Applications of Braid Theory

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

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

Hemispheric Chirality
Sunspot whorls
Filaments
Applications of Braid Theory 3

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

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

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

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,

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

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.

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)

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

Sigmoids are often precursors of
Coronal Mass Ejections

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.

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

(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.

Gibson & Fan model

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

For closed curves:
Writhe

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

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?

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)?

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

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 = ‐0.752
local writhe
= 0.374
Nonlocal writhe = ‐1.5
local writhe
= 0.374

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

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

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!

Kinked Loop: Sine Height profile
heig ght
Sine height profile z ∼ SinHπ tL
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
horizontal position
Twisted Sine Shape
0.3
0.2
0.1
Writh he
0
−0.1
−0.2
−0.3
Q=-pê2
Q=-3pê8
Q=-pê4ê
Q=-pê8
0 0.2 0.4 0.6 0.8 1
height Hunits of footpoint separationL
h=0.4 Ө = ‐π/4

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