Global Distribution of Current Helicity

The global distribution ofα in latitude and longitude is shown in Figure4.11on days 10, 100, and 910 of simulation run L4. From the initial potential field on day 0 (with α = 0), a pat- tern of intermixed positive and negativeαhas developed by day 10, simply due to photospheric shearing—this is before the first active region emergence. After about 100 days a clear latitudinal trend inα emerges, although there is still significant local variation in both strength and sign, in

4.7 Evolution of Current Helicity 78

Figure 4.12: Height distribution ofαin a latitudinal plane atφ= 180◦, on days 10, 100, and 910 of run L4. Left column shows contours ofα(white for positive and black for negative, saturation level±20×10−8_m−1_{), with green lines showing selected coronal magnetic field lines. Right}

column shows height profile of|α|, averaged over latitude in bins (of height1inz). Vertical bars show one standard deviation.

accordance with observations. Comparing Figures4.11(d), (e), and (f), it can be seen how the meanαat low latitudes (0◦to about50◦) develops into the observed hemispheric trend, although with considerable scatter as observed on the real Sun. However, at high latitudes the sign ofαis reversed. These polar reversals correspond to the East-West PILs at the polar crown boundaries, and move steadily poleward through the simulation as the polar crowns reduce in size towards polar field reversal (we are approaching solar maximum). This opposite sign ofα is caused by differential rotation of the predominantly North-South field lines at this latitude, and will be con- sidered further in Chapter5.

4.7 Evolution of Current Helicity 79

Figure 4.13: Global distribution ofα on day 283 for (top) observed sign of emerging twist (run A2), and (bottom) opposite sign of emerging twist (run Am2). Left column shows contours of

αat heightz = 1(white for positive and black for negative, saturation level±20×10−8m−1). Right column shows latitudinal profile, averaged over longitude in2◦ latitude bins. Vertical bars show one standard deviation.

Figure4.12shows how the distribution ofαvaries with height, in a latitudinal cut atφ= 180◦, for the same sequence of three days in run L4 as Figure4.11. This illustrates how non-zero values of

αare largely concentrated in closed field regions, and are not uniformly distributed in the corona. On open field lines, helicity is free to be lost through the top boundary, so that such regions are close to potential. In the right-hand column of Figure4.12, we see how the meanαdecreases with height, owing to the concentration of closed field structures lower in the corona.

In simulation L4 we insert bipoles with the “correct” sign of twist β in each hemisphere, i.e. matching the observed mean sign ofα in each hemisphere (Figure4.9). Does our coronal field have the correct sign of current helicity at active latitudes simply as a direct result of this inserted sign of twist? This is not the case. We have already described in Section4.7.2several sources of helicity in the model which do not depend on the sign ofβ. In Figure4.13, we compare the global distribution ofαon day 283 in runs A2 and Am2, i.e. with opposite signs ofβ. We see that in run Am2, with the “wrong” sign ofβin each hemisphere, the sign ofαis indeed reversed between0◦ and30◦ latitude. This originates from newly-emerging bipoles with the opposite sign ofβ, such as the bipole “B” in the green box in Figure4.13. However, above30◦ latitude, broadly the same

4.7 Evolution of Current Helicity 80

distribution ofαis obtained in both runs. This is because shearing by surface motions dominates the helicity production here, independent of the sign of emerging twist. Furthermore, the mean magnitude of the reversed-sign α at low latitudes in run Am2 is only about5×10−8m−1, as compared to10×10−8m−1 in run A2. So emerging twist is not the only factor responsible for production of α, even at active latitudes. This point will be further illustrated in Chapter 5 in relation to the physical mechanism responsible for the hemispheric pattern in filament chirality. Finally we consider the magnitude ofαvalues in our simulation. Figure4.11shows mean values ofαat active latitudes of the order10−7m−1. The actual maximum and minimum values recorded on day 910 of the simulation were2.24×10−6m−1 and−1.84×10−6m−1. A key result is that these values are much higher than those estimated from linear force-free extrapolations. Such solutions suffer a constraint on the maximumα in order to obtain a decay with height (Aulanier and Demoulin, 1998), requiring that α < 2π/Lx (the “first resonant value”), where Lx is the

horizontal length of the periodic box. The linear force-free model of an observed filament by

Aulanier, Srivastava, and Martin(2000) hasα= 2.3×10−8_m−1_{, and for the solutions ofMackay,}

Longbottom, and Priest(1999) this first resonant value was atα= 4.24×10−8m−1. By contrast, studies using nonlinear force-free extrapolations from vector magnetograms using the Grad-Rubin type method (Amari et al.,1997) find locally higher values ofα(e.g.,Bleybel et al.,2002). They are also more realistic because they allow variableαwithin a single region, as in our simulations. For a particular active region,R´egnier, Amari, and Kersal´e(2002) found maximum values of the order10−6_m−1_{, consistent with the results of our simulations.}

In summary, we have demonstrated the development of current helicity,α, in non-potential simu- lations over many solar rotations of the coronal magnetic field. We find a clear latitudinal pattern ofαthat persists throughout the simulation, although locally there is significant scatter and inter- mixing of both signs ofα, even within individual bipoles. The remaining chapters of this thesis will apply the model to study filaments and magnetic flux ropes, both of which depend crucially on twisted and sheared magnetic structures.

Chapter 5

The Hemispheric Pattern of Filaments

In this chapter we model the large-scale magnetic field structure of solar filaments (also known as prominences). Filaments provide an ideal method to directly observe the non-potential magnetic field structure in the low corona, because they are cool, dense structures suspended in the hotter, rarified corona. There are two complementary aims: firstly to test our global non-potential coronal simulations (Chapter4) against observations of real filaments, and secondly to explain a large- scale organisation in filament magnetic chirality. This hemispheric pattern, which underlies the long-term evolution of solar magnetic fields and of the solar dynamo, has been well observed but not yet satisfactorily explained by theory.

The layout of this chapter is as follows. Section5.1presents a summary of observations of filament magnetic fields, and of the hemispheric chirality pattern. Then in Section5.2we review previous theoretical work on the origin of filament chirality and the hemispheric pattern. For this study we have used an observed data set of 255 filaments, and this is described in Section5.3 along with the technique for determining the chirality of our observed filaments. Section5.4 presents the key results of this chapter, where we compare the skew of the simulated magnetic field with the observed chiralities of 109 filaments, for four simulation runs with different emerging bipole twist. When the sign of bipole twist matches the observed hemispheric sign of active region helicity, the simulations perform exceptionally well, producing the correct chirality type for up to 96% of filaments, including both filaments with the dominant chirality and minority cases. We then go on to consider the explanation for this performance. In Section 5.5 we describe eight physical mechanisms identified to be responsible for producing skew in our simulations. Section 5.6looks at the relative importance of each mechanism, and formulates an explanation for why the hemispheric pattern occurs in our simulation.

5.1 Observational Background 82

5.1

Observational Background

It has been known since the earliest magnetograph observations that filaments lie above polarity inversion lines (PILs) in the photospheric magnetic field (Babcock and Babcock,1955). In this thesis we consider only “quiescent” filaments—i.e. those which are long-lived and stable—which occur at a wide range of latitudes on the Sun, including within active regions.Tandberg-Hanssen

(1974, p.117) suggests the subdivision of low-latitude filaments into “Type A” and “Type B”, according to whether they overlie a PIL within a single bipolar magnetic region or a PIL between two neighbouring regions. Tang(1987) studied the magnetic location of 330 quiescent filaments from two periods (the declining phase of Cycle 20 and the maximum of Cycle 21), and found55% to be Type B in the first period and66%in the second. By extending the classification scheme,

Mackay, Gaizauskas, and Yeates(2008, see also Section3.7) show that actually many of the “Type A” filaments form between flux regions originating from multiple bipole emergences. Less than 8%of filaments in their sample form wholly within a single bipolar region. In this chapter we will demonstrate how the chirality of a filament depends on the location where it forms. This chirality property is considered next.