HEMISPHERIC HELICITY TREND FOR SOLAR CYCLE 24

The concept of magnetic helicity was introduced to solar physics in the 1980s (Heyvaerts & Priest 1984; Berger & Field 1984) and has attracted great attentions since then. It is a physical quantity that measures the topological complexity of magnetic field such as the degree of linkage or twistedness in the field (Moffatt 1985; Berger & Field 1984) and has been considered important in modeling many solar phenomena such as coronal mass ejections (Zhang & Low 2005; Zhang et al. 2006; Zhang & Flyer 2008). The helicity of magnetic fields may be characterized by several different parameters (Moffatt 1978) such as magnetic helicity (H_m) and current helicity (H_c). However, only the vertical component of current helicity density h_c (and the local twist α, etc.) can be practically computed by using vector magnetograms.

Seehafer (1990) was the first to statistically study the sign of magnetic helicity of solar active regions (ARs) using magnetograms. He estimated current helicity h_c of 16 ARs by using extrapolation of measured photospheric magnetic fields and concluded that in ARs the current helicity is predominantly negative in the northern hemisphere and positive in the southern hemisphere. This tendency is the so-called hemispheric helicity sign rule. In the following two decades, many researchers (Pevtsov et al. 1995, 2001, 2008; Abramenko et al. 1997; Bao & Zhang 1998; Hagino & Sakurai 2004; Zhang 2006) have studied and confirmed this rule by using data sets obtained with different instruments located in different places of the world, e.g., the University of Hawaii Haleakala Stokes Polarimeter and The Marshall Space Flight Center Vector Magnetograph in the US, the Solar Magnetic Field Telescope (SMFT) in China, and the Mitaka Solar Flare Telescope and The Okayama Astrophysical Observatory Solar Telescope in Japan. It is believed that the usual hemispheric helicity sign rule holds for all three solar cycles observed (that is, solar cycles 21, 22, 23).

However, there is also some debate about this rule. For instance, Bao et al. (2000) found that h_c in their data showed an opposite hemispheric preference at the beginning of solar cycle 23. Hagino & Sakurai (2005) also reported that the hemispheric helicity sign rule may not be satisfied in the solar minimum phase. Choudhuri et al. (2004) developed a model that predicts deviations from the usual hemispheric rule at the beginning of a solar cycle. However, Pevtsov et al. (2001) argued that the usual hemispheric helicity sign rule still holds for the first four years of solar cycle 23, although by nature it is a weak rule with significant scatter. Pevtsov et al. (2008, p. 722) further compared data from four different instruments and concluded that "the notion that the hemispheric helicity rule changes sign in some phases of solar cycle is not supported at a high level of significance."

Apart from these arguments, Zhang (2006) did a statistical study using 17,200 vector magnetograms obtained by SMFT. She separated her data into two parts, the weak fields (100 G <|B_z| < 500 G) and the strong fields (|B_z| >1000 G). She calculated α and h_c of weak and strong fields separately and found that the weak magnetic fields follow the usual hemispheric helicity sign rule but strong fields do not. She interpreted this as the reason why Bao et al. (2000) found that the h_c in their data violates the usual hemispheric helicity sign rule whereas α not.

Since its launch in 2006 September, Hinode has provided us with high spatial resolution vector magnetograms for both the descending phase of solar cycle 23 and the ascending phase of solar cycle 24. This gives us a unique chance in this Letter to use these so-far most accurate vector magnetic field measurements to shed light on the above arguments. We organize our Letter as follows. In Section 2, we describe the observations and data reduction. In Section 3, we present our analysis and results. We conclude with a discussion in the last section.

We used vector magnetograms obtained by the Spectro-polarimeter (SP) on board Hinode (Kosugi et al. 2007). SP/Hinode obtains line profiles of two magnetically sensitive Fe lines at 630.15 and 630.25 nm and nearby continuum, using a 016 × 164'' slit. There are four mapping modes of operation: normal map, fast map, dynamics, and deep magnetogram (Tsuneta et al. 2008). In this study, we only use the normal maps and fast maps. The resolution of these magnetograms is about 032 pixel⁻¹ for fast maps and 016 pixel⁻¹ for normal maps.

For the period we studied, that is, from 2006 November to 2010 September, in total 190 ARs appeared on the Sun; that is, from NOAA 10921 to NOAA 11110. However, not every AR has been observed by SP/Hinode. We searched those ARs observed by SP/Hinode using following criteria. (1) If more than one magnetograms have been obtained for the same AR, then we only use the one that is closest to the disk center. (2) Both the longitude and latitude of the AR when observed are within 40° from the disk center. This gives a total number of 64 ARs to form the sample, including 30 ARs in solar cycle 23 and 34 ARs in solar cycle 24.

The SP data are calibrated and inverted at the Community Spectro-polarimetric Analysis Center (CSAC, http://www.csac.hao.ucar.edu/). The inversion is based on the assumption of the Milne–Eddington atmosphere model and a nonlinear least-square fitting technique is used to fit analytical Stokes profiles to the observed profiles. The inversion gives 36 parameters including the three components of magnetic field (field strength B, field inclination γ, and field azimuth ϕ), the stray light fraction (1 − f, where f is the filling factor), and so on. The 180° azimuth ambiguity was resolved by setting the directions of the transverse fields most closely to a current-free field, an approach that was used in most other studies.

We calculated two different helicity parameters, α_z and α_hc, for these 64 ARs. α_z is the mean value of local twist, defined as

$$\begin{equation} \alpha _{z}=\overline{(\bigtriangledown \times \textbf {\emph {B}})_{z}/B_{z}}. \end{equation} \tag{ 1 }$$

α_hc is the normalized mean current helicity density, obtained by

$$\begin{equation} \alpha _{{\rm hc}}= \frac{\sum (\bigtriangledown \times \textbf {\emph {B}})_{z} B_{z}}{\sum B_{z}^{2}}. \end{equation} \tag{ 2 }$$

Both the averaging and integral are done over the whole magnetogram. The definition here gives the parameter α_hc the same units of α_z and is the same as the α_g parameter discussed in Tiwari et al. (2009). In our calculation we have only used points whose total wavelength-integrated polarization is larger than 10⁻², which is about three times of the polarization noise level (Lites et al. 2008). This is a criterion applied to all helicity parameter calculations, in addition to all other criteria we apply in following analysis.

In calculating α_z and α_hc, we have used two different representations of magnetic field measurement. One is related to "flux density," where the longitudinal magnetic field B_z = f · Bcos (γ) and the transverse magnetic field $B_t = \sqrt{f} \cdot B \sin (\gamma)$. The other is the "field strength" where B_z = Bcos (γ) and B_t = Bsin (γ). Hereafter, we present the first type as B¹_z, B¹_t and the second type as B²_z, B²_t. Correspondingly helicity parameters are also hereafter presented as α¹_z, α¹_hc and α²_z, α²_hc, respectively. In most previous studies researchers used the helicity parameters of the first type, that is, based on the flux density measurement of magnetic field. Due to the precise measurement of SP on board Hinode, an accurate measurement of filling factor and hence of field strength becomes possible. Thus, in this Letter we calculate the helicity parameter of the second type too, in order to check whether our results depend on the type of magnetic field measurement or not.

Figure 1 presents the variation of α¹_z (left panels) and α²_z (right panels) with the solar latitude for the 30 ARs in the descending phase of solar cycle 23 (top panels), the 34 ARs in the ascending phase of solar cycle 24 (middle panels), and the total 64 ARs (bottom panels). Here, α¹_z and α²_z are calculated only using points with |B¹_z| > 100 G or |B²_z| > 100 G. The solid lines indicate the results of least-square linear fits.

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Similarly, Figure 2 gives the variation of α¹_hc (left panels) and α²_hc (right panels) with the solar latitude for the 30 ARs in solar cycle 23 (top panels), the 34 ARs in solar cycle 24 (middle panels), and the total 64 ARs (bottom panels). The α¹_hc and α²_hc are also calculated only using points with |B¹_z| > 100 G or |B²_z| > 100 G.

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Values of dα/dθ from the linear fittings are also shown in Figures 1 and 2, in the unit of 10⁻⁹ m⁻¹ deg⁻¹. Here we see that for the 30 ARs of solar cycle 23, dα/dθ for α¹_z, α²_z, α¹_hc, and α²_hc are all positive. Out of these 30 ARs, only 8 (27%) ARs of α¹_z and 14 (47%) ARs of α¹_hc obey the usual hemisphere sign rule. This means that ARs in the descending phase of solar cycle 23 do not follow the usual hemispheric helicity sign rule. This is consistent with Tiwari et al. (2009) who made a similar conclusion from a sample combining data from three instruments.

Contrary to that in solar cycle 23, for the 34 ARs of solar cycle 24, 20 (59%) ARs of the α¹_z and 20 (59%) ARs of the α¹_hc obey the usual hemisphere sign rule. dα/dθ for α¹_z, α²_z, α¹_hc, and α²_hc are all negative. This means that ARs in the ascending phase of solar cycle 24 follow the usual hemispheric helicity sign rule, contrary to the prediction made in Choudhuri et al. (2004). Note that ARs in the descending phase of solar cycle 23 do show a deviation from the usual hemispheric helicity sign rule. We speculate that the physical process described in Choudhuri et al. (2004), that is, poloidal flux lines getting wrapped around a toroidal flux tube rising through the convection zone to give rise to the helicity, may still apply, but a phase shift may be required in the dynamo model.

For all of the 64 ARs, 28 (44%) ARs of α¹_z and 34 (53%) ARs of α¹_hc follow the usual hemisphere sign rule. As a whole, these 64 ARs still follow the usual hemispheric helicity sign rule, with dα/dθ for α¹_z, α²_z, α¹_hc, and α²_hc all negative. This is consistent with the results from most previous studies, that is, most ARs follow the usual hemispheric helicity sign rule.

An interesting observation is that, despite for the fact that we have used the so-far most accurate measurement of vector magnetic field given by SP/Hinode, the hemispheric helicity sign rule, either indicated by the 34 ARs in solar cycle 24 or by the 64 ARs as a whole, is still weak with large scatters. As evidence, we see from Figures 1 and 2 that the magnitudes of the correlation coefficients between the latitude and the helicity parameters are all low, with the maximum magnitude only being 0.21. This seems to indicate that the large scatter is an inherent property of the rule, not caused by the measurement errors. This is consistent with the prediction in Longcope et al. (1998), where helicity is considered to be produced in the process of magnetic flux tubes rising through the solar convection zone and being buffeted by turbulence with a non-vanishing kinetic helicity (Σ-effect).

When calculating α¹_z, α²_z, α¹_hc, and α²_hc in Figures 1 and 2 we have only used points with |B¹_z| > 100 G or |B²_z| > 100 G. Next, we progressed further to calculating α¹_z, α²_z, α¹_hc, and α²_hc for |B¹_z| or |B²_z| > 200, 300, 400 G, and so on until for |B¹_z| or |B²_z| > 2000 G. This not only allows us to check whether our results depend on the selection of the |B_z| threshold, but also gives us a chance to examine how the hemispheric helicity sign rule might vary with the increase of field strength.

Results of the obtained dα/dθ with different |B_z| thresholds are plotted in Figure 3 for all four helicity parameters. We see here that when changing the |B_z| threshold from 100 G to 200 G or even to 500 G, the sign of dα/dθ does not change. This suggests that our above conclusion, that is, ARs in the descending phase of solar cycle 23 do not follow the usual hemispheric helicity sign rule and the ARs in the ascending phase of solar cycle 24 do, is not very sensitive to the |B_z| threshold we selected.

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At the same time, we see from the middle and bottom panels of Figure 3 that, when the |B_z| threshold goes to high values such as 1200 G for α¹_z and α¹_hc or 1800 G for α²_z and α²_hc, the sign of dα/dθ becomes opposite to those with low |B_z|. This suggests that the weak and strong fields have opposite helicity signs, as first pointed out in Zhang (2006) and later confirmed by Su et al. (2009).

Zhang (2006) only points out that strong and weak fields have opposite helicity sign, here we use two examples to show that this actually indicates that on average sunspot umbra and penumbra show opposite helicity sign. In Figure 4 we show the continuum image (top left panel) and the electric current map (top right panel) of NOAA 10940, which appeared on 2007 February 1 of the solar cycle 23. Circles in these two images show where the distances to the sunspot center (r) are 5'', 10'', 15'', and 20'', respectively. The middle right panel shows the variation of electric current $J_{z}^1=\mu _0 (\bigtriangledown \times \textbf {\emph {B}})_{z}^1$ with r, where the dots give the values of J¹_z and the double-shelled curve shows the mean value of J¹_z with a bin of 1'' in r. Similarly, the bottom left and right panels, respectively, show the h¹_c = J_z¹B¹_z and α¹_z values and their averages with a bin of 1'' in r. We see clearly here that the inner most fields (where r < 5'') have a positive average value of h¹_c or α¹_z and the average value becomes negative when r > 5''.

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The mean value of |B_z| in the central umbra (r ⩽ 5'') is 2976 G and is 970 G for fields in 5'' < r ⩽ 20''. The mean value of α¹_z in r ⩽ 5'' is 5.033 × 10⁻⁸ m⁻¹ and is −0.717 × 10⁻⁸ m⁻¹ for regions in 5'' < r ⩽ 20''. The mean value of h¹_c in r ⩽ 5'' is 3.942 × 10⁻² G² m⁻¹ and is −0.543 × 10⁻² G² m⁻¹ for regions in 5'' < r ⩽ 20''. For the whole AR, with |B_z| > 100 G, α¹_z = −3.274 × 10⁻⁹ m⁻¹ and α¹_hc = −1.332 × 10⁻⁸ m⁻¹. This means that the sign of the whole AR is dominated by the sign of weak field (penumbra), as also pointed out in Zhang (2006).

Figure 5 gives another example, NOAA 11084, observed on 2010 July 2 of the solar cycle 24. As in Figure 4, the top two panels present the continuum image of the sunspot and the corresponding electric current distribution. Here the circles represent where r is 5'', 10'', and 15'', respectively. Similar trend as that in Figure 4 can be seen from the bottom panels of h¹_c and α¹_z. Here the average h¹_c or α¹_z values change their sign at about 4''. The mean value of |B_z| in r ⩽ 5'' is 2382 G and is 713 G in 5'' < r ⩽ 20'' region. The mean value of α¹_z in r ⩽ 5'' is −1.300 × 10⁻⁸ m⁻¹ and is 2.950 × 10⁻⁸ m⁻¹ in 5'' < r ⩽ 20'' region. The mean value of h¹_c in r ⩽ 5'' is −0.901 × 10⁻² G² m⁻¹ and is 0.315 × 10⁻² G² m⁻¹ in 5'' < r ⩽ 20''. For the whole AR, α¹_z = 3.599 × 10⁻⁸ m⁻¹ and α¹_hc = 1.910 × 10⁻⁸ m⁻¹ with |B_z| > 100 G. We see here again that the inner umbra and outer penumbra has the opposite helicity sign and the helicity sign of the whole AR is dominated by the sign in penumbra.

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Note that Chatterjee et al. (2006) modeled the penetration of a poloidal field into a toroidal rising flux tube through turbulence diffusion. One important prediction of their model is the existence of a ring of reverse current helicity on the periphery of ARs. Our observations seem consistent with their prediction.

Using high-quality magnetograms taken with SP/Hinode we examined the hemispheric helicity sign rule in the descending phase of solar cycle 23 and the ascending phase of solar cycle 24. We studied two helicity parameters, α_z and α_hc, of 64 ARs, 30 belonging to solar cycle 23 and 34 belonging to solar cycle 24. We also examined how the hemispheric helicity sign rule depends on the selection of field points and whether strong and weak fields have opposite helicity signs as previously reported.

Our analysis gives following results. (1) The 34 ARs in the ascending phase of the solar cycle 24 follow the so-called hemispheric helicity sign rule. (2) The 30 ARs in the descending phase of the solar cycle 23 do not follow the usual hemispheric helicity sign rule. (3) When combining all 64 ARs as one sample, the usual hemispheric helicity rule is indicated as in most other observations. (4) Even though we have used the so-far most accurate measurement of vector magnetic field given by SP/Hinode, the observed hemispheric helicity sign rule is still weak with large scatters. (5) The data show evidence of opposite helicity signs between strong and weak fields, and this indicates that the helicity parameters change sign from the inner umbra to the outer penumbra.

We argue that results (1), (3), and (4) are consistent with the model by Longcope et al. (1998), and result no. (5) is consistent with the model by Chatterjee et al. (2006). Results nos. (1) and (2) seem to suggest that the result of Choudhuri et al. (2004) has merit in its physical picture, but it may need to be modified based on which phase of the solar cycle the deviations from the hemispheric rule take place. From our observations we speculate that both the Σ-effect (Longcope et al. 1998) and the dynamo (Choudhuri et al. 2004; Chatterjee et al. 2006) have contributed in the generation of helicity, and in both models turbulence in the convection zone has played an important role.

We thank the referee for helpful comments and suggestions. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in cooperation with ESA and NSC (Norway). Hinode SOT/SP Inversions were conducted at NCAR under the framework of the Community Spectro-polarimetric Analysis Center (CSAC, http://www.csac.hao.ucar.edu/). This work was partly supported by the National Natural Science Foundation of China (grant no. 10921303), Knowledge Innovation Program of Chinese Academy of Sciences (grant no. KJCX2-EW-T07), and National Basic Research Program of MOST (grant no. 2011CB811401).