Summary of component equations

For the ease of reference, Eqs. (29), (82), (87), (99), (105), (), and (119) are repeated here:

$$p_1 + p_0' \xi_{\psi} = - \gamma p_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (120)

$$Q_{\psi} =\mathbf{B}_0 \cdot \nabla \xi_{\psi},$$ (121)

$$Q_s = \frac{\vert \nabla \Psi \vert^2}{B^2_0} (\mathbf{B}_0 \cdot \nabla \xi_s - S \xi_{\psi}),$$ (122)

$$Q_b = \left( - \frac{2 B^2_0}{\vert \nabla \Psi \vert^2} \kappa_{... ..._0^2 \xi_s + B_0^2 \mathbf{B}_0 \cdot \nabla \left( \frac{\xi_b}{B^2_0} \right)$$ (123)

$$- \omega^2 \rho_0 \xi_{\psi} = - \nabla \Psi \cdot \nabla P_1 +{\textmu}_0^{- 1} \vert \nabla \P... ...hbf{B}_0 \cdot \nabla \left( \frac{Q_{\psi}}{\vert \nabla \Psi \vert^2} \right)$$

$$+ ({\textmu}_0^{- 1} \vert \nabla \Psi \vert^2 S - B_0^2 \sigma) Q_s + 2{\textmu}_0^{- 1} \kappa_{\psi} Q_b .$$ (124)

$$- \omega^2 \rho_0 \vert \nabla \Psi \vert^2 \xi_s = - (\mathbf{B}... ...1} B^2_0 \mathbf{B}_0 \cdot \nabla Q_s + 2{\textmu}_0^{- 1} \kappa_s B_0^2 Q_b,$$ (125)

$$\omega^2 \rho_0 \xi_b =\mathbf{B}_0 \cdot \nabla (p_1 + p'_0 \xi_{\psi}),$$ (126)

where $p_0' \equiv d p_0 / d \Psi$, $\sigma \equiv \mathbf{B}_0 \cdot \mathbf{J}_0 / B_0^2$, $\kappa_s \equiv \ensuremath{\boldsymbol{\kappa}} \cdot (\mathbf{B}_0 \times \nabla \Psi) / B_0^2$, which is usually called the geodesic curvature, $\kappa_{\psi} \equiv \ensuremath{\boldsymbol{\kappa}} \cdot \nabla \Psi$, which is usually called the normal curvature.