Self-Similar Magnetic Arcades

We study self-similar analytical solutions for force-free magnetic field in azimuthal symmetry and arcade topology. We assume the existence of a poloidal magnetic field, anchored on a heavy spherical conductor. The field is changed by shearing the foot points of the arcade due to differential rotation. This rotation gives rise to a toroidal component in the magnetic structure which reacts by expanding the poloidal flux outwards. This could be a slow process at the early stages, however it becomes very fast at the final stages when the poloidal flux expands to infinity. We address the question of the pressure environment confining the arcade, a pressure profile proportional to $r^{-4}$ is particularly interesting as it allows finite twist before the field expands to infinity. Finally, some time evolution estimates are made to demonstrate the limitations of this study.