Non-Vanishing Angular Momentum Density in the Photospheric Horizontal Motions Induces Magnetic Helicity in the Solar Corona in Agreement with the “Hemisphere Rule”

The presented work revisits the Statistical Injection of Condensed Helicity (STITCH) model by Antiochos et al. 2013; Mackay et al. 2014; Dahlin et al. 2022. The model includes a statistical description of the small-scale circulation motions in the solar photosphere and accounts for their effect on the magnetic helicity in the solar corona.

The statistical effect of small-scale circulation motions may be quantified following well-known analogy between the density of the magnetic moment of microscopic currents in magnetic media, on one hand, and the angular momentum density, on the other hand. Within the framework of magnetostatics the magnetic moment density of the microscopic currents (referred to as magnetization) is a statistical quantity characterizing the magnetized medium. Its gradient in non-uniform medium results in macroscopic magnetization current. Similarly, the angular momentum density may be involved as the statistical characteristic of the small-scale horizontal motions in the photosphere. In this application only the vertical component of the angular momentum density,  ζ, matters, which is ideologically close to the parameter used in the STITCH model.

Analogously to the magnetization current in magnetostatics, the horizontal gradient in ζ  would result in large-scale horizontal motion.  Indeed, for a uniform isotropic turbulence, the chaotic small-scale and high-frequency velocity would cancel in average. However, with any horizontal gradient in  ζ  the larger rotational velocity of a stronger nearby vortex is not fully balanced by the opposite rotation of a smaller vortex, thus resulting in the averaged larger-scale velocity.

This velocity is perpendicular to the horizontal gradient of vertical magnetic field, B_R or, equivalently, it is aligned with the level contours, B_R=const, of the vertical field. Such motion drags the footpoints of the field lines of the coronal magnetic field, thus resulting in generation and accumulation of the magnetic helicity. The average velocity field parameterized in terms of gradients in  ζ  may be used as the boundary condition for an analytical or numerical model of the solar corona. Particularly, it is implemented in the SWMF code of the University of Michigan and used to pump helicity and the magnetic free energy of the active region to bring it to the threshold of eruption

Assuming that the sign of  ζ   is the same as that of projection of the solar angular velocity vector on the radial direction, it should be mostly positive in the northern hemisphere and mostly negative in the southern hemisphere. The rate of magnetic helicity production appears to be proportional to the negative of  ζ.   Hence, the described mechanism may result in the magnetic helicity in the solar corona such that the negative magnetic helicity dominates in the northern hemisphere and the positive magnetic helicity dominates in the southern hemisphere. The latter conclusion agrees with the so called “hemisphere rule” as confirmed by statistical analysis of observations.