The dependence of the helicity bound of force-free magnetic fields on   boundary conditions
  This paper follows up on a previous study showing that in an open atmosphere such as the solar corona the total magnetic helicity of a force-free field must be bounded and the accumulation of magnetic helicity in excess of its upper bound would initiate a non-equilibrium situation resulting in an expulsion such as a coronal mass ejection (CME). In the current paper, we investigate the dependence of the helicity bound on the boundary condition for several families of nonlinear force-free fields. Our calculation shows that the magnitude of the helicity upper bound of force-free fields is non-trivially dependent on the boundary condition. Fields with a multipolar boundary condition can have a helicity upper bound ten times smaller than those with a dipolar boundary condition when helicity values are normalized by the square of their respective surface poloidal fluxes. This suggests that a coronal magnetic field may erupt into a CME when the applicable helicity bound falls below the already accumulated helicity as the result of a slowly changing boundary condition. Our calculation also shows that a monotonic accumulation of magnetic helicity can lead to the formation of a magnetic flux rope applicable to kink instability. This suggests that CME initiations by exceeding helicity bound and by kink instability can both be the consequences of helicity accumulation in the corona. Our study gives insights into the observed associations of CMEs with the magnetic features at their solar surface origins. 
Mei Zhang and Natasha Flyer