Two-dimensional analysis of trapped-ion eigenmodes

The first fully two-dimensional eigenmode analysis of the trapped-ion instability in an axisymmetric toroidal geometry is presented. The poloidal structure is taken into account by Fourier-expanding the perturbed electrostatic potential, φ, in θ. Assuming that the perturbation varies mildly over a typical ion-banana-width, ρ_bi, each poloidal harmonic is expressed as a truncated Taylor series in the minor radius to account for the radial structure. The resulting set of coupled ordinary second-order differential equations is solved numerically by the method of finite elements. The formalism is also applicable to the radially local and one-dimensional radial approximations. Results obtained in these limits are presented and found to be in reasonable agreement with previous calculations. In low shear plasmas, the full two-dimensional calculation is in qualitative agreement with the one-dimensional radial approximation. However, for higher shear the two-dimensional calculation yields a somewhat different picture for the radial structure of the instability. The analysis presented is limited to long radial wavelength and electrostatic perturbations.