Abstract Starting from a dipole field and a given distribution of footpoint displacement of field lines on the photosphere, we find axisymmetric, force-free field solutions in spherical coordinates that have the same distribution of normal field on the photosphere and magnetic topology as the dipole field. A photospheric shear is introduced in the azimuthal direction in a region that strides across the equator and ends at latitude λs. The footpoint displacement has a sine distribution in latitude and a peak amplitude of m. The magnetic energy E, azimuthal flux F, and magnetic helicity HT in the solar corona are then calculated for each force-free field solution. It is found that for a given shear region range λs, all of the three quantities increase monotonically with increasing m. In particular, both F and HT have a linear dependence on m. When m reaches a certain critical value mc, the force-free field loses equilibrium, leading to a partial opening of the field and the appearance of a current sheet in the equatorial plane. At this point, E, F and HT reach their maximum values, Ec, Fc and HTc. Ec increases, and Fc and HTc decrease with decreasing λs. It is found that Ec is always smaller than the open field energy, in agreement with the Aly conjecture. Of the three critical parameters, Ec has the weakest dependence on λs. Therefore, if one is interested in the transition of a magnetic configuration from a stable state to a dynamic one, the magnetic energy is probably the most appropriate marker of the transition.

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https://doi.org/10.1088/1009-9271/6/1/009