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Solar Physics

, 294:167 | Cite as

Correction to: On Fields and Mass Constraints for the Uniform Propagation of Magnetic-Flux Ropes Undergoing Isotropic Expansion

  • Daniel Benjamín BerdichevskyEmail author
Correction
Part of the following topical collections:
  1. Flux-Rope Structure of Coronal Mass Ejections

Correction to: Solar Phys. (2013) 284:245 – 259   https://doi.org/10.1007/s11207-012-0176-5

When expressing the steady current density \(\mathbf{J} _{\mathrm{c}}\) (convection-current) it is easy to check that, up to \(O(R _{\mathrm{FRcore}} / R_{ \mathrm{cFR} } ) ^{2}\), its evaluation (\(\mathbf{J} _{\mathrm{c}} = \nabla\times\mathbf{H}\), see e.g. Jackson 1963) gives an axial component that is
$$J _{\mathrm{c}x}(\mathit{modified})= A(t)/\mu _{0} \bigl\{ H B_{x} \bigl[1 - 2 \rho/R _{\mathrm{cFR}} \bigl(\cos(\varphi) - | \sin(\varphi)| \bigr)\bigr] + \Delta j _{cx}\bigr\} $$
with
$$\Delta j _{\mathrm{c}x} = \rho/R _{\mathrm{cFR}} \bigl(\cos(\varphi) - | \sin(\varphi)|\bigr) J _{1} \bigl( A(t,\rho,\varphi)\bigr)/A(t,\rho, \varphi) $$
correcting/completing the \(\mathbf{J} _{\mathrm{c}}\) expression presented before Equation 8 in the paper. The meaning for each symbol used follows:
  • \(H\ (=+1\mbox{ or}-1)\) is the handedness of the flux rope (FR).

  • \(\mathbf{H}\ (= \mathbf{B}/ \mu _{0})\) is the magnetic field intensity, and \(\mathbf{B}\) the magnetic field density.

  • \(\mu _{0}\) is the magnetic permeability of the vacuum. (We assume no magnetization.)

  • \(R _{\mathrm{FRcore}}\) is the cross section radius of the FR modeled.

  • \(R _{\mathrm{cFR}}\) is the distance from the center of the Sun to the axis of the flux rope modeled.

  • \(A(t) = j _{0} / R _{\mathrm{FRcore}}\), where \(j _{0}\) is the first node of the grade 0 cylindrical Bessel function ‘\(\mathcal{J}_{0}(A(t,\rho,\varphi))\)’ of the first kind.

  • \(B _{x} = B _{0} (t _{0} /t)^{2} \mathcal{J} _{0}\) (\(A(t,\rho,\varphi)\)) is the axial field of the flux-rope, where \(B _{0}\) is the magnetic field strength parameter in the MHD flux-rope solution at the initial time \(t _{0}\) in its evolution. Finally, the argument of the Bessel function \(\mathcal{J}_{0(1)}\) is
    $$A(t, \rho, \varphi) = A(t)\rho \bigl[1 + \rho/R_{\mathrm{cFR} } \bigl(\cos( \varphi) - |\sin(\varphi)|\bigr)\bigr], $$
    which is given in Equation 7 of the paper. (\(\rho , \varphi\) are polar radial and angle coordinates.)
The other two corrected components are
$$\begin{aligned} J _{c \rho} =& - A(t)/\mu _{0} H B _{\phi} \rho/R_{ \mathrm{cFR} } \bigl[\sin(\varphi) + \partial _{\varphi} |\sin( \varphi)|\bigr]\\ J _{c \phi} =& A(t)/\mu _{0} H B _{\phi} \bigl[1 - 2 \rho/R_{ \mathrm{cFR} } \bigl(\cos(\varphi) - |\sin(\varphi)| \bigr)\bigr] \end{aligned}$$
where \(B _{\phi} = HB _{0} (t _{0} /t) ^{2} J _{1}\) (\(A(t,\rho,\varphi)\)) is the polar component of the magnetic field density \(\mathbf{B}\).

The relevance of the previously missing term \(\Delta j _{cx}\) is that it dominates the axial current contribution within the FR plasma located in the region defined by \(0.93 R _{\mathrm{FRcore}} < \rho < 1.07R _{\mathrm{FRcore}}\).

It is straightforward to check that \(\mathbf{div}(\mathbf{J}_{\mathrm{c}}) = 0\) is well satisfied up to \(n=1\) in a perturbation expansion in \(\varepsilon ^{n}\) of our 3D MHD analytical solution of a FR evolution, where \(\varepsilon \equiv R _{\mathrm{FRcore}} / R_{ \mathrm{cFR} }\) is a small quantity.

Notes

References

  1. Jackson, D.: 1963, Classical Electrodynamics, Section 5.8, 3rd edn. Wiley, New York. Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College ParkUSA

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