2.6 Twist and helicity in sunspots’ magnetic field

Let us define the angle of twist of a sunspot’s magnetic field, Δ, as the angle between the magnetic field vector B at a given point of the sunspot and the radial vector that connects that particular point with the sunspot’s center, r (Equation (47View Equation)):
Δ = cos−1 -Br--- . (26 ) |B||r|

Note that in Section 1.3.2 this angle Δ is precisely the quantity that was being minimized when solving the 180°-ambiguity (Equation (9View Equation)). However, minimizing it does not guarantee that Δ will be zero. This is, therefore, the origin of the twist: a deviation from a purely radial (i.e., parallel to r) magnetic field in the sunspot. Figure 19View Image shows maps of the twist angle Δ for two different sunspots at two different heliocentric angles. These two examples illustrate that the magnetic field vector is radial throughout most of the sunspot, but there are regions where significant deviations are observed. These deviations could be already seen in the arrows in Figures 5View Image and 6View Image representing the magnetic field vector in the plane of the solar surface. In addition, in these two examples the sign of the twist (wherever it exists) remains constant for the entire sunspot.

Twisted magnetic fields in sunspots have been observed for a very long time, going back to the early works of Hale (1925, 1927) and Richardson (1941), who observed them in Hα filaments. They established what is known as Hale’s rule, which states that sunspots in the Northern hemisphere have a predominantly counter-clockwise rotation, whereas it is clockwise in the Southern hemisphere. However, sunspots violating Hale’s rule are common if we attend only at H α filaments (Nakagawa et al., 1971). A better estimation of the twist in the magnetic field lines can be obtained from spectropolarimetric observations. To our knowledge, the first attempts in this direction were performed by Stepanov (1965).

Twist in sunspots can also be studied by means of the α-parameter in non-potential force-free magnetic configurations: ∇ × B = αB. Another commonly used parameter is the helicity: R H = V A ⋅ BdV, where B is the magnetic field vector and A represents the magnetic vector potential. As demonstrated by Tiwari et al. (2009a) the value of α corresponds to twice the degree of twist per unit axial length. In addition, α and H posses the same sign. Thus, any of these two parameters can be also employed to study the sign of the twist in the magnetic field vector. Using these parameters Pevtsov et al. (1994) and Abramenko et al. (1996) found a good correlation (up to 90%) between the sign of the twist and the hemisphere where the sunspot appear (Hale’s rule).

Recent works, however, find large deviations from Hale’s rule (Pevtsov et al., 2005; Tiwari et al., 2009b). It has been hypothesized that these deviations from Hale’s rule might indicate a dependence of the twist with the solar cycle (Choudhuri et al., 2004). Other possible explanations for the twist of the magnetic field in sunspots, in terms of the solar rotation and Coriolis force, have been offered by Peter (1996) and Fan and Gong (2000). In particular, the former work is also able to explain the deviations from Hale’s rule observed in H α filaments. However, a definite explanation is yet to be identified. This might be more complicated than it seems at first glance because different twisting mechanisms might operate in different regimes and atmospheric layers. This is supported by recent spectropolarimetric observations that infer a twist in the magnetic field that can change sign from the photosphere to the chromosphere (Socas-Navarro, 2005a).

View Image

Figure 19: Same as Figure 7View Image but for twist angle of the magnetic field in the plane of the solar surface: Δ (Equation (26View Equation)).

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