Energy buildup of partly open bipolar field by photospheric shear motion

The Aly conjecture on the maximum energy of force-free fields with a given normal component distribution at the lower boundary was recently extended from fully open force-free fields to partly open ones in such a way that in the frame of ideal MHD, it is impossible to store more magnetic energy in the corona by photospheric shear motions at the base of any part of the closed flux of a force-free field than that of the field in which the sheared flux opens, while the unsheared one remains closed. This paper provides another example in support of such an extension, in which the unsheared field is a partly open bipolar field, and the photospheric shear is specified in terms of the footpoint displacements. A new numerical approach is proposed to implement a given footpoint displacement for two-dimensional force-free fields. It is found that the magnetic energy buildup of the system depends on the distribution of the displacement at the photosphere: the more the displacement concentrates to the center of the closed arcade, the larger the energy buildup will be acquired. During the shear, the unsheared field remains closed, whereas the sheared one tends to open in its outermost part. In any case, the maximum energy cannot exceed that of the corresponding partly open field, in which the sheared field is assumed to be open, but the unsheared one remains closed.