Comment on ‘‘Validity of the field line resonance expansion’’ [Phys. Fluids B 4, 2713 (1992)]

In a recent paper Hansen and Goertz (hereafter HG) [Phys. Fluids B 4, 2713 (1992)] considered the coupling between fast and Alfven modes in a cold plasma containing a uniform magnetic field (B0x) extending between two perfectly reflecting plane boundaries at constant x. The equilibrium medium is invariant in only one direction (ŷ), and (importantly) the density may vary along the equilibrium field lines, ρ0(x,z). HG sought solutions of the coupled governing PDEs (partial differential equations) for linear perturbations of the form exp i(kyy−ωt). The solution has been studied previously [Planet. Space Sci. 22, 483 (1974); J. Geophys. Res. 79, 1024 (1974)] in the case when ρ0 does not vary along the background field lines, when each Fourier mode in x decouples from the others and may be considered separately—reducing the problem to an ODE (ordinary differential equation). In this case a logarithmic singularity exists at the resonant field line where ω2=k2xV2A(z), VA being the Alfven speed (V2A=B20/4πρ0). H...