Braids, Twist, Writhe, & Solar Activity

Transcription

1 Braids, Twist, Writhe, & Solar Activity The Sun Hemispheric Chirality Writhe Applications of Braid Theory 1

2 Coronal Heating Parker model (1983): The interior structure t re of coronal loops is braided. When neighbouring tubes are misaligned by ~ 30 degrees, reconnection removes a crossing, releasing magnetic energy into heat a nanoflare. Linton, Dahlburg and Antiochos 2001 Applications of Braid Theory 2

3 Hemispheric Chirality Sunspot whorls Filaments Applications of Braid Theory 3

4 Coronal Magnetic Structures often display kinked or helical structure. Yohkoh soft x-ray image of Southern hemisphere sigmoid X Ray Bi Brightenings i (Sigmoids) id

5 For closed curves and fields: Magnetic Helicity 1. For two linked torii of flux F 1 and F 2, and internal twists T 1 and T 2 2. Helicity is an ideal MHD invariant. Applications of Braid Theory 5

6 Magnetic Helicity of a subvolume of space Applications of Braid Theory 6

7 global and regional helicities Let space be divided into N regions. Let H i be the helicity in region i (relative to vacuum field). Then for regions separated by parallel planes or concentric spheres,

8 Helicity between two planes can be expressed as a sum of winding numbers, over all pairs of field lines:

9 What if curves go up and down? 1. Cut the curves into segments at turning points in height. 2. Sum winding numbers for each pair of segments (if opposite directions, multiply by 1). Heresegments2and3give Here segments 2 and 3 give winding angle of 4p.

10 Solar Helicity Observations Current helicity j z /B z from vector magnetograms (Abramenko et al 1997; Pevtsov & Latushko 2002) Effects of differential rotation on active regions (van Ballegooijen et al 1998; B & Ruzmaikin 2000; Devore 2000; Green et al 2002; Nindos et al 2003) Helicity flow through photosphere (Kusano et al 2002; Tian 2003; Démoulin & B 2003; Chae et al 2004; Kusano et al 2005; Pariat et al 2006; Longcope 2007) Magnetograms plus best fit force free free extrapolations (Démoulin et al 2002; Aulanier et al 2002, 2005)

11 Sigmoids Soft X ray brightenings g first identified in Yohkoh (Rust & Kumar 1996). Most (91%) active regions containing sigmoids also display filaments (Pevtsov 2002). Sigmoids are mostly S shaped in the South and shaped in the North

12 Sigmoids are often precursors of Coronal Mass Ejections

13 Flux Tube Models Twisted Flux tubes have often been used in models of solar filaments and Coronal Mass Ejections. Could the Sigmoid x ray brightenings show the shape of these tubes? Low & Berger 2003: magnetostatic solutions with helical symmetry embedded in external field Unfortunately, these give the wrong sign! However, surrounding current concentrations can have the correct sign.

14 Sideways helical tubes? No! the sign is wrong! shape for positive helicity opposite to hemispheric law.

15 (2004, 2005) Kliem & Török simulation Follow nonlinear evolution of kink instability in Titov Démoulin model Intense currents develop along field Intense currents develop along field lines below flux tube which resemble sigmoids of the correct sign.

16 Gibson & Fan model

17 Linking, Twist, and Writhe Calugareanu 1961: Linking number = Twist + Writhe

18 For closed curves: Writhe

19 A trefoil knot and its tantrix curve Let A = the area enclosed by the tantrix curve. Then (Fuller 1971)

20 Writhe (by popular belief) Kink instability: internal twist converted to writhe (Ricca & Moffatt 1995, Rust 1996) Stretch Twist Fold Dynamos: large scale positive writhe helicity, small scale negative twist helicity (Gilbert 2003) Outer Convection Zone: coriolis force on rising tubes creates large scale positive writhe helicity, small scale negative twist helicity (in North) bihelical fields (Blackman & Brandenburg) S effect (Longcope & Pevtsov): helicity source in active regions?

21 Helicity Decomposition Magnetic Helicity for (thin) closed flux tube with axial flux Φ: But how do we define the writhe of a curve with endpoints on a boundary plane (or boundary sphere)?

22 Answer: write the writhe of a framed curve as the difference between winding number and twist of the curves on the tube. When averaged over the tube, this is independent of framing. Applications of Braid Theory 22

23 This methods divides up a curve into pieces at its maxima and minima, then computes the local writhe of each piece (winding twist), and the nonlocal writhes (just winding) between pieces. Total writhe = local writhe = Nonlocal writhe = 1.5 local writhe = 0.374

24 Example: the tangent to a vertical helix has constant latitude The local writhe term equals the area The local writhe term equals the area between the tantrix curve and the North pole

25 Writhe can also be defined for loops by extending the loop and calculating the corresponding enclosed tantrix area (same results!)

26 Normal and Anomalous writhe How an elastic rod buckles: Normally, Negative twist S shape (as seen from above) Positive twist shape Solar sigmoids: Two positive twist sigmoids anomalous!

27 Kinked Loop: Sine Height profile heig ght Sine height profile z SinHπ tl horizontal position Twisted Sine Shape Writh he Q=-pê2 Q=-3pê8 Q=-pê4ê Q=-pê height Hunits of footpoint separationl h=0.4 Ө = π/4

28 Two loops with identical Writhe = 0.2 Normal Anomalous Sigmoids on the sun seem to be of the anomalous type