Evolution of relative helicity in active regions

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_pv 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_ov 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 vh_o (h_o=A_oB_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.