Erratum: “Effect of Electrical Resistivity on the Damping of Slow Sausage Modes” (2020, ApJ, 897, 120)

ApJ, 897, 120) Michaël Geeraerts , Tom Van Doorsselaere , Shao-Xia Chen, and Bo Li 1 Centre for mathematical Plasma Astrophysics (CmPA), KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium; michael.geeraerts@kuleuven.be 2 Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, People’s Republic of China Received 2022 July 11; published 2022 September 14


Introduction
In the published article we studied slow sausage modes in a cylindrical flux tube in linearized resistive magnetohydrodynamics (MHD) using cylindrical coordinates (r, j, z). We assumed that electrical resistivity is small, such that the diffusion of the background magnetic field could be neglected and the approximation of a sharp boundary separating two homogeneous plasmas inside and outside the tube could be made. The perturbed MHD quantities f 1 were taken of the form The bar is dropped next for simplicity. We then derived the solution of the compression R(r): where C 1 , C 2 , C 3 , and C 4 are constants; J 0 is the complex-valued Bessel function of the first kind of order 0; H 0 stands for either H 0 1 ( ) or H 0 2 ( ) depending on the argument of the function (the amplitude of the oscillations having to decrease as r → ∞, corresponding to a wave propagating energy outwardly from the cylinder); and From R, all the other MHD quantities could be derived.
To derive the dispersion relation of the modes, four boundary conditions were needed at the interface between the two plasmas. The boundary conditions that were used are the following: with ξ r the radial component of the plasma displacement, P 1 the perturbed total pressure, B the magnetic field, E the electric field, v r the component of the plasma velocity along its normal, and n the unit normal to the boundary. Here [f] denotes the jump of a quantity f at the boundary.

Correction 1
The fourth boundary condition, Equation (7), is actually not correct for our model. The correct form of this boundary condition in this case is However, this is equivalent to Equation (6) for time-varying B and thus cannot be used as an independent fourth boundary condition. Instead, the following condition needs to be used, as it must hold at any position in a resistive plasma (Roberts 1967): As in our case we have n = 1 r and B = B 0z 1 z + B 1 , Equation (9) becomes [B z ] = 0, with B z the z-component of B. However, we note that our approximation of the interface remaining discontinuous entails that there remains a nonvanishing jump in the background magnetic field B 0 instead of a continuous smoothing of B 0 in a narrow diffusion layer around the initial interface. Equation (9) is hence not fulfilled by the background magnetic field B 0 , similarly to the case of an ideal plasma (in which case the left-hand side of Equation (9) defines μ 0 J, with J the surface current on the interface). Only the perturbed magnetic field will thus satisfy Equation (9), yielding The dispersion relation obtained by imposing the boundary conditions (4), (5), (6), and (9) is then altered (with respect to the one described in the published article, namely, Equation (18) therein) and becomes Here F 1 , F 2 , F 3 and F 4 are very long expressions in ω. Equation (11) should thus replace Equation (18) in the published article. We call the left-hand side of Equation (11) the dispersion function.
To find the ideal limit (i.e., η → 0) of this dispersion relation, the asymptotic approximations for large arguments of the Bessel and Hankel functions J 0 , Abramowitz & Stegun 1965, Chapter 9) can be used for the functions having κ + a as an argument in Equation (11), as indeed |κ + | → ∞ when η → 0. From the asymptotic form of J n for large arguments it can be seen that, for a complex nonreal z˜for which z arg(˜) remains constant while z  ¥ |˜| , we have ) . Hence, for this mode we use Equations (12) and (14).
As η → 0, the first term on the left-hand side of Equation (11) is O(η), whereas the second, third, fourth, and fifth terms are O h ( ). The last term is 0 because F 4 is identically 0 in this limit. For surface modes this leads to the following limit of the dispersion relation ) . This relation is almost the dispersion relation from ideal MHD given, for example, by Equation (8) in Edwin & Roberts (1983), but there is a discrepancy due to the model not being entirely physical. Indeed, there remains a jump in the background magnetic field at r = a over time such that the boundary condition (9) is not fulfilled by the background. In the special case where B 0zi = B 0ze the model is physical, as there is no diffusion of the background magnetic field, and condition (9) is fulfilled by the background. We see that in that case Equation (16) reverts exactly to the ideal dispersion relation. The plots resulting from the corrected dispersion relation (11) and corresponding to Figures 2 and 3 of the published article are very similar to those, and our qualitative discussion of them in Section 4 remains entirely valid. The correct plots resulting from the dispersion relation (11) are shown in Figures 2 and 3.

Correction 2
A second minor correction concerns the ideal limit discussed in Section 3.2 in the published article and is required owing to the fact that we never gave an explicit definition of m. It should first be clarified what is meant by " ." We can take as the definition for z˜of a number z Î  the principal value of the square root of z˜, defined as the solution w Î  to z w 2 = with w Arg In Section 3.2 we mentioned that for η → 0 we have The plus sign must be used except if arg 2 k p  -( ) as η → 0. In that case the argument of the square root function " " in Equation (18) lies on the other branch of the multivalued square root function with respect to the argument of the external " " in the true expression of κ − in Equation (3). For completeness we note that for η → 0, which is used in order to find the ideal limit of dispersion relation (11) in the previous section. As a result, the expression for R(r) in Equation (2) becomes If Equation ( )in the part r < a of the solution, which can be rewritten as I 0 (m i r) if m 0 i 2 > (i.e., for a surface mode). Hence, we retrieve the ideal solution in its form given by Edwin & Roberts (1983).

Correction 3
In Section 5 of the published article we investigated the long-wavelength limit of the dispersion relation. Whereas the method employed in that section was correct, the new dispersion relation found here does not allow us to calculate in advance the expression of the frequency ω for the limit k z a → 0, rendering the method followed in the published article useless. It turns out to be actually quite complicated to calculate the long-wavelength limit of this dispersion relation for slow modes owing to their having a frequency close to ω Ci . As mentioned in the published article, it can be shown that κ ± → 0 when k z a → 0. However, as κ + → ∞ when η → 0, the derivation of a long-wavelength limiting form of the dispersion relation using truncated Taylor series like in Section 5 of the published article is only applicable for extremely small values of k z a. These are so small that they are not relevant for realistic conditions of slow modes in photospheric pores, as the small value of resistivity keeps κ + a several orders of magnitude higher than them. For observations of slow sausage modes in pore conditions (such as discussed, e.g., by Grant et al. 2015;Moreels et al. 2015;Freij et al. 2016) we indeed have κ + a ? 1. The slow-mode oscillations observed in photospheric pores do not have a long wavelength, but even for values of k z a typically considered in the long-wavelength limit in coronal loops (say, 10 −2  k z a = 1) the condition κ + a ? 1 is still valid. It therefore does not seem useful to further attempt to find an analytical expression for the dispersion relation in the long-wavelength limit with η ≠ 0, and the results presented in Section 5 in the published article are best ignored.