Toroidal elliptic operator in magnetic surface coordinate system

In the magnetic surface coordinate system, by using $\partial \Psi / \partial \theta = 0$, the toroidal elliptic operator in Eq. (415) is reduced to

$$\triangle^{\star} \Psi = \frac{R^2}{\mathcal{J}} \left[ \left( \P... ...rac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \theta \right)_{\theta} \right],$$ (467)

and the GS equation, Eq. (421), is reduced to

$$\frac{R^2}{\mathcal{J}} \left[ \left( \Psi_{\psi} \frac{\mathcal{... ...\theta} \right] = - \mu_0 R^2 \frac{d P}{d \Psi} - \frac{d g}{d \Psi} g (\Psi),$$ (468)

Equation (468) is the GS equation in magnetic surface coordinate system.