5.6 The rise of kink unstable magnetic flux tubes and the origin of delta-sunspots

Most of the solar active regions are observed to have very small twists, with an averaged value of α ≡ ⟨Jz∕Bz ⟩ (the averaged ratio of the vertical electric current over the vertical magnetic field) measured to be on the order of 0.01 Mm–1 (see Pevtsov et al., 19952001Jump To The Next Citation Point). However there is a small but important subset of active regions, called the δ-sunspots, which are observed to be highly twisted with α reaching a few times 0.1 Mm–1 (Leka et al., 1996Jump To The Next Citation Point), and to have unusual polarity orientations that are sometimes reversed from Hale’s polarity rule (see Zirin and Tanaka, 1973Zirin, 1988Jump To The Next Citation PointTanaka, 1991Jump To The Next Citation Point). These δ-sunspots are compact structures where umbrae of opposite polarity are contained within a common penumbra. They are found to be the most flare productive active regions (see Zirin, 1988Jump To The Next Citation Point). Through careful analysis of the evolution of flare-active δ-sunspot groups, Tanaka (1991Jump To The Next Citation Point) proposed a model of an emerging twisted flux rope with kinked or knotted geometry to explain the observed evolution of these regions.

Motivated by the observations of δ-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., 1996Jump To The Next Citation Point1998Jump To The Next Citation Point1999Jump To The Next Citation PointFan et al., 1998bJump To The Next Citation Point1999Jump To The Next Citation Point). For an infinitely long twisted cylindrical flux tube with axial field Bz(r), azimuthal field B𝜃(r) = q(r)rBz (r), and plasma pressure p(r) in hydrostatic equilibrium where dp∕dr = − (B2 ∕4πr) − d(B2 + B2 )∕dr 𝜃 z 𝜃, a sufficient condition for the flux tube to be kink unstable is (see Freidberg, 1987)

( ′)2 ′ r- q- + 8-πp-< 0 (28 ) 4 q B2z
to be true somewhere in the flux tube. In Equation (28View Equation) the superscript ’ denotes the derivative with respect to r. This is known as Suydam’s criterion. Note that condition (28View Equation) is sufficient but not necessary for the onset of the kink instability and hence there can be cases which are kink unstable but do not satisfy condition (28View Equation). One such example are the force-free twisted flux tubes which are shown to be always kink unstable without line-tying (i.e. infinitely long) (Anzer, 1968), but for which p′ = 0. Force-free fields are the preferred state for coronal magnetic fields under low plasma-β conditions and are not a likely state for magnetic fields in the high-β plasma of the solar interior. Linton et al. (1996Jump To The Next Citation Point) considered the linear kink instability of uniformly twisted cylindrical flux tubes with q = B 𝜃∕rBz being constant, confined in a high β plasma. They found that the equilibrium is kink unstable if q exceeds a critical value qcr = a−1, where a− 2 is the coefficient for the r2 term in the Taylor series expansion of the equilibrium axial magnetic field Bz about the tube axis: −2 2 Bz (r ) = B0 (1 − a r + ⋅⋅⋅). 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 q 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 − 1 qcr = a 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, 1979Linton et al., 1996).

The non-linear evolution of the kink instability of twisted magnetic flux tubes in a high-β plasma has been investigated by 3D MHD simulations (Linton et al., 19981999Fan et al., 1998bJump To The Next Citation Point1999Jump To The Next Citation Point). Fan et al. (1998b1999Jump To The Next Citation Point) modeled the rise of a kink unstable flux tube through an adiabatically stratified model solar convection zone.

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Figure 28: mpg-Movie (314 KB) The rise of a kink unstable magnetic flux tube through an adiabatically stratified model solar convection zone (result from a simulation in Fan et al. (1999) with an initial right-handed twist that is 4 times the critical level for the onset of the kink instability). In this case, the initial twist of the tube is significantly supercritical so that the e-folding growth time of the most unstable kink mode is smaller than the rise time scale. The flux tube is perturbed with multiple unstable modes. The flux tube becomes kinked and arches upward at the center where the kink concentrates, with a rotation of the tube orientation at the apex that exceeds 90°.
View Image

Figure 29: A horizontal cross-section near the top of the upward arching kinked loop shown in the last panel of Figure 28Watch/download Movie. The contours denote the vertical magnetic field Bz with solid (dotted) contours representing positive (negative) B z. The arrows show the horizontal magnetic field. One finds a compact bipolar region with sheared transverse field at the polarity inversion line. The apparent polarity orientation (i.e. the direction of the line drawn from the peak of the positive pole to the peak of the negative pole) is rotated clockwise by about 145° from the +x direction (the east-west direction) of the initial horizontal flux tube.

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 28Watch/download Movie). 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 29View Image) as often seen in δ-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 δ-sunspots (Tanaka, 1991Leka et al., 19941996Jump To The Next Citation PointLópez Fuentes et al., 2003Jump To The Next Citation Point). Observations have found with both consistent and opposing cases (Leka et al., 1996López Fuentes et al., 2003). A recent study (Tian et al., 2005) which includes a large sample (104) of complex δ-configuration active regions shows that 65 – 67% of these δ-regions have the same sign of twist and writhe, supporting the model of kinked flux tubes.


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