Toroidal hydromagnetic waves in an axi-symmetric magnetic field

Hydromagnetic wave equations are derived for toroidal Alfven waves in a background, axi-symmetric magnetic field. In the case, where the spatial variations in the wave equations are along the background magnetic field, the equations can be re-cast in Klein-Gordon equation forms, with cut-off frequency omega(c). For harmonic, constant frequency solutions, the Klein Gordon equation reduces to a Sturm-Liouville equation. We compute the eigenvalues and eigenfunctions appropriate for the Earth's dipole magnetic field, which are relevant for geomagnetic pulsations. It is suggested that the breather type Alfven eigenmodes, obtained from the analysis may describe ultra low frequency Alfvenic pulsations (e. g. Pc and Pi pulsations) observed in space and on the ground with frequencies ranging from a few milli-Herz (mHz) to a few Hz). The wave equations for the velocity and magnetic field perturbations can be re-written in terms of Elsasser variables. The resultant wave equations are a special case of the wave mixing equations for Alfvenic fluctuations used to describe locally incompressible turbulence in the solar wind. The canonical wave energy equation for the backward and forward waves, and wave reflection and transmission coefficients are briefly discussed. The equations can also be cast in the form of a Dirac equation, in the Weyl spinor form, in which the mass in the Dirac equation is identified with the wave mixing coefficient for the backward and forward waves.