Abstract
The evolution dH R/dt of relative helicity H R provides a gauge-invariant measure of the helicity flow across the open surface S o 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 h mvp=A p⋅v contributed by the vector potential A p of a reference potential field B p, where v is the fluid velocity, but by another cross-helicity h mvo=A o⋅v contributed by the vector potential A o of the open field B o in the region as well. Only under two conditions, (1) A p=A ⊥ (A ⊥ is the transverse component of A o), (2) v z=0 (v z is the longitudinal component of v) or A z=0 (A z is the longitudinal component of A o), can h mvo be merged into h mvp to give the pioneering dH R/dt equation shown in Equation (4) of Berger (1984). Results show that h mvo originates from v⋅∇h o (h o=A o⋅B o 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 dH R/dt in active regions is synthesized and presented.
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Ma, Z. Evolution of relative helicity in active regions. Solar Physics 211, 189–198 (2002). https://doi.org/10.1023/A:1022405129558
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DOI: https://doi.org/10.1023/A:1022405129558