Starting from a dipole field and a given distribution of footpoint displace- ment 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 photo- sphere 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 γ_s.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 H_T in the solar corona are then calculated for each force-free field solution.It is found that for a given shear region range γ_s,all of the three quantities increase monotoni- cally with increasing (?)_m.In particular,both F_(?) and H_T have a linear dependence on (?)m.When (?)_m reaches a certain critical value (?)_(mc),the force-free field loses equilib- rium,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 H_T reach their maximum values, E_c,F_((?)c) and H_(Tc).E_c increases,and F_((?)c) and H_(Tc)decrease with decreasing λ_s.It is found that E_c is always smaller than the open field energy,in agreement with the Aly conjecture.Of the three critical parameters,E_c has the weakest dependence on λ_s.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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