Resonant absorption of the slow sausage wave in the slow continuum (Corrigendum)

D. J. Yu; T. Van Doorsselaere; M. Goossens

doi:10.1051/0004-6361/201630355e

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Erratum

This article is an erratum for:
[https://doi.org/10.1051/0004-6361/201630355]

Issue		A&A Volume 626, June 2019


Article Number		C2
Number of page(s)		1
Section		The Sun
DOI		https://doi.org/10.1051/0004-6361/201630355e
Published online		12 June 2019

A&A 626, C2 (2019)

Resonant absorption of the slow sausage wave in the slow continuum (Corrigendum)

D. J. Yu, T. Van Doorsselaere and M. Goossens

Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, 3001 Leuven, Belgium

Key words: magnetohydrodynamics (MHD) / Sun: photosphere / Sun: oscillations / errata, addenda

Erratum for: A&A 602, A108 (2017), https://doi.org/10.1051/0004-6361/201630355

The derivation of the damping rate in the long wavelength limit by Yu et al. (2017) is incorrect. This was already noted independently by Sadeghi and Karami (2019) and by Roberts (2019, priv. comm.). What follows is the correct derivation. The sign of P₀ (A.7) in Appendix A was incorrect. Equation (41) should read

$\begin{matrix} T_{0} & = ω_{r}^{2} (ω_{r}^{2} - ω_{Ae}^{2}) \times \\ {\frac{(ω_{r}^{2} - 2 ω_{ci}^{2}) k_{i}^{2} R^{2} [1 - \frac{3}{16} k_{i}^{2} R^{2}] ln (k_{e} R)}{(ω_{r}^{2} - ω_{si}^{2}) (ω_{r}^{2} - ω_{Ai}^{2}) (ω_{r}^{2} - ω_{Ci}^{2})} \\ + \frac{(ω_{r}^{2} - 2 ω_{Ce}^{2}) k_{i}^{2} R^{2}}{2 (ω_{r}^{2} - ω_{se}^{2}) (ω_{r}^{2} - ω_{Ae}^{2}) (ω_{r}^{2} - ω_{Ce}^{2})}} \end{matrix}$ $\begin{aligned} T_0&= \omega _r^2(\omega ^2_r-\omega _{Ae}^2)\times \nonumber \\&\bigg \{ \frac{(\omega _r^2-2\omega _{ci}^2)k_i^2R^2\big [1-\frac{3}{16}k_i^2R^2\big ]\ln ({k_eR})}{(\omega _r^2-\omega _{si}^2)(\omega _r^2-\omega _{Ai}^2)(\omega _r^2-\omega _{Ci}^2)}\nonumber \\&+\frac{(\omega _r^2-2\omega _{Ce}^2)k_i^2R^2}{2(\omega _r^2-\omega _{se}^2) (\omega _r^2-\omega _{Ae}^2)(\omega _r^2-\omega _{Ce}^2)}\bigg \}\nonumber \end{aligned}$

and for Eq. (43)

$\begin{matrix} T_{0} & = \frac{4 ω_{Ai}^{6}}{χ^{2} ω_{Ci}^{2} ω_{si}^{2} (ω_{Ci}^{2} - ω_{Ae}^{2}) k_{z}^{2} R^{2} ln (k_{z} R)}, \end{matrix}$ $\begin{aligned} T_0&= \frac{4\omega _{Ai}^6}{\chi ^2\omega _{Ci}^2\omega _{si}^2(\omega _{Ci}^2-\omega _{Ae}^2)k_z^2R^2\ln (k_zR)} ,\nonumber \end{aligned}$

where we have only retained the first term (see Eq. (C.1) below). Equations (44) and (45) should then read

$\begin{matrix} γ_{0} & = - \frac{π χ^{2} (l / R)}{8 | ω_{Ce} - ω_{Ci} |} \frac{ω_{Ci}^{6} {(ω_{Ci}^{2} - ω_{Ae}^{2})}^{2}}{ω_{Ai}^{8}} {(k_{z} R)}^{4} {ln}^{2} (k_{z} R), \end{matrix}$ $\begin{aligned} \gamma _0&= -\frac{\pi \chi ^2(l/R)}{8|\omega _{Ce}-\omega _{Ci}|} \frac{\omega _{Ci}^6(\omega _{Ci}^2-\omega _{Ae}^2)^2}{\omega _{Ai}^8}(k_zR)^4\ln ^2(k_zR),\nonumber \end{aligned}$

$\begin{matrix} γ_{0} & = - \frac{π χ^{2}}{8} \frac{l}{R} \frac{ω_{Ci}^{9}}{ω_{Ai}^{8}} {(k_{z} R)}^{4} {ln}^{2} (k_{z} R) . \end{matrix}$ $\begin{aligned}\gamma _0&= -\frac{\pi \chi ^2}{8}\frac{l}{R} \frac{\omega _{Ci}^9}{\omega _{Ai}^8}(k_zR)^4\ln ^2(k_zR).\nonumber \end{aligned}$

These expressions for γ₀ are the same as those obtained by Roberts (2019) by using a different approach.

Equations (C.1), (C.3), and (C.4) in Appendix C should read

$\begin{matrix} T_{0} & = \frac{4 ω_{Ai}^{6}}{χ^{2} ω_{Ci}^{2} ω_{si}^{2} (ω_{Ci}^{2} - ω_{Ae}^{2}) k_{z}^{2} R^{2} ln (k_{z} R)} \\ + \frac{3 ω_{Ai}^{8}}{2 χ^{3} ω_{si}^{4} {(ω_{Ci}^{2} - ω_{Ae}^{2})}^{2} k_{z}^{2} R^{2} {ln}^{2} (k_{z} R)} \\ - \frac{ω_{Ci}^{4} ω_{Ai}^{2} (ω_{Ci}^{2} - 2 ω_{Ce}^{2})}{χ ω_{si}^{2} (ω_{Ci}^{2} - ω_{Ae}^{2}) (ω_{Ci}^{2} - ω_{se}^{2}) (ω_{Ci}^{2} - ω_{Ce}^{2}) ln (k_{z} R)}, \end{matrix}$ $\begin{aligned} T_0&= \frac{4\omega _{Ai}^6}{\chi ^2\omega _{Ci}^2\omega _{si}^2(\omega _{Ci}^2-\omega _{Ae}^2) k_z^2R^2\ln (k_zR)}\nonumber \\&+\frac{3\omega _{Ai}^8}{2\chi ^3\omega _{si}^4(\omega _{Ci}^2-\omega _{Ae}^2)^2 k_z^2R^2\ln ^2(k_zR)}\nonumber \\&-\frac{\omega _{Ci}^4\omega _{Ai}^2(\omega _{Ci}^2-2\omega _{Ce}^2)}{\chi \omega _{si}^2(\omega _{Ci}^2-\omega _{Ae}^2)(\omega _{Ci}^2-\omega _{se}^2) (\omega _{Ci}^2-\omega _{Ce}^2)\ln (k_zR)}\nonumber , \end{aligned}$

$\begin{matrix} T_{0} & = \frac{4 ω_{Ai}^{6}}{χ^{2} ω_{Ci}^{4} ω_{si}^{2} k_{z}^{2} R^{2} ln (k_{z} R)} + \frac{3 ω_{Ai}^{8}}{2 χ^{3} ω_{si}^{4} ω_{Ci}^{4} k_{z}^{2} R^{2} {ln}^{2} (k_{z} R)} \\ - \frac{ω_{Ci}^{2} ω_{Ai}^{2}}{χ ω_{si}^{2} (ω_{Ci}^{2} - ω_{se}^{2}) ln (k_{z} R)}, \end{matrix}$ $\begin{aligned} T_0&= \frac{4\omega _{Ai}^6}{\chi ^2\omega _{Ci}^4\omega _{si}^2 k_z^2R^2\ln (k_zR)}+\frac{3\omega _{Ai}^8}{2\chi ^3\omega _{si}^4\omega _{Ci}^4 k_z^2R^2\ln ^2(k_zR)}\nonumber \\&-\frac{\omega _{Ci}^2\omega _{Ai}^2}{\chi \omega _{si}^2(\omega _{Ci}^2-\omega _{se}^2) \ln (k_zR)} ,\nonumber \end{aligned}$

and

$\begin{matrix} \frac{γ_{0}}{ω_{Ci}} = \frac{π χ (l / R) (\frac{ω_{Ci}^{4}}{ω_{si}^{2} ω_{Ai}^{2}}) k_{z}^{2} R^{2} ln (k_{z} R)}{\frac{4 ω_{Ai}^{2}}{ω_{Ci}^{2}} - \frac{8 ω_{Ai}^{6}}{χ ω_{Ci}^{4} ω_{si}^{2} k_{z}^{2} R^{2} ln (k_{z} R)} - \frac{3 ω_{Ai}^{8}}{χ^{2} ω_{si}^{4} ω_{Ci}^{4} k_{z}^{2} R^{2} {ln}^{2} (k_{z} R)} + \frac{2 ω_{Ci}^{2} ω_{Ai}^{2}}{ω_{si}^{2} (ω_{Ci}^{2} - ω_{se}^{2}) ln (k_{z} R)}} \cdot \end{matrix}$ $\begin{aligned} \frac{\gamma _0}{\omega _{Ci}}= \frac{\pi \chi (l/R)\bigg (\frac{\omega _{Ci}^4}{\omega _{si}^2\omega _{Ai}^2}\bigg )k_z^2R^2\ln (k_zR)}{\frac{4\omega _{Ai}^2}{\omega _{Ci}^2}-\frac{8\omega _{Ai}^6}{\chi \omega _{Ci}^4\omega _{si}^2 k_z^2R^2\ln (k_zR)} -\frac{3\omega _{Ai}^8}{\chi ^2\omega _{si}^4\omega _{Ci}^4k_z^2R^2\ln ^2(k_zR)} +\frac{2\omega _{Ci}^2\omega _{Ai}^2}{\omega _{si}^2(\omega _{Ci}^2-\omega _{se}^2) \ln (k_zR)}}\cdot \nonumber \end{aligned}$

Fig. 5.

Damping rate |γ₀|/ω versus k_zR for slow sausage surface mode (ss) where l/R = 0.1. The other parameters are the same as in the previous figures. For the linear cusp velocity, we compared the analytical formula, Eq. (35), with the simplified formula in the long wavelength limit, Eq. (45).

It should be noted that the values of the quantity |γ ₀|/ω in Fig. 5 are small as can be understood from the equation for |γ ₀|/ω for the values of k _z R ≤ 0.4 used in Fig. 5.

We note also that eight in the second paragraph after Eq. (10) and 8 in the caption of Fig. 1 should read four.

Acknowledgments

We are grateful to B. Roberts for pointing out the error in our manuscript. Equations (41), (43)–(45), (A.7), (C.1), (C.3), and (C.4) were wrong in our paper.

References

Yu, D. J., Van Doorsselaere, T., & Goossens, M. 2017, A&A 602, A108 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
Sadeghi, M. & Karami, K. 2019, ArXiv e-prints [arXiv:1903.04171] [Google Scholar]

© ESO 2019

All Figures

	Fig. 5. Damping rate \|γ₀\|/ω versus k_zR for slow sausage surface mode (ss) where l/R = 0.1. The other parameters are the same as in the previous figures. For the linear cusp velocity, we compared the analytical formula, Eq. (35), with the simplified formula in the long wavelength limit, Eq. (45).
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[1] Yu, D. J., Van Doorsselaere, T., & Goossens, M. 2017, A&A 602, A108 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]

[2] Sadeghi, M. & Karami, K. 2019, ArXiv e-prints [arXiv:1903.04171] [Google Scholar]