**Principal Investigator**
Robert L Dewar **Project **k12, r21

Department of Theoretical Physics and **Machine**
VP

Plasma Research Laboratory,

Research School of Physical Sciences and Engineering

**Co Investigators** Sean
A Dettrick, Henry J Gardner, Sally S Lloyd and Helen B Smith

Department of Theoretical Physics and Plasma Research Laboratory,

Research School of Physical Sciences and Engineering

**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 the 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 1020 charged 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 CM5 and VP2200. 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 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 are developing a parallelised Monte Carlo code which will 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 and should be amenable to experimental measurement.

**What are the basic questions addressed?**

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 future of the
work?**

Progress was achieved on a number of fronts in 1995. The GOURDON field-line-tracing code was substantially re-coded and documented. Investigations into the efficiency of the integration algorithm were also made, with particular reference to the utilisation of a new symplectic scheme (to preserve the Hamiltonian nature of the magnetic field in regions of field-line chaos). The DESCUR surface mapper acts as an interface between the GOURDON code and the MHD equilibrium codes. This interface was documented and improved and extended to cover another field-line-tracing code (called HELIAC). a commonly used straight-field-line mapper (the JMC code) was rewritten to be more user friendly (and was subsequently exported back to its home institute in Germany). A code written by W.A. Cooper of the CRPP, Lausanne, Switzerland, to evaluate linear ballooning stability (as well as its own mapper) was ported to the VP2200 and applied to the new MHH3 reactor design giving results which were disappointing for the design but important for the physics. The HINT equilibrium code, of T. Hayashi, National Institute for Fusion Science, Japan, was also successfully ported to the VP2200. This code is capable of calculating MHD equilibria in the presence of magnetic islands and stochastic regions and will be a backbone for the analysis of experimental data from the H-1NF. The Monte Carlo transport code under continuing development on the Connection Machine CM-5 supercomputer has been used to demonstrate some short-comings in a well known analytic model for particle fluxes from non-axisymmetric toroidal confinement devices. It has also been used to self consistently calculate the radial electric field in the plasma column of one configuration of the H-1 Heliac. Further work to improve the robustness of the simulation, and to satisfy its benchmarks is presently underway.

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

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. NSTAB fixed-boundary convergence studies can last 3 hours (for 40,000 cycles) on the VP2200. These space and time requirements necessitate the use of a supercomputer. 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.

**Publications**

*Reduction of Bootstrap Current in the Modular
Helias-like Heliac Stellarator*, P. R.
Garabedian and H. J. Gardner, Physics of Plasmas, **2,** 2020
(1995).

*Evolution of Magnetic Islands in a Heliac,*
T. Hayashi, T. Sato, H. J. Gardner and J. D. Meiss, Physics of
Plasmas,** 2**, 752 (1995).

*Hamiltonian Maps for Heliac Magnetic Islands*,
M. G. Davidson, R. L. Dewar, H. J. Gardner and J. Howard, Australian
Journal of Physics, **48**, 871 (1995).

*Magnetic Fusion Research in Australia: Opportunities
and Benefits* H. J. Gardner, in Proceedings
"Nuclear Science and Engineering in Australia 1995",
Lucas Heights, NSW, Oct. 1995 (Australian Nuclear Association,
ISBN 0949188085,1995) 117-119.

*Global Internal Modes in the H-1 Heliac *W.
A. Cooper and H. J. Gardner, in Proceedings of the 22nd European
Physics Society Conference on Controlled Fusion and Plasma Physics,
Bournemouth, UK, (Eur. Conf. Abs. Vol. 19c, Part II, Eds. B.E.
Keen, P.E. Stott, J. Winter, 1995) 145-148.

*Gourdon Manual and Report*,
H. B. Smith, ANU report (to be published)