Exact analytical solution of the toroidal singular mode equations

摘要

A subset of the complete fluid equations governing the inner region of low-(eta) singular modes in toroidal geometry is solved exactly and analytically. The subset includes toroidal curvature as well as ion and electron diamagnetic and E x B drifts, but not viscosity or thermal conductivity. The Fourier-transformed equations are reduced to a second-order ODE with singular points at 0 and (infinity). Exact solutions are found in terms of confluent hypergeometric functions. A dispersion relation (Delta)(prime) = (Delta)(Q) is derived from the matching condition between inner and outer solutions in configuration space, with (Delta)(prime) the ratio of small to large asymptotic coefficients in the ideal outer region and (Delta)(Q) a complicated function of the scaled growth rate Q obtained from the inner region analysis.