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Astron. Astrophys. 325, 1199-1212 (1997)

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2. The model

2.1. Kink waves

We first consider a vertical thin FT in equilibrium and possessing a circular cross section. (A FT is thin when the pressure scale height is larger than the radius of the circular FT cross section.) The FT is assumed to have an untwisted magnetic field B. For a vertical thin FT horizontal force balance reduces to pressure equilibrium,


In Eq. (1) [FORMULA] and [FORMULA] are the external and internal gas pressure, respectively. If all displacements are small then the propagation of a kink mode oscillation of the above FT is described by (Spruit, 1981)


Here [FORMULA] is the horizontal displacement at the height z and time t, [FORMULA] the density inside the FT, [FORMULA] the density of the field-free external plasma, g the gravitational acceleration and [FORMULA] the Alfvén speed inside the FT. The first term on the right-hand-side describes the effects of gravitation and stratification by means of buoyancy; the second term includes the restoring forces of the magnetic field. To account for the back-reaction of the external plasma onto the FT, the orthogonal force acts on [FORMULA] (the sum of the external and internal density) instead of on [FORMULA] alone, where [FORMULA] represents the increase of inertia for a potential flow around a circular cylinder.

Spruit solved Eq. (2) for isothermal atmospheres, for which the pressure scale height H and the Alfvén velocity [FORMULA] are independent of height z and the internal and external atmospheres are governed by the same pressure scale height H so that [FORMULA]. Under this assumption the coefficients of Eq. (2) are height independent and the solution reads (Spruit, 1981)


where [FORMULA] signifies the real part of a complex function. The corresponding dispersion relation is found to be


In Eq. (4) k is the wavenumber, [FORMULA] the cut-off frequency and [FORMULA] the plasma beta. For propagating waves - the waves of greatest interest for the heating of the upper atmosphere - the wavenumber k must be real. This condition is only fulfilled for [FORMULA].

2.2. The atmosphere in the presence of a kink wave

A kink wave excitation changes the direction of the magnetic field vector, the horizontal velocity inside the FT and the position of the FT, each as a function of height and time. We assume, for simplicity, that the oscillations do not affect the surroundings. Hence no velocities are induced outside the FT in our model. In order to isolate the signature of kink modes we also neglect other external motions, in particular granular flows. We emphasize, however, that Eq. (2) takes into account the back-reaction of the external material onto the FT. Note also that the polarized Stokes parameters, whose reaction to kink waves we are mainly interested in, are unaffected by velocities outside the FT, the exception being the blue-red asymmetry of the Stokes parameters (Solanki 1989). Equation (2) describes only isolated kink waves, i.e. it neglects the coupling between these and longitudinal waves. This assumption should be of little consequence for our results since we consider only the photosphere, whereas according to Ulmschneider et al., (1991) the coupling between the wave modes becomes significant only in the chromosphere.

We employ the solution for isothermal atmospheres (Eq.  3). With this approximation we avoid problems arising from partial reflections in higher layers. Effects due to departures from isothermality are expected to be minor over the height range of formation of the spectral lines. This approximation requires the wavelength to be small compared to the temperature scale height. A lower limit to the wavelength is provided by the tube radius due to the thin-tube approximation.

The displacement of the tube due to the wave is given by Eq. (3),


where [FORMULA], [FORMULA] km and [FORMULA] is the FT displacement's amplitude at [FORMULA] km, i.e. at [FORMULA], the lower end of the calculation domain. Taking the derivative with respect to time gives us the (horizontal) velocity,


([FORMULA] denotes the velocity amplitude at [FORMULA] km), while the derivative with respect to z is a measure of the inclination of the FT axis relative to the vertical direction. The angle [FORMULA] between the magnetic field on the axis of the perturbed tube and the vertical is then given by


Here the wavenumber has been replaced using Eq. (4). In addition, [FORMULA] also depends on the radial coordinate r, since the field is increasingly inclined at larger r (due to the expansion of the FT with height). Note that the cross section is not affected by a linear kink wave. The additional inclination of the magnetic field away from the FT axis is


where [FORMULA] is the FT radius at z. The radius is calculated by means of flux conservation. Accordingly


is the inclination of the magnetic field to the vertical at a distance r from the tube's central axis.

Fig. 1 shows a vertical cross section through the deformed FT at 4 phases for a given wave ([FORMULA] Hz, [FORMULA] km s-1). Fig. 2 exhibits [FORMULA], [FORMULA], v and [FORMULA] along the (deformed) axis of the FT at the same 4 phases. At the considered frequency the phase difference between v and [FORMULA] is roughly [FORMULA], while [FORMULA] and [FORMULA] are approximately [FORMULA] out of phase. This agrees qualitatively with the more elaborate calculations of Ulmschneider et al. (1991). The phase difference between local maximum velocity and local maximum inclination depends on wave frequency. Fig. 3 illustrates the strong change in the phase relation close to the cut-off frequency. For high wave frequencies ([FORMULA]) the sine term dominates in Eq. (7), giving rise to a phase difference of approximately [FORMULA] for an upward propagating wave. Near the cut-off frequency the cosine dominates in Eq. (7), so that velocity and inclination are nearly [FORMULA] out of phase even for [FORMULA].

[FIGURE] Fig. 1. Vertical cut through a flux tube (FT) along a plane containing its axis and the direction of the displacement due to the kink wave. The dotted line represents the boundary of the unperturbed FT with [FORMULA] km at the lower end of the calculation domain, the solid line that of the FT distorted by a kink wave with [FORMULA] Hz ([FORMULA] km) and [FORMULA] km s-1, where [FORMULA] is the wave frequency, [FORMULA] the wavelength and [FORMULA] the velocity amplitude at the lower end of the computational domain. The 4 frames correspond to phases at which the displacement vanishes or is largest at the estimated height ([FORMULA] km) of formation of Fe i 5250 Å. The height [FORMULA] corresponds to continuum optical depth unity in the quiet sun at 5000 Å ([FORMULA])

[FIGURE] Fig. 2. The horizontal displacement [FORMULA], the horizontal velocity v, the inclination [FORMULA] and the vertical gradient of the horizontal velocity [FORMULA] at the FT axis vs. height z for the same wave as in Fig. 1. Each parameter is normalized by a factor that is indicated in the plots in brackets. The signs have been chosen such that positive magnitudes always indicate the direction away from the observer. Note that inside the flux tube the visible spectral lines we consider here are formed roughly within a range of about 150 km around a height of [FORMULA] km

[FIGURE] Fig. 3. The phase difference between local maximum velocity [FORMULA] and local maximum inclination [FORMULA] of the FT axis vs. wave frequency. The vertical dashed line indicates the cut-off frequency, [FORMULA]

In order to obtain realistic line profiles we follow the same procedure as Solanki & Roberts (1992), i.e. we use empirically obtained atmospheres to describe the unperturbed FT, on which we superimpose the wave calculated in an isothermal atmosphere. The undoubted relevance of non-isothermal effects for line formation is thus taken into account, whereas such effects are assumed to play no role for the wave propagation. For the internal atmosphere we have used the plage FT model of Solanki & Brigljevi (1992) with [FORMULA] G imposed at [FORMULA], in accordance with measurements of Rüedi et al. (1992). The external atmosphere is described by the empirical quiet-sun model of Maltby et al. (1986). The isothermal atmosphere for which the kink wave is calculated corresponds to the parameters of the plage FT model at a height of roughly [FORMULA] km. The pressure scale height is then [FORMULA] km, the plasma beta [FORMULA] and the cut-off frequency [FORMULA] close to but smaller than 0.013 Hz (corresponding to a wave period of roughly 8 min).

2.3. Radiative transfer

We first intersect the FT with a grid of inclined mutually parallel rays (or lines-of-sight) lying in a plane containing the velocity vector and the vertical symmetry axis of the FT (compare with Bünte et al., 1993). Only waves oscillating in the plane spanned by the FT axis and the line-of-sight are considered. Such waves are not easily visible at disc centre. Therefore, we simulate observations at different values of the heliocentric angle [FORMULA], the angle between the line-of-sight and the solar surface normal. We restrict ourselves to the rays in this central plane, since each wave also requires a full phase coverage, i.e. line profiles must be calculated at different phases in the wave, so that the computational load is significant. Test calculations suggest that this restriction should not affect our conclusions. The number of rays varied between 25 (for [FORMULA]) and over 100 (for [FORMULA]). It was dictated by the requirement that enough rays intersect the FT within the height range of line formation. Test calculations with more rays produced no significant change in the line profiles. Along each ray the data relevant to the radiative transfer are calculated as described by Bünte et al. (1993). Care is taken to keep the difference between the optical depth of neighbouring grid points constant. This increases the geometrical density of grid points at critical locations such as the FT boundaries. It is also ensured that there are sufficient grid points (typically at least 10) over the sometimes quite small height range over which a ray passes through the FT interior. For the present calculations we consider an array of FTs, a situation typical of active region plage. For simplicity we assume the whole array of FTs to oscillate in phase. Note that in this geometry a ray may enter a magnetic region more than once.

Stokes profiles are calculated in LTE with the code described by Solanki et al. (1992), which employs the Stokes Profile Synthesis Routine package (SPSR, Rees et al., 1989, Murphy & Rees, 1990, cf. Solanki, 1987 for details on other parts of the codes).

We concentrate here on Stokes V, the difference between right and left circular polarization, and Stokes Q parameters, the difference between linear polarization parallel and perpendicular to the limb in the polarization coordinates used here. In the selected geometry Stokes U signals are only due to magneto-optical effects and are not discussed further. Finally, Stokes I only exhibits a minute influence of the waves and is also not considered. In order to detect even subtle influences of kink waves on the Stokes profile we compare the perturbed profiles of each spectral line with profiles calculated in the unperturbed FT, which we call the reference line profiles.

Table 1 lists the three calculated spectral lines. Here [FORMULA] is the solar wavelength of the transition, [FORMULA] its effective Landé factor (note, however, that Fe i [FORMULA] and Fe i [FORMULA] are Zeemann triplets), and [FORMULA] is the excitation potential of its lower level. The [FORMULA] values (oscillator strengths) are taken from Thévenin (1989) and Solanki et al. (1992). Fe i [FORMULA] is well known and often observed in magnetic features, Fe i [FORMULA] is a stronger, more saturated spectral line, whose Stokes Q and V profiles should react more strongly to velocity gradients. It has been modeled by Bünte et al. (1993) in an only partially successful attempt to reproduce the centre-to-limb variation of its Stokes V asymmetry. Finally, Fe i [FORMULA] is extremely Zeeman sensitive and is the most popular infrared line for magnetic measurements. It is formed considerably deeper in the atmosphere than the other lines. Each of the selected lines is expected to react differently to velocity gradients.


Table 1. Atomic data of Fe i of the calculated spectral lines

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998