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

In this paper we develop the hydromagnetic wave equations for toroidal Alfvén waves in a background axi-symmetric magnetic field. In the case where spatial variations are directed along the ambient magnetic field direction, the equations can be cast in a Klein-Gordon form in which the adiabatic-geometric amplitude factor of the perturbations varies as √ρL5sin5θ along a magnetic field line (where θ is colatitude and L the L-shell number) and the cut-off frequency, associated with the Klein-Gordon form, displays an astonishing variation with distance along a field line (see Eqs. 35 and 37 of the text), in the case of a dipole magnetic field. We compute the eigenvalues and eigenfunctions for the Earth's dipole field which are relevant to geomagnetic pulsations.