1. Introduction to Helicity
Mitchell Berger
EGU Vienna 2017

2. New Directions in Basic Helicity Theory
Absolute Measures of Helicity (B & Hornig 2017)
Mapping the Field Line Helicity (Yeates & Hornig 2014, 2016; Russell, Yeates, Hornig & Wilmot- Smith 2015)
Canonical Helicity (You 2012)

4. Helicity in Open volumes
B & Field 84.
True Field
Reference Field
Equivalently, we set the helicity of the potential
field to zero (and assume helicity is bilinear).

5. 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 . Example (each tube has unit flux):
H = H1 + H2 + 2H12.
H1 =
H12 = - 2.5
H2 = - 0.2
Total helicity = - 2.6

6. Helicity of flux ropes = Twist plus Writhe
Helix with three turns:
Writhe = 2.68
Writhe = 0.46

7. W=-0.72 Tw=0 L=-1
W=-0.72 Tw=6 L=5

8. Kinked Loops: Writhe depends on height!
Two loops with identical Writhe = -0.2

9. Helicity Flow through the photosphere

10. Helicity Flow through the photosphere

11. Helicity Flow through the photosphere

12. 4. Helicity Balance of the Heliosphere

13. Helicity Balance of the Heliosphere

14. On the largest scales, differential rotation injects helicity into each hemisphere.

15. Absolute Measures of Magnetic Helicity
Several Authors have explored alternative measures of helicity in open volumes (e.g. Hornig, Prior, Low, Berger & Hornig).
Alternative Reference Fields
Unique Vector Potentials
Field Representations

16. Poloidal – Toroidal Formula
Physical Meaning: Poloidal and Toroidal Fields link each other, but not themselves. B85, Low 2010
(Similar for spheres)

17. Linking of Poloidal flux (red) by Toroidal flux (blue)