Helicity Conservation in JOREK

Magnetic helicity is defined as:

\[\begin{equation*} H=\mathbf{A}\cdot\mathbf{B}. \end{equation*}\]

It can be shown that in ideal MHD, the volumic integral of $H$ is a conserved quantity if the components of $\mathbf{B}$ and $\mathbf{v}$ normal to the boundary vanish (see for example, Section 2.2 of the book Nonlinear Magnetohydrodynamics by D. Biskamp).

In resistive MHD, $H$ is not exactly conserved but it is dissipated at a rate proportional to the plasma resistance (see Section 7.3.1 of Biskamp). During a fast MHD relaxation (like the thermal quench phase of a disruption, for example), it is expected that $H$ will be dissipated much more slowly than magnetic energy, which sets constraints to the global dynamics (see for example this 2019 PPCF paper by A. Boozer).

In the reduced MHD version of JOREK, an acceptable expression for the vector potential is:

\[\begin{equation*} \mathbf{A} = \psi \nabla \phi - F_0 \frac{Z}{R} \nabla R. \end{equation*}\]

Indeed,

\[\begin{equation*} \nabla \times \mathbf{A} = \nabla\psi\times \nabla\phi + F_0 \nabla\phi \end{equation*}\]

which corresponds to the expression for $\mathbf{B}$ (see JOREK Reduced MHD Models), and

\[\begin{equation*} \partial_t \mathbf{A} = \partial_t \psi \nabla\phi \end{equation*}\]

which corresponds to the inductive part of the electric field.

The corresponding expression for the helicity is:

\[\begin{equation*} \begin{split} H &= \mathbf{A} \cdot \mathbf{B} \\ &= (\psi \nabla \phi - F_0 \frac{Z}{R} \nabla R) \cdot (\nabla\psi\times \nabla\phi + F_0 \nabla\phi) \\ &= F_0 \frac{\psi}{R^2} - F_0 \frac{Z}{R^2} \partial_Z \psi \\ &= F_0 \frac{\psi}{R^2} - \frac{F_0}{R^2} [\partial_Z (Z \psi) - \psi] \\ &= 2 F_0 \frac{\psi}{R^2} - \frac{F_0}{R^2} \partial_Z (Z \psi). \end{split} \end{equation*}\]

Let us now consider $\frac{d}{dt}\int H dV$, where the integral is a volumic integral over the computation domain (the latter being fixed in time):

\[\begin{equation*} \begin{split} \frac{d}{dt}\int H dV &= \int \partial_t H dV \\ &= \int [2 F_0 \frac{\partial_t \psi}{R^2} - \frac{F_0}{R^2} \partial_Z (Z \partial_t \psi)] dV \\ &= \int [2 F_0 \frac{\partial_t \psi}{R} - \frac{F_0}{R} \partial_Z (Z \partial_t \psi)] d\phi dS \\ \end{split} \end{equation*}\]

where $dS$ is an infinitesimal surface element in the poloidal plane.

The second term can be readily integrated along $Z$ to give boundary terms which vanish if we have $\partial_t \psi = 0$ at the boundary, which is the case for a fixed boundary simulation.

To calculate the first term, we make use of the poloidal flux equation:

\[\begin{equation*} \partial_t\psi = R[\psi,u] - F_0 \partial_\phi u + \eta j - R^2 \nabla \cdot \left( \frac{\eta_{num}}{R} \nabla_{\text{pol}} j \right) \end{equation*}\]

The term in $\partial_\phi u$ will vanish when integrated over $\phi$. As for the Poisson bracket:

\[\begin{equation*} \begin{split} [\psi,u] &= \partial_R \psi \partial_Z u - \partial_Z \psi \partial_R u \\ &= \partial_Z (u \partial_R \psi) - \partial_R (u \partial_Z \psi) \end{split} \end{equation*}\]

and both of these terms will vanish by integration over $R$ or $Z$ if we have $u=0$ at the boundary, which is the case for a fixed boundary simulation, i.e., the electric potential is constant on an ideally conducting wall corresponding to a vanishing normal velocity component into the wall.

The hyper-resistivity term becomes a boundary integral due to the divergence theorem:

\[\begin{equation*} \int - \nabla \cdot \left( \frac{\eta_{num}}{R} \nabla_{\text{pol}} j \right)dV = -\oint \frac{\eta_{num}}{R} \mathbf{n}\cdot\nabla_{\text{pol}} j dS_{bnd} \end{equation*}\]

where $\mathbf{n}$ is a unit vector normal to the JOREK boundary (pointing outwards the domain). Finally the time derivative of the total helicity is

\[\begin{equation*} \begin{split} \frac{d}{dt}\int H dV &= 2 F_0 \int \frac{\eta j}{R} d\phi dS -2F_0\oint \frac{\eta_{num}}{R} \mathbf{n}\cdot\nabla_{\text{pol}} j dS_{bnd} \\ \end{split} \end{equation*}\]

We therefore recover the classical result that helicity is dissipated at a resistive rate.