### Berger

```Review of Magnetic Helicity and Field Topology
1.
2.
3.
Helicity: basic properties, open
geometries
Observations of helicity and
helicity flux
Twist and Writhe
Mitchell Berger
L=5
Open volumes
True Field
Equivalently, we set the helicity of the potential
field to zero (and assume helicity is bilinear).
Reference Field
Helicity Dissipation
Helicity Dissipation
Reconnection of two flux tubes converts
mutual helicity to self (twist + writhe) helicity
Linton, Dahlburg and Antiochos 2001
Relaxation
(Nandy, Hahn, Canfield, & Longcope 2003)
Relaxation
But … relaxation to linear force-free state requires
1. Confinement (The only linear force-free field in
infinite space outside a sphere is a potential field)
e.g. relaxation in one flux tube, or under one helmet streamer
2.
Killing all subhelicities (through widespread
reconnection – but does line-tying make this less
likely than in laboratory plasmas?)
Decompositions of Magnetic Helicity
1. Fourier Spectra
2. Poloidal-Toroidal
3. Regions of Space
4. Self and Mutual Helicity
5. Twist and Writhe
1. Fourier Spectra
Physical Meaning: Represent field as sum of circularly polarized
modes. Each mode has self linking, but there is no net linking between
modes.
Fourier helicity spectra do not
always detect helical structure …
Helicity spectrum is identically zero!
Example – twisted ring of flux
Asgari-Targhi & B 2009
2. Poloidal - Toroidal
Physical Meaning: Poloidal and Toroidal Fields link each
other, but not themselves. B85, Low 2010
3. Self and Mutual Helicity
Suppose we divide the coronal magnetic field into two pieces. In each
piece, the field lines begin and end at the photosphere. We can write
the helicity as a sum of self helicities H1 and H2, and mutual helicities
H12 :
H = H1 + H2 + 2H12.
H1 = 1.5
H2 = -1.2
H12 = 0.22
1
2
2
2
1
2
Individual twist helicity may be difficult to observe. The total mutual
helicity of a collection of threads may be easier.
Example: a set of sheared loops (each of unit flux)
H = 6.1
A simple prominence model
Self Helicities
Mutual Helicities
barbs
spine
Hb = 6.1
Hs = -5.1
Ha = -2.2
Hbs = -2.9
Hba = -5.5
Hsa = -12.9
Here the self helicity of the barbs is of opposite sign to the
helicities of the spine and overlying arcade
Self Helicity of one flux tube
 The self helicity arises as a sum of twist within the tube and writhe
of the tube axis. Choosing the volume to just contain the flux
tube gives only the twist helicity. Implications for onset of kink
instability for fat flux tubes. Longcope & Malanushenko 2008;
Malanushenko et al 2009
Helicity Flow through the photosphere
Helicity Observations
 Current helicity jz/Bz from vector magnetograms
(Abramenko et al 1997; Pevtsov & Latushko 2002)
 Effects of differential rotation on active regions (van
Ballegooijen et al 1998; Devore 2000; Green et al 2002; Nindos et al 2003)
 Helicity flow through photosphere (Kusano et al 2002; Tian 2003;
Démoulin and Berger 2003; Chae et al 2004; Kusano et al 2005; Longcope,
Ravindra, & Barnes 2007; Kazachenko et al 2010)
 Magnetograms plus best-fit force-free extrapolations
(Démoulin et al 2002; Aulanier et al 2002; Georgoulis & LaBonte 2007)
Helicity Flow through the photosphere

A combination of LCT and time evolution of vector magnetograms can
reconstruct the velocity field, giving the best observation of helicity flow
(Welsh et al 2004; Longcope 2004; Kusano et al 2005)

There is a gauge-invariant and physically meaningful way of mapping a helicity
flow density (Pariat, Démoulin &B 2005; Pariat, Nindos, Démoulin & B 2006) – leads
to much more coherent maps.
3. Helicity flow into active regions more coherent than previously thought.
Helicity Flow through the photosphere
On the largest scales, differential rotation injects helicity
into each hemisphere.
Helicity Flow through the photosphere
Helicity Flow through the photosphere
Getting the helicity flux right
Some non-helical motions can have large
fluctuations in helicity flux, creating noise
Helicity Flux negative
Helicity Flux positive
Single footpoint moving in a straight line
When practical, a different formula for helicity flux is much less noisy!
The θij braiding terms measure braiding of the tubes connected
to foot points i and j (unless these belong to same tube).
The θii spinning terms measure twisting of the tube connected to
foot point i. (Longcope, Ravindra, & Barnes 2007)
Field line helicity flux can be mapped!
(In practice, each of the two footpoints must be treated separately)
Twist and Writhe
 Universal language for describing tubes ribbons, and
curves
 Biology: DNA and proteins (Fuller 1971, Ricca & Maggioni
2006)
 Engineering: elastic rods (van der Heijden & Thompson 2000)
 Fluid Mechanics: magnetic tubes and vortex filaments (Ricca
1995)
W=-0.72 Tw=0 L=-1
W=-0.72 Tw=6 L=5
Supercoiled DNA
(R. Friddle)
Helicity Decomposition
 Magnetic Helicity for (thin) flux tube with axial flux Φ:
 Writhe can be determined by subtracting twist from helicity
Writhe
Helix with three turns:
Writhe = 2.68
Writhe = 0.46
Writhe of Magnetic Fields
 Kink instability: internal twist converted to writhe (Ricca
& Moffatt 1995, Rust 1996, Turok, Berger and Kliem 2010)
 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)
  effect (Longcope & Pevtsov): helicity source in active regions?
Winding Number method
A simpler and more efficient method for calculating writhe
divides the writhe into local and nonlocal terms (Berger &
Prior 2006):
This methods divides up a curve into pieces at its maxima and
minima, then computes the “local writhe” of each piece, and
the “nonlocal writhes” between pieces.
local writhe
= -0.45
Nonlocal = winding number
between the two pieces = 0.02
local writhe
= -1.36
Vertical Helical Tube with n turns
Helix Shape
2.5
Writhe
2
n
n
n
3
2
1
1.5
1
0.5
Writhe = 2.68
Writhe = 0.46
0
0
0.2
0.4
0.6
0.8
radius units of footpoint separation
1
Kinked Loop: Sine Height profile
Two loops with identical Writhe = -0.2
You need height to calculate writhe!
Following the writhe of an erupting
filament.
```