Magnetic field expression in flux coordinate system

Any axisymetrical magnetic field consistent with the equilibrium equation $\nabla p_0 =\mathbf{J}_0 \times \mathbf{B}_0$ can be written in the form

$$\mathbf{B}_0 = \triangledown \Psi \times \triangledown \phi + g (\Psi) \triangledown \phi,$$

where $\Psi \equiv A_{\phi} R$. In the straight-line magnetic surface coordinates system $(\psi, \theta, \zeta)$, the contra-variant form of the equilibrium magnetic field is expressed as

$$\mathbf{B}_0 = - \Psi' [\nabla \zeta \times \nabla \psi + q (\psi) \nabla \psi \times \nabla \theta],$$ (165)

where $\Psi' = d \Psi / d \psi$. The covariant form of the equilibrium magnetic field is given by

$$\mathbf{B}_0 = \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \... ...- \frac{B_0^2}{\Psi'} \mathcal{J}- g q \right) \nabla \theta + g \nabla \zeta .$$ (166)

where $\mathcal {J}$ is the Jacobian of $(\psi, \theta, \zeta)$ coordinate system.