Solar Physics

, 294:167

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

Correction
Part of the following topical collections:
1. Flux-Rope Structure of Coronal Mass Ejections

Correction to: Solar Phys. (2013) 284:245 – 259

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.

## References

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