Motivated by the observations of -sunspots, MHD calculations of the evolution of highly twisted, kink unstable magnetic flux tubes in the solar convection zone have been carried out (Linton et al., 1996, 1998, 1999; Fan et al., 1998b, 1999). For an infinitely long twisted cylindrical flux tube with axial field , azimuthal field , and plasma pressure in hydrostatic equilibrium where , a sufficient condition for the flux tube to be kink unstable is (see Freidberg, 1987)et al. (1996) considered the linear kink instability of uniformly twisted cylindrical flux tubes with being constant, confined in a high plasma. They found that the equilibrium is kink unstable if exceeds a critical value , where is the coefficient for the term in the Taylor series expansion of the equilibrium axial magnetic field about the tube axis: . This result is consistent with Suydam’s criterion. They further argued that an emerging, twisted magnetic flux loop will tend to have a nearly uniform along its length since the rise speed through most of the solar convection zone is sub-Alfvénic and torsional forces propagating at the Alfvén speed will equilibrate quasi-statically. Meanwhile expansion of the tube radius at the apex as it rises will result in a decrease in the critical twist necessary for the instability. This implies that as a twisted flux tube rises through the solar interior, a tube that is initially stable to kinking may become unstable as it rises, and that the apex of the flux loop will become kink unstable first because of the expanded tube cross-section there (Parker, 1979; Linton et al., 1996).
The non-linear evolution of the kink instability of twisted magnetic flux tubes in a high- plasma has been investigated by 3D MHD simulations (Linton et al., 1998, 1999; Fan et al., 1998b, 1999). Fan et al. (1998b, 1999) modeled the rise of a kink unstable flux tube through an adiabatically stratified model solar convection zone.
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In the case where the initial twist of the tube is significantly supercritical such that the e-folding growth time of the most unstable kink mode is smaller than the rise time scale, they found sharp bending of the flux tube as a result of the non-linear evolution of the kink instability. During the onset of the kink instability, the magnetic energy decreases while the magnetic helicity is approximately conserved. The writhing of the flux tube also significantly increases the axial field strength and hence enhances the buoyancy of the flux tube. The flux tube rises and arches upward at the portion where the kink concentrates, with a rotation of the tube orientation at the apex that exceeds 90° (see Figure 28). The emergence of this kinked flux tube can give rise to a compact magnetic bipole with polarity order inverted from the Hale polarity rule (Figure 29) as often seen in -sunspots. The conservation of magnetic helicity requires that the writhing of the tube due to the kink instability is of the same sense as the twist of the field lines. Hence for a kinked emerging tube, the rotation or tilt of the magnetic bipole from the east-west polarity orientation defined by the Hale’s polarity rule should be related to the twist of the tube. The rotation or tilt should be clockwise (counterclockwise) for right-hand-twisted (left-hand-twisted) flux tubes. This tilt–twist relation can be used as a means to test the model of kinked flux tubes as the origin of -sunspots (Tanaka, 1991; Leka et al., 1994, 1996; López Fuentes et al., 2003). Observations have found with both consistent and opposing cases (Leka et al., 1996; López Fuentes et al., 2003). A recent study (Tian et al., 2005) which includes a large sample (104) of complex -configuration active regions shows that 65 – 67% of these -regions have the same sign of twist and writhe, supporting the model of kinked flux tubes.
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