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Astron. Astrophys. 331, 392-404 (1998)

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4. Results: eigenfrequencies for a uniform flux tube: comparison with Cally (1986)

The characteristic uniform equilibrium quantities that we use in this paragraph will be chosen as the axial values of the non-uniform equilibrium, to make the comparison between homogeneous and inhomogeneous tubes possible. As a consequence our uniform equilibrium quantities are not exactly the same as Cally's. He also included an external magnetic field, which is neglected in our equilibrium. Therefore we exclude the possibility of fast and slow magneto-acoustic waves in the external region and consider only acoustic oscillations. The density contrast between the external and internal density equals 2.

The values and order of characteristic speeds for our uniform model are:


when scaled to the axial Alfvén speed. The subscript 'e' or 'i' denotes the external or internal region. The tube speed is defined by [FORMULA].

To compare with the characteristic speeds Cally employs, we have to multiply with a scaling factor [FORMULA] to get:


Our results may therefore differ from Cally's results, but the overall picture will be satisfyingly similar.

4.1. Sausage modes: [FORMULA]

Fig. 2 shows numerically calculated leaky and non-leaky sausage modes. It is to be compared with Cally's dispersion diagram Fig. 1a for [FORMULA]. Notice the logarithmic scale for the imaginary part in our figure.

[FIGURE] Fig. 2. Eigenspectrum for the sausage modes ([FORMULA]) for a uniform photospheric flux tube with characteristic velocities [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. This spectrum can be compared with Fig. 1a in Cally (1986). Notice the logarithmic scaling for the imaginary part of the eigenfrequencies. The solid lines in the upper figure are respectively the slow, the sound and the Alfvén frequency. The different kind of modes, as discussed in the text are denoted by different letters.

The following modes can be distinguished (letters corresponding to those on Fig. 1 in Cally):

a) The frequency of this mode only has an imaginary part. So it can be considered as a surface perturbation diffusing into the external region. As [FORMULA] it becomes one of Cally's 'cubic' modes. In the long wavelength limit, one can obtain a cubic equation in [FORMULA], where [FORMULA] is the longitudinal phase velocity (Cally Eq. 5.14):


where [FORMULA] (d is the ratio of the external to the axial density). Either one or all of the roots are real. For the photospheric values discussed above, we obtain three real roots for [FORMULA]: [FORMULA] or [FORMULA] or [FORMULA] and thus two real and 1 imaginary root for v. This one is the cubic mode whose phase speed converges to the imaginary cubic root ([FORMULA].

b) A single non-leaky slow, surface mode ([FORMULA] mode in Cally's notation), which converges to one of the two real cubic roots as [FORMULA] ([FORMULA]. Cally also considered the thin tube approximation where R (radius of the tube) is assumed small compared with the characteristic longitudinal length scale of the waves:



When [FORMULA] and [FORMULA] is purely imaginary, as is the case here, the phase speed v converges to [FORMULA] from below and therefore this thin tube mode corresponds to this single surface mode.

[FORMULA] Two of an infinite anti-Sturmian set of body-modes (indicated by [FORMULA] modes in Cally). We use a capital letter (C) to indicate that this set includes more modes than shown. In contrast with Cally we do find a very small imaginary part for these body modes. This would mean that at least some of these body modes are also leaky. In fact, there are an infinite number of non-leaky body modes (Cally 1985), with a possible finite number at the start of the sequence being slightly leaky. We seem to have found some of these leaky modes. For both [FORMULA] and [FORMULA] the imaginary part vanishes. In these asymptotic regions the two body modes are non-leaky. Their finite decay time is so large compared to their period that they seem like wave-solutions. As [FORMULA] they converge to an infinite set of thin tube modes accumulating to [FORMULA] from above.

d) A single non-leaky, surface mode ([FORMULA] mode in Cally's notation), which converges to the second real, cubic root as [FORMULA] ([FORMULA]).

e, f & g) Cally distinguishes another class of solutions, which are mostly found in the high frequency, long wavelength limit and are called 'trig' modes. Asymptotically the trig modes produce an infinite number of solutions described by Eq. 5.6 in Cally or by:



The asymptotic ([FORMULA]) imaginary part and the spacing between the real parts can be calculated for the photospheric values discussed above:



Therefore the trig modes are a set of infinite, nearly equally spaced eigenfrequencies which lie asymptotically along a horizontal line in the lower half [FORMULA] -plane. Cally describes the eigenfrequencies to approach the real axis asymptotically as [FORMULA], with [FORMULA] increasing linearly with [FORMULA].

These three modes are respectively the first, second and third trig mode. They are leaky modes whose period and decay time are of the same order for [FORMULA] and when [FORMULA] increases the period decreases whilst the decay time becomes long. The spacing between the real parts (as indicated by the vertical full line between f and g in Fig. 2) of subsequent trig modes and the asymptotic imaginary part (indicated with horizontal full line) is as predicted.

4.2. Kink modes: [FORMULA]

Fig. 3 shows the dispersion diagram for the kink ([FORMULA]) modes. The modes are completely analogous as in Fig. 2 and can be compared with Cally's Fig. 1b.

[FIGURE] Fig. 3. Idem as in Fig. 2, but now for [FORMULA] (kink-modes). It can be compared with Fig. 1b in Cally (1986).

The analogy with Cally stands except for the fact that Cally only finds the cubic mode (a) for [FORMULA], while we do find it for the full [FORMULA] -range.

Apart from a few differences, the same interpretation can be made as for the sausage modes. The body kink modes found here do not have an imaginary part, in contrast with some of the body sausage modes (C) we found.

For [FORMULA], the phase speed in the thin tube approximation


as [FORMULA], which is the kink speed [FORMULA] (d is the ratio of the external and internal density). For the photospheric values used above:


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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998