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Erratum
This article is an erratum for:
[https://doi.org/10.1051/0004-6361/201629122]

Issue
A&A
Volume 603, July 2017
Article Number C2
Number of page(s) 6
Section The Sun
DOI https://doi.org/10.1051/0004-6361/201629122e
Published online 03 July 2017

Erratum for: A&A 594, A98 (2016), DOI: 10.1051/0004-6361/201629122

It has come to our attention that incorrect units were inadvertently used for the axes and colour scales of a number of figures in our paper (Yeates & Hornig 2016). Specifically:

  • 1.

    The units of field line helicity \hbox{${\cal A}$} are quoted asMaxwells [Mx], but the values for \hbox{${\cal A}$} shown inFigs. 3, 5, 7, 10, 11e and 12 in theoriginal paper were actually in Mx R⊙-2\hbox{$\textrm{Mx}\,R_\odot^{-2}$}. They should be multiplied by (6.96 × 1010)2 toreach the true value in Mx.

  • 2.

    The units of total helicity H, and of dH/ dt are quoted as Mx2, but the values were actually in Mx2 R⊙-4\hbox{$\textrm{Mx}^2\,R_\odot^{-4}$}. This applies to Figs. 4c, d, 8c, d and 11d. They should be multiplied by (6.96 × 1010)4 to reach the true value in Mx2.

Corrected versions of the affected figures are included here. This does not in any way affect the qualitative results or conclusions of the paper, but we would like to correct the error for the benefit

of future readers who may wish to make quantitative comparisons.

To put the corrected values into context, DeVore (2000) estimates that the typical helicity content of a single interplanetary magnetic cloud is of the order 2 × 1042 Mx2, which is similar to that shed in our axisymmetric flux rope (Fig. 8c). And Berger & Ruzmaikin (2000) estimate a net transfer rate of helicity due to solar rotation of up to 5 × 1043 Mx day-1 in each hemisphere, at solar maximum. This is also consistent with our results, although a more detailed comparison with a fully data-driven simulation remains to be performed in the future.

thumbnail Fig. 3

Illustration of the dipolar simulation with ν0 = 0.36 × 10-5 s-1 on days 0, 1, 2, and 20. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 0.5 G), and projected coronal magnetic field lines traced from height r = R⊙ are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 5 × 1019 Mx with white indicating \hbox{${\cal A}\approx 0\,\mathrm{Mx}$}.

thumbnail Fig. 4

Various integrated quantities as a function of time, for the dipolar simulations with different ν0 (indicated by line styles). Panel a) shows the open magnetic flux ∫r = r1 | Br | dΩ, panel b) shows ∫D | j | dV, and panel c) shows HN (asterisks) and HS (circles). Panel d) shows the terms in Eq. (18) for the northern hemisphere, with asterisks denoting S0, circles S1, squares Seq, and diamonds SV.

thumbnail Fig. 5

Latitudinal distribution of field line helicity for the dipolar simulations, showing how the peak value tends to zero as the friction parameter ν0 is successively doubled. A log-log fit shows that 𝒜max~ν0-0.8\hbox{${\cal A}_{\rm max}\sim \nu_0^{-0.8}$}.

thumbnail Fig. 7

Illustration of the quadrupolar simulation with ν0 = 0.36 × 10-5 s-1 on days 0, 20, 66, and 68. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 2 G), and projected coronal magnetic field lines traced from height r = 1.2R⊙ are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 1021Mx with white indicating \hbox{${\cal A}\approx 0\,\mathrm{Mx}$}.

thumbnail Fig. 8

Various integrated quantities as a function of time, for the quadrupolar simulations with different ν0 (indicated by line styles). The format is the same as Fig. 4. For clarity, panel d) shows only the run with ν0 = 0.36 × 10-5 s-1, and only for the northern hemisphere, although the hemispheres are no longer symmetric. The peak value of S1 during the flux rope eruption is not shown, and is much larger, about − 2.7 × 1043 Mx2 day-1.

thumbnail Fig. 10

Projected magnetic field lines in the period A (left column) and B (right column) simulations, on days 0, 60, 120, and 180. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 10 G), and projected coronal magnetic field lines traced from height r = r0 are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 2.5 × 1021 Mx. Animated versions of these sequences are available online.

thumbnail Fig. 11

Various integrated quantities as a function of time, for the non-axisymmetric simulations (periods A and B). Panel a) shows the total photospheric magnetic flux ∫r = r0 | Br | dΩ, panel b) shows the total open flux ∫r = r1 | Br | dΩ, panel c) shows ∫D | j | dV, panel d) shows HN (asterisks) and HS (circles), and panel e) shows the root-mean-square field line helicity (∫𝒜2D dV/∫D dV)1/2\hbox{$(\int_D{\cal A}^2\,\mathrm{d}V/\int_D\,\mathrm{d}V)^{1/2}$}. The vertical grey lines indicate times of strong flux rope ejections, as explained in the text.

thumbnail Fig. 12

Example of a flux rope ejection from period A. From top to bottom, the rows show days 122, 124, 126, and 128. The left column shows the (absolute) running daily difference of horizontal field B⊥:=(Bθ2+Bφ2)1/2\hbox{$B_\perp := (B_\theta^2 + B_\phi^2)^{1/2}$} at the outer boundary r = r1. The middle column shows the distribution of \hbox{${\cal A}$} on r = r0 (saturated at ± 1022 Mx), and the right column shows the distribution of Tw at r = r0 (saturated at ± 40). The dashed lines in the left column show the neutral line where Br(r1,θ,φ) = 0. Black circles in the other columns identify footpoints of field lines traced down from locations at r = r1 where the running difference of B⊥ exceeds 0.05 G day-1. Animated versions of this figure for both periods A and B are available online.

Movies

Movie 1 of Fig. 10 Access here

Movie 2 of Fig. 10 Access here

Movie 1 of Fig. 12 Access here

Movie 2 of Fig. 12 Access here

References

  1. Berger, M. A., & Ruzmaikin, A. 2000, J. Geophys. Res., 105, 10481 [NASA ADS] [CrossRef] [Google Scholar]
  2. DeVore, C. R. 2000, ApJ, 539, 944 [NASA ADS] [CrossRef] [Google Scholar]
  3. Yeates, A. R., & Hornig, G. 2016, A&A, 594, A98 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]

© ESO, 2017

All Figures

thumbnail Fig. 3

Illustration of the dipolar simulation with ν0 = 0.36 × 10-5 s-1 on days 0, 1, 2, and 20. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 0.5 G), and projected coronal magnetic field lines traced from height r = R⊙ are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 5 × 1019 Mx with white indicating \hbox{${\cal A}\approx 0\,\mathrm{Mx}$}.
In the text
thumbnail Fig. 4

Various integrated quantities as a function of time, for the dipolar simulations with different ν0 (indicated by line styles). Panel a) shows the open magnetic flux ∫r = r1 | Br | dΩ, panel b) shows ∫D | j | dV, and panel c) shows HN (asterisks) and HS (circles). Panel d) shows the terms in Eq. (18) for the northern hemisphere, with asterisks denoting S0, circles S1, squares Seq, and diamonds SV.
In the text
thumbnail Fig. 5

Latitudinal distribution of field line helicity for the dipolar simulations, showing how the peak value tends to zero as the friction parameter ν0 is successively doubled. A log-log fit shows that 𝒜max~ν0-0.8\hbox{${\cal A}_{\rm max}\sim \nu_0^{-0.8}$}.
In the text
thumbnail Fig. 7

Illustration of the quadrupolar simulation with ν0 = 0.36 × 10-5 s-1 on days 0, 20, 66, and 68. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 2 G), and projected coronal magnetic field lines traced from height r = 1.2R⊙ are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 1021Mx with white indicating \hbox{${\cal A}\approx 0\,\mathrm{Mx}$}.
In the text
thumbnail Fig. 8

Various integrated quantities as a function of time, for the quadrupolar simulations with different ν0 (indicated by line styles). The format is the same as Fig. 4. For clarity, panel d) shows only the run with ν0 = 0.36 × 10-5 s-1, and only for the northern hemisphere, although the hemispheres are no longer symmetric. The peak value of S1 during the flux rope eruption is not shown, and is much larger, about − 2.7 × 1043 Mx2 day-1.
In the text
thumbnail Fig. 10

Projected magnetic field lines in the period A (left column) and B (right column) simulations, on days 0, 60, 120, and 180. Greyscale shading on r = r0 shows Br (white positive, black negative, saturated at ± 10 G), and projected coronal magnetic field lines traced from height r = r0 are coloured (red/blue) according to \hbox{${\cal A}$}, saturated at ± 2.5 × 1021 Mx. Animated versions of these sequences are available online.
In the text
thumbnail Fig. 11

Various integrated quantities as a function of time, for the non-axisymmetric simulations (periods A and B). Panel a) shows the total photospheric magnetic flux ∫r = r0 | Br | dΩ, panel b) shows the total open flux ∫r = r1 | Br | dΩ, panel c) shows ∫D | j | dV, panel d) shows HN (asterisks) and HS (circles), and panel e) shows the root-mean-square field line helicity (∫𝒜2D dV/∫D dV)1/2\hbox{$(\int_D{\cal A}^2\,\mathrm{d}V/\int_D\,\mathrm{d}V)^{1/2}$}. The vertical grey lines indicate times of strong flux rope ejections, as explained in the text.
In the text
thumbnail Fig. 12

Example of a flux rope ejection from period A. From top to bottom, the rows show days 122, 124, 126, and 128. The left column shows the (absolute) running daily difference of horizontal field B⊥:=(Bθ2+Bφ2)1/2\hbox{$B_\perp := (B_\theta^2 + B_\phi^2)^{1/2}$} at the outer boundary r = r1. The middle column shows the distribution of \hbox{${\cal A}$} on r = r0 (saturated at ± 1022 Mx), and the right column shows the distribution of Tw at r = r0 (saturated at ± 40). The dashed lines in the left column show the neutral line where Br(r1,θ,φ) = 0. Black circles in the other columns identify footpoints of field lines traced down from locations at r = r1 where the running difference of B⊥ exceeds 0.05 G day-1. Animated versions of this figure for both periods A and B are available online.
In the text

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