Magnetic Vector Potential

The ansatz for the magnetic vector potential is (this ansatz is not relevant for the derivation of the JOREK equations; details about the Gauge missing):

\[\begin{equation*} \mathbf{A}=\nabla\alpha\times\nabla\phi+\psi\nabla\phi \end{equation*}\]

where $\partial_3\alpha\equiv 0$ is assumed. The curl of $\mathbf{A}$ is given by

\[\begin{equation*}\begin{split} \nabla\times\mathbf{A} &= \nabla\times(\nabla\alpha\times\nabla\phi)+\nabla\times(\psi\nabla\phi) \\ &= -\Delta^*\alpha\nabla\phi+\frac{1}{R^2}\nabla_\text{pol}\underbrace{\partial_3\alpha}_{\equiv 0} + \nabla\psi\times\nabla\phi + \psi\underbrace{\nabla\times\nabla\phi}_{\equiv 0} \\ &= -\Delta^*\alpha\nabla\phi+\nabla\psi\times\nabla\phi \end{split}\end{equation*}\]

As this has to be in agreement with the magnetic field expression $\mathbf{B}=F\nabla\phi+\nabla\psi\times\nabla\phi$, the following condition can be derived:

\[\begin{equation*} F\equiv-\Delta^*\alpha \end{equation*}\]