Derivation of the Current Expression

The derivation of the expression $j=\Delta^*\psi$ is sketched in the following. Starting point is the definition of the magnetic field

\[\begin{equation} \mathbf{B} = F\nabla\phi+\nabla\psi\times\nabla\phi \end{equation}\]

The plasma current is given by

\[\begin{equation} \mathbf{j} = \nabla\times\mathbf{B}. \end{equation}\]

So,

\[\begin{equation}\begin{split} \mathbf{j} &= \nabla\times(F\nabla\phi)+\nabla\times(\nabla\psi\times\nabla\phi) \\ &= \nabla F\times\nabla\phi + F\underbrace{\nabla\times\nabla\phi}_{\equiv 0} + \nabla\times\left[\left(\partial_1\psi~\mathbf{a}^1+\partial_2\psi~\mathbf{a}^2+\partial_3\psi~\mathbf{a}^3 \right)\times\mathbf{a}^3 \right] \\ &= \nabla F\times\nabla\phi + \nabla\times\left[-\frac{1}{R}\partial_1\psi~\mathbf{a}^2+\frac{1}{R}\partial_2\psi~\mathbf{a}^1 \right] \\ &= \nabla F\times\nabla\phi + \frac{1}{R^2}\partial_{1,3}\psi~\mathbf{a}_1+\frac{1}{R^2}\partial_{2,3}\psi~\mathbf{a}_2-\frac{1}{R}\left[\partial_1\left(\frac{1}{R}\partial_1\Psi\right)+\frac{1}{R}\partial_{2,2}\psi\right]\mathbf{a}_3 \\ &= \nabla F\times\nabla\phi + \frac{1}{R^2}\partial_{1,3}\psi~\mathbf{a}_1+\frac{1}{R^2}\partial_{2,3}\psi~\mathbf{a}_2-\frac{1}{R^2}\Delta^*\psi~\mathbf{a}_3 \\ &= \nabla F\times\nabla\phi + \frac{1}{R^2}\partial_3(\nabla_\text{pol}\psi)-\Delta^*\psi~\nabla\phi \\ &= \left(\frac{1}{R}\partial_2 F+\frac{1}{R^2}\partial_{1,3}\psi\right)\mathbf{a}^1+\left(-\frac{1}{R}\partial_1 F+\frac{1}{R^2}\partial_{2,3}\psi\right)\mathbf{a}^2-\Delta^*\psi~\mathbf{a}^3 \end{split}\end{equation}\]

The physical toroidal component of the current is given by

\[\begin{equation} j_\phi = \mathbf{j}\cdot\mathbf{e}_\phi = -\partial_1\left(\frac{1}{R}\partial_1\psi\right)-\frac{1}{R}\partial_{2,2}\psi = -\frac{1}{R}\Delta^*\psi \end{equation}\]

and the JOREK variable $j$ is given by

\[\begin{equation} j = -R~j_\phi = \Delta^*\psi \end{equation}\]