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Astron. Astrophys. 345, 986-998 (1999)

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2. The model

This 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. Overview

Fig. 1 provides an overview of the model flux tube and fixes Cartesian coordinates [FORMULA], of which z describes the height in the atmosphere. A part of the axially symmetric flux-tube boundary which separates the inner magnetized from the outer field-free atmosphere is represented by the shaded surface around z. In a first step, the static equilibrium flux tube is determined by horizontal pressure balance (using the zeroth-order thin flux-tube approximation, e.g. Ferriz Mas et al. 1989)


where [FORMULA] and p are respectively the outer and inner zeroth-order, i.e. unperturbed, gas pressure, and [FORMULA] is the zeroth-order vertical magnetic field. Both the pressure and the magnetic field decrease with increasing height and magnetic flux conservation causes the flux tube to expand with height.

[FIGURE] Fig. 1. Illustration of the model flux tube and a plane containing rays parallel to the line-of-sight. The shaded surface represents the boundary between the outer, field-free and inner, magnetized plasma. As an illustration a plane intersecting the flux tube at the location [FORMULA] is shown. The plane contains mutually parallel rays pointing towards the observer located at heliocentric angle [FORMULA].

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 [FORMULA] ([FORMULA] marks the layer at which optical depth [FORMULA] at [FORMULA] in the quiet sun). The flux-tube radius at [FORMULA] km (the lower boundary of the calculation domain) is [FORMULA] km, resulting in a radius of 100 km at [FORMULA]. The upper boundary of the domain lies at [FORMULA] 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 y-z planes equally spaced in the x-direction. In Fig. 1 a plane located at [FORMULA] is shown, where [FORMULA] is the distance to the flux-tube axis along the x direction. Each plane contains a number of mutually parallel rays (lines-of-sight) pointing towards the observer. Each ray is inclined by the heliocentric angle [FORMULA] to the vertical. The atmosphere along each ray is determined on a grid with constant [FORMULA]-spacing (see Bünte et al. 1993).

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 I (total intensity), Stokes V (net circular polarization) as well as Stokes Q and U (net linear polarizations). In a first part of the subsequent analysis we investigate the Stokes profiles which stem from a fixed plane, i.e. for a given [FORMULA]. Then signals resulting from spatially and later also temporally averaged line profiles are considered.

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 x (Sect. 2.3).

2.2. Torsional waves

Torsional waves in axially symmetric flux tubes are best described in cylindrical coordinates r (radial distance from flux-tube axis), [FORMULA] (azimuthal angle) and z (height, see Fig. 1.). We consider linear, azimuthally symmetric (i.e. with no explicit [FORMULA]-dependence) waves in the thin flux-tube approximation (e.g. Ferriz Mas et al. 1989). Zhugzhda (1996) found a way to close the linearized system of equations including radial expansion terms up to second order (Ferriz Mas et al. 1989). Those equations are particularly simple for a non-rotating and untwisted flux tube. In this case the azimuthal components of the momentum and induction equations separate out from the remaining magneto-hydrodynamic equations and read


respectively. Here, [FORMULA] and [FORMULA] are the zeroth order components of the density and vertical magnetic field, respectively, and [FORMULA] and [FORMULA] are first order disturbances to the azimuthal components of the velocity and magnetic field, respectively. Finally, [FORMULA] stands for [FORMULA] with a being an arbitrary coordinate. For an isothermal atmosphere (i.e. [FORMULA] and [FORMULA], with H the pressure scale height) Eqs. (2) and (3) possess the following solution:


where t is the time (or phase) and [FORMULA] the (constant) Alfvén speed. Eq. (7) is the dispersion relation between the frequency [FORMULA] and wavenumber [FORMULA] of a pure Alfvén wave. The torsional wave described by Eqs. (4) and (5) is determined by specifying the wave frequency [FORMULA] and angular velocity [FORMULA] (which determines the constant [FORMULA] through [FORMULA] in Eq. 6). Note that the phase shift (Eq. 6) between velocity and azimuthal field is constant and agrees with the expectations for upward propagating Alfvén waves. It also agrees with the case of high frequency kink waves (Paper I), which is responsible for some of the similarities in observational signature.

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 [FORMULA] km s-1 and the scale height is [FORMULA] km. These parameter values correspond to those of the equilibrium flux tube at the lower boundary of the estimated height range of line formation ([FORMULA] km). We again justify this approximation by noting that the deviation from an isothermal atmosphere within the height range of line formation generally is rather small. Strictly speaking, the use of isothermal torsional waves limits the wavelength to be smaller than the temperature scale height. For oscillations with larger wavelengths the temperature stratification, e.g. in the upper atmosphere, becomes important. Partial reflection caused by a temperature increase or effects due to merging flux tubes may influence the wave properties in the height range of line formation. But note, that we are only interested in the principal changes of the atmosphere due to torsional waves and neglect to model comprehensively the wave propagation. We therefore do not take the restriction to short wavelength too serious and go beyond this limit. Larger wavelengths are of interest because they provide a constant phase with height and allow us to separate the effects introduced by the wave frequency.

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 properties

In Sect. 3 we first investigate the effect of torsional waves on polarized profiles generated in a single plane (see Fig. 1). Torsional waves cause r-dependent changes within the flux tube which give rise to y- and x-dependences from the vantage points of an observer located in the y-z plane. It is therefore necessary to discuss the changes induced by the wave and the symmetries the changes may possess along individual rays. Because the flux tube harbouring the torsional wave is assumed to be vertical, the wave-induced changes, [FORMULA] and [FORMULA], lie in a horizontal plane. So in a first step, in order to simplify explanations, we consider only a single horizontal plane and work out in it the horizontal velocity and magnetic components parallel and perpendicular to a hypothetical horizontal line-of-sight. Only after that do we take into account that the rays are inclined to the horizontal.

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 [FORMULA]. Of the wave's velocity [FORMULA] only its component in the y-direction, [FORMULA], is relevant (because only [FORMULA], the line-of-sight velocity, enters the transfer equation). Fig. 2a shows that [FORMULA] changes sign between the planes at [FORMULA] (at a fixed time t):


[FIGURE] Fig. 2a and b. Illustration of how azimuthal disturbances of a torsional wave are projected onto a horizontal plane. The circles represent horizontal cuts through the flux tube seen in Fig. 1 and the vertical lines parallel to y symbolize two intersections of these cuts with planes at locations [FORMULA]. This figure illustrates that the component of the horizontal velocity, [FORMULA] frame a and the projection in the y-direction of the azimuthal component of the magnetic field, [FORMULA] frame b are proportional to [FORMULA] and do not depend upon y.

In other words, an observer sees a line-of-sight velocity in one half of the flux tube ([FORMULA]) that is directed oppositely to that in the other half ([FORMULA]). This result is independent of [FORMULA] (except [FORMULA]). Half a wave period later [FORMULA] reverses its direction again giving rise to a change in sign of [FORMULA],


where [FORMULA] is the wave period. In addition, [FORMULA] is independent of y for a fixed [FORMULA], as follows from


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 [FORMULA] is the angle between [FORMULA] and [FORMULA] as well as between r and [FORMULA]. The magnitude of [FORMULA] is consequently proportional to [FORMULA] and the line profiles formed in the outermost parts of the flux tube are expected to exhibit the largest reaction to the wave.

The situation for the magnetic field is far more complex than for the velocity, since in addition to the wave-induced [FORMULA] component time independent [FORMULA] and [FORMULA] components are also present, all of which affect the polarization state. Consider first the azimuthal component, [FORMULA], of the magnetic field generated by the torsional wave (Fig. 2b). Note that according to Eq. (6) [FORMULA] is directed oppositely to [FORMULA]. Eqs. (8) to (10) found for [FORMULA] are also valid for [FORMULA]. [FORMULA] (which influences Stokes Q and U) has the same sign on [FORMULA] and [FORMULA] whereas it changes sign along y. The dominant component in the photosphere is [FORMULA]. It is almost an order of magnitude larger than the other components. In order to estimate the relative significance of [FORMULA] and [FORMULA] we first note that at the height of line formation ([FORMULA] km) and at the flux-tube boundary the expansion of the magnetic field with height results in [FORMULA]. The field inclination due to the wave [FORMULA] is a factor of 2 smaller there (assuming a velocity amplitude of [FORMULA]1 km s-1) because of the comparatively high Alfvén speed (Eq. 6). Hence the wave superimposes relatively small changes [FORMULA] onto the static field ([FORMULA] and [FORMULA]). Note that the radial field [FORMULA] has the opposite symmetry properties relative to x and y ([FORMULA] behaves like [FORMULA] and [FORMULA] like [FORMULA]) and it modifies the symmetry noted above because [FORMULA].

In the Stokes formalism the orientation of the magnetic field enters the radiative transfer through the angles [FORMULA] (the inclination between field vector and line-of-sight) and [FORMULA] (azimuth, measured in a plane perpendicular to the line-of-sight). Examples of [FORMULA] and [FORMULA] 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 ([FORMULA]). 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 [FORMULA]. Fig. 3a shows that in the static case [FORMULA] is smaller than [FORMULA] for [FORMULA] (i.e. for locations of the flux tube nearer the observer), whereas [FORMULA] for [FORMULA] (located in the flux tube away from the observer). This reflects the combined effects of [FORMULA] and [FORMULA]. Note that [FORMULA] is the same on both halves of the flux tube (thick and thin curves corresponds to [FORMULA] and [FORMULA] in Fig. 3). The sign of [FORMULA], however, corresponds to the sign of [FORMULA] which is due to [FORMULA]. The changes caused by the wave can be judged from the difference between the solid and dotted lines.

[FIGURE] Fig. 3a and b. Dependence of [FORMULA] and [FORMULA] on y and [FORMULA], where [FORMULA] is the angle between the magnetic field vector and the line-of-sight, and [FORMULA] is the magnetic azimuth relative to the line-of-sight. Displayed is the situation along 2 horizontal cuts through a flux tube observed at an angle of [FORMULA]o to the vertical. The thick and thin lines correspond to [FORMULA] and [FORMULA], respectively. The solid curves display the time-independent magnetic field with components [FORMULA] and [FORMULA] (the thick and thin curves are identical in frame a ) The dotted curves result when the twist due to a torsional wave, [FORMULA], is included. Frame a shows that the broad range of [FORMULA] along y is caused by [FORMULA] whereas the wave only affects [FORMULA] weakly (indicated by the difference between the thick and thin dotted lines). It follows from frame b that the sign of [FORMULA] is coupled to the sign of [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999