Plasma Research Laboratory, **Machine** VP

Research School of Physical Sciences and Engineering

**Co-Investigators** H J Gardner, R B Tumlos, M G Shats and D L Rudakov

Plasma Research Laboratory,

Research School of Physical Sciences and Engineering

**The Effect of Symmetry-Breaking Errors on the Heliac Magnetic
Configuration**

A new type of plasma confinement machine, the H-1 heliac, was commissioned in the Plasma Research Laboratory in late 1992. The heliac magnetic configuration, first demonstrated in the small machine SHEILA at ANU, employs magnetic fields to hold very hot, ionized gas in the shape of a helix bent around into a torus. This new configuration promises high plasma pressure limits, freedom from violent disruption of the plasma, elimination of the need to drive a steady current in the plasma, and simplicity of construction. H-1 is the first toroidal heliac of a reasonable size for hot plasma experiments, and this computing project is one of several in support of this experiment. For an ideal confinement geometry, a magnetic field line, if followed for a sufficient distance, will cover a surface enclosing a volume topologically equivalent to a torus. Other field lines cover other surfaces which ideally would all be nested inside each other. This can break down if the field line joins up with itself, when the twist per turn is a rational number. The otherwise smooth surface breaks up into a chaotic tangle of field lines, which may even wind its way out of the plasma region. When there is perfect three-period symmetry in the magnet windings, only rational numbers with a factor of three in the numerator can cause this effect, so the design of H-1 emphasises avoidance of these. The presence of small errors can break the symmetry of the configuration, and can induce the formation of magnetic islands and chaos for all other rational numbers, degrading the confinement properties. We plan to identify the most damaging sources of error so that action can be taken to minimize them. Beyond these details important to the success of H-1, we hope that this work will lead to some more general results on chaos. We believe that the H-1 device coupled with the computing power of the VP2200 will provide a unique opportunity for comparison of experimental and theoretical results.

**What are the basic questions addressed?**

What are the optimum operating regimes for the H-1 heliac, maximizing plasma confinement, and minimizing the effect of errors? What precision is required in the construction of future, larger machines? Are there some dimensions which are particularly critical? Can we compensate for some types of error by adjusting other windings, or by adding correction windings? Can we use the H-1 heliac for experimental verification of some of the theory of chaos and non-linear dynamics?

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

In 1993, the magnetic surfaces were investigated experimentally, and compared in detail with the computations. The resolution and agreement between the computed and measured results is of the highest standard. In 1994 an innovative, high resolution magnetic surface mapping apparatus was installed to extend the work to higher levels of detail, and to allow the experimental investigation of magnetic islands and ergodic magnetic structure. This `rotating wire grid tomography' apparatus is permanently mounted in the machine to allow an ongoing program of magnetic field line mapping. Both the tomographic inversion of the experimental data, on a grid size of >500x500, and the magnetic field line tracing need to be performed on a supercomputer. Initial experimental results obtained in 1994 indicate a very high resolution (0.5mm), and comparison with VP simulations for different magnetic configurations than those examined in 1993 gave consistent results. A scatter/gather tomographic algorithm suitable for the VP has been developed, but several other algorithms, including the `Arithmetic Reconstruction Technique', are being evaluated before implementation on the VP. General theoretical support for this work is being increased, in the mathematical field of flux conserving maps. The possibility of a theoretical and experimental comparison of the Lyapunov exponents of magnetic field line trajectories is being evaluated.

**What computational techniques are used and why is a supercomputer
required?**

Magnetic field line tracing is performed by explicit integration by higher order (5-8) single evaluation predictor-corrector techniques to minimize computationally expensive derivative calculations. Hastings-type power series are found to be most efficient on the VP for elliptic integral evaluations, and a prime factor decomposition FFT routine provides the spectrum.

The peak computing power of the VP2200 makes it possible for exploratory work to be performed almost interactively, followed up by more detailed runs in batch mode. Because of the lack of symmetry, six times more computing power is required than with symmetry. In practice, a further factor of 10 is required because we are usually interested in fine detail that precedes the onset of chaos.

**Publications**

*Experimental investigation of the magnetic structure of the H-1
heliac,* M G Shats, D L Rudakov, B D Blackwell, L E Sharp, R B Tumlos, S M
Hamberger and O I Fedyanin Nuclear Fusion, **34,** 1653-1661 (1994).

*Experimental study of Magnetic Islands on the H-1 heliac,* M G Shats, D L
Rudakov, B D Blackwell, L E Sharp, R B Tumlos and S M Hamberger, Proceedings of
the 21st European Conference on Controlled Fusion and Plasma Physics,
Montpellier, European Physical Society, Ed. G Thomas, Geneva 436-439 (1994).