Evaluating the $Σ$-effect Model of the Solar Hemispherical Helicity Bias via Direct Numerical Simulations

The Solar Hemispherical Helicity Rule(s) (SHHR) is a term used to represent a bias observed in proxies for the magnetic helicity in active regions at the solar surface. The SHHR states that predominantly negative magnetic helicity is observed in active regions in the northern hemisphere, whereas predominantly positive is found in the southern. The $Σ$-effect model of \cite{longcope1998flux} is one of the most cited models for the explanation of the SHHR. In this model, the magnetic structures derive the bias in their magnetic helicity from the kinetic helicity of the turbulent convection through which they travel, where the latter is handed owing to the rotational influence of the star. The original paper built an elegant mathematical model for the dynamics of thin flux tubes influenced by parameterized helical turbulence. Here, we attempt to explore the conceptual ideas of this original simplified model using fully-nonlinear, three-dimensional, Cartesian-domain simulations of isolated, finite cross-sectional, twisted magnetic flux structures rising though rotating, overshooting, turbulent compressible convection. We look for evidence of a correlation between the kinetic helicity content of the turbulence and the evolving magnetic helicity of the structures. We find little evidence of such a relationship, and do not even find any clear hemispheric dependence. Although these simulations are far from a perfect representation of the ideas, this work raises many questions about the potential efficacy of the $Σ$-effect in reality.