## 2. The modelThis section introduces the model which underlies the present calculations. Basically, the method agrees with that used for the investigation of kink waves by Ploner & Solanki (1997, henceforth called Paper I) and details can be found there. Here we concentrate on aspects unique to torsional waves. We begin with an overview of the 3-D geometrical situation (Sect. 2.1), proceed with the description of torsional waves (Sect. 2.2) and end with basic symmetry considerations of torsional waves in a flux tube (Sect. 2.3), which turn out to be important for the interpretation of the synthesized line profiles. ## 2.1. OverviewFig. 1 provides an overview of the model flux tube and fixes
Cartesian coordinates , of which
where and
For both the internal and external atmosphere we employ empirical models in order to obtain realistic polarized line profiles. The internal atmosphere used here is the plage flux-tube model of Solanki & Brigljevic (1992), while the external atmosphere is the empirical quiet-sun model of Maltby et al. (1986). Note, however, that the perturbation is calculated for an isothermal atmosphere (see Sect. 2.2). Following Rüedi et al. (1992) the magnetic field strength is chosen to be 1500 G at ( marks the layer at which optical depth at in the quiet sun). The flux-tube radius at km (the lower boundary of the calculation domain) is km, resulting in a radius of 100 km at . The upper boundary of the domain lies at km. In a second step, the perturbations to the magnetic and velocity
vectors due to the torsional wave are added to the zeroth-order
quantities of the inner atmosphere (see Sect. 2.2). The flux tube is
then intersected by Finally, the equations of polarized radiative transfer are
numerically integrated along each ray using the Stokes formalism. This
calculation provides us with the line profiles in Stokes For details of the calculation of atmospheric quantities along the
rays or the subsequent integration of the radiative transfer equation
we refer the interested reader to Bünte et al. (1993) and
Paper I. The major change relative to Paper I consists of
the inclusion of the 3-D flux-tube structure, which is dictated by the
nature of torsional waves whose line-of-sight velocity component is
largest at large ## 2.2. Torsional wavesTorsional waves in axially symmetric flux tubes are best described
in cylindrical coordinates respectively. Here, and
are the zeroth order components of
the density and vertical magnetic field, respectively, and
and
are first order disturbances to the
azimuthal components of the velocity and magnetic field, respectively.
Finally, stands for
with where As in Paper I we disturb the equilibrium flux tube, whose
stratification is described by a realistic model atmosphere, with an
isothermal torsional wave. The employed Alfvèn speed is
km s Additional limitations are introduced by the thin flux-tube approximation. The radial expansion of the equations underlying this approximation forces us to consider wavelengths that are large compared to the flux-tube radius. Note that this radius increases exponentially with height, so that this requirement is increasingly poorly fulfilled in the upper atmosphere. However, as mentioned above the less realistically modeled upper part of the flux tube does not significantly influence the spectral lines, which obtain their main contribution at smaller height. ## 2.3. Symmetry propertiesIn Sect. 3 we first investigate the effect of torsional waves on
polarized profiles generated in a single plane (see Fig. 1). Torsional
waves cause The circles in Fig. 2 represent horizontal cuts through the flux
tube (compare with Fig. 1). For illustrative purposes they are
intersected by two planes symbolized by the two vertical lines located
at . Of the wave's velocity
only its component in the
In other words, an observer sees a line-of-sight velocity in one half of the flux tube () that is directed oppositely to that in the other half (). This result is independent of (except ). Half a wave period later reverses its direction again giving rise to a change in sign of , where is the wave period. In
addition, is independent of i.e. along a horizontal ray the line-of-sight velocity remains
constant within the flux tube. In Eq. (10) we have made use of the
fact that is the angle between
and
as well as between The situation for the magnetic field is far more complex than for
the velocity, since in addition to the wave-induced
component time independent
and
components are also present, all of which affect the polarization
state. Consider first the azimuthal component,
, of the magnetic field generated by
the torsional wave (Fig. 2b). Note that according to Eq. (6)
is directed oppositely to
. Eqs. (8) to (10) found for
are also valid for
.
(which influences Stokes In the Stokes formalism the orientation of the magnetic field enters the radiative transfer through the angles (the inclination between field vector and line-of-sight) and (azimuth, measured in a plane perpendicular to the line-of-sight). Examples of and are displayed in Fig. 3 for a flux tube with (dotted lines) and without (solid lines) a twist such as that introduced by a torsional wave (). In order to illustrate the main effects clearly, all quantities have been assumed to be height independent when making these figures (but not in the rest of the paper). In the plotted case the flux tube is seen at . Fig. 3a shows that in the static case is smaller than for (i.e. for locations of the flux tube nearer the observer), whereas for (located in the flux tube away from the observer). This reflects the combined effects of and . Note that is the same on both halves of the flux tube (thick and thin curves corresponds to and in Fig. 3). The sign of , however, corresponds to the sign of which is due to . The changes caused by the wave can be judged from the difference between the solid and dotted lines.
© European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |