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.


Principal Investigator

Robert Dewar

Department of Theoretical Physics & Plasma Research Lababoratory,




David Singleton


Jerome Lewandowski

Paul Cuthbert

Department of Theoretical Physics & Plasma Research Laboratory,


Mikael Persson

Chalmers University of Technology

Gothenburg, Sweden



s55 - VPP, PC



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

The linear stability of resistive drift waves in H1-NF has been studied using an initial-value code, indicating that the local growth rate is strongly dependent on the electron temperature gradient parameter and on the local properties of the confining magnetic field. A hybrid gyrokinetic-fluid code to study ion drift waves in stellarator geometry has been developed. The code, valid for arbitrary 3-dimensional plasmas, retains exact finite ion Larmor radius

Appendix A -


effects (using the collisionless gyro-kinetic equation) and includes electron dynamics (using fluid equations). Numerical results for H1-NF have been reported.

Two different configurations of the ANU H-1 heliac have been investigated for stability against ideal MHD ballooning modes. These instabilities are known to limit the maximum pressure which can be obtained in a confinement device, and as such are an important consideration for the development of a fusion reactor. Investigations with the VPP have led to a greater understanding of the conditions under which ballooning instabilities develop. The role of the low magnetic shear in H-1 has been examined, and a model has been developed to give a more intuitive understanding of the behaviour of ballooning modes. Calculations have also been performed for ballooning instabilities in the Large Helical Device (LHD), currently under construction in Japan. The structure of ballooning instabilities in this device was investigated, and an equilibrium case was found where both non-localised and localised ballooning instabilities coexisted.

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.


R. L. Dewar, Spectrum of the Ballooning Schrödinger Equation, Plasma Phys. Control. Fusion 39 453-470 (1997).

R. L. Dewar, P. Cuthbert, J. L. V. Lewandowski, H. J. Gardner, D. B. Singleton, M. Persson, and W. A Cooper, Calculation of Global Modes via Ballooning Formalisms, Journal of the Korean Physical Society (Proc. Suppl.), 31 S115­S118 (1997).

R.L. Dewar, Global Unstable Ideal MHD Continuum "Modes" in 3-D Geometries, in Theory of Fusion Plasmas, Soc. Ital. di Fisica, Editrice Compositori, Bologna, pp. 247­252, (1997).

R.L. Dewar, Reduced Form of MHD Lagrangian for Ballooning Modes, Journal of Plasma and Fusion Research, 73 1123-1134 (1997).

- Appendix A



P. Cuthbert, J. L. V. Lewandowski, H. J. Gardner, M. Persson, D. B. Singleton, R. L. Dewar, N. Nakajima, W. A. Cooper, Toroidally localized and nonlocalized ballooning instabilities in a stellarator, submitted to Physics of Plasmas.

J. L.V. Lewandowski, Collisional Drift Waves in Stellarator Geometry, Physics of Plasmas 4 4023 (1997).

J. L.V. Lewandowski, Resistive Drift Waves in a Toroidal Heliac, Journal of the Physical Society of Japan 66 1401 (1997).

J. L.V. Lewandowski, A Simple Model for Collisional Drift Waves, Canadian Journal of Physics 75 891­906 (1997).

M. Persson, and J. L.V. Lewandowski, Localisation of Drift Waves in a Helically Symmetric Stellarator Model, Plasma Physics and Controlled Fusion 39 1941-1946 (1997).









Appendix A -