Klein-Gordon equations for toroidal hydromagnetic waves in an axi-symmetric field

In this paper we develop the hydromagnetic wave equations for toroidal Alfv ´ en waves in a background axi- symmetric magnetic field. In the case where spatial vari- ations are directed along the ambient magnetic field direc- tion, the equations can be cast in a Klein-Gordon form in which the adiabatic-geometric amplitude factor of the per- turbations varies as p L 5 sin 5 along a magnetic field line (where is colatitude and L the L-shell number) and the cut- off frequency, associated with the Klein-Gordon form, dis- plays an astonishing variation with distance along a field line (see Eqs. 35 and 37 of the text), in the case of a dipole mag- netic field. We compute the eigenvalues and eigenfunctions for the Earth's dipole field which are relevant to geomagnetic pulsations. the effects of plasma inhomogeneity and background mag- netic field geometry as developed by Allan and Knox (1979); Walker (1980), and Taylor and Walker (1984). These works provide numerical solutions to the "full wave" equations in a dipole magnetic field geometry.