Magnetic Relaxation for MHD Equilibria
๐ MRX paper (PDF) | ๐ Boundary preprint | ๐ป Code
A fusion plasma near steady state is in magnetohydrostatic (MHS) equilibrium: $(\text{curl } B) \times B = \nabla p$, $\text{div } B = 0$. MRX is able to compute configurations have magnetic islands and chaotic regions.
A non-nested NCSX equilibrium computed with MRX, colored by rotational transform (left) and pressure (right). A 3/5 island chain has formed through magnetic reconnection.
Relaxation
MHS states are minima of the magnetic energy along admissible variations $\delta B = \text{curl}(v \times B)$, which preserve field topology. MRX follows these orbits.
Magnetic energy landscape with constant-helicity sheets and the orbit of admissible variations.
Targeted resistive steps can be used to change the topology in a controlled way.
Poincarรฉ sections of a NCSX equilibrium before and after reconnection.
The code
MRX is written in JAX, using Finite Element Exterior Calculus. An inexact Newton method computes high-resolution finite-$\beta$ equilibria in minutes on a single GPU. Just clone and pip install!
Different meshes on a poloidal plane with their equilibria configurations.
Papers
MRX: A Differentiable 3D MHD Equilibrium Solver Without Nested Flux Surfaces
Tobias Blickhan, Julianne Stratton, Alan A. Kaptanoglu
arXiv preprint, 2025
Computational Boundary Specification in 3D Fixed-Boundary Magnetohydrodynamic Equilibrium Modeling
Alan A. Kaptanoglu, Tobias Blickhan
arXiv preprint, 2026
Acknowledgements
This work was supported by the Simons Collaboration on Hidden Symmetries and Fusion Energy and a Google TPU Research Award.