Alfv\'en Polar Oscillations of Relativistic Stars

We study polar Alfv\'en oscillations of relativistic stars endowed with a strong global poloidal dipole magnetic field. Here we focus only on the axisymmetric oscillations which are studied by evolving numerically the two-dimensional perturbation equations. Our study shows that the spectrum of the polar Alfv\'{e}n oscillations is discrete in contrast to the spectrum of axial Alfv\'{e}n oscillations which is continuous. We also show that the typical fluid modes, such as the f and p modes, are not significantly affected by the presence of the strong magnetic field.