Introduction

We set out to solve a linear system. Electromagnetic waves are launched into a confined toroidal plasma by means of a plasma facing antenna mounted inside the vacuum vessel. The field and density perturbations produced by the plasma are small compared to background quantities and take place on a time scale that is much faster than diffusive or transport times with frequencies that are typically in the GHz range. (Although quasilinearly[2,3], the waves can create macroscopic changes in plasma quantities, and while of great interest in itself, this does not violate the assumption of linearity.) Several effects conspire to make a simple linear boundary value problem, much more difficult.

While Maxwell's equations form a straight-forward hyperbolic system. After Fourier transforming it in time, and introducing a single wave frequency, $\omega$, they become elliptic (Eq. (1a)). The inclusion of the plasma response in the form of the perturbed current transforms the problem into an integro-differential equation. One approximation is to reduce the perpendicular range of wave-particle interactions. That is, the dielectric response is only accurately retained for wavelengths larger than the ion gyroradius. When the wavelength is smaller than the ion gyroradius, the anti-Hermitian part of the dielectric that is responsible for damping the wave is modified to model the correct level of damping at those small scales, and this leads to the finite Larmor radius (FLR) approximation used in this paper. Recently developed codes[4], made possible with massively parallel processor (MPP) architectures, avoid this sacrifice, but at the cost of needing thousands of CPU hours for relatively modest resolutions. In the direction parallel to the equilibrium magnetic field, the waves interact with ions and electrons streaming along field lines, and this creates a strong coupling along this dimension. By using a Fourier basis in this direction, we are able to use the developments of homogeneous wave theory[5,6] in formulating the plasma response, and retain a degree of physical intuition. Finally, when exploring quasilinear effects and feedbacks on the equilibrium, it is necessary to evaluate the plasma dielectric response for non-Maxwellian particle distributions. The evaluation of the susceptibility expressions in reference [7] must now be done numerically. If in addition, we wish to ``close the loop'' with Fokker-Planck calculations of these distributions in the presence of the wave fields, then MPP calculations are necessary to evolve the weakly coupled system in a reasonable amount of time.