Direct Prediction of Nonlinear Tearing Mode Saturation Using a Variational Principle*
It is shown that the variational principle of multi-region relaxed magnetohydrodynamics (MRxMHD) can be used to predict the stability and nonlinear saturation of tearing modes in strong guide field configurations without resolving the dynamics and without explicit dependence on the plasma resistivity. While the magnetic helicity is not a good invariant for tearing modes, we show that the saturated tearing mode can be obtained as an MRxMHD state of a priori unknown helicity by appropriately constraining the current profile. The predicted saturated island width in a tearing-unstable force-free slab equilibrium is shown to reproduce the theoretical scaling at small values of Δ′Δ′ and the scaling obtained from resistive magnetohydrodynamics simulations at large Δ′Δ′.
We acknowledge very helpful discussions with Amitava Bhattacharjee, Per Helander, Robert Dewar, Adelle Wright, Alessandro Zocco, and Jonathan Graves. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was also supported by a grant from the Simons Foundation/SFARI (No. 560651, A.B.). This work (A.K. and Z.S.Q.) is partly funded by Australian ARC project No. DP170102606.