3D MHD Equilibrium and Stability and Simulation of Neoclassical Plasma Transport


The study of plasma (fully ionized matter) and its interaction with electromagnetic fields is fundamental both to our understanding of a basic material of the universe and to important applications such as the quest for controlled fusion energy.

Four decades of intensive experimental research world-wide have shown that obtaining hot, well-controlled plasma in the laboratory requires its production and containment inside a toroidal magnetic field of sufficient strength and dimensions.

At present there are two main classes of experiment being investigated: tokamaks and stellarators. The former, while theoretically simpler due to their axisymmetry, are prone to violent instabilities and may not be suitable as commercially-viable fusion reactors. The situation is very different for the stellarator class of experiment where Australia has recently become a major player internationally with the upgrade of the H-1 Heliac to National Facility status (the H-1NF) through the federal government's Major National Research Facility Program.

The computation of the physical properties of a plasma (of some 1020charged particles) and its self-consistent interactions with magnetic and electric fields is a grand-challenge of modern science - particularly when a detailed comparison with experiment is needed. A high priority area of experimentation on the enhanced H-1NF will be the achievement of fusion relevant conditions of plasma temperature and pressure and the measurement of plasma fluctuations and turbulent transport under these conditions to confirm or refute the theoretical predictions.

A theoretical program studying the physics of the H-1NF Heliac has been underway for some time using the ANUSF supercomputers. The use of these computers has been crucial in laying the groundwork for the successful H-1NF bid.

The first step in modelling a plasma experiment is to make detailed calculations of the external magnetic field. Thus, in fusion laboratories around the world, much use is made of large engineering software packages to calculate the


Principal Investigator

Robert L. Dewar

Dept of Theoretical Physics and Plasma Research Lab.


Henry J. Gardner

Department of Computer Science




Sean A. Dettrick

Sally S. Lloyd

Dept of Theoretical Physics and Plasma Research Lab.




k12, r21 - VPP, PC




basic vacuum magnetic flux surface geometry by a technique known as field-line-tracing. The Biot-Savart law is used in these field-line-tracing codes and the magnetic field coils are usually either modelled as a collection of linear current elements or circular filaments.

In the theory of magnetohydrodynamics (MHD) the plasma is pictured as being a conducting magnetofluid obeying the field equations of electromagnetism and hydrodynamics. MHD theories have been very successful in describing the equilibrium and stability properties of magnetically confined plasmas, although it is only with the advent of supercomputers that it has been possible to apply them to fully three dimensional (3D) geometries such as the H-1 Heliac. The 3D equilibrium calculations are often used as the means of construction of special ("straight-magnetic-field-line") coordinates systems in which theoretical and computational analyses of the plasma stability and transport can be carried out. In particular, the MHD equilibrium calculations provide the background magnetic field in which test particles are propagated, in the drift kinetic approximation, to model neoclassical transport. We have developing a parallelised Monte Carlo code which can estimate the self consistent electric field which results from the ambipolar diffusion of test particle distributions of ions and electrons. The magnitude of this electric field turns out to be crucial for the plasma confinement time.

The aims of this project are to determine:

What instabilities limit the confinement of Heliacs and other fusion devices?

What is the influence of magnetic islands and magnetic stochasticity on a confined plasma?

What radial electric field is consistent with ambipolar transport of the plasma particles?



What are the results to date and the future of the work?

A Monte Carlo code for simulation of neoclassical transport in the H-1NF heliac has been ported from the CM5 supercomputer to the Silicon Graphics PowerChallenge and the Fujitsu VPP300 supercomputers and a DEC Alpha-farm. A highly-portable parallel implementation of the code has been developed using the Message Passing Interface and Fortran 90. The transport code has been used to self consistently calculate the radial electric field in the plasma column of the H-1NF Heliac and electric fields and diffusion coefficients calculated by the code have been compared with experiment. Excellent agreement has been found between the code and measured electric field profiles for the recently-discovered high confinement mode of H-1NF.


- Appendix A



An approximate adiabatic invariant has been used to compare the geometry of charged particle drift surfaces in the H-1NF heliac with those calculated by a guiding center code. Surprisingly good agreement between the two methods has been observed. The approximate adiabatic invariant is being used to examine orbit behaviour over the full operating regime of H-1NF.

The HINT magnetohydrodynamic equilibrium code, together with a new technique for scanning the parameter space of the magnetic field configurations of the H-1NF heliac, has been used to investigate whether true self-healing of magnetic islands can occur in a heliac. The HINT code is being improved to significantly speed the relaxation of pressure in regions of the plasma containing magnetic islands. The HINT code has also been used to compare the details of the internal current distribution in the H-1 Heliac with measurements taken using Rogowski coils. The experimental measurements seem to have the same qualitative nature as the simulations but there also appears to be signs of secondary (non-equilibrium) currents in the plasma column.

What computational technques are used?

The Adams-Bashforth algorithm is most commonly used for the field-line-tracing. The computational techniques used are hybrid spectral and finite difference methods, an accelerated conjugate-gradient method of steepest descent and Monte Carlo methods with a stochastic differential equation for the collision operator. To model a three-dimensional plasma with any accuracy one needs a large number of spatial grid points and Fourier modes. A convergence run of the VMEC hybrid spectral equilibrium code uses 392 modes on 1568 grid points for each of 153 plasma surfaces. For a stability analysis the Fourier dimension must be increased to 2720 modes. The Monte Carlo algorithm of the neoclassical transport code is intrinsically parallel. Once again the large number of Fourier harmonics needed to describe the H-1 magnetic field strength necessitate a large amount of computing power.


Ichiguchi K, Nakajima N and Gardner H J, Free-Boundary Studies for the Large Helical Device, Nuclear Fusion 36, 1157-1166 (1996).

Cooper W A and Gardner H J, in Proceedings of the 3rd Aust.-Japan Work. on Plasma Theory (Eds. S A Dettrick and H J Gardner, ANU, ISBN 0 7315 2450 0), 6-9 (1996).

Dettrick S A and Gardner H J , Parallel Implementation of Monte Carlo Neoclassical Transport Calculations for Stellarators, in Proceedings of the 3rd Aust.-Japan Work. on Plasma Theory (Eds. S A Dettrick and H J Gardner, ANU, ISBN 0 7315 2450 0), 15-20 (1996).

B D Blackwell, G G Borg, P Cuthbert, R L Dewar, S A Dettrick, H J Gardner, S M Hamberger, J Howard, S R Hudson, J V L Levandowski, S L Lloyd, M Persson, D L Rudakov, L E Sharp, D B Singleton, M G Shats and G B Warr, Fluctuation and Internal Current Studies in the H-1 Heliac, Paper IAEA-F1-CN-64/CP-7, 16th Int. Conf. on Plasma Physics and Controlled Fusion, Montreal, October 1996. To be be published in Plasma Physics and Controlled Nuclear Fusion Research.

S A Dettrick and H J Gardner, Proceedings of the 3rd Australia-Japan Workshop on Plasma Theory ANU, ISBN 0 7315 2450 0 (1996).