## 2. The model## 2.1. Kink wavesWe 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 In Eq. (1) and 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 is the horizontal displacement at the
height Spruit solved Eq. (2) for isothermal atmospheres, for which
the pressure scale height where signifies the real part of a complex function. The corresponding dispersion relation is found to be In Eq. (4) ## 2.2. The atmosphere in the presence of a kink waveA 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 , km and is the FT displacement's amplitude at km, i.e. at , the lower end of the calculation domain. Taking the derivative with respect to time gives us the (horizontal) velocity, ( denotes the velocity amplitude at
km), while the derivative with respect to
Here the wavenumber has been replaced using Eq. (4). In
addition, also depends on the radial coordinate
where is the FT radius at is the inclination of the magnetic field to the vertical at a
distance Fig. 1 shows a vertical cross section through the deformed FT
at 4 phases for a given wave ( Hz,
km s
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 G imposed at , 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 km. The pressure scale height is then km, the plasma beta and the cut-off frequency close to but smaller than 0.013 Hz (corresponding to a wave period of roughly 8 min). ## 2.3. Radiative transferWe 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 , 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 ) and over 100 (for ). 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 Table 1 lists the three calculated spectral lines. Here
is the solar wavelength of the transition,
its effective Landé factor (note,
however, that Fe i and Fe i
are Zeemann triplets), and
is the excitation potential of its lower level.
The values (oscillator strengths) are taken
from Thévenin (1989) and Solanki et al. (1992). Fe i
is well known and often observed in magnetic
features, Fe i is a stronger, more saturated
spectral line, whose Stokes
© European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |