A fully three-dimensional spectral code has been developed and implemented. It computes three-dimensional equilibria using the variational approach, by minimizing the potential energy subject to appropriate constraints. A second minimization, with an additional constraint, is used to examine the stability of solutions.
The magnetic field representation allows for non-nested flux surfaces. Thus it can be used to study solutions with islands and variation of the potential energy with respect to topology changes.
A spectral representation in the toroidal and poloidal angles, coupled with a special choice of collocation points in the radial direction results in greatly enhanced resolution.
A fast iterative method was developed, with the number of iterations required for convergence independent of the mesh size. Residuals converge exponentially to the round-off error, allowing the potential energy to be computed to the eight-digit accuracy required for nonlinear stability analysis. A small amount of artificial viscosity may be needed for convergence in cases where low-order resonances are present in the plasma region.
Numerical results show convergence to known axially symmetric equilibrium solutions, as well as close agreement with eigenvalue calculations in helically symmetric stellarator configurations. Solutions exhibiting island formations are found to have lower potential energy than that of the nearby nested case. Internal modes are found to localize inside the resonant surface, with the eigenfunction having sharp gradients at the resonant surface.
1. Magnetic Field Representation
Our formulation of the variational principle of magnetohydrodynamics extends the approach developed in [l], [2] to a slightly larger set of solutions by choosing a magnetic field representation which allows for non-nested flux surfaces.
The magnetic field B, defined in a toroidal region and satisfying v B = 0 can be represented in the form (see [3])
where s and + are single-valued functions corresponding to the toroidal and poloidal flux, respectively. The potentials 8 and u are angle-like variables, and one of them can be chosen arbitrarily.
It is important to note that given a divergence-free field B, and a fixed choice for u, there is still a degree of freedom in the determination of the potentials 6