Surface differential operators

The MHD eigenmode equations (154) and (161) involve two surface operators, $\mathbf{B}_0 \cdot \nabla$ and $(\mathbf{B}_0 \times \nabla \Psi / B_0^2) \cdot \nabla$ (they are called surface operators because they involve only differential on magnetic surfaces). Next, we provide the form of the two operators in flux coordinate system $(\psi, \theta, \zeta)$. Using Eq. (165), the $\mathbf{B}_0 \cdot \nabla$ operator (usually called magnetic differential operator) is written

$$\mathbf{B}_0 \cdot \nabla = - \Psi' \mathcal{J}^{- 1} \left( \frac{\partial}{\partial \theta} + q \frac{\partial}{\partial \zeta} \right) .$$ (168)

Using the covariant form of the equilibrium magnetic field [Eq. (166)], the $(\mathbf{B}_0 \times \nabla \Psi / B_0^2) \cdot \nabla$ operator is written

$$\frac{\mathbf{B}_0 \times \nabla \Psi}{B_0^2} \cdot \nabla = \lef... ...a} + \Psi' g \frac{\mathcal{J}^{- 1}}{B_0^2} \frac{\partial}{\partial \theta} .$$ (169)