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

Abstract

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.

Figures (20)

• Figure 1: Volume renderings of the typical initial conditions used in these simulations. (a) An initial magnetic flux tube, with its magnetic intensity rendered in green, is placed at $x_c=3.0$, $z_c=1.25$ in a statistically stationary realization of overshooting convection, where a convection zone (CZ) overlays a radiative (or convectively-stable) zone (RZ). The convective vertical velocities, $w$, are rendered with downflows in red and upflows in blue. The side view (b) shows that the magnetic flux tube initially sits clearly in region of overshooting convection just below the CZ. (c) An example of a three tube magnetic initial condition. Tubes are placed at the same height as previously, but equally spaced in the periodic $x$ direction.
• Figure 2: Several magnetic field lines (blue) of the magnetic initial condition, zoomed in on the tube within the computational domain. The magnetic field lines exhibit right-handed helicity (bearing in mind that the axial field is oriented in the positive axial $y$ direction) exhibited by the red arrows, and all loop around the central axis of the tube (black straight line in the tube) the same number of times regardless of their position within the tube, rendering the winding number of this initial condition well-defined.
• Figure 3: Diffusive decay of $\mathcal{W}$ versus time for five different parameter pairs ($Ro, \phi$) in the absence of convection. Parameters of the simulations plotted encompass $Ro=0.5,1.0$ with $\phi=+90^\circ$ and $\phi=-90^\circ$, as well as a case with no rotation ($Ro=\infty$). The colored lines associated with each case here will pervade throughout this paper. The nearly identical decay of $\mathcal{W}$ at various rotational influences suggests that the decay of $\mathcal{W}$ in the absence of convection is purely diffusive and is independent of rotation. The red line shows the average non-convective decay of $\mathcal{W}$ which will be used as a reference for later plots.
• Figure 4: Example of the typical evolution of a simulation. Volume renderings of both the magnetic intensity, $\bf{B}^2$ (green=yellow tones), and the vertical velocity, $w$ (shown with downflows in red tones and upflows in blue). In the volume rendering, the opacity is set to emphasize high (absolute) values of each field. Time evolves from the top row to the bottom for the two views shown (angled view and top down view). This case is shown here simply as a clear example of a magnetic flux tube rising through rotationally-influenced convection, but is the case with $Ro=1.0$ in the southern hemisphere ($\phi=-90^{\circ}$) used later in this paper. The times shown are $t=1.04, 1.27, 1.45$ (top to bottom).
• Figure 5: A further example of the general evolution of a simulation, this time shown via the magnetic fieldlines. This case corresponds exactly to the parameters and times of the previous figure, Fig. \ref{['fig:sh']} ($Ro=1.0, \phi=-90^{\circ}, t=1.04, 1.27, 1.45$. Several randomly chosen magnetic field lines from the many used in the calculation of $\mathcal{W}$ are shown in red. Several randomly chosen field lines of the initial condition are also shown in blue for reference. The black line shows the axis of the tube, discerned as the peak magnetic intensity in the $x-z$ plane at any $y$. The grey lines on the coordinate axis planes show the projection of the flux tube axis onto the respective planes. These examples demonstrate the clear deformation of the magnetic flux tube as it rises through the rotationally-influenced convection.