Abstract
The evolution dHR/dt of relative helicity HR provides a gauge-invariant measure of the helicity flow across the open surface So of an active region. With the incompressible approximation, reformulation of the evolution reveals that it is determined not only by the widely used cross-helicity hmvp=Ap⋅v contributed by the vector potential Ap of a reference potential field Bp, where v is the fluid velocity, but by another cross-helicity hmvo=Ao⋅v contributed by the vector potential Ao of the open field Bo in the region as well. Only under two conditions, (1) Ap=A⊥ (A⊥ is the transverse component of Ao), (2) vz=0 (vz is the longitudinal component of v) or Az=0 (Az is the longitudinal component of Ao), can hmvo be merged into hmvp to give the pioneering dHR/dt equation shown in Equation (4) of Berger (1984). Results show that hmvo originates from v⋅∇ho (ho=Ao⋅Bo is the helicity density of the open field) and should also be considered in dealing with the development of relative helicity in active regions. Finally, the equation to calculate dHR/dt in active regions is synthesized and presented.
Published Version
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