1. Magnetic Helicity in Emerging Active Regions: A Statistical Study
Yang Liu, Peter W. Schuck

2. Topics to be discussed.
Buildup of magnetic relative helicity in two emerging active regions;
Hemispheric helicity rule: test with HMI data;
Test with Demoulin and Berger’s hypothesis (Demoulin & Berger, 2003) for helicity flux computation;
Computation of helicity flux density: which is better: G_A or G_theta?

3. 1. Energy and helicity buildup: Calculation of energy and helicity fluxes
emerging term
shear term
shear term
emerging term
Bh, Bn [obs], and Vh, Vn [obs + DAVE4VM (Schuck 2008)]

4. Evolution of AR11072: emerging and developing to be a simple active region.

5. Bz + Vt
Bz + Vz (green: upflow, red: downflow)
AR11072:
Separation motion is detected;
Upflow is associated with emerging flux, and surrounds the leading sunspot;
Simple bipolar active region.
Vector B

6. Summary:
Helicities from the shear-term and emergence-term have the same sign;
Shear-term dominant;
Upflow and downflow inject oppsite sign helicities. Both are small.

7. Evolution of AR11158: 1. emerging; 2
Evolution of AR11158: 1. emerging; 2. separation of leading and following fields; 3. rotation of sunspots; 4. shear motion along the polarity inversion line.

8. Bz + Vt
Bz + Vz (green: upflow, red: downflow)
AR11158:
Spinning sunspots;
Shear motion along the PIL;
Upflows surround sunspots;
Highly sheared magnetic field near the PIL.
Vector B

9. Summary of AR11158:
Same sign of helicity from shear-term and emergence-term;
Shear-term dominants;
Upflow and downflow contribute helicity with opposite signs.

10. Energy in the two active regions (left: AR11072, right: AR11158):
shear-term and emergence-term are consistent in phase;
Both terms well correspond to the flux emergence;
Emergence-term is dominant.

11. 1. Energy and helicity buildup: conclusions
Shear-term contributes most helicity in the corona;
The helicities from shear-term and emergence-term have the same sign;
Upflow and downflow contribute helicity with the opposite signs;
Energy flux well corresponds with the flux emergence;
Emergence-term and shear-term energy fluxes are consistent in phase;
Emergence-term contributes more energy.

12. 2: Hemispheric helicity rule
Methodology and data
Select emerging active regions;
Compute the helicity flux;
Integral over time of the emergence process to obtain the helicity accumulated in the corona in the active region.

13. A weak hemisphere-preference is found in this 56-active-region sample
A weak hemisphere-preference is found in this 56-active-region sample. 56% of the active regions follow the rule, while 44% against.

14. Fitting the data with |H| ~ a * Flux ^ alpha yields alpha = 1.91;
Roughly estimate the turn of the twist of the flux by |H| ~ N * Flux^2. The average turn N =

15. 2. Hemispheric helicity rule: conclusions
It is found in a sample of 56 active regions that 56% of them obey the hemispheric rule while 44% violate it;
Fitting the data with a formula |H| = a * Flux^alpha yields alpha = 1.91;
Turn of the flux twist is roughly in average.

16. 3: Test Demoulin & Berger’s hypothesis (2003)—DB03 hypothesis
Using DAVE (Schuck 2006)u

17. AR11072: Both energy flux and helicity from the tracking velocity [u] (DAVE; blue curves) don’t agree with the total fluxes (black curves). This indicates that the DB03 hypothesis is incorrect.

18. AR11158: Blue curves don’t agree with the black curves, indicating that the DB03 hypothesis is incorrect.

19. 3. DB03 hypothesis: conclusion
Both cases indicate that the energy and helicity fluxes computed from the tracking-footpoint velocity does not equal to the total fluxes. This indicates that the DB03 hypothesis is incorrect.

20. 4. Computation of helicity density: Two helicity density proxies

21. Calculate the vector potential

22. Boundary condition for the Green’s function.
The periodic Green’s function (G_A-FFT; G_theta-FFT).
The free-space Green’s function (G_A-FS; G_theta-FS).

23. Vector B
Test with MHD data:
As Pariat et al (2005) pointed out that G_A proxy contains fake signals.

24. Test with HMI data: G_A has fake signals.

25. When the boundary is chosen consistently, both G_A and G_theta yield identical helicity fluxes, as well as two components of the helicity flux (shear-term and emergence-term).

26. Helicity flux versus zero-padded boundary
Helicity flux versus zero-padded boundary. N refers to the factor by which the original data is expanded and zero-padded. The zero-paddede data us N x Lx by N x Ly, where Lx and Ly are the width and height of the field of view of the original data. The solid line with asterisks represents the helicity flux computed from G_A-FFT, while the horizontal dashed line refers to the helicity flux from G_A-FS from the original data. The helicity flux from G_A-FFT converges to that from G_A-FS while increasing the padding areas.

27. 4. Computation of helicity flux: conclusions
Helicity flux density proxy G_A contains fake signal, as Pariat et al. pointed out (2005);
This fake signal is cancelled out completely when computing the total helicity flux, as well as the two components of the flux (shear-term and emergence-term), by integral over the entire area of the region;
The difference of the helicity fluxes computed from helicity density proxies G_A and G_theta is not due to the fake signal that G_A introduces as Pariat et al (2005) suggested, but rather due to inconsistency of the boundaries chosen in the Green’s function;
The helicity flux computed from G_A-FFT converges to that from G_A-FS with increase of the padding area.