Subscriber Authentication Point
Free Access
 Issue A&A Volume 575, March 2015 A130 11 The Sun https://doi.org/10.1051/0004-6361/201424611 11 March 2015

Online material

Appendix A: Graphical investigation of the dispersion equation

In this section we present the graphical investigation of the dispersion equation. We start from the case where ζ> 1. In Fig. A.1 the graph of function Y = f(ϖ) is shown for κ> 0. The upper horizontal line Y = 4/(ζ − 1) corresponding to ζ> 1 crosses this graph twice, so there are two positive roots to Eq. (51). However, only one of them satisfies the inequality (53). Hence, there is only one trapped wave mode with positive frequency when κ> 0.

 Fig. A.1 Graphical investigation of the dispersion equation for κ> 0. The solid lines show the graph of function Y = f(ϖ), while the dashed lines have equations Y = 4/(ζ − 1). The upper dashed-dotted line corresponding to ζ> 1 intersects the solid lines at two points corresponding to two roots to the dispersion equation. However, only one of these roots satisfies the condition ϖ< | κ |. The vertical dotted line indicates the root satisfying the condition ϖ< | κ |. The lower dashed-dotted line corresponding to ζ< 1 intersects the solid lines at two points corresponding to roots of the dispersion equation. However, none of them satisfies the condition ϖ< | κ |. Open with DEXTER

In Fig. A.2 the graph of function Y = f(ϖ) is shown for . Now the upper horizontal line Y = 4/(ζ − 1) does not cross the graph of function Y = f(ϖ). Hence, there are no roots of Eq. (51) when . In Fig. A.3 the graph of function Y = f(ϖ) is shown for . Again the upper horizontal line Y = 4/(ζ − 1) does not cross the graph of function Y = f(ϖ). Hence, there are no roots of Eq. (51) when . Summarizing, we conclude that there are no propagating wave modes when − 1 <κ< 0.

 Fig. A.2 Graphical investigation of the dispersion equation for . The solid lines show the graph of function Y = f(ϖ), while the dashed-dotted lines have equations Y = 4/(ζ − 1). The upper dashed-dotted line corresponding to ζ> 1 does not intersect the solid lines, which implies that there are no propagating wave modes when . Depending on the value of ζ< 1, the lower dashed-dotted line intersects the solid lines either at two or at four points, so there are up to four positive roots of the dispersion equation. However, only one of them satisfies the condition ϖ< | κ |. The vertical dotted line indicates the root satisfying the condition ϖ< | κ |. Open with DEXTER

 Fig. A.3 Graphical investigation of the dispersion equation for . The solid lines show the graph of function Y = f(ϖ), while the dashed-dotted lines have equations Y = 4/(ζ − 1). The upper dashed-dotted line corresponding to ζ> 1 does not intersect the solid lines, which implies that there are no propagating wave modes when . Depending on the value of ζ< 1, the lower dashed-dotted line intersects the solid lines either at two or at four points, so there are up to four positive roots of the dispersion equation. The root corresponding to the intersection with the right branch of graph of function Y = f(ϖ) does not satisfy the condition ϖ< | κ |, while all other roots satisfy this condition. The vertical dotted lines indicate the roots satisfying the condition ϖ< | κ |· Open with DEXTER

Finally, Fig. A.4 displays the graph of function Y = f(ϖ) for κ< − 1. It is quite similar to Fig. A.1. In particular, the upper horizontal line Y = 4/(ζ − 1) again crosses the graph of function Y = f(ϖ) twice, so there are two positive roots to Eq. (51). However, now both roots satisfy the inequality (53). Hence, there are two trapped wave modes with positive frequency when κ< − 1 but, as is shown in Sect. 4.1, the wave with the higher frequency is a quasi-mode that is subject to resonance absorption.

 Fig. A.4 Graphical investigation of the dispersion equation for κ< − 1. The solid lines show the graph of function Y = f(ϖ), while the dashed lines have equations Y = 4/(ζ − 1). The upper dashed-dotted line corresponding to ζ> 1 intersects the solid lines at two points corresponding to two roots to the dispersion equation. Both these roots satisfy the condition ϖ< | κ |. The lower dashed-dotted line intersects the solid lines at two points, so there are two positive roots. But only the smaller root satisfies the condition ϖ< | κ |. The vertical dotted lines indicate the roots satisfying the condition ϖ< | κ |. Open with DEXTER

Now we proceed to studying the case where ζ< 1. Since now 4/(ζ − 1) < − 4, while f(ϖ) → − 1 as ϖ → ∞, in Fig. A.1 the lower horizontal line Y = 4/(ζ − 1) corresponding to ζ< 1 crosses the graph of function Y = f(ϖ) twice, so there are two positive roots of Eq. (51). However, none of them satisfies the inequality (53). Hence, there are no trapped wave modes when κ> 0.

Depending on the value of ζ< 1, the lower horizontal line Y = 4/(ζ − 1) in Fig. A.2 can cross the graph of function f(ϖ) either two or four times. Hence, there can be up to four positive roots to Eq. (51). However, only one of them satisfies the inequality (53). Hence, there is exactly one trapped wave mode with the positive frequency when .

 Fig. A.5 Dependence of fm on κ is shown by the solid line. The horizontal dashed-dotted line shows fm = 4/(ζ − 1), and the vertical dotted line indicates κm. Open with DEXTER

In Fig. A.3 the graph of function Y = f(ϖ) is shown for . For ϖ> 0 this graph consists of three branches. An intersection of the horizontal line Y = 4/(ζ − 1) with the right branch corresponds to a root that does not satisfy the inequality (53). The horizontal line intersects the left branch for any value ζ< 1. The corresponding root of Eq. (51) satisfies the inequality (53) and thus it is the frequency of a trapped mode. We denote the maximum value of function f(ϖ) in the interval (κ + 1, − κ) as fm(κ). This function monotonically decreases from 0 to − ∞ when κ varies from − 1 to . The dependence of fm on κ is shown in Fig. A.5. When 4/(ζ − 1) >fm(κ)

the dashed-dotted line does not intersect the middle branch, so in this case there is only one trapped wave mode. Otherwise there are three trapped wave modes. We see in Fig. A.5 that the condition 4/(ζ − 1) >fm(κ) is satisfied when , and it is not satisfied when κ ∈ ( − 1,κm). Hence, there are three trapped wave modes when κ ∈ ( − 1,κm), and only one when . The dependence of κm on ζ is shown in Fig. 3.

Finally, in Fig. A.4 the graph of function Y = f(ϖ) is shown for κ< − 1. The horizontal line intersects this graph at two points. But only the smaller root satisfies the condition ϖ< | κ |.

Appendix B: Proof that the frequencies of the decelerated kink wave and the quasi-mode do not coincide with the internal Alfvén frequency

In this section we prove that the frequencies of the two trapped wave modes that exist when ζ> 1 and κ< − 1 do not coincide with the internal Alfvén frequency. We can see in Fig. A.3 that the function f(ϖ) is monotonically increasing in the interval (0, − κ − 1). Let us prove that (B.1)This inequality can be rewritten as (B.2)We have (B.3)because ζ> 1 and κ − 1 < − κ. Hence, g(κ) is a monotonically increasing function. For | κ | ≫ 1(B.4)Since g(κ) is monotonically increasing, this result implies that the inequality (B.2) is correct for any κ< − 1. Then the same is true for the inequality (B.1), which can be rewritten as (B.5)Since f(ϖ) is monotonically increasing in the interval (0, − κ − 1), this inequality implies that ϖAi<ϖ. Since ϖ<ϖ+, it follows that also ϖAi<ϖ+.