**Principal Investigator**
Robert Dewar **Project** s55

Plasma Research Laboratory and **Machine** VP

Department of Theoretical Physics

Research School of Physical Sciences and Engineering

**Plasma Turbulence in 3-dimensional Magnetically
Confined Plasmas**

The goal of nuclear fusion power research is to use strong magnetic fields to insulate a plasma (hot gas) of hydrogen isotopes, at a temperature of hundreds of millions of degrees, from walls near room temperature. The enormous temperature gradient creates a highly non-equilibrium situation where various self-organisation phenomena tend to occur, destroying smooth density and temperature distributions and instead creating turbulent structures that allow heat and particles to escape faster than desired ("anomalous transport").

The aim of advanced magnetic confinement designs, such as the H-1 Heliac (to be upgraded to a national facility, H-1NF, in which temperatures up to ten million degrees will be obtained), is to thwart this tendency towards formation of turbulence by designing twisted magnetic field configurations that make instability energetically unfavourable. The simulation of such a device forms a grand challenge to theory and computation because the full three-dimensional geometry, both of the equilibrium and the perturbation, must be accurately taken into account.

The project consists of three basic phases - linear stability analysis, weakly nonlinear evolution of coupled modes, and simulation of developed turbulence. In each of these there is a choice of plasma model - whether to use a simple magnetohydrodynamic (MHD) fluid model, a more sophisticated fluid model or a full particle (gyrokinetic) model. The aim is to develop efficient computational models, verifying at each step that they can produce a converged result for the heliac geometry when the spatial resolution is increased.

**What are the basic questions addressed?**

Can a WKB representation represent global modes with sufficient accuracy, and is this more efficient than a "brute force" spectral expansion? What are the dominant linear instabilities predicted for H-1 (MHD ballooning modes, drift waves ...)? What are the details of their spectrum and eigenfunctions? What forms a good mode-coupling model? What are the nonlinear consequences for transport of energy and particles?

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

The first phase of the project has been devoted to developing a parallel implementation of an MHD ODE eigenvalue solver on the CM-5, and comparing the global eigenvalues obtained by the WKB ballooning method with those obtained by Dr W A Cooper, of the Centre de Recherche en Physique des Plasmas, EPFL, Switzerland using the TERPSICHORE global eigenvalue code. At least for the test case used (less extremely non-axisymmetric than H-1), the WKB ballooning method has been found to give very good agreeement with TERPSICHORE, while greatly reducing memory requirements. Calculation and visualisation of eigenmode structure and application of the method to the H-1 geometry is underway. A gyrokinetic code is also being adapted to heliac geometry.

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

The fundamental computation in the "WKB ballooning method" consists in solving a linear 2nd order, one-dimensional boundary-value eigenproblem on a magnetic field line - a simple exercise in principle. However the ODE coefficients (derived from an equilibrium configuration also calculated on a supercomputer) must be calculated at up to 30,000 points on a field line. Each point requires summing between 600 and 1,100 Fourier components and calculation of coefficients on the full set of points takes tens of cpu-seconds.

The supercomputing needs come from computing sufficient of these localised one-dimensional eigensolutions to construct a three-dimensional arrray from which global eigenmodes and their growth rates can be constructed. The growth rate is derived from the average pitch of a helical trajectory (ray path) in a 3D parameter space, obtained by interpolation from the array of field line eigensolutions. Hence, thousands of eigenproblems must be solved to ascertain the global structure of these trajectories. Since the calculation of each coefficient and each eigenvalue is independent, the problem is easily parallelisable and also involves large arrays suitable for vectorisation..

**Publications**

*Gyro-kinetic Calculations in 3-Dimensional Geometry,*
J.L.V. Lewandowski, M. Persson, R.E. Waltz. and D. Singleton,
20th AINSE Plasma Science and Technology Conference, Flinders
University, February 13-14, 1995, Conference Handbook pp14.

*WKB-Ballooning vs Global Expansion Methods for
Short-Wavelength MHD Waves in Stellarators, *R.L.
Dewar, D.B. Singleton, H.J. Gardner, J. Lewandowski and W.A. Cooper,
20th AINSE Plasma Science and Technology Conference, Flinders
University, February 13-14, 1995, Conference Handbook pp53.

*Ballooning Representation Approach to Low-Frequency
Instabilities in Stellarators, *R.L. Dewar,
H.J. Gardner, J. Lewandowski, M. Persson, D.B. Singleton, W.A.
Cooper and H.A. Nordman, Tenth International Conference on Stellarators
(IEA Technical Committee Meeting), Madrid, 22-26 May 1995, Proceedings,
eds. E. Ascasibar & D. Aranda pp 257-260.

*Ballooning Modes: Semiclassical Quantization in
a Non-Axisymmetric Torus*, R.L. Dewar,
D.B. Singleton, and W.A. Cooper, Proceedings of the 17th Symposium
on Plasma Physics and Technology, Prague, June 13-16, 1995 pp
1-3.

*Spectrum of ballooning instabilities in a stellarator*,
W.A. Cooper, D.B. Singleton. and R.L. Dewar, Phys. Plasmas, submitted