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In this paper we continue our theoretical studies on addressing what are thepossible consequences of magnetic helicity accumulation in the solar corona.Our previous studies suggest that coronal mass ejections (CMEs) are naturalproducts of coronal evolution as a consequence of magnetic helicityaccumulation and the triggering of CMEs by surface processes such as fluxemergence also have their origin in magnetic helicity accumulation. Here we usethe same mathematical approach to study the magnetic helicity of axisymmetricpower-law force-free fields, but focus on a family whose surface fluxdistributions are defined by self-similar force-free fields. Thesemi-analytical solutions of the axisymmetric self-similar force-free fieldsenable us to discuss the properties of force-free fields possessing a hugeamount of accumulated magnetic helicity. Our study suggests that there may bean absolute upper bound on the total magnetic helicity of all bipolaraxisymmetric force-free fields. And with the increase of accumulated magnetichelicity, the force-free field approaches being fully opened up, withParker-spiral-like structures present around a current-sheet layer as evidenceof magnetic helicity in the interplanetary space. It is also found that amongthe axisymmetric force-free fields having the same boundary flux distribution,the one that is self-similar is the one possessing the maximum amount of totalmagnetic helicity. This possibly gives a physical reason why self-similarfields are often found in astrophysical bodies, where magnetic helicityaccumulation is presumably also taking place.

Magnetic Helicity of Self-Similar Axisymmetric Force-free Fields