## Abstract

Can the ubiquitously magnetic solar atmosphere have any effect on solar global oscillations? Traditionally, solar atmospheric magnetic fields are considered to be somewhat less important for the existence and characteristic features of solar global oscillations (*p*,*f* and the not-yet-observed *g*-modes). In this paper, I demonstrate the importance of the presence of magnetism and plasma dynamics for global resonant oscillations in the solar atmosphere. In particular, in the lower part of the solar atmosphere there are both coherent and random components of magnetic fields and velocity fields, each of which contribute on its own to the line widths and frequency variations of solar global acoustic waves.

Changes in the coherent large-scale atmospheric magnetic fields cause frequency shifts of global oscillations over a solar cycle. The random character of the continuously emerging, more localized, magnetic carpet (i.e. small-scale, possibly even sub-resolution, loops) gives rise to additional frequency shifts. On the other hand, random and organized surface and sub-surface flows, like surface granulation, meridional flows or differential rotation, also affect the coupling mechanism of global oscillations to the lower magnetic atmosphere. The competition between magnetic fields and flows is inevitable. Finally, I shall discuss how solar global oscillations can resonantly interact with the overlaying inhomogeneous lower solar atmosphere embedded in a magnetic carpet. Line width broadening and distorsion of global acoustic modes will be discussed. The latter is suggested to be tested and measured by using ring-analysis techniques.

## 1. Introduction

The solar atmosphere from its lower boundary, the photosphere, through the chromosphere, transition region (TR) up to its open-ended upper region, the corona, is magnetically coupled. This coupling is obvious from concurrently taken images of the various solar atmospheric layers as a function of height (or temperature) and of a magnetogram taken at the same time. Magnetic field concentration, e.g. an active region, will show up as strong brightenings at various locations in UV, EUV and X-ray images, indicating the coupling of the all-pervading magnetic field. One can also observe, at the same time, how dynamic and inhomogeneous this coupling is on broad time and spatial scales.

The traditional technique of seismology is that seismic motion, i.e. displacement or velocity amplitudes, is observed at distant locations on the surface of Earth and analysing these data the internal structure, e.g. density distribution along the ray path, is constructed. The same approach is used to obtain information about the internal (and otherwise invisible) structure of the Sun, within the framework of helioseismology. Helioseismology is concerned with solar global oscillations that are mostly acoustic in nature and are trapped mainly in the interior of the Sun. For more details on helioseismology, see the paper by Thompson (2006). However, recent high-resolution satellite technology provides us with unprecedented spatial and time resolution images of the magnetically structured solar atmosphere. These images show clearly that the various magnetic structures, e.g. coronal loops, arcades, spicules, oscillate on wide spatial and time-scales. Note, that the observed oscillations are within the magnetic structure, in contrast to solar global oscillations. It is natural to raise the question, whether from the observations of such oscillations one would be able to derive diagnostic information, e.g. density along the loop, fine structure, curvature, etc. The idea of *coronal seismology* was put forward by Roberts *et al*. (1984) in a seminal paper. An interesting and very promising application is to investigate oscillations in the lower part of the solar atmosphere. The various magnetic structures serve as excellent waveguides to the propagation of perturbations. Typical lower atmospheric waveguides, in a first approximation, could be considered as isolated magnetic flux tubes with practically no magnetic field in their environment. Another feature that distinguishes these waveguides from their coronal counterparts is that the density and pressure scale heights inside the waveguide are comparable to their radial dimensions, leading to the introduction of the Klein–Gordon term even for the simplest linear governing equation of wave perturbations (Roberts 2004).

To carry out lower atmospheric seismology, one of the first tasks is to understand what is the role of the presence of the *boundary layer* between the solar interior and the corona. When compared to the extent of the interior (the solar radius is about 700 Mm) or of the corona, the transition between the solar interior and the corona occurs in a rather narrow layer. This boundary layer, that includes the photosphere, chromosphere and TR, is around 2–3 Mm thick and contains *coherent and random magnetic and velocity fields*, giving a very difficult task to describe even in approximate terms the wave perturbations. Random flows (e.g. turbulent granular motion), coherent flows (meridional flows or the near-surface component of the differential rotation), random magnetic fields (e.g. the continuously emerging tiny magnetic fluxes or magnetic carpet) and coherent fields (large loops and their magnetic canopy region) each have their own effect on wave perturbations. Some of these effects may be more important than the others. The magnitude of these corrections has to be estimated one by one and it is suspected that, unfortunately, they all may contribute to line widths or frequency shifts of the global acoustic oscillations on a rather equal basis. In helioseismology, corrections from this boundary layer are called the surface term (Basu 2002), and in many helioseismologic modellings the surface term is taken in some *ad hoc* functional form.

One of the main goals of the present paper is to summarize our knowledge about this ‘surface term’ at least as far as wave perturbations are concerned. We review the progress made on how global solar oscillations may couple to the boundary layer; how the presence of the lower atmospheric boundary layer shifts the frequencies of global acoustic oscillations and contributes to their line widths. We discuss some simple theoretical examples to demonstrate how the random and coherent fields, both magnetic and velocity, can influence the wave coupling of internal oscillations with the solar atmosphere. We extend the term of *coronal seismology* by applying similar basic principles to the lower atmospheric boundary layer and we introduce *atmospheric seismology*.

Before we embark on these discussions we wish to make one more point. Once the techniques of helioseismology had improved such that measured frequencies of global oscillations could be compared (with high precision) to theoretically predicted values, small though systematic discrepancies were found. This has initiated research to improve our understanding of the interior. For example, a vast amount of effort has been devoted to model the internal differential rotation, to understand the physics at the bottom of the convection zone, in the tachocline. However, somehow the upper boundary of the solar interior, i.e. the surface term or boundary layer at the lower atmosphere, was slightly neglected. In spite of the discovery of the correlation between changes of global acoustic frequencies and the solar cycle (e.g. Woodard & Noyes 1985; Woodard & Libbrecht 1991), the focus of attention has generally not turned towards the importance of the atmospheric or boundary layer (magnetic) fields. Today, there is an overwhelming evidence for the above-mentioned correlation. I shall only mention a few arbitrarily chosen ones. Global Oscillations Network Group (GONG) observations for a single multiplet (e.g. *l*=50, *m*=9) indicate a strong correlation between the line width, magnetic flux and sunspot numbers (Komm *et al*. 2000). Birmingham Solar-Oscillations Network observations of line widths variations of low-angular degree *p*-modes during the fall of Solar Cycle 22, averaged over 2.6–3.6 mH, showed 24±3% mean increase in the modal line width from activity minimum to maximum as a function of the 10.7 cm radio flux, which is an excellent proxy for low-atmospheric magnetic field (Chaplin *et al*. 2000). Finally, a very recent example is by Dziembowski & Goode (2005) who interpreted Solar and Heliospheric Observatory (SOHO) Michelson Doppler Imager (MDI) data on oscillation frequency changes between 1996 and 2004, focusing on differences between the activity minimum and maximum of Solar Cycle 23. They found that both the *f*- and *p*-mode frequencies are correlated with general measures of the Sun's magnetic activity.

In this paper, I argue the case that the frequency and line width changes are attributed to random and coherent fields in the boundary layer. I will demonstrate the importance of the results achieved on the coupling of global perturbations to the lower solar atmosphere.

### (a) Solar global oscillations

Solar global oscillations driven by pressure forces (*p*-modes) are acoustic waves trapped in the solar interior within a resonant cavity. The cavity is a region where a wave can propagate, but is bounded from above and below by regions where it cannot propagate any longer. An outward propagating wave is reflected inward from the solar upper surface, or boundary layer, because of a sudden decrease in the plasma density, while the lower boundary of the cavity is formed by the increasing sound speed due to temperature rises. If the frequency of a global oscillation mode is above a certain value (i.e. the acoustic cutoff frequency), the wave leaks out from the cavity into the atmosphere above. Such leakage can influence the dynamics of the lower (and upper) atmosphere, as was found by De Pontieu *et al*. (2004). They showed that oscillatory spicules are driven by the leakage of global oscillations along inclined thin flux tubes. An interesting consequence of this leakage is when these photospheric motions propagate even further into the lower corona, causing loop oscillations (De Pontieu *et al*. 2005).

The solar *f*-mode, a gravity wave driven by buoyancy forces at the solar surface, resides right below the solar photosphere and can be regarded as a trapped mode in a cavity of zero depth, so that the upper and lower turning points coincide. This mode is similar to the well-known surface modes propagating along the free surface of a deep ocean.

Early theoretical studies (Ulrich 1970; Leibacher & Stein 1971) have predicted that the eigenfrequencies of solar global oscillations follow separated parabolic ridges as a function of the horizontal wavenumber. In the simplest possible model, in a plane parallel geometry a semi-infinite plasma with a free surface may represent the solar interior. Assuming a compressible plasma with polytropic temperature increase as a function of depth from its surface, one can solve the eigenvalue problem of this configuration. The solutions are parabolic ridges for the *p*- and *f*-modes. Soon after the parabolic ridges of frequencies were predicted, Deubner (1975) observed these separated ridges of power. High-resolution measurements show deviations from the ridges. The above semi-infinite model can be improved by assuming, say, an isothermal atmosphere above the solar interior. The eigenfrequencies of such two-layer solar model are shown in figure 1*a* and the eigensolutions of a few eigenmodes can be seen in figure 1*b* (courtesy B. Pintér).

In order to improve our model, the next step is to consider a three-layer model where there is a narrow (approximately *L*=2 Mm thick) boundary layer between the solar interior and the hot isothermal solar atmosphere. Again, solving for the frequencies and eigenfunctions of this geometry, one recovers the internal *p*/*f*-modes; however, they have a more pronounced tail in the boundary layer and in the atmosphere (figure 2). In this three-layer model, the boundary layer *g*-modes also appear.

### (b) Atmospheric magnetic field

So far no magnetic fields have been considered. We recall the observations that the global acoustic frequencies show to be small, but significant and systematic correlations with the solar cycle. Although the acoustic modes are strongly evanescent in the atmosphere, changes of magnetic fields and mean temperature at the boundary layer (i.e. lower atmosphere) could play a rather important role in the determination of their frequencies. Of course, the mass inertia of the solar atmosphere is negligible compared to its interior counterpart. However, in general, when one solves an eigenvalue problem, the eigenfrequencies and eigensolutions can be rather sensitive to the boundary conditions, not just to the conditions in the domain itself. Since magnetic or flow fields in a boundary layer (i.e. photosphere–chromosphere–TR) can change the mean elasticity of the boundary itself or alter the upper turning points, these effects could contribute small, but clearly present, corrections to the eigenfrequencies.

The pioneering papers by Roberts & Campbell (1986) and Campbell & Roberts (1989; CR89 hereafter), opened a new series of studies of the coupling of solar global oscillations to the lower solar atmosphere. The influence of a chromospheric (i.e. boundary layer) magnetic field on *p*- and *f*-mode frequencies was evaluated theoretically for a simple and elegant model of the solar plasma, consisting of a polytrope solar interior, above which is an isothermal atmosphere. The atmosphere is permeated by a horizontal magnetic field. The eigenfrequencies of this model are shown in figure 3.

Frequency changes and shifts in phase factors due to the presence of a magnetic atmosphere were calculated analytically in the long-wavelength limit, and numerically for arbitrary wavelength. It was shown that, at low to moderate degree *l*, an increase in chromospheric magnetic field leads to a frequency increase for the *n*=1 *p*-mode, whereas the overtones (*n*=2,3, etc.) suffer a frequency decrease. It was shown that at high *l*, all the *p*-modes suffer a frequency decrease. The modelling assumption about the horizontal magnetic field was improved by considering different magnetic field profiles. For example, Evans & Roberts (1990; ER90 hereafter), Jain & Roberts (1994*b*) and Rosenthal (1995) have examined the role of atmospheric magnetic fields with various profiles as well as temperature effects on the global oscillation frequencies. Jain & Roberts (1993, 1994*c*) and Hindman & Zweibel (1994) have also studied the influence of hot chromosphere and corona on the eigenoscillations. It was found that if the chromospheric magnetic field strength and chromospheric temperature are increased simultaneously, then the frequencies of the *p*-modes are shifted in a qualitatively similar fashion to that found in the observations, i.e. the frequency shift increases with frequency until about 3.9 mHz, when the shift decreases dramatically.

A promising model was put forward by Evans & Roberts (1991, 1992). The effect of the magnetic canopy on the solar acoustic modes was extended to allow for variations in height of the magnetic canopy. Analytical solutions in the limit of long horizontal wavelength were obtained; the solutions exhibit explicitly the dependence of frequency shifts on magnetic field strength, wavenumber and canopy height. Frequency shifts are principally due to changes in canopy height. Full numerical solutions were also presented. It was found that changes in chromospheric magnetism can be manifested in *p*- and *f*-mode datasets gathered at different phases of the solar cycle. These predictions of solar-cycle variability in high-degree *p*-mode frequencies from a simple model of the magnetic canopy, which permeates the solar atmosphere, were compared with the observations of Libbrecht & Woodard (1990). Good agreement was found with the observed frequency shifts for modes of frequency less than 4 mHz, through a mechanism in which an increasing magnetic field induces ‘stiffening’ of the Sun's chromosphere (see also Wright & Thompson 1992).

Before we embark on studies of flow fields on global oscillations there is one more case of a magnetic boundary layer in a static plasma to be recalled. If the characteristic lengths of perturbations are small compared to the gravitational scale heights one can approximate the boundary layer by a simple surface (e.g. there is a jump in density or temperature). In this case, propagations could be both parallel to the atmospheric magnetic field lines (e.g. Miles & Roberts 1992; Miles *et al*. 1992) or the horizontal wavenumber of the perturbations may have a finite angle to the field lines (e.g. Uberoi & Narayanan 1986; Gonzales & Gratton 1991; Jain & Roberts 1994*a*). These studies are relevant to *f*-modes.

### (c) Random magnetic carpet

Similarly to granulation, the magnetic field in the solar atmospheric boundary layer is also very dynamic. High-resolution magnetograms reveal that outside active regions the solar surface is covered with a mixed polarity network, which is termed *magnetic carpet* (Title & Schrijver 1998). The structure of this small-scale field changes rapidly on very short spatial and time-scales and flux continuously emerges and disappears nearly homogeneously over the surface. Interestingly, the smallest magnetic structures show apparently no correlation with the solar cycle (Hagenaar *et al*. 2003), and, therefore, they are believed to originate in a separate, small-scale, dynamo process possibly close to the surface. Erdélyi *et al*. (2005) investigated the influence of this disorganized, small-scale atmospheric field on the *f*-mode frequencies. In a first approximation, the magnetic carpet was modelled as a time-independent, stochastic field. Since, depending on their spherical degree, some *f*-modes may have a lifetime comparable to the characteristic replacement time (of the order of tens of hours) of the magnetic carpet, this limits the validity of their study. The magnetic field was taken to be independent of time, because first they wanted to assess the effect of random magnetic field alone and in the case of a time-dependent field one would have to deal with generated flows in the initial state. They found that a time-independent random magnetic field can significantly increase the *f*-mode frequencies, in contrast with random steady flows which tend to have an opposite effect (Murawski & Roberts 1993*a*,*b*). Observations of *p*-modes show a more complex picture, and it is found that for some spherical degrees there is actually a frequency decrease as a function of the sunspot number—a good proxy for magnetic flux. In reality, both magnetic and velocity fields are very dynamic at the solar surface, and, therefore, studying their interaction could be of crucial importance in interpreting the observed frequency shifts.

Bi *et al*. (2003) studied the influence of magnetic perturbations inside the Sun on the low-*l* solar *p*-mode oscillations. They described the various possibilities of frequency shifts for a time-dependent source of magnetohydrodynamics (MHD) turbulence. For the magnetic perturbation contribution, they obtained the frequency shifts of modes with different degrees as a function of the spectrum of fluctuating magnetic field. The frequency shift was found to increase with the strength of magnetic fields in the solar interior, and its temporal behaviour closely follows the phase of the synthetic solar activity cycle. This analysis indicates that the magnetic activities cause shifts of up to 0.3 μHz. It is shown that the mode frequency, which is sensitive to the effect of magnetic fields, can also be used as a diagnostic tool for the presence of turbulent magnetic fields in the convection zone.

### (d) Flow fields

Flows at the boundary layer in the lower atmosphere may be random or coherent. By inverting the observational data of solar global oscillations one could potentially reconstruct the global flow structures. Large-scale sub-surface flows were found by this technique (e.g. Braun & Fan 1998). They measured the mean frequencies of acoustic waves propagating towards and away from the poles of the Sun from observations made with MDI on board SOHO and the ground-based GONG. Significant frequency shifts between poleward- and equatorward-travelling waves measured over solar latitudes 20–60°, which is consistent with the Doppler effect of a poleward meridional, flow of the order of 10 m s^{−1}. From the variation of the frequency shifts of *p*-modes (with degree *l* between 72 and 882), as a function of the lower turning point depth, they inferred the speed of the meridional flow, averaged over these latitudes, over a range in depth extending over the top half of the solar convection zone. Interestingly, there was no evidence for a significant equatorward return flow within this depth range. Howe *et al*. (2000) have completed an analysis of the first 35 GONG Months (1 GONG Month=36 days), covering the last solar minimum and the rising phase of Solar Cycle 23. The mode parameters have been estimated from 33 time-series, each of 3 GONG Month duration, with centres spaced by 1 GONG Month. They reported on the temporal evolution of the rotational splitting coefficients up to 15th order. The coefficients do not correlate well with any surface magnetic flux measure yet considered, but Howe *et al*. found small though significant trends in their temporal evolution. Inverting the coefficients for two-dimensional rotation information and looking at deviations from the mean produces a picture of a systematic zonal flow migrating towards lower latitudes during the rising phase of the cycle. This flow is probably associated with the torsional oscillation. Similar trends were seen in the 1986–1990 BBSO (Big Bear Solar Observatory) data. These large-scale flows are important for at least two reasons. If there are slowly varying large-scale flows in the boundary layer (or around), they change the physical properties of the coupling mechanism between the interior and the atmosphere, resulting in changes in the eigenfunctions and eigenfrequencies. However, if there is damping of global oscillations (i.e. if line widths could be measured with sufficient accuracy), due to some dissipative mechanism present in the magnetic plasma, one could measure changes in the damping (i.e. in line widths) as a function of time in a slowly varying steady state. Inverting such measurements would give a clue as to the sub-surface flow structure.

To the best of our knowledge, Erdélyi *et al*. (1999) were the first who studied the effect of a sub-surface motion on magnetoacoustic-gravity (MAG) surface waves, representing the *f*-mode in a model of the solar interior–solar atmosphere interface. The main characteristics of their isothermal atmosphere was a magnetic (though constant) plasma-*β*, while in the sub-surface interior region there was a uniform and homogeneous equilibrium flow. They found that the flow causes a shift of the forward and backward propagating MAG modes, which in certain cases bifurcate. Erdélyi & Taroyan (1999, 2001) and Taroyan (2003) generalized the model by allowing the temperature to increase linearly with depth in the sub-surface zone. They derived the dispersion relation and analytical formulae for the frequencies of *p*- and *f*-modes in the limit of small wavenumbers. Numerical solutions were presented for other cases.

The influence of short-scale motion (i.e. granulation) modelled as a random flow on *f*-mode frequencies was first evaluated by Murawski & Roberts (1993*a*,*b*; see also Murawski & Goossens 1993; Ghosh *et al*. 1995; Gruzinov 1998 and Medrek *et al*. 1999). Erdélyi *et al*. (2004) have re-visited this problem. The *f*-mode is essentially a surface wave; hence, the mode frequencies are less likely to be influenced by the solar stratification. Most probably, the discrepancies are the result of near-surface mechanisms, such as interactions with surface or sub-surface magnetic fields and flows. Erdélyi *et al*. (2004) followed the general approach of Murawski & Roberts, which is a valuable one, but corrected certain errors which appeared in that paper. The simple model used by Murawski & Roberets and Erdélyi *et al*. gives a deviation of the *f*-modes from the theoretically predicted parabolic ridges which agrees qualitatively with observations. They found that turbulent background flows can reduce the eigenfrequencies of global solar *f*-modes by several per cent, as found in observations at high spherical degree. Extensive numerical simulations of the outer parts of the Sun carried out by, for example, Rosenthal *et al*. (1999) demonstrated and quantified the influence of turbulent convection on solar oscillation frequencies.

In what follows we discuss in detail the modelling efforts when both magnetic *and* flow field are present in or around the boundary layer. In §2, the governing equations are discussed, followed by the solutions for a steady convective zone in §3. In §§4 and 5, two particular magnetic profiles are studied, representing weak and strong magnetic activities in the boundary layer and in the solar atmosphere. The dispersion relation is derived in each case. Analytical solutions are obtained for the corrections to the eigenfrequencies in the limit of long wavelength approximation. Section 6 is devoted to a discussion of resonant coupling of solar global oscillations to the atmospheric boundary layer. We conclude in brief in §6.

## 2. Governing equations

In order to model mathematically the coupling of solar global oscillation to the boundary layer and the solar atmosphere, we consider initially a two-layer model. To make analytical progress we introduce a constant horizontal flow in the lower (internal) region *z*>0, while the upper atmospheric regions of the model are embedded in a *uniform* magnetic field. The models can be grouped into two categories: (i) an *homogeneous and uniform* atmospheric magnetic field that may represent the solar atmosphere at high solar activities; we call this the case of a strong magnetic field; or (ii) an *homogeneous though non-uniform* magnetic field, where the magnetic field strength decays to zero far away from the solar surface. This latter model may be applicable to the solar atmosphere at low magnetic activity (i.e. at solar minimum); we call it the weak field approximation. Such solar models, without a flow, have been considered by CR89 and ER90. We outline the dispersion relations and show the solutions analytically in the limit of small wavenumber, for both cases. The obtained dispersion relations are also evaluated numerically for arbitrary wavelengths. The results are in good agreement with the analysis based on helioseismic measurements of the sub-surface meridional flows (e.g. Braun & Fan 1998).

The linearized governing equations in compressible MHD are in standard notation(2.1)(2.2)(2.3)(2.4)where all equilibrium quantities, denoted by the index 0, depend only on depth *z* and is the local sound speed. The governing equations are supplemented by the solenodial condition, . Further, we express all perturbed vector and scalar quantities in the formrespectively, where *f*_{x}, *f*_{z}, *q* are functions of *z*, and consider equations (2.1)–(2.4) in the convection zone and magnetic boundary layer separately.

## 3. The convection zone

In the convection zone (*z*>0), we suppose the magnetic field to be absent and to have a uniform homogeneous steady flow along the *x*-axis, i.e. . Then from equations (2.1)–(2.4) we obtain(3.1)(3.2)Here is the Doppler shifted frequency, and *γ*_{p} is the adiabatic index in the region *z*>0. Elimination of *u*_{z} yields a second-order ordinary differential equation(3.3)where the prime denotes the derivative with respect to *z*. In the special case of a linear temperature profile, i.e. when the sound speed is given by(3.4)Equation (3.3) has the solution(3.5)where and *M* and *U* are the confluent hypergeometric functions (Abramowitz & Stegun 1965); *C*_{1} and *C*_{2} are arbitrary constants. The parameter *a* is given by(3.6)and(3.7)is the polytropic index.

We require the kinetic energy density (*ρ*_{0}*u*^{2})/2 to be finite as *z*→∞. This implies that *C*_{1}=0 in equation (3.5), resulting in(3.8)

## 4. Boundary layer with **B**_{0}=const. (strong field)

**B**

In the transitional layer (*z*<0), the steady flow is absent and a uniform magnetic fieldis present. The magnetohydrostatic equation is given by(4.1)From equations (2.1)–(2.4), we obtain the equations(4.2)(4.3)where is the local Alfvén speed and is the MHD cusp speed.

We assume the boundary layer to be isothermal. Then equation (4.3) reduces to(4.4)where

(4.5)Here *c*_{sc} is the constant sound speed in the layer, *v*_{Ac} is the Alfvén speed at the base of the boundary layer and *γ*_{c} is the adiabatic index in the region *z*<0. Equation (4.4) possesses solutions of the form (e.g. ER90)(4.6)where(4.7)In the boundary layer, the perturbation of the magnetic energy density is . Using equations (2.1)–(2.4) we may show thatSince we are interested in trapped modes, both the perturbations of the kinetic and magnetic energy densities must vanish in the limit *z*→−∞. Using the relation (Abramowitz & Stegun 1965)(4.8)it is easy to show that *E*_{B} will vanish as *z*→−∞ only if *α*_{2}=0. Thus, the solution in the region *z*<0 takes the form (e.g. ER90)(4.9)

### (a) Dispersion relation for strongly magnetized boundary layer and atmosphere

In order to obtain the dispersion relation, observe that the vertical component of the Lagrangian displacement and the Lagrangian perturbation of total pressure are continuous across the interfaces *z*=0. This can be expressed by the following equations:(4.10)where and the square brackets denote the jump of the enclosed quantity across the interface. For *ξ*_{z} we have(4.11)Using equations (2.1)–(2.4) and (4.11) we obtain(4.12)Substituting (4.11), (4.12) and the obtained expressions (3.8), (4.2) for *Δ* and (3.1), (4.9) for *u*_{z} in continuity conditions (4.10), we obtain two homogeneous linear equations for the constants *α*_{1}, *C*_{2}. The condition for the existence of a non-trivial solution yields the dispersion relation(4.13)where(4.14)and *ρ*_{0p} and *ρ*_{0c} are the limits of the density *ρ*_{0} when *z*→+0 and *z*→−0, respectively. In the absence of an equilibrium flow, one recovers the dispersion relation derived by ER90.

### (b) Small wavenumber approximation

The continuity of the total equilibrium pressure across the interface yieldswhere *p*_{0p} and *p*_{0c} are the limits of the gas pressure *p*_{0} when *z*→+0 and *z*→−0, respectively. Therefore, we can writewhere , *T*_{p} is the temperature at the photospheric top of the convection zone and *T*_{c} is the temperature in the boundary layer. Following ER90 and introducing , , *K*=*kz*_{0} and , we can rewrite the dispersion relation (4.13) in the form(4.15)The quantity *φ* defined by (4.14) takes the form

Using the result (Abramowitz & Stegun 1965)and assuming that *m* is not an integer we may show that(4.16)where we have introduced(4.17)

(4.18)

We can treat (4.16) analytically along similar lines to those laid in CR89 and ER90. First suppose that *a* does not tend to −1,0 or a positive integer and *Ω* does not tend to zero as *K*→0. Using the formula (4.16) it is easy to see that in this limit the left-hand side of the dispersion relation (4.15) reduces to 0. To calculate the limit of the right-hand side we note thatand (Abramowitz & Stegun 1965)Then the dispersion relation (4.15) takes the form(4.19)where *Ω*_{*} is the limit of *Ω* as *K*→0. Thus, . If , then *a*=−1, which is not allowed; gives complex roots, which are of no interest here. Suppose now that *a*→*n*−1, *n*=1,2,3, …, and that(4.20)where *θ*_{n} and *s* are constants yet to be determined. Assuming the limit *Ω*_{n} of *Ω* as *K*→0 to be non-zero, we find the limit of the right-hand side of (4.15)(4.21)If *s*<*m*+1, then the left-hand side of (4.15) reduces to 0 as *K*→0 and we obtain the same solutions that we found when *a* did not tend to −1,0 or a positive integer. If *s*>*m*+1, then the left-hand side of (4.15) reduces to as *K*→0 and solutions to the dispersion relation in this case are . gives complex roots. In the case of , we arrive atwhich does not fit with the assumption that *a* tends to 0 or a positive integer as *K*→0. Thus, *s*=*m*+1. Then calculating the left-hand side we find that the dispersion relation (4.15) reduces to(4.22)

From (4.22) we determine the behaviour of the *n*=1,2, …, *p*-modes in the limit of small *K*:(4.23)where *Ω*_{n} is determined from(4.24)for *n*=1,2, ….

There remains the case *a*→−1, corresponding to the *f*-mode. Set(4.25)with *f*_{0} and *s* to be determined. Suppose that 1<*m*<2. Then we can write(4.26)where(4.27)Substituting (4.25) and (4.26) in (4.15) we obtain *s*=*m*+1 (as before) and(4.28)Thus, the frequency behaviour of the *f*-mode in the limit of small *K* is given by the formula(4.29)

From (4.23) and (4.28), we see that in the small wavenumber limit, the flow has a more dominant influence on *p*/*f*-mode frequencies than the magnetic field.

### (c) Numerical results and discussion

To solve the dispersion relation (4.13) numerically, we adopt a marginally stable stratification of the convection zone by taking *m*=1/(*γ*_{p}−1) and suppose that *γ*_{p}=*γ*_{c}=5/3. The temperature in the isothermal atmosphere is assumed to correspond to that at the temperature minimum, yielding *T*_{p}=*T*_{c}=4170 K. From the continuity condition for the total equilibrium pressure across the interface, we arrive at the following expression for the ratio , where the pressure *p*_{p}=86.82 N m^{−2}. The horizontal wavenumber *k* is related to the spherical harmonic degree by the formula , where . The results are presented in terms of spherical harmonic degree *l* and cyclic frequency *ν* (related to the angular frequency by the relation *ω*=2*πν*).

In figure 4*a–c*, the frequency difference (μHz), expressing the flow effect in the absence of a magnetic field (*B*=0), is plotted as a function of the spherical harmonic degree *l* and the flow *V* (km s^{−1}) for the (*a*) *f*-mode and (*b*, *c*) *n*=1,2 *p*-modes, respectively. The flow *V* changes in the interval (−0.1,0.1) The kinetic energy density of the *f*-mode in the field-free case is given by the expression . To satisfy the finite kinetic energy density condition as *z*→−∞, we require that (see also ER90). This gives the condition *l*<3550. The *p*-modes leak into the atmosphere after reaching their cutoff frequency. In figure 4*a–c*, the harmonic degree *l* is continued until the cutoff frequency is reached (for *p*-modes) or until the finite kinetic energy condition is fulfilled (for the *f*-mode). For *n*=1,2 *p*-modes and for the *f*-mode, the magnitude of *Δ*_{V}*ν* increases with *l* and *V*, starting to decrease at high degrees. The largest frequency shift (approx. 30 μHz) occurs at about *l*=2000 when *V*=±0.1 km s^{−1} for the *f*-mode. The frequency shift *Δ*_{V}*ν* for the *n*=1 *p*-mode reaches its maximum value (approx. 20 μHz) at *l*=1100 when *V*=±0.1 km s^{−1}. The *n*=2 *p*-mode has its largest frequency shift (approx. 14 μHz) at *l*=750 when *V*=±0.1 km s^{−1}.

In figure 5*a–c*, the frequency difference (μHz), expressing the flow effect in the presence of a uniform magnetic field with *B*=30 G, is plotted as a function of *l* and *V* (km s^{−1}) for (*a*) the *f*-mode and (*b*, *c*) *n*=1,2 *p*-modes, respectively. Note that in the presence of a uniform magnetic field there is no cutoff frequency, since the right-hand side of (4.9) vanishes when *z*→−∞ for any frequency (ER90). Comparing figures 4 and 5 one can conclude that the *flow influence on both p- and f-modes is slightly stronger* in the presence of the given magnetic field. The same behaviour for

*p*-modes can be observed in figure 6

*a–c*, where this effect becomes more pronounced with increasing radial order

*n*. Also observe from figure 5

*a–c*that in the presence of the magnetic field the magnitude of the frequency shift continues to increase for high values of

*l*, unlike the case with no magnetic field.

A similar equilibrium, though without flows, was used by ER90 to study the effect of a chromospheric magnetic field on the *p*- and *f*-modes. Comparing the results of ER90 with our results, we observe that the flow influence on the *p*- and *f*-modes is more significant than the magnetic field influence in the small wavenumber limit. However, this is not true for an arbitrary wavelength. The magnetic field influence might be stronger than the flow influence or vice versa, depending on their characteristic values. Frequencies with higher *l* are more sensitive to atmospheric effects and, hence, the magnitude of the frequency shift caused by the magnetic field grows more rapidly.

Measurements of *p*-mode frequencies (Braun & Fan 1998) show that there is a frequency shift between poleward- and equatorward-travelling waves measured over solar latitudes 20–60°, which is consistent with the Doppler effect of a poleward meridional flow of the order of 10 m s^{−1}. The measurements show frequency shifts of the order of 1–3 μHz for about *l*≈477. In order to compare our results in figure 6*a–c*, we have plotted the frequency shift for *V*=10 m s^{−1} as a function of the frequency *ν*(*l*, *B*, 0). We have taken three values of the harmonic degree: *l*=70, 270 and 470. Cases with *B*=0 and 10 G are considered. Figure 6*c* shows the frequency shift predicted by our simple model is in good agreement with the measurements indicated above. However, these measurements are done only for low frequencies. Our model predicts a decrease in the magnitude of the frequency shift with increasing radial order *n* and cutoffs at high frequencies when no magnetic effects are taken into account.

## 5. Boundary layer with *v*_{a}=const. (weak field)

In the lower atmosphere (*z*<0), the steady flow is absent and a unidirectional magnetic field is present. The magnetohydrostatic equation is given by equation (4.1), while perturbations are governed by equations (4.2) and (4.3). We again assume the transitional layer to be isothermal with the temperature equal to that at the top of the field-free medium, i.e. in *z*<0 we take *c*_{s}(*z*)=*c*_{0}. Also, we suppose that the *Alfvén speed is constant*. With these assumptions, the plasma parameter is constant and the governing equation reduces to (e.g. CR89)(5.1)whereare the magnetically modified adiabatic exponent and pressure scale height, respectively (in the absence of a magnetic field, *Γ*=*γ* and *H*_{B}=*H*_{0}). The solutions can be written in the formWe suppose that and choose the plus sign. This corresponds to evanescent disturbances in the chromosphere. So we have *u*_{z}=exp(*λz*), where

### (a) Boundary conditions and the dispersion relation

From continuity of the normal component of Lagrangian displacement (−i*u*_{z}/*ω* in *z*<0 and −i*u*_{z}/*ω*_{D} in *z*>0) across *z*=0, we have(5.2)Substituting *u*_{z} and *Δ*, we find (with ) that(5.3)From continuity of the equilibrium total pressure across *z*=0, we have the relation(5.4)We have(5.5)and(5.6)From these results it follows that(5.7)and(5.8)where *ξ*_{z} is the normal component of the Lagrangian displacement. Finally, fromwe obtainor(5.9)Substituting *u*_{z} and *Δ*(5.10)we obtain the dispersion relation for the case of weak field approximation(5.11)

In the absence of flow *ω*_{D}=*ω* and (5.11) reduces to the dispersion relation found in CR89.

### (b) Limit of long wavelength

The dispersion relation (5.11) may be written as(5.12)where(5.13)It is convenient to treat the *p*- and *f*-modes separately.

#### (i) *p*-Modes

We write the dispersion relation in the form(5.14)where(5.15)and expand the right-hand side in a series in (*kz*_{0})^{1/2} to yield(5.16)Here *a*_{8}, *a*_{4} and *a*_{0} are constants.

Observe that this expansion of the right-hand side breaks down if is close to zero or unity. The case corresponds to the *f*-mode and is considered below. We see that remains finite as (*kz*_{0})^{1/2}→0. For this to be compatible with equation (5.16), it is necessary that *Γ*(−*a*) tends to infinity in such a way that remains finite. In particular, we require that(5.17)From the definition of *a* under adiabatic conditions (see (3.6)), we see that(5.18)To obtain a correction to this result, set(5.19)with *p*_{V1}, *p*_{V2}, *δ*_{0} and *s* to be determined. After some algebra, we obtain *s*=*m*+3/2 and(5.20)(5.21)and(5.22)Thus,(5.23)or(5.24)with *p*_{V1}, *p*_{V2} and *δ*_{0} defined above.

#### (ii) f-Mode

Carrying out a similar analysis, we can write the solutions to the dispersion relation for the *f*-mode in the form(5.25)with *f*_{V1}, *f*_{V2}, *f*_{0} and *s* to be determined. After some algebra we obtain(5.26)and(5.27)or(5.28)

### (c) Numerical results for weak fields

For numerical investigations we have taken the sound speed in the isothermal atmosphere to correspond to that at the temperature minimum, yielding *c*_{0}=6.76 km s^{−1} for adiabatic index *γ*=5/3. The scale height *H*_{0} is then 100 km and the polytropic index *m* is 3/2. For conditions typical of the temperature minimum, we may take *β*=(180/*B*_{0})^{2}, where *B*_{0} is the field at the base of the magnetic atmosphere, measured in gauss (see also Campbell & Roberts 1989). Figures 7–10 are obtained by solving the full dispersion relation for the weak magnetic field case. Cutoffs in the figures are denoted by dotted lines.

## 6. Resonant coupling of solar oscillations to the atmosphere

So far, we have mainly presented results of a very simple solar model, where we pointed out that both steady states and atmospheric magnetic fields could be important when evaluating helioseismic observational data. One key feature of the boundary layer was that the profile of the magnetic field was selected such that the Alfvén speed was *constant* in the boundary layer. In the following we relax this condition on the Alfvén speed.

A three-layer model with an intermediate zone, where the magnetic field, together with the Alfvén speed, varies continuously from zero was introduced by Tirry *et al*. (1998) (figure 11). The importance of the continuum is that global modes may interact resonantly to local boundary layer Alfvén and/or slow oscillations at the height, where their frequency matches the frequency of the global mode, hence the model can be used to investigate the effects of *resonant coupling between global modes and local atmospheric MHD oscillations*. Pintér & Goossens (1999) discussed the case of parallel propagation in this model. For propagation parallel to the magnetic field, the global oscillation modes couple to slow continuum modes only, and this was found to occur for a rather large range of realistic parameters. In addition to the damping of global oscillation modes due to resonant absorption, it was also found that the interaction of global eigenmodes with slow continuum modes leads to an unanticipated behaviour in the global eigenmodes. The rather strange behaviour in the slow continuum involves the disappearance, appearance and splitting and merging of global modes. Additionally, frequency shifts of global modes due to the magnetic field were examined. Pintér *et al*. (submitted) extended the analysis to non-parallel propagation. They demonstrated that obliquely propagating global modes can couple also to local MHD Alfvén *and* slow continuum modes. They investigated the magnetic effects on global mode frequencies, especially the frequency shifts and damping rates caused by the resonant interaction with local slow and Alfvén waves. Due to the presence of a number of characteristic frequencies in the model, they found not only the *f*- and *p*-modes, but also Lamb modes, with frequencies near the characteristic cutoff frequencies. Atmospheric gravity modes also appeared as solutions to the linear MHD equations. A theoretical study of the influence of the orientation of wave vector also raises the question how to measure the angle between the direction of the wave propagation and the atmospheric magnetic field lines. Although the number of observed oscillations is rapidly increasing, the resolution of detection has to be enhanced to obtain more features of the waves, such as the direction of their propagation with respect to the local magnetic field lines.

A straightforward application of the resonant coupling of acoustic oscillations in a *steady state* is the rotational splitting of helioseismic modes influenced by a magnetic atmosphere. Pintér *et al*. (2001*b*) studied the splitting of sectoral (*m*=±*l*) helioseismic eigenmodes. The solar interior was in a steady state, with sub-photospheric plasma flow along the equator representing solar rotation. The Cartesian geometry employed restricted the study to sectoral modes with *l*≥50, which guarantees that the modes do not penetrate deeply into the solar interior and, therefore, experience an approximately uniform rotation. The mean increase of with *B* for the *p*-modes they studied, for *l*=100, is around 370 nHz, which is a 0.41% relative increase. For GONG and MDI data, the observational error of measuring due to rotational splitting is better than 0.25%. Hence, the effect obtained in the present model is on the verge of detectability, and ought to be detectable by combining a number of modes. On the other hand, there are other competing effects—such as those due to zonal flows—which are of about the same order. One possible way of helping to differentiate between the several competing shifts would be to evaluate the modelled rotational splitting for all *m*. This requires a move to spherical geometry.

Another application of the resonant coupling is the damping of helioseismic modes in a steady state (Pintér *et al*. 2001*a*). The frequencies and the line widths of eigenmodes are affected by sub-surface flow *and* atmospheric magnetic fields. A key contribution to the effects comes from the universal mechanism of resonant absorption. When both atmospheric magnetic field and sub-surface flows are present, a complex picture of competition between these two effects was found. Their table 1 shows the sensitivity of the line width of the *f*- and *p*_{1}-modes to an equilibrium flow varying between [−0.1*c*_{s},0.1*c*_{s}]. The ratio is given for different values of *l*. The line width of the *f*-mode increases or decreases linearly with *V* in the interval , while that of the *p*-mode always increases, also linearly, in the given interval of *V* and *l*. For larger *l*, the effects are more complicated, and the line width of *p*-modes can also decrease with *V*. In addition, for larger values of *V*, the dependence becomes nonlinear.

In the present paper I have tried to give an overview of studies on the coupling of global solar acoustic oscillations to the lower magnetized solar atmosphere. In analogy to critical layers in fluid dynamics, I introduced the terminology of *boundary layer* for this *narrow transitional layer (embracing the photosphere, chromosphere and TR)* between the solar interior and the solar corona. The effects of both magnetic fields and flows on the acoustic oscillations were investigated. The fields were split into their coherent and random parts. Each of these components has, in its own peculiar way, influences on the coupling of solar global oscillations to the atmosphere. At the moment it is hard to determine which effect is dominant or which one is less relevant. The Solar Dynamics Observatory to be launched soon may shed light on this exciting and rapidly evolving question of coupling of solar acoustic oscillations to the solar atmosphere.

## Acknowledgments

The author acknowledges M. Kéray for patient encouragement and the financial support obtained from the Royal Society to make his presentation and this follow-up paper. The author is also grateful to the NSF, Hungary (OTKA, Ref. No. TO43741) and thanks Drs B. Pintér and Y. Taroyan for preparing some of the figures. Finally, the author would like to thank the Referee for his careful reading and suggestions.

## Footnotes

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

- © 2005 The Royal Society