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.