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These simulations have considered twisted, buoyant toroidal flux tubes at the base of the solar convection zone with an initial field strength of 105 G, being 10 times the equipartition field strength, and thus have neglected the effect of convection. The main finding from these simulations is that the twist of the tube induces a tilt at the apex of the rising -tube that is opposite to the direction of the observed mean tilt of solar active regions, if the sign of the twist follows the observed hemispheric preference. It is found that in order for the tilt driven by the Coriolis force to dominate, such that the emerging -tube shows a tilt consistent with Joy’s law of active region mean tilt, the initial twist rate of the flux tube needs to be smaller than about a half of that required for the tube to rise cohesively. Under such conditions, the buoyant flux tube is found to undergo severe flux loss during its rise, with less than 50% of the initial flux remaining in the final -tube that rises to the surface (see Figures 26a and 26b).
Furthermore, it is found that the Coriolis force drives a retrograde flow along the apex portion, resulting in a relatively greater stretching of the field lines and hence stronger field strength in the leading leg of the tube. With a greater field strength, the leading leg is more buoyant with a greater rise velocity, and remains more cohesive compared to the following leg (see Figure 26a). Figure 26c shows selected field lines threading through the coherent apex cross-section of the final -tube, resulting from the simulation of a weakly twisted buoyant tube described in Fan (2008, see the LNT run in that paper). It can be seen that the field lines in the leading side are winding about each other smoothly in a coherent fashion, while the field lines in the following side are significantly more frayed. By evaluating the local twist rate given by , where is the electric current density, along each of the selected field lines as a function of depth, Fan (2009a) found that field lines in the leading leg show more coherent values of , whereas the field lines in the following leg show significantly larger fluctuations and mixed signs of local twist (see Figure 27).
Although the mean value of averaged over the field lines is not systematically greater along the leading leg compared to the following (Fan, 2009a), the greater buoyancy and hence higher rise velocity of the leading leg can give rise to a greater upward helicity flux in the leading polarity comparing to the following as a result of the emergence of the -tube (Fan et al., 2009). Furthermore, based on a simplified model of active region flux emergence into the corona by Longcope and Welsch (2000), Fan et al. (2009) show that a stronger field strength in the leading tube also produces a faster rotation of the leading polarity sunspot driven by torsional Alfvén waves along the flux tube. This also contributes to a greater helicity injection rate in the leading polarity of an emerging active region. Observational study of bipolar emerging active regions by Tian and Alexander (2009) have found that the helicity injection rate is about 3 – 10 times greater in the (compact) leading polarity than the (fragmented) following polarity.
Jouve and Brun (2007) have also carried out anelastic MHD simulations in a rotating spherical shell geometry to study the buoyant rise of an axisymmetric toroidal flux ring in an isentropically stratified (non-convecting) envelope. They have considered a even greater initial field strength of 1.8 × 105 G for the initial toroidal flux ring. As was discussed in Fan (2008), the poleward deflection of the rise trajectory of the tube due to the Coriolis force is far more severe for an axisymmetric toroidal ring (where the whole ring is moving away from the rotation axis of the Sun) than for a localized 3D -shaped tube (see Section 3.1 in Fan, 2008). Thus an initial field strength of 1.8 × 105 G is needed for an axisymmetric toroidal ring to rise nearly radially (Jouve and Brun, 2007). The simulations of Jouve and Brun (2007) also recovered the previous results from Cartesian simulations that if the flux tube is not twisted, it splits into two counter rotating vortices before reaching the top of the envelope.
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