1 \cdot \grad U_1 \\
&+ B^2_0 \qty(\bm{b_0} \cdot \grad \frac{J_{\parallel 1}}{B_0}
– \bm{b_0} \cp \grad \psi_1 \cdot \grad \frac{J_{\parallel 0} + J_{\parallel 1}}{B_0}) \\
&+ 2 \bm{b_0} \cp \bm{\kappa_0} \cdot \grad P_1 \\
\pdv{\psi_1}{t} =& – \frac{1}{B_0}
\bm{b_0} \cdot \grad \phi_1 \\
\pdv{t} P_1 =& – \frac{\bm{b_0}}{B_0} \cp \grad \phi_1 \cdot \grad\qty(P_0 + P_1) \\
U_1 =& \frac{1}{B_0} \nabla_{\perp}^2 \phi_1 \\
J_{\parallel 1} =& – \frac{1}{\mu_0} B_0 \nabla^2_\perp \psi_1 \\
\end{aligned}
\end{equation}
去除\eqref{eq:3field_1}中所有非线性项之后方程化成如下形式：
\begin{equation}
\label{eq:3field_2}
\begin{aligned}
\rho_0 \pdv{U_1}{t}
=&
B^2_0 \qty(\bm{b_0} \cdot \grad \frac{J_{\parallel 1}}{B_0}
– \bm{b_0} \cp \grad \psi_1 \cdot \grad \frac{J_{\parallel 0}}{B_0})
+ 2 \bm{b_0} \cp \bm{\kappa_0} \cdot \grad P_1 \\
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:3field_3}
\begin{aligned}
\pdv{\psi_1}{t} =& – \frac{1}{B_0}
\bm{b_0} \cdot \grad \phi_1 \\
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:3field_4}
\begin{aligned}
\pdv{t} P_1 =& – \frac{\bm{b_0}}{B_0} \cp \grad \phi_1 \cdot \grad P_0 \\
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:3field_5}
\begin{aligned}
U_1 =& \frac{1}{B_0} \nabla_{\perp}^2 \phi_1 \\
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:3field_6}
\begin{aligned}
J_{\parallel 1} =& – \frac{1}{\mu_0} B_0 \nabla^2_\perp \psi_1 \\
\end{aligned}
\end{equation}
共有5个方程，为$U_1,\,\psi_1,\,P_1,\,\phi_1,\,J_{\parallel 1}$五个变量的齐次方程组，为了方程有非平凡解，系数矩阵$D = 0$。用$-iw$代替$\pdv{t}$，$i\bm{k}$代替$\grad$，从\eqref{eq:3field_2}式可得：
\begin{equation}
\label{eq:3field_7}
\begin{aligned}
-i \omega \rho_0 U_1
=&
B^2_0 \qty(\bm{b_0} \cdot i \bm{k} \frac{J_{\parallel 1}}{B_0}
– \bm{b_0} \cp i \bm{k} \psi_1 \cdot \grad \frac{J_{\parallel 0}}{B_0})
+ 2 \bm{b_0} \cp \bm{\kappa_0} \cdot i \bm{k} P_1 \\
\end{aligned}
\end{equation}
同理可以得到其他四个式子，可以用$\phi$表示的其他变量：
\begin{equation}
\label{eq:3field_8}
\begin{aligned}
\psi_1 =& \frac{k_\parallel}{\omega B_0}\phi_1 \\
P_1 =& \frac{1}{\omega B_0} \qty(\bm{b_0} \cp \bm{k})\phi_1 \cdot \grad P_0 \\
U_1 =& – \frac{k^2_\perp}{B_0}\phi_1 \\
J_{\parallel 1} =& \frac{k_\parallel k^2_\perp}{\mu_0 \omega} \phi_1
\end{aligned}
\end{equation}
将\eqref{eq:3field_8}代入\eqref{eq:3field_7}有：
\begin{equation*}
\begin{aligned}
-i\omega \rho_0\qty(-\frac{k^2_\perp}{B_0})\phi_1
=& iB^2_0 k_\parallel \frac{1}{B_0}
\frac{k_\parallel k^2_\perp}{\mu_0 \omega} \phi_1 \\
&- \frac{B^2_0 i k_\parallel}{\omega B_0}
\qty[\bm{b_0} \cp \bm{k} \cdot
\grad{\frac{J_{\parallel 0}}{B_0}}]\phi_1 \\
&+ \qty[\frac{2}{\omega B_0}\qty(\bm{b_0} \cp \bm{k})]
\cdot \grad P_0 i \qty(\bm{b_0} \cp \bm{\kappa_0}) \cdot \bm{k} \phi_1
\end{aligned}
\end{equation*}
所有项$\cdot \frac{\omega B_0}{\rho_0}$，并且有$v_A^2 = \frac{\omega_A^2}{k^2} = \frac{B_0^2}{\mu_0 \rho_0}$可得：
\begin{equation*}
\begin{aligned}
\omega^2 k_\perp^2 \phi_1 =
&\frac{\omega_A^2 k_\parallel^2 k_\perp^2}{k^2} \phi_1
– \frac{B_0^2}{\rho_0} k_\parallel
\qty[\bm{b_0} \cp \bm{k} \cdot \grad
\frac{J_{\parallel 0}}{B_0}] \phi_1
+ \frac{2}{\rho_0}
\qty[\qty(\bm{b_0} \cp \bm{\kappa_0}) \cdot \bm{k}]
\qty[\qty(\bm{b_0} \cp \bm{k}) \cdot \grad P_0] \phi_1
\end{aligned}
\end{equation*}
所有项$\cdot \frac{\mu_0 \rho_0}{B^2_0 k^2 k_\perp^2}$，并且有$\phi_1$的系数相等：
\begin{equation*}
\begin{aligned}
\frac{\omega^2}{\omega^2_A}
= \frac{k_\parallel^2}{k^2}
– \frac{\mu_0 k_\parallel}{k_\perp^2 k^2}
\qty[\bm{b_0} \cp \bm{k} \cdot
\grad{\frac{J_{\parallel 0}}{B_0}}]
– \frac{2 \mu_0}{B_0^2 k_\perp^2 k^2}
\qty[\qty(\bm{b_0} \cp \bm{k}) \cdot \bm{\kappa_0}]
\qty[\qty(\bm{b_0} \cp \bm{k}) \cdot \grad P_0]
\end{aligned}
\end{equation*}
右边第二项为Peeling/Kink项，第三项为Balloning项。
