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Astron. Astrophys. 322, 995-1006 (1997)

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4. Numerical results

In this section, we present numerically obtained continuous spectra for coronal magnetic flux tubes with bases situated in high-beta chromospheric/photospheric plasma regions.

The equilibrium and the necessary metric tensor elements are determined with the equilibrium code PARIS. The spectral equations (26) or (30) are then solved on each flux surface by means of bicubic Hermite finite elements. Convergence studies have been made for each example to make sure that the results are reliable.

For all the examples shown in this section the boundary shape [FORMULA] is chosen as


where the parameter [FORMULA] determines the expansion of the loop at [FORMULA] (the top of the flux tube). This form of the boundary shape was chosen to illustrate the effect of expansion of the flux tube on the continuous spectrum. It is not meant to be a truly realistic flux tube shape. As mentioned in Sect. 2, we force the radial magnetic field component [FORMULA] to vanish at both ends of the flux tube. Hence, the magnetic field is purely vertical there. The shape (34) has been chosen accordingly to guarantee that [FORMULA] at both ends. The equilibrium profiles F and [FORMULA] are chosen as


so that F is zero for [FORMULA] and [FORMULA] and it reaches its maximum of 1 at [FORMULA]. The values of the equilibrium parameters of Eq. (8) are given by [FORMULA] and [FORMULA], respectively. The pressure at the boundary is given by [FORMULA]. Unless stated otherwise, the aspect ratio [FORMULA] and [FORMULA].

In the next subsection we examine the two-dimensional continua for a typical flux tube. Effects of expansion are considered in Sect. 4.2and continua for a flux tube with an additional density stratification are studied in Sect. 4.3.

4.1. Example: an expanded flux tube

In this subsection we present and discuss the continuous spectra for a typical expanded flux tube. We choose the following values of the parameters: the pressure scale height [FORMULA], the length of the gravitational region [FORMULA], and the density ratio [FORMULA]. In Fig. 2a and b the longitudinal dependence of the equilibrium density, the total magnetic field, and the parameter [FORMULA] are shown. The former is plotted at the boundary flux surface ([FORMULA]) while the latter two profiles are plotted on axis ([FORMULA]). In Fig. 2c the magnetic twist [FORMULA] and the pressure function [FORMULA] are plotted versus [FORMULA]. Here, the twist of magnetic field lines is defined by


The small increase of the plasma [FORMULA] in the top region is due to the fact that the pressure is constant in this region while the magnetic pressure is a monotonically decreasing function of [FORMULA] on the interval [FORMULA]. The latter decrease is a consequence of the increase of the cross-section of the flux tube and flux conservation. Since [FORMULA] remains smaller than 0.1 this pecularity does not influence the continuum frequencies very much.

[FIGURE] Fig. 2a-c. Equilibrium profiles: a Density on the [FORMULA] boundary surface. b Total pressure and plasma beta on axis. c Twist of the magnetic field and the pressure function [FORMULA] versus [FORMULA].

In Fig. 3 the branches of the continuous spectrum are plotted as a function of [FORMULA] for the azimuthal mode number [FORMULA]. A straight field line flux coordinate grid with 120 radial and 40 longitudinal grid points has been used to obtain good accuracy. With an orthogonal grid we found spurious unstable continuous spectra which were eliminated by using a straight field line grid. Only by using many more longitudinal grid points could we get rid of the instabilities introduced by the orthogonal flux grid. Also, the lowest frequency continuum branch showed some deviations with respect to the straight field line coordinate grid. However, for the higher frequencies branches the difference between results for the two grids becomes small.

[FIGURE] Fig. 3. Part of the [FORMULA] continuous spectrum belonging to the equilibrium configuration plotted in Fig. 2.

Two types of continuum branches can easily be distinguished in Fig. 3. The five strongly wiggling branches are Alfvén-like, whereas the other branches are slow-like. This distinction is based on the wave polarization, as will be discussed below. Coupling between the Alfvén-like branches, due to flux tube expansion and gravitational stratification, leads to avoided crossings and the formation of gaps. These gaps are larger for lower frequencies indicating that the lower frequency branches experience a stronger coupling. For the slow-like branches, this coupling is so strong that the branches become almost straight at low frequencies. This behavior of the slow-like branches, which is observed in all our examples, is due to the gravitational stratification of the plasma density. For smaller stratification (larger values of H), the low frequency slow-like branches become similar to the higher frequency ones of Fig. 3. Coupling between Alfvén-like and slow-like branches is hardly visible. However, a very weak coupling is present between the 'intersecting' slow-like branch and lowest Alfvén-like branch at [FORMULA] and [FORMULA].

It is to be understood that the continuous spectrum is obtained by projecting all branches in Fig. 3 onto the [FORMULA] -axis. By doing so it becomes clear that the entire [FORMULA] -axis is covered by continuum frequencies, except for small regions around [FORMULA] and [FORMULA]. These regions form gaps in the continuous spectrum. It was shown by Beliïn et al. (1996a) that these gaps, which are truly due to the two-dimensionality of the considered equilibrium, facilitate the existence of discrete global waves with frequencies within the gaps. Since these waves are global, they should be easier to detect than continuum waves. Because the frequency of these waves depends on detailed equilibrium parameters, observation of these waves may provide a powerful diagnostic tool.

We now focus our attention on the proper part of the continuum waves. By 'proper' we mean the non-singular part of the eigenfunctions, i.e., the variation within the resonant flux surface rather than across the flux surface. In Fig. 4, we have plotted the [FORMULA] -dependence for three continuum waves that are all resonant (singular) on the [FORMULA] flux surface. Note that we have plotted the parallel and perpendicular displacement components themselves instead of the components defined in Eq. (29).

Based on the amplitude ratios of the parallel and perpendicular components the waves are either called slow-like, when the parallel amplitude is larger, or Alfvén-like when the perpendicular amplitude is larger. Hence, the modes displayed in Fig. 4b and 4c are slow-like and the mode shown in Fig. 4a is Alfvén-like. The flux surface ([FORMULA]) is chosen rather arbitrarily, but the amplitude ratio of continuum waves on other flux surfaces corresponding to the same branches is very similar to that of Fig. 4. This is the basis of classifying the corresponding branches as either Alfvén-like or slow-like.

[FIGURE] Fig. 4a-c. [FORMULA] -dependence of three continuum waves corresponding to the frequencies indicated by the crosses in Fig. 3: a [FORMULA], b [FORMULA], c [FORMULA]. In each frame the parallel as well as the perpendicular displacement components are shown.

4.2. Effects of expansion

In this subsection we study the effects of expansion on the profile of the continuum branches and the polarization of the corresponding waves. The larger the expansion, i.e., the larger values of [FORMULA] in Eq. (34), the larger is the geodesic curvature [FORMULA] of the magnetic field lines. Hence, we can expect a larger coupling between Alfvén and slow magnetosonic continuum waves for larger flux tube expansions because of the coupling terms involving [FORMULA] of the matrix [FORMULA] of Eq. (32). We consider two cases, one in which the flux tube is a straight cylinder (constant cross-section), and another one with a strong expansion ([FORMULA]). We take the same equilibrium as used in the previous subsection. However, leaving the other equilibrium parameters unchanged would give [FORMULA] values above 0.15 at the top of the flux tube. To keep [FORMULA] at the top of the flux tube at the same value of Fig. 2, we have reset the scale height parameter H to [FORMULA]. We have checked that this decrease of H only has a marginal influence on the continuous spectrum and the wave properties as compared to the case of the previous subsection ([FORMULA]).

In Fig. 5 some parts of the continuous spectra are shown for (a) [FORMULA] and (b) [FORMULA]. Since the averaged magnetic field strength for [FORMULA] is smaller than for [FORMULA] and [FORMULA] and the order of magnitude of the continuum frequencies scales with the magnetic field strength, the [FORMULA] continuum branches have lower frequencies. We, therefore, plotted a smaller frequency range for [FORMULA]. Qualitatively, the continuum branches have not changed much compared to the branches in Fig. 3. However, if we look at the polarization of the waves, in particular the Alfvén-like waves, we see that expansion of the flux tube has a large impact. Whereas for [FORMULA] the ratio [FORMULA] is less than [FORMULA], it is of the order one for [FORMULA].

[FIGURE] Fig. 5a and b. Parts of the [FORMULA] continuous spectra for a [FORMULA] and b [FORMULA]. Equilibrium parameters are the same as those used to compute Fig. 3 except for H which has be seen set to [FORMULA] in the calculation of the continuous spectrum for [FORMULA].
[FIGURE] Fig. 6a and b. [FORMULA] -dependence of the two continuum waves with frequencies indicated by the triangles of Fig. 5: Alfvén-like continuum wave for a [FORMULA] and [FORMULA], b Alfvén-like continuum wave for [FORMULA] and [FORMULA].

From this we may conclude that the polarization of coronal waves depends sensitively on the expansion of flux tubes, whereas gravity has less effect. The implication for observations of MHD waves in the chromospheric/photospheric as well as the coronal regions is that one should determine both the parallel and the perpendicular velocity to establish the identity of the wave. This information could be obtained, for example, from non-thermal line broadening.

4.3. Additional non-gravitational density stratification

In this subsection, the continuous spectrum for a flux tube with an additional density stratification is discussed. Except for the density, the equilibrium configuration is the same as the one used for Fig. 3. The additional density stratification, displayed in Fig. 7, is obtained with the parameters [FORMULA] and [FORMULA] of Eq. (12). Such a stratification can model the transition region where the sharp temperature transition gives rise to an additional decrease in density.

[FIGURE] Fig. 7. The density as a function of [FORMULA]. Compared to Fig. 2 an additional non-gravitational density stratification is included.

Because the density is two orders of magnitude larger near the ends of the flux tube compared with the example displayed in Fig. 2, the averaged Alfvén velocity is smaller and, consequently, the continuum frequencies are lower. This can be seen in the Fig. 8b which shows a large density of continuum branches in a small frequency interval just above [FORMULA].

[FIGURE] Fig. 8a and b. Parts of the [FORMULA] continuous spectrum belonging to the equilibrium configuration plotted in Fig. 7.

Fig. 8 also reveals that the branches are flatter as compared to the continuum branches shown in Fig. 3 so that the continuous spectrum contains more gaps.

The flattening of the continuum branches indicates that the continuous spectrum becomes less dependent on the twisting of the magnetic field so that it starts to resemble the behavior of line-tied continua. Recall that the continuous spectrum of a line-tied one-dimensional cylinder is proportional to [FORMULA], i.e., it is independent of the transverse magnetic field component (Halberstadt & Goedbloed 1994). For the present two-dimensional equilibria, this expression should be replaced by an average over the magnetic field lines. Since in our examples the average of [FORMULA] is a very weakly varying function of [FORMULA], the flattening of the continuum branches is consistent with the continuous spectrum becoming more 'line-tied'.

The additional density increase does not alter the coupling between Alfvén and slow magnetosonic waves qualitatively, as can be seen in Fig. 9. The amplitude ratios of the displacement components of the slow-like and Alfvén-like continuum waves are of the same order as in Fig. 4.

[FIGURE] Fig. 9a and b. [FORMULA] -dependence of two continuum waves corresponding to the frequencies indicated by triangles in Fig. 8: a Alfvén-like continuum wave for [FORMULA], b slow-like continuum wave for [FORMULA].

From these results it follows that a density stratification contributes more to the coupling between similar continuum waves than between dissimilar ones.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998