Abstract: The axisymmetric magnetic field configurations can usually be solved by  the variable separation method. Even for the more complex magnetic field configurations, such as the field with current source term, the variable separation method can also get satisfactory results. In this paper, we explain the orthogonal relationship between the magnetic potential function and the stream function. By using the variable separation method, we get axisymmetric magnetic field of the solar corona (with two magnetic flux ropes within the corona) in equilibrium states under the force-free condition. We numerically visualize the correspoding stream function solution. Finally, we conclude the feasibility of the variable separation method to solve the complex axisymmetric equilibrium magnetic field.

Key words: variable separation method, magnetic field, axisymmetry