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Non-Neutralized Electric Current of Active Regions Explained as a Projection Effect

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Abstract

Active regions (ARs) often possess an observed net electric current in a single magnetic polarity. We show that such “non-neutralized” currents can arise from a geometric projection effect when a twisted flux tube obliquely intersects the photosphere. To this end, we emulate surface maps of an emerging AR by sampling horizontal slices of a semi-torus flux tube at various heights. Although the tube has no net toroidal current, its poloidal current, when projected along the vertical direction, amounts to a significant non-neutralized component on the surface. If the tube emerges only partially as in realistic settings, the non-neutralized current will 1) develop as double ribbons near the sheared polarity inversion line, (2) positively correlate with the twist, and 3) reach its maximum before the magnetic flux. The projection effect may be important to the photospheric current distribution, in particular during the early stages of flux emergence.

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Acknowledgements

This work is supported by NSF award 1848250 and NASA award 80NSSC190263. MCMC acknowledges support from NASA award 80NSSC19K0088. This work was initiated during a scientific visit to the Institute for Astronomy by MCMC with travel support from NSF award 1848250. We thank T. Török, Y. Liu, and V. S. Titov for their valuable comments. Visualization softwares include the Plotly Python package.

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Appendix

Appendix

In FG03, the flux tube is kinematically lifted from subsurface (\(z<0\)) at a constant velocity \({\boldsymbol{v}} _{0}=v_{0} \hat{\boldsymbol{z}} \). As the tube intersects the \(z=0\) plane, its magnetic field \({\boldsymbol{B}} _{0}= {\boldsymbol{B}} (z=0)\) and \({\boldsymbol{v}} _{0}\) induce an ideal electric field \({\boldsymbol{E}} \propto- {\boldsymbol{v}_{0}} \times{ \boldsymbol{B}} _{0}\), which is used as an evolving boundary condition for the domain of interest (\(z>0\)). The initial potential field in the \(z>0\) volume interacts with the newly emerged flux, resulting in a sigmoidal flux rope that eventually erupts.

For simplicity, we place the origin at \({\boldsymbol{r}} _{0}=(0,0,0)\) in the Cartesian coordinate (Figure 1). A horizontal slice at \(z_{0}>0\) represents the photospheric condition when the tube is centered at a depth of \(h=-z_{0}\).

We define two auxiliary variables: \(\rho\) for the projected distance of a point \({\boldsymbol{r}} \) to the origin on the \(y=0\) plane, and \(w\) for the distance to the tube axis:

$$ \begin{aligned} \rho& = \left(x^{2} +z^{2} \right)^{1/2}, \\ w & = \left[( \rho- R )^{2} + y^{2} \right]^{1/2}. \end{aligned} $$
(11)

In our convention, both the Cartesian and the spherical coordinate systems are right-handed (Figure 1). The coordinates \((x, y, z)\) and \((r, \theta, \phi)\) are related by

$$ \begin{aligned} x & = r \sin{\theta} \sin{\phi}, \\ y & = r \cos{\theta}, \\ z & = r \sin{\theta} \cos{\phi}. \end{aligned} $$
(12)

Their unit vectors \(( \hat{\boldsymbol{x}} , \hat{\boldsymbol{y}} , \hat{\boldsymbol{z}} )\) and \(( \hat{\boldsymbol{r}} , \hat{\boldsymbol{\theta}} , \hat{\boldsymbol{\phi}} )\) are related by

$$ \begin{aligned} \hat{\boldsymbol{x}} & = \sin{\theta} \sin{\phi} \, \hat{\boldsymbol{r}} + \cos{\theta} \sin{\phi} \, \hat{\boldsymbol{\theta}} + \cos{\phi} \, \hat{\boldsymbol{\phi}} , \\ \hat{\boldsymbol{y}} & = \cos{\theta} \, \hat{\boldsymbol{r}} - \sin{\theta} \, \hat{\boldsymbol{\theta}} , \\ \hat{\boldsymbol{z}} & = \sin{\theta} \cos{\phi} \, \hat{\boldsymbol{r}} + \cos{\theta} \cos{\phi} \, \hat{\boldsymbol{\theta}} - \sin{\phi} \, \hat{\boldsymbol{\phi}} . \end{aligned} $$
(13)

In the spherical coordinate, the three components of the magnetic field \({\boldsymbol{B}} \) are

$$ \begin{aligned} B_{r} & = B_{t} \frac{q R \cos{\theta}}{r \sin{\theta}} \exp\left(-w^{2}/a^{2}\right), \\ B_{\theta}& = B_{t} \frac{q (r - R \sin{\theta})}{r \sin{\theta}} \exp\left(-w^{2}/a^{2}\right), \\ B_{\phi}& = B_{t} \frac{a}{r \sin{\theta}} \exp\left(-w^{2}/a^{2} \right). \end{aligned} $$
(14)

The three Cartesian components are

$$ \begin{aligned} B_{x} & = B_{t} \frac{q x y + a z}{\rho^{2}} \exp \left(-w^{2}/a^{2}\right), \\ B_{y} & = B_{t} q \frac{R - \rho}{\rho} \exp\left(-w^{2}/a^{2} \right), \\ B_{z} & = B_{t} \frac{q y z - a x}{\rho^{2}} \exp\left(-w^{2}/a^{2} \right). \end{aligned} $$
(15)

In the spherical coordinate, the three components of the current density \({\boldsymbol{j}} \) are

$$ \begin{aligned} j_{r} & = \frac{2 B_{t} R \cos\theta}{a \rho} \exp \left(-w^{2}/a^{2}\right), \\ j_{\theta }& = \frac{ 2 B_{t} (r - R \sin{\theta})}{a \rho} \exp\left(-w^{2}/a^{2} \right), \\ j_{\phi }& = - \frac{B_{t} q}{\rho} \left( \frac{2w^{2}}{a^{2}} - \frac{R}{\rho} - 1 \right) \exp\left(-w^{2}/a^{2}\right). \end{aligned} $$
(16)

The three Cartesian components are

$$ \begin{aligned} j_{x} & = - \frac{B_{t}}{\rho^{2}} \left\{ q \left[ \frac{2 w^{2}}{a^{2}} - \frac{R}{\rho} - 1 \right] z - \frac{2 x y}{a} \right\} \exp\left(-w^{2}/a^{2}\right), \\ j_{y} & = \dfrac{2 B_{t} (R-\rho)}{a \rho} \exp\left(-w^{2}/a^{2} \right) , \\ j_{z} & = \frac{B_{t}}{\rho^{2}} \left\{ q \left[ \frac{2 w^{2}}{a^{2}} - \frac{R}{\rho} - 1 \right] x + \dfrac{2 y z}{a} \right\} \exp\left(-w^{2}/a^{2}\right). \end{aligned} $$
(17)

A torus coordinate \((w, \psi, \phi)\) is related to the Cartesian coordinate as

$$ \begin{aligned} x & = (R + w \cos{\psi}) \sin{\phi}, \\ y & = - w \sin{\psi}, \\ z & = (R + w \cos{\psi}) \cos{\phi}. \end{aligned} $$
(18)

Here \(w\) is the radius with respect to the tube axis, and \(\phi\) is the toroidal angle with respect to the center of the torus. They are identically defined as in the spherical coordinate. The new coordinate \(\psi\) defines the poloidal angle, which is the polar angle with respect to the tube axis. It is 0 in the \(x\)\(z\) plane, and increase in the same direction as \(\theta\) (Figure 1).

The unit vectors \(( \hat{\boldsymbol{w}} , \hat{\boldsymbol{\psi}} )\) and \(( \hat{\boldsymbol{r}} , \hat{\boldsymbol{\theta}} )\) are related by

$$ \begin{aligned} \hat{\boldsymbol{w}} & = \frac{r^{2} - R \rho}{wr} \, \hat{\boldsymbol{r}} - \frac{Ry}{wr} \, \hat{\boldsymbol{\theta}} , \\ \hat{\boldsymbol{\psi}} & = \frac{Ry}{wr} \, \hat{\boldsymbol{r}} + \frac{r^{2} - R \rho}{wr} \, \hat{\boldsymbol{\theta}} . \end{aligned} $$
(19)

In this definition, the axial component is toroidal (\(\hat{\boldsymbol{\phi}} \)). The non-axial components include the radial (\(\hat{\boldsymbol{w}} \)) and poloidal (\(\hat{\boldsymbol{\psi}} \)) contributions. Using Equations (14), (16), and (19), we find

$$ \begin{aligned} B_{w} & = 0, \\ B_{\psi}& = \frac{q w B_{t}}{\rho} \exp\left(-w^{2}/a^{2}\right), \end{aligned} $$
(20)
$$ \begin{aligned} j_{w} & = 0, \\ j_{\psi}& = \frac{2wB_{t}}{a\rho} \exp\left(-w^{2}/a^{2}\right). \end{aligned} $$
(21)

This indicates that the non-axial \({\boldsymbol{B}} \) and \({\boldsymbol{j}} \) are purely poloidal.

Using Equations (5), (6), (13), and (16), we can evaluate the contributions of poloidal and toroidal components in \(B_{z}\) and \(j_{z}\):

$$ \begin{aligned} {\boldsymbol{B}} _{\mathrm{P}} \cdot \hat{\boldsymbol{z}} & = B_{t} \frac{q y z}{\rho^{2}} \exp\left (-w^{2}/a^{2} \right), \\ {\boldsymbol{B}} _{\mathrm{T}} \cdot\hat{\boldsymbol{z}} & = - B_{t} \frac{a x}{\rho^{2}} \exp\left(-w^{2}/a^{2}\right), \end{aligned} $$
(22)
$$ \begin{aligned} {\boldsymbol{j}} _{\mathrm{P}} \cdot \hat{\boldsymbol{z}} & = \frac{2 B_{t} yz}{a \rho^{2}} \exp \left(-w^{2}/a^{2}\right), \\ {\boldsymbol{j}} _{\mathrm{T}} \cdot\hat{\boldsymbol{z}} & = \frac{B_{t} q x}{\rho^{2}} \left( \frac{2w^{2}}{a^{2}} - \frac{R}{\rho} - 1 \right) \exp\left(-w^{2}/a^{2}\right). \end{aligned} $$
(23)

The semi-torus tube is axisymmetric about the \(y\)-axis. However, on a cross section with constant \(\phi\), variables are not axisymmetric about the tube axis. For example, for \(j_{\phi}\), the inner, lower portion of the RC sheath appears to be stronger than other parts (Figure 7(a)).

Figure 7
figure 7

Properties of a FG03 flux tube on a cross section (\(q=0.8\), \(\phi=x=0\)). (a) Map of toroidal current density \(j_{\phi}\) (left) and Lorentz force \({\boldsymbol{f}} \) (right). The contours are for \(j_{\phi}= -0.45\), 0, and 2. The background colors show the force amplitude; the arrows show the force vectors at \(w=0.12\). The tube axis is marked with a cross. (b) Map of the length \(L\) (left) and the number of turns \(T\) around the axis (right) for field lines passing through the \(x=0\) plane and end at \(\Delta z=-0.5\). Horizontal dotted line shows the photosphere for \(h=-0.37\) when the observed magnetic flux reaches maximum. A portion of field lines will cross the photosphere four times; one such field line is shown (for \(x>0\)). The wedge-shaped shaded region denotes their intersections with \(x=0\).

We evaluate the Lorentz force density \({\boldsymbol{f}} = {\boldsymbol{j}} \times{\boldsymbol{B}} \) on the cross section \(\phi=0\). As both the radial field (\(B_{w}\)) and the radial current density (\(j_{w}\)) are zero, \({\boldsymbol{f}} \) must be purely radial. The force points away from the axis so the tube tends to expand. It is strongest at the lower portion of the DC core (Figure 7(a)).

We trace field lines in both directions from the \(\phi=0\) plane to investigate their lengths \(L\) and turns of twist \(T\) around the axis (Figure 7(b)). The latter is evaluated from the change of \(\psi\) in the torus coordinate. For \(q=0.8\), the axis of the semi-torus has \(L = \pi R = 1.57\); field lines passing through the lower edge are more than twice in length due to twist. Field lines close to the axis have \(T= q R / 2 a = 1.67\); those passing slightly below the axis have a larger \(T\) due to increased \(L\).

For the \(q=0.8\) case, we find that a portion of the field lines may intersect the photosphere four times (Figure 8(b)). These field lines cross the \(\phi=0\) plane in a wedge-shaped region below the tube axis, where \(L\) and \(T\) are both large. This is a result of competition between the curvatures given by the flux tube axis and by the twist in the flux tube. No such field lines are present for the \(q=0.2\) and 0.5 cases.

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Sun, X., Cheung, M.C.M. Non-Neutralized Electric Current of Active Regions Explained as a Projection Effect. Sol Phys 296, 7 (2021). https://doi.org/10.1007/s11207-020-01742-9

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