A linear least squares fit to the data of as a function of latitude (Figure 19a) found that , where is latitude in degrees, and that the r.m.s. scatter of from the linear fit is (Longcope et al., 1998). The observed systematic in solar active regions may reflect a systematic field line twist in the subsurface emerging flux tubes.
If the measured values are a direct consequence of the emergence of twisted magnetic flux tubes from the interior, then it would imply subsurface emerging tubes with a field line twist of (Longcope and Klapper, 1997), where denotes the angular rate of field line rotation about the axis over a unit axial distance along the tube. Several subsurface mechanisms for producing twist in emerging flux tubes have been proposed (see e.g. the review by Petrovay et al., 2006). The twist may be due to the current helicity in the dynamo generated toroidal magnetic field, from which buoyant flux tubes form at the base of the convection zone (Gilman and Charbonneau, 1999), or it may be acquired during the rise of the flux tubes through the solar convection zone (Longcope et al., 1998; Choudhuri, 2003; Choudhuri et al., 2004; Chatterjee et al., 2006).
Longcope et al. (1998) explain the origin of the observed twist in emerging active region flux tubes as a result of buffeting by the helical turbulence in the solar convection zone during the rise of the tubes. Applying the dynamic model of a weakly twisted thin flux tube (Section 2.1), Longcope et al. (1998) modeled the rise of a nearly straight, initially untwisted tube, buffeted by a random velocity field representative of the turbulent convection in the solar convection zone, which has a nonzero kinetic helicity due to the effect of solar rotation. The kinetic helicity causes helical distortion of the tube axis, which in turn leads to a net twist of the field lines about the axis in the opposite sense within the tube as a consequence of the conservation of magnetic helicity. This process is termed the -effect by Longcope et al. (1998). Quantitative model calculations of Longcope et al. (1998) show that the -effect can explain the hemispheric sign, magnitude, latitude variation, and the r.m.s. dispersion of the observed of solar active regions. Furthermore, because the aerodynamic drag force acting on flux tubes by the convective flows has an dependence, where is the tube radius, this model predicts that the resulting twist generated should also have an dependence, suggesting a systematic trend for greater twist in smaller flux tubes.
As discussed in Section 5.1.2, -shaped emerging loops are themselves acted upon by the Coriolis force, developing a “tilt” of the loop. This helical deformation of the tube axis will then also induce a twist of the field lines of the opposite sense within the tube as a consequence of conservation of magnetic helicity. Calculations based on the weakly twisted thin flux tube model show that this twist generated by the large scale tilting of the emerging -loop resulting from the Coriolis force has the right hemispheric sign and latitude dependence, but is of too small a magnitude to account for the observed twist in solar active regions (Longcope and Klapper, 1997; Fan and Gong, 2000).
Another interesting and natural explanation for the origin of twist in emerging flux tubes is the accretion of the background mean poloidal field onto the rising flux tube as it traverse through the solar convection zone (Choudhuri, 2003; Choudhuri et al., 2004; Chatterjee et al., 2006). In a Babcock–Leighton type dynamo, the dispersal of solar active regions with a slight mean tilt angle at the surface generates a mean poloidal magnetic field. The mean tilt angle of solar active regions is produced by the Coriolis force acting on the rising flux tubes (see Section 5.1.2). In the northern hemisphere, when a toroidal flux tube rises into a poloidal field that has been created due to the tilt of the same type of flux tubes emerged earlier, the poloidal field gets wrapped around the flux tube will produce a left-handed twist for the tube. This is illustrated in Figure 20.
Using a circulation-dominated (or flux-transport) Babcock–Leighton type mean-field dynamo model, Choudhuri et al. (2004) did a rough estimate of the twist acquired by an emerging flux tube rising through the solar convection zone. Figure 21 shows the resulting butterfly diagram indicating the sign of of the emerging regions as a function of latitude and time.
It is found that at the beginning of a solar cycle, there is a short duration where the sign of is opposite to the preferred sign for the hemisphere. This is because of the phase relation between the toroidal and poloidal magnetic fields produced by this types of solar dynamo models. At the beginning of a cycle, the mean poloidal magnetic field in the convection zone is still dominated by that generated by the emerging flux tubes of the previous cycle, and toroidal flux tubes of the new cycle emerging into this poloidal field gives rise to a right-handed (left-handed) twist of the tube in the northern (southern) hemisphere. However, for the rest of the cycle starting from the solar maximum, the poloidal magnetic field changes sign and the twist for the emerging tubes becomes consistent with the hemispheric preference. For the whole cycle, it is found that about 67% of the emerging regions have a sign of consistent with the hemispheric rule. The rough estimate also shows that the magnitude of the values produced by poloidal flux accretion is consistent with the observed values, and that there is an dependence on the radius of the emerging tube, i.e. smaller sunspots should have greater values. This prediction is also made by the -effect mechanism.
Given the frozen-in condition of the magnetic field, it is expected that the accreted poloidal flux be confined in a sheath at the outer periphery of the rising tube, and that in order to produce a twist within the tube, some form of turbulent diffusion needs to be invoked (Chatterjee et al., 2006). By solving the induction equation in a co-moving Lagrangian frame following the rising flux tube and using several simplifying assumptions, Chatterjee et al. (2006) modeled the evolution of the magnetic field in the rising tube cross-section as a result of poloidal flux accretion and penetration due to a field strength dependent turbulent diffusivity. They found that with plausible choices of assumptions and parameter values an value comparable to the observations is obtained.
When a buoyant magnetic flux tube formed at the base of the solar convection zone from the dynamo generated (pre-dominantly) toroidal magnetic field, it should already obtain an initial twist due to the weak poloidal mean field contained in the magnetic layer. This initial twist will then be further augmented or altered due to poloidal flux accretion and also due to the -effect as the tube rises through the solar convection zone. MHD simulations of the formation and rise of buoyant magnetic flux tubes directly incorporating the mean field profiles from dynamo models (for both the fields at the base and in the bulk of the convection zone) as the initial state is necessary to quantify the initial twist and contribution from poloidal field accretion. Such simulations should be done for dynamo mean fields at different phases of the cycle to access the cycle variation of the twist in the emerging flux tubes.
Observationally, looking for any systematic variations of the the value (or twist) of solar active regions with the solar cycle phase is helpful for identifying the main mechanisms for the origin of the twist. So far results in this area have been inconclusive (Bao et al., 2000; Pevtsov et al., 2001). On the other hand, observational studies of the correlation between active region twist (as measured by ) and tilt angles have revealed interesting results (Holder et al., 2004; Tian et al., 2005; Nandy, 2006). The -effect predicts that the twist being generated in the tube is uncorrelated to the local tilt of the tube at the apex (Longcope et al., 1998). However, due to the Coriolis force, active region -tubes acquire a mean tilt that has a well defined latitudinal dependence as described by the Joy’s law. The mean twist generated by the -effect also has a latitudinal dependence that is consistent with the observed hemispheric rule. Thus, due to the mutual dependence of their mean values on latitude there should be a correlation between the tilt angle and twist of solar active regions and the correlation is expected to be positive if one assigns negative (positive) sign to a clockwise (counter-clockwise) tilt. However, Holder et al. (2004) found a statistically significant negative correlation between the twist and tilt for the 368 bipolar active regions studied, opposite to that expected from the mutual dependence on latitude of mean twist and mean tilt of active regions. Removing the effect of the mutual dependence of the mean tilt and mean twist on latitude (by either determining the correlation at a fixed latitude, or by subtracting off the fitted mean tilt and mean twist at the corresponding latitude), they found that the negative correlation is enhanced. Furthermore, it is found that the negative correlation is mainly contributed by those active regions (174 out of 368 regions) that deviate significantly from Joy’s law (by ), while regions that obey Joy’s law to within show no significant correlation between their twist and tilt. A separate study by Tian et al. (2005) found that a sample of 104 complex -configuration active regions, more than half of which have tilts that are opposite to the direction prescribed by Joy’s law, show a significant negative correlation between their twist and tilt (after correcting for their definition of the sign of tilt which is opposite to the definition used in Holder et al. (2004)). The results of Holder et al. (2004) and Tian et al. (2005) both indicate that there is a significant population (about one half in the case of Holder et al. (2004)) of solar active regions whose twist/tilt properties cannot be explained by the -effect together with the effect of the Coriolis force alone. These active regions are consistent with the situation where the buoyant flux tube form at the base of the solar convection zone has acquired an initial twist, such that as it rises upward due to buoyancy into an -tube, it develops a writhe that is of the same sense as the initial twist of the tube. In the extreme case, the twist can be so large that the flux tube becomes kink unstable (see Section 5.6). The resulting tilt at the apex due to the writhe has the negative correlation with the twist as described in the above observations.
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