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

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?

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)]

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

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

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.

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.

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

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

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.

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.

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.

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.

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 = 0.055.

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 0.055 in average.

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

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.

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

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.

4. Computation of helicity density: Two helicity density proxies

Calculate the vector potential

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).

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

Test with HMI data: G_A has fake signals.

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).

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.

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.