Energy Buildup of Multipolar Magnetic Fields by Photospheric Shear Motion

Using an axisymmetrical ideal magnetohydrodynamic (MHD) model in spherical coordinates, we present a numerical study of the energy buildup of coronal magnetic fields by photospheric shear motions. The unsheared potential field consists of a dipolar and an octopolar component with a neutral point somewhere at the equator. The separatrix crossing the neutral point divides the field into four topologically disconnected regions: a central arcade astride the equator, a bipolar field on each side of the central arcade, and an overlying arcade above the separatrix. A specific type of photospheric shearing is then introduced at the base of the central arcade, and the energy of the resultant sheared field is calculated. It is found for this special case that the magnetic energy of the sheared force-free field cannot exceed that of the corresponding partly open field in which the central and overlying arcades become fully open but the bipolar fields at the flank remain closed. This inspires us to extend Aly's conjecture 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 closed flux opens but the remainder remains closed.