Two-dimensional spatial structure of the dissipative trapped-electron mode

The complete two-dimensional structure of the dissipative trapped-electron mode over its full width, which may extend over several mode-rational surfaces, is discussed. The complete intergrodifferential equation is studied in the limit κ_rρ_i < 1, where ρ_i is the ion gyroradius, and κ_r, the radial wavenumber, is regarded as a differential operator. This is converted into a matrix equation which is then solved by standard numerical methods. Solutions obtained are in reasonably good agreement with one-dimensional analytic solutions, in the limits where such results are expected to be valid. More significantly, the present approach can readily treat many physically important cases for which purely analytic solutions are difficult to obtain. The results indicate that the differential equation formulation of the eigenmode equation is valid only for long wavelength modes (κ_θρ_i ≲ 0.3, with κ_θ being the poloidal wavenumber). For such cases it is found that shear stabilization estimates obtained from the one-dimensional radial solution are quite inaccurate for modes overlapping only a small number of mode-rational surfaces, but become more accurate for modes overlapping many mode-rational surfaces.