Department of Theoretical Physics and **Machine** CM, VP

Plasma Research Laboratory,

Research School of Physical Sciences and Engineering

**Co-Investigators** S A Dettrick and H J Gardner

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**

Magnetic fusion experiments study the interaction of a very hot, ionised
gas (plasma), consisting of some 10^{20} charged particles, with an
externally-applied magnetic field. In order to model these experiments, one
would ideally like to solve the equations of motion for each individual
particle--a task still well beyond the capabilities of present-day
supercomputers. A way forward is to combine a fluid-level analysis of plasma
equilibrium and stability with Monte Carlo simulations of the confinement of
plasma particles.

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 at the Plasma Research Laboratory. 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?**

Can we design the world's best fusion reactor? What instabilities limit the confinement of Heliacs and other fusion devices? What is the best way to handle singular magnetic surfaces? What radial electric field is consistent with ambipolar diffusion of the ions and electrons?

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

In collaboration with Professor P R Garabedian of New York University a new MHD equilibrium and stability code, NSTAB, has been combined with an estimate of the neoclassical `bootstrap' current in a design study for a fusion reactor. The MHH3 helical-axis stellarator which has resulted from this study is similar to the H-1 Heliac but requires an exotic, `wobbly' coil set as shown in the figure. It appears to be a breakthrough in fusion reactor design and offers the prospect of the smallest stellarator reactor discovered to date. The physics assumptions behind this design study will be addressed during the experimental program of the H-1 Heliac.

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

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, accepted for publication in
Physics of Plasmas.

*Ballooning Optimized Pressure Profiles in Toroidal Heliacs*, W A Cooper
and H J Gardner, in Nuclear Fusion, **34**, 729 (1994).

*Faraday Rotation as a Density Diagnostic in Helical Axis Stellarators*, H
J Gardner and J Howard in Plasma Physics and Controlled Fusion, **36**, 245
(1994).

*Evolution of Magnetic Islands in a Heliac, T Hayashi*, T Sato, H J
Gardner and J D Meiss, accepted for publication in Physics of Plasmas.

*Behaviour of Magnetic Islands in 3D MHD Equilibria of Helical Devices,* T
Hayashi, T Sato, N Nakajima, K Ichiguchi, P Merkel, J Nuhrenberg, U Schwenn, H
J Gardner, A Bhattacharjee and C C Hegna, in National Institute for Fusion
Science Report, NIFS-305, (Sept 1994).