Plasma Turbulence in 3-Dimensional Magnetically Confined Plasmas

Principal Investigator

Robert L. Dewar

Plasma Research Lab & Department of Theoretical Physics


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.


David Singleton


J. L.V. Lewandowski

P. Cuthbert

Plasma Research Lab & Dept of Theoretical Physics


M. Persson

Chalmers University of Technologhy

Gothenburg, Sweden


s55 - VPP, PC

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

A drift wave model assuming cold ions and (quasi) adiabatic electrons in helically-symmetric geometry has been derived. The stable and unstable part of the wave spectrum have been determined using a shooting code. This code and the GKS gyrokinetic code have been generalized to full 3-D geometry.

A total of six different configurations of the ANU H-1 plasma confinement device were studied using the VPP. Each of these studies involved solving for ideal MHD ballooning mode growth

rates in a phase space consisting of 124 x 100 x 10 points. At each point in the phase space, a differential eigenvalue equation must be solved by iterating the eigenvalue, which represents the growth rate of the ballooning mode. The large number ofcalculations required takes around 100 hours CPU time for each configuration studied. A general formalism for calculating amplitude and phase of global modes using the WKB approximation has been developed and applied to the same torsatron case as used previously for spectral studies.

What computational techniques are used?

The fundamental computation in the "WKB ballooning method" consists of solving a linear 2nd order, one-dimensional boundary-value eigenproblem on a magnetic field line a simple exercise in principle, at least when a simple fluid model is used. 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.


Cooper, W. A., Singleton, D. B. and Dewar, R. L. Spectrum of ballooning instabilities in a stellarator, Phys. Plasmas 3, 1996, 275­280; Erratum Phys. Plasmas 3, 1996, 3520.

Dewar, R. L. Ballooning-Mode Schrödinger Equation Revisited, Proceedings of the 3rd Australia-Japan Workshop on Plasma Theory and Computation (Robertson, NSW, 15-17 Nov. 1995), Eds. Dettrick, S.A. and Gardner, H.J., ANU, ISBN 0 7315 2450 0 (1996), 21­25.

Dewar, R. L. Spectrum of the Ballooning Schrödinger Equation, Plasma Phys. Control. Fusion 33 (1997) in press.

Forty-minute Review/Tutorial talk, at The First Asia-Pacific Plasma Theory Conference, Taejeon, Korea, August 21­23, 1996. To be published as a refereed paper in a supplementary edition of the Journal of the Korean Physical Society, 1997.

Dewar, R. L. Global Unstable Ideal MHD Continuum "Modes" in 3-D Geometries, Proc. Joint Varenna-Lausanne Workshop on Theory of Fusion Plasmas, Varenna, Italy 26­30 August 1996, in Theory of Fusion Plasmas (Società Italiana di Fisica, Editrice Compositori, Bologna, 1997) to be published 1997.

Lewandowski J. L. V. and Persson M., Australian J. Phys., 49, 1996, 1121.

Persson M., Lewandowski, J. L. V. and Nordman H., Phys. Plasmas 3, 1996, 3129.

Lewandowski, J. L. V., ANZ Physicist 33, 1996, 185.

- Appendix A