Photospheric Injection of Magnetic Helicity: Connectivity-Based Flux Density Method

Abstract

Magnetic helicity quantifies the degree to which the magnetic field in a volume is globally sheared and/or twisted. This quantity is believed to play a key role in solar activity due to its conservation property. Helicity is continuously injected into the corona during the evolution of active regions (ARs). To better understand and quantify the role of magnetic helicity in solar activity, the distribution of magnetic helicity flux in ARs needs to be studied. The helicity distribution can be computed from the temporal evolution of photospheric magnetograms of ARs such as the ones provided by SDO/HMI and Hinode/SOT. Most recent analyses of photospheric helicity flux derived a proxy to the helicity-flux density based on the relative rotation rate of photospheric magnetic footpoints. Although this proxy allows a good estimate of the photospheric helicity flux, it is still not a true helicity flux density because it does not take into account the connectivity of the magnetic field lines. For the first time, we implement a helicity density that takes this connectivity into account. To use it for future observational studies, we tested the method and its precision on several types of models involving different patterns of helicity injection. We also tested it on more complex configurations – from magnetohydrodynamics (MHD) simulations – containing quasi-separatrix layers. We demonstrate that this connectivity-based proxy is best-suited to map the true distribution of photospheric helicity injection.

Appendix: Analytical Solutions for G θ

In the following, we use the same notation as defined in Section 3. The flux-transport velocity fields are given by Equations (11) – (15). The total magnetic flux in the positive (negative) polarity is called Φ₊ (resp. −Φ₋, with Φ_±>0). For generality purpose, Φ₋ can be different from Φ₊, and no specific assumption is made concerning the magnetic field configuration.

1.1 A.1 Two Separating Magnetic Polarities

We consider two opposite magnetic polarities separating in the x-direction at constant speed and without any rotation. The flux-transport velocity field is given by Equation (11). Because the velocity field is constant in each polarity, the terms of Equation (4) associated to (M,M′) in the same polarity are zero as a consequence of u′=u.

In this case, we have u−u′=∓2U ₀ e _x when ±B _n(M)>0 and ∓B _n(M′)>0, which leads to

$$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}}= \pm 2U_{0} \mathbf{M}' \mathbf{M} \cdot \mathbf{e}_{y}. $$

(25)

Thus, Equation (4) leads to

$$ G_{\theta }\bigl(M(\mathbf {x})\bigr)=\mp \frac{U_{0} B_{\mathrm {n}}}{\pi} \biggl(\int_{M' \ \mathrm{in} \ P_{\mp}} B_{\mathrm {n}}' \frac{\mathbf{M}'\mathbf{M}}{|\mathbf{M}'\mathbf{M}|^{2}} \,\mathrm {d} {\mathcal{S}} ' \biggr) \cdot \mathbf{e}_{y},\quad \textrm{for} \ \pm B_{\mathrm {n}}(M)>0. $$

(26)

This integral can be computed by analogy to the electric field created by a 2D distribution of charge, $\sigma (M)= B_{\mathrm {n}}'(M)$, of an infinite cylinder of radius R (of vertical axis crossing the z=0 plane at point O _∓), using Gauss theorem, i.e.,

$$ \int_{M' \ \mathrm{in} \ P_{\mp}} B_{\mathrm {n}}' \frac{\mathbf{M}'\mathbf{M}}{|\mathbf{M}'\mathbf{M}|^{2}} \,\mathrm {d} {\mathcal{S}} ' = \mp \frac{\mathbf{O}_{\mp} \mathbf{M}}{|\mathbf{O}_{\mp} \mathbf{M}|^{2}}\ \Phi_{\mp}. $$

(27)

Hence, we find that the helicity flux density is given by Equation (12).

1.2 A.2 One Polarity Rigidly Rotating Around the Other

In this model, the negative polarity rigidly rotates around the positive one. The velocity field is given by Equation (13). There are four cases to consider.

• c1. If B _n(M)>0 and B _n(M′)>0, then u−u′=0 and the associated term of Equation (4) is null.

• c2. If B _n(M)>0 and B _n(M′)<0, then u−u′=−Ωe _z ×O ₊ M′ and

  $$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}} = - \bigl(\mathbf{M}'\mathbf{M} \cdot \mathbf{O}_{+}\mathbf{M}'\bigr) \Omega. $$

  (28)

  The helicity flux density is then (using O ₊ M′=O ₊ M−M′M)

  

  (29)

  which, by regrou** terms, leads to Equation (14).

• c3. If B _n(M)<0 and B _n(M′)>0, then u−u′=Ωe _z ×O ₊ M and

  $$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}}=\bigl(\mathbf{M}'\mathbf{M} \cdot \mathbf{O}_{+}\mathbf{M}\bigr) \Omega, $$

  (30)

  which, using Equation (27) leads to

  $$ G_{\theta }\bigl(M(\mathbf {x})\bigr) = - \frac{\Omega B_{\mathrm {n}}}{2\pi} \Phi_{+}, \quad \textrm{for} \ \bigl(B_{\mathrm {n}}(M)<0, B_{\mathrm {n}}\bigl(M'\bigr)>0 \bigr). $$

  (31)

• c4. If B _n(M)<0 and B _n(M′)<0, u−u′=Ωe _z ×M′M and

  $$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}}= \big|\mathbf{M}'\mathbf{M}\big|^{2} \Omega, $$

  (32)

  leading to

  $$ G_{\theta }\bigl(M(\mathbf {x})\bigr) = \frac{\Omega B_{\mathrm {n}}}{2\pi} \Phi_{-}, \quad \textrm{for} \ \bigl(B_{\mathrm {n}}(M)<0, B_{\mathrm {n}}\bigl(M'\bigr)<0 \bigr). $$

  (33)

  Then the total helicity flux density within the region B _n(M)<0 is therefore

  $$ G_{\theta }\bigl(M(\mathbf {x})\bigr) = - \frac{\Omega B_{\mathrm {n}}}{2\pi}\ (\Phi_{+} - \Phi_{-}),\quad \textrm{for} \ B_{\mathrm {n}}(M)<0. $$

  (34)

  Note that in the particular case of two magnetic-flux-balanced polarities, Φ₊=Φ₋ and G _θ =0 for B _n(M)<0.

1.3 A.3 Two Counter-rotating Magnetic Polarities

In this model, the positive polarity rotates clockwise around its center, while the negative polarity rotates counterclockwise around its center. The velocity field is given by Equation (15). There are four cases to consider, which, by symmetry, reduce to two cases.

• c1. If B _n(M)>0 and B _n(M′)>0, we have u−u′=−Ωe _z ×M′M, which leads to

  $$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}}= -\big|\mathbf{M}'\mathbf{M}\big|^{2} \Omega, $$

  (35)

  giving

  $$ G_{\theta }\bigl(M(\mathbf {x})\bigr) = \frac{\Omega B_{\mathrm {n}}}{2\pi} \Phi_{+}, \quad \textrm{for} \ \bigl(B_{\mathrm {n}}(M)>0, B_{\mathrm {n}}\bigl(M'\bigr)>0 \bigr). $$

  (36)

• c2. If B _n(M)>0 and B _n(M′)<0, we have u−u′=Ωe _z ×(M′M−O ₊ M−O ₋ M), which leads to

  $$ \mathbf{M}'\mathbf{M} \times \bigl(\mathbf {u}-\mathbf {u}' \bigr) \big\vert_{\mathrm{n}}=\big|\mathbf{M}'\mathbf{M}\big|^{2} \Omega - \bigl( \mathbf{M}'\mathbf{M} \cdot ( \mathbf{O}_{+}\mathbf{M} + \mathbf{O}_{-} \mathbf{M}) \bigr) \Omega, $$

  (37)

  giving (using O ₊ M=O ₊ O ₋+O ₋ M)

  

  (38)

  for (B _n(M)>0,B _n(M′)<0).

  The total helicity flux density in the positive polarity is obtained by summing Equations (36) and (38) and supposing Φ₊=Φ₋=Φ₀ to simplify

  $$ G_{\theta }\bigl(M(\mathbf {x})\bigr)= - \frac{\Omega B_{\mathrm {n}}}{2\pi} \frac{\mathbf{O}_{-}\mathbf{M} \cdot \mathbf{O}_{+}\mathbf{O}_{-}}{|\mathbf{O}_{-}\mathbf{M}|^{2}} \Phi_{0}, \quad \textrm{for} \ B_{\mathrm {n}}(M)>0. $$

  (39)

Following the same derivation as above for B _n(M)<0, we find Equation (16).