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

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5. Results: eigenfrequencies for a non-uniform flux tube

We use a generalized Lou-type (1990) equilibrium, as in Goossens and Poedts (1992) and Keppens (1996). This ideal, axisymmetric 1D MHD equilibrium is characterized by 2 parameters [FORMULA] and d. The sharpness of the transition between the magnetized and the unmagnetized external region is controlled by [FORMULA]. And d measures the density ratio between the external and the axial density. In the results discussed here, the choice of parameters is:

  • [FORMULA]: a rather smooth transition from internal non-homogeneous magnetized region to the external non-magnetic surroundings
  • [FORMULA]: the external density is twice the axial density value.

If we scale the equilibrium quantities to the cylinder radius R, the total axial field [FORMULA] and the axial density [FORMULA], the analytical distribution of the pressure is (the subscript '0' denotes equilibrium variables):


where [FORMULA] is the dimensionless constant photospheric plasma pressure outside the flux tube.

The density profile is given by:




The magnetic field [FORMULA] (there is no twist in the magnetic field) is then determined from the static equilibrium condition, and varies smoothly from its axial value [FORMULA] to zero at the flux tube radius.

Figs. 4 and 5 show the profiles of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] for both [FORMULA] (Fig. 4) and [FORMULA] (Fig. 5)

[FIGURE] Fig. 4. Profile of Alfvén and slow frequency and of the cut-off frequencies for [FORMULA] and [FORMULA] and for a non-uniform photospheric flux tube with equilibrium parameters [FORMULA] and [FORMULA]. The regions where to expect oscillatory and exponential behaviour are indicated by 'osc.' and 'exp.'. The horizontal lines (b, C, d, and h) indicate modes found at these frequencies. Their letters correspond with those in and in the text.

[FIGURE] Fig. 5. Idem as Fig.4, but now for [FORMULA]. The horizontal lines indicate modes found at these frequencies (b, C, h). The letters correspond with those found in Fig. 9 and in the text.

Depending on the ordering of the eigenfrequency with respect to [FORMULA], [FORMULA], [FORMULA] and [FORMULA], we can determine where (in radial position) to expect oscillatory or exponential behaviour in the eigenfunctions, as indicated (by osc. and exp.) on Figs. 4 and 5. These predictions are valid for real eigenfrequencies, but even for complex frequencies, these figures still might give an idea about the kind of eigenfunctions to expect.

5.1. Sausage modes: [FORMULA]

Fig. 6 gives the complete sausage spectrum for a non-uniform flux-tube.

[FIGURE] Fig. 6. Eigenspectrum for the sausage modes ([FORMULA]) for a non-uniform photospheric flux tube. The equilibrium is characterized by a smooth transition ([FORMULA]) from the inhomogeneous and magnetized internal region to the uniform, unmagnetized, external region. The density contrast between the external and axial density is 2 (d). Different letters stand for different kinds of modes as discussed in the text.

For the trig modes, the picture found for the uniform flux tube remains. An infinite Sturmian set of leaky, fast, equally spaced (in real part) modes with an asymptotically constant imaginary part. When compared with the uniform tube, the real part of the frequency is decreased somewhat and the spacing between the modes is smaller. The imaginary part has a larger absolute value in the non-uniform case. The trig modes show a mainly oscillatory behaviour (as expected when looking at Fig. 4). The external solution increases with distance from the tube due to the leaky character of these modes. The first of these trig modes enters the Alfvén continuum for [FORMULA]. Since there is no Alfvén resonance for [FORMULA], this singularity has no consequences for the eigenvalue and the eigenfunction.

The cubic mode (a) is also (partly) found. The diffusion of this perturbation happens even faster than in the uniform case due to a slightly larger (in absolute value) imaginary part of the eigenfrequency.

We followed the fast surface mode (d) in the transition from uniform to non-uniform tubes. This transition was made by changing the radial percentage of non-uniformity from 0% (entirely homogeneous internal region and discontinuity at the boundary) to 100% (inhomogeneous internal region). The eigenfunctions for the inhomogeneous tube mode are shown on Fig. 7.

[FIGURE] Fig. 7. The real part of the radial displacement and of the pressure perturbation (imaginary part is zero) as a function of the radial coordinate for the eigenmode [FORMULA] ([FORMULA], [FORMULA]). This mode corresponds to a fast, surface mode in the case of a uniform tube. Due to the inhomogeneity the perturbation is shifted towards the Alfvén resonance and is rounded and looses therefore its typical surface-like behaviour. This eigenmode lies in the Alfvén continuum, but not in the slow continuum.

When the degree of non-uniformity is increased from 0% to 100%, the typical surface perturbation at the boundary-discontinuity in the uniform case is transformed continuously into a perturbation with a peak in amplitude at the Alfvén resonant surface. This resonance moves inwards as the percentage of non-uniformity increases. For [FORMULA], the Alfvén resonance does not lead to jumps in the eigenfunctions however.

Fig. 4 (horizontal line indicated by d) shows that in addition to the Alfvén resonance at the eigenfrequency [FORMULA] ([FORMULA]), there is a dip in the [FORMULA] -profile so that oscillating behaviour may be expected in this radial region. This oscillating behaviour probably provides for a continuous derivative at the peak of amplitude of the perturbation.

Due to this continuous derivative and the radial inwards shift of the perturbation, this mode looses the typical surface-like behaviour and might be characterized as a body-mode. Due to the non-uniformity and the abscence of a discontinuity the clear distinction between surface and body has faded, making the characterization of the modes more difficult.

This mode has a purely real frequency, and is thus a non-leaky mode. The external solution is therefore an exponential decreasing one.

The set of body-modes (C) is recovered in this inhomogeneous case. They are lying in the Alfvén continuum, but not in the slow continuum. But since [FORMULA], there are no jumps in the eigenfunction due to the Alfvén resonance. Fig. 4 shows that the behaviour of the eigenfunctions is expected to be similar for these eigenfrequencies as in the uniform case, except that the external region of exponential behaviour now reaches up to the resonant point. Once again the Alfvén resonant point in the inhomogeneous tube takes over the role of the boundary in the homogeneous tube. We could still describe this set of modes as body-modes, although the oscillating behaviour is now restricted to a smaller region.

The set of these slow modes is still anti-Sturmian, now accumulating towards the maximum of the slow continuum. They are generally non-leaky.

There is (at least) one mode (indicated by 'h'), which has no clear explanation. Our suspicion is that it is part of this infinite set of modes (C). This would mean that, as in the case of a uniform tube, this set would start with a finite number of leaky modes. Due to the leaky character, the eigenfunctions look different than for the other (non-leaky) modes in the set. Therefore the connection is not easily made. But when looking at the case where [FORMULA], where all the modes of this set are leaky due to the Alfvén resonance, the corresponding 'h'-mode is recognised as the first mode of this anti-Sturmian set. The correspondence of these two 'h'-modes for different m -values is assured by making a continuous transition from one m -value to the other.

Also the slow surface mode (b) from the uniform case has a corresponding mode in inhomogeneous tubes. This mode now lies in both the Alfvén and slow continuum. In contrast with the uniform case, the eigenfrequency has an imaginary part, comparable in absolute value with the real part. This imaginary part is due to the slow resonance (no jumps due to Alfvén resonance when [FORMULA]) and leaking. This imaginary part makes the predictions based on Fig. 4 less certain since only the real part of the frequencies is considered there. Nevertheless we suspect from Fig. 4 a similar story as for the fast surface mode. We expect (mainly) exponential behaviour in the region external to the Alfvén resonance, (mainly) oscillatory in between the two resonances and again (mainly) evanescence internal to the slow resonance. These features are recognized in the eigenfunctions in Fig. 8. Again the exponentially increasing solution external to the tube is due to leaky propagation of the perturbations in the external medium.

[FIGURE] Fig. 8. The real and imaginary part of the radial displacement and of the pressure perturbation as a function of the radial coordinate for the eigenmode [FORMULA] ([FORMULA], [FORMULA]). This eigenmode lies in both the Alfvén and slow continuum. And shows jumps over the slow resonance at [FORMULA]. The Alfvén resonance is at [FORMULA], but does not provide for jumps ([FORMULA]). Oscillatory behaviour is expected in between the two resonances and exponential behaviour elsewhere. In the external region the solution is increasing exponentially due to the leaky character of this mode.

5.2. Kink modes: [FORMULA]

We move over to the kink-modes ([FORMULA]). The spectrum can be seen in Fig. 9. The main difference is that now there are jumps over the Alfvén resonance that may play a role. All eigenfrequencies in the Alfvén continuum now have an imaginary part and therefore give a complex [FORMULA] with both a real and an imaginary part. Thus the external solution is propagating and exponentially increasing with distance from the tube.

[FIGURE] Fig. 9. Idem as in Fig. 6, but now the spectrum of the kink-modes ([FORMULA]).

For the so-called trig modes (e, f, g) with frequencies with real part above both continua, we expect and find a similar picture as for the sausage trig modes, as they are not affected by the Alfvén resonance. Only the first trig mode 'e' moves into the Alfvén continuum from [FORMULA]. As seen on Fig. 5 the internal region where evanescent behaviour is expected becomes larger as the real part of the frequency decreases. Nevertheless the character of the mode shows hardly any difference for the first trig modes from that of the others. Except for the fact that the eigenfunctions are influenced by an Alfvén resonance.

The mode (b) that lies in both the Alfvén and slow continuum and that corresponds to the slow surface mode in the uniform case is also recovered. Looking at Fig. 5 and comparing the horizontal line indicated by 'b' with the one in Fig. 4, we expect similar eigenfunctions as in the [FORMULA] case (Fig. 8), except that around [FORMULA] they now show a small jump (in the imaginary part of the radial displacement) due to the Alfvén resonance.

The set of infinite modes (C) in the Alfvén continuum converging towards the top of the slow continuum is still present, as it was for [FORMULA]. But these eigenfrequencies now have an imaginary part, probably due to the effect of the Alfvén resonance. When looking at Fig. 4 and 5 (horizontal line indicated by 'c'), once more we expect little difference in the internal solutions with the case where [FORMULA]. Except that the eigenfunctions are now highly influenced by the jumps due to the Alfvén resonance. The external solution is now exponentially increasing with distance from the tube.

The mode indicated by 'h' is probably the first mode of this infinite anti-Sturmian set. It shows one node in the solution for [FORMULA] and none in the solution for [FORMULA] (Fig. 10). As expected from Fig. 5, the eigenfunctions have a mainly exponential behaviour, supplemented with jumps due to the Alfvén resonance.

[FIGURE] Fig. 10. The radial displacement and the pressure perturbation as a function of the radial coordinate for the eigenmode [FORMULA] ([FORMULA], [FORMULA]). This mode is the one indicated by 'h' in Fig. 9. It is the first mode of an infinite, anti-Sturmian set of modes accumulating towards the top of the slow continuum. These eigenfunctions show clear jumps at [FORMULA] due to the Alfvén resonance.

The 'd'-mode, corresponding to the fast surface mode in the uniform tube case, is not refound in the case where [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998