3D MHD Equilibrium and Stability

For some decades there has been an international scientific and engineering program to study the containment of plasmas by toroidal magnetic fields with the aim of developing a fusion power reactor. Australia presently makes a major contribution to this program through the H-1NF Heliac which is a "stellarator" experiment located at ANU and is funded through the Major National Research Facility Program. Stellarators are an alternative to the better known "tokamak" types of experiments but have the great advantage of not needing large currents within the plasma in order to generate helical field lines. On the debit side, the lack of axial symmetry of stellarators makes their theory more complicated and simulation more computationally expensive than for tokamaks. In particular, the simply nested magnetic surfaces found in tokamaks can be broken and give rise to magnetic islands and regions of chaotic magnetic field lines. Activities under this project cover two distinct areas: (1) Magnetic Islands in the H-1NF Heliac and (2) 3D Resistive MHD Resistive Stability (with R.G. Storer, Flinders University, S.A.)

Principal Investigator

Henry Gardner
Theoretical Physics



Facilities Used



Bob Dewar
Sally Lloyd
Ben McMillan
Theoretical Physics

David Singleton
Supercomputer Facility

RFCD Codes


Significant Achievements, Anticipated Outcomes and Future Work

Following extensive parameter studies with an accelerated version of HINT, we have a strong inference that magnetic island self-healing can be expected for Heliacs. Detailed comparisons with local theory based on expressions involving the resistive Mercier criterion have been more problematic and are still under investigation.

The development of a fully three dimensional resistive MHD stability code is underway. As part of this development the Jacobi-Davidson method has been used to compute a number of spectral eigenvalues at once. The code has been satisfactorily benchmarked in axisymmetric systems such as tokamaks. Additional new features of the resistive tokamak MHD spectrum seem to have been identified. Comparisons of our incompressible model with a compressible code used elsewhere have identified intriguing similarities and differences.


Computational Techniques Used

The HINT code, developed by T. Hayashi, solves time-dependent, resistive MHD equations on a specially-shaped coordinate grid. Relaxation along field lines is much slower than perpendicular to them and is treated using a special interpolation algorithm which has been developed as part of this project (pressure is averaged along field-lines and interpolated back to the fixed grid).

The SPECTOR3D resistive stability code, which we are developing, computes matrix elements corresponding to a particular formulation of the incompressible, resistive, linearised MHD stability equations. Initially inverse iteration was used to compute eigenvalues of the system but this has been changed to implement the Jacobi Davidson method.

The VMEC equilibrium code, developed by SP Hirshman, is also extensively used to compute ideal MHD equilibrium. This code uses a form of steepest descent to find the minimum energy state of a plasma. The formulation is a combination of spectral, in the magnetic surfaces, and finite differences across the surfaces. SPECTOR3D has a similar, hybrid, representation of the plama.

Local MHD stability is computed using the JMC code developed by J. Nuehrenberg. This code calculates surface averages of equilibrium quantities.

Field line tracing, and surface fitting, codes are used to set up an equilibrium modelling study from the real, H1NF magnetic field.


Publications, Awards and External Funding

R. Storer and H. Gardner, Resistive Magnetohydrodynamic Modelling for 3D Stellarator Geometry, Accepted for publication in Computer Physics Communications (December, 2001)