Low mode approximation in the axion magnetohydrodynamics

In this paper, we study the evolution of interacting large scale magnetic and axionic fields. Based on the new induction equation accounting for the contribution of spatially inhomogeneous axions, we consider the evolution of a magnetized spherical axion structure. Using the thin layer approximation, we derive the system of the nonlinear ordinary differential equations for harmonics of poloidal and toroidal magnetic fields, as well as for the axion field. In this system, we account for up to four modes. Considering this small and dense axion clump to be in a solar plasma, we numerically simulate the evolution of magnetic fields. We obtain that the behavior of magnetic fields depends on the initial fields configuration. Moreover, we find an indication on a magnetic field instability in the magnetohydrodynamics with inhomogeneous axions.