## Abstract

Solar global observations suggest that the frequency and the line width of helioseismic acoustic eigenmodes vary with the solar cycle. One reason for the measured changes could be the variation of the global atmospheric magnetic fields. We model global solar oscillations in a plane-parallel, three-layer model within the framework of linear dissipative magnetohydrodynamics, and study the effects of a homogenous, horizontal atmospheric magnetic field on global oscillations. We find magnetoacoustic *f*- and *p*-modes and also atmospheric gravity modes (*g*-modes) among the eigenoscillations of the system. We conclude that changes in the atmospheric magnetic field can, significantly, shift the frequencies and vary the line width of global oscillation modes.

## 1. Introduction

Recent observations revealed many details of the highly complex atmosphere and interior of the sun. We try to set up models as realistic as possible, but it is hard to fully understand the outcomes of complex models, without having a deep understanding of how the simplest models behave, which can be considered as the building blocks of more realistic models. In this paper, we use a fairly simple, plane-parallel model with three layers, and study how a resonant coupling of global eigenoscillations to local atmospheric magnetohydrodynamic (MHD) oscillations affects the frequencies of the global modes.

## 2. The model

The lower layer is a polytrope, free of magnetic field and represents the solar interior. The upper layer has a constant temperature and a horizontal magnetic field with constant Alfvén speed (e.g. Campbell & Roberts 1989). This semi-infinite layer is a basic representation of the solar corona, emphasizing the presence of magnetic canopies (e.g. the regions of the upper parts of the coronal magnetic loops, see Taroyan *et al*. 2004). The intermediate layer is 2 Mm thick, and it is also called the transition layer, as the temperature and the horizontal magnetic field increase there with height. More detailed descriptions can be found in Pintér & Goossens (1999).

The physical processes in the model are governed by the linearized ideal MHD equations, except in a thin layer around a height where the local frequency of an MHD Alfvén or slow wave matches the global frequency of an eigenoscillation. Those modes interact resonantly there, and dissipative forces become important. Dissipations make the global oscillations damped, which mathematically can be expressed with complex mode frequencies, the imaginary part of the frequency measuring the damping rate.

The solutions of the linear ideal MHD equations for the left and right to the dissipative layer can be connected by the so-called connection formulae, first suggested by Sakurai *et al*. (1991) (see Goossens & Ruderman 1995; Goossens *et al*. 1995; Erdélyi 1997).

Ruderman *et al*. (1997) and Ballai *et al*. (1998) developed the nonlinear theory of slow resonant layers (see Ruderman & Erdélyi 2000). The Reynolds number at the resonant position is estimated and found that, in accordance with criteria obtained by Ruderman *et al*. (1997) and Ballai *et al*. (1998), the motion in the slow dissipative layer is strongly nonlinear. Inspired by this observation, the connection formulae for strongly nonlinear slow dissipative layer, derived by Ruderman (2000), were used.

It is shown that there is an overlapping region below and above the resonant layer, where both the ideal MHD equations and the approximations used to solve the dissipative equations are valid. There, the solutions can be matched by the continuity of the Lagrangian displacement and the Eulerian perturbation of the total (kinetic and magnetic) pressure. More can be read about the concept of resonant absorption in Goossens *et al*. (1995), Stenuit *et al*. (1995), Erdélyi (1997), where cylindrical geometry is used, and in Tirry *et al*. (1998), where Cartesian geometry is used.

We study waves confined to the interior and the atmosphere. The relating boundary condition is that the total (kinetic and magnetic) energy of the perturbations tends to zero with height. At the surfaces between the different layers, the Lagrangian displacement and the Eulerian perturbation of the total pressure are continuous. A dispersion relation, *f*(*l*, *ν*)=0, can be derived from the governing MHD equations, where *l* is the harmonic degree and *ν* is the global mode frequency. The dispersion relation together with the boundary conditions given above can be solved as an eigenvalue problem for the frequency.

## 3. Results

The magnetic effects are illustrated in this study, for the simple case where the horizontal wave propagates parallel to the magnetic field lines. This wave is resonantly coupled to local *slow* MHD modes if its frequency is in the slow continuum, i.e. *ν*<*ν*_{c}, where *ν*_{c} is the slow frequency taken in the upper layer of the model. (Oblique propagation is studied in detail in Pintér *et al*. submitted.) Figure 1 displays the magnetic effects on the global oscillations. Panel *a* shows the *frequency shift* of the *f*- and the second *p*-modes. The strength of the magnetic field is given by *B*_{c}, which is the field strength taken at the top of the transition layer. The *f*-mode frequency has a gap in the spectrum, indicating that the mode does not exist for a certain interval of the magnetic field strength. The reason is that the *f*-mode frequency is below the lower magnetoacoustic cutoff frequency and above the slow continuum; hence, the *f*-mode is a leaky mode for that interval of *B*_{c}. The second *p*-mode exists only for strong enough magnetic field ( for *l*=50). For weaker field (and for free field case), the *p*_{2}-mode is a leaky mode, because its frequency is above the upper magnetoacoustic cutoff frequency. The magnetic field increases the frequencies of both the *f*- and the *p*_{2}-modes.

The *frequency width*, *Γ*≡−2*ν*_{im}, of the *f*-mode is plotted as a function of *B*_{c} in panel *b*. The mode has non-zero frequency width for strong magnetic fields, for which the mode frequency is in the slow continuum (compare to panel *a*). The frequency width has a maximum value (*Γ*_{max}∼41 nHz taken for *B*_{c}∼114 G for *l*=50). The frequency of the *p*_{2}-mode is above the slow continuum, hence it is not coupled resonantly to local slow modes. As a result, it has no frequency width.

Besides the magnetoacoustic *f*- and *p*-modes, we also find gravity modes among the eigenmodes, of which the frequencies are below the Brunt–Väisälä frequency. The frequencies of these gravity modes are always in the slow continuum; consequently, these *g*-modes are damp due to resonant absorption.

## 4. Conclusion

The simple solar model, used in this study, is powerful enough to study the basic effects of an atmospheric magnetic field on global oscillations. The magnetic field can increase the frequency of global oscillations by tens or hundreds of nanohertz for lower spherical degree, *l*. The shift can be of the order of microhertz for modes with higher *l*.

Global oscillations with frequency in the slow continuum are resonantly coupled to local MHD slow waves. This results in wave damping and, consequently, a non-zero frequency width due to dissipative effects. Another important consequence of the resonant coupling is that the amplitude of the global modes, which modes are evanescent in the atmosphere, becomes large in the resonant layer, and by this they are relatively easily observable. The phenomenon of resonant absorption may make even gravity modes observable at fairly high regions in the solar atmosphere. However, the large line width of the gravity modes suggests that they can exist above the photosphere for a much shorter time than magnetoacoustic modes.

The model can be improved by adding flow effects to it, which has been done by Erdélyi & Goossens (1996), see Erdélyi & Taroyan (2001), and also by replacing the steady magnetic and/or flow fields by random fields (Erdélyi *et al*. 2004, 2005).

## Acknowledgments

The author acknowledges the financial support by PPARC (UK).

## Footnotes

One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.

- © 2005 The Royal Society