Exact analytical solution of the toroidal singular mode equations

A subset of the complete fluid equations governing the inner region of low-η 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 ∞. Exact solutions are found in terms of confluent hypergeometric functions. A dispersion relation Δ' = Δ(Q) is derived from the matching condition between inner and outer solutions in configuration space, with Δ' the ratio of small to large asymptotic coefficients in the ideal outer region and Δ(Q) a complicated function of the scaled growth rate Q obtained from the inner region analysis