A&A 466, 377?388 (2007) DOI: 10.1051/0004-6361:20041632 c?ESO 2007 Astronomy & Astrophysics Global oscillations in a magnetic solar model II. Oblique propagation star B. Pint?r 1 ,R.Erd?lyi 2 , and M. Goossens 3 1 Solar System Physics Group, Institute of Mathematical and Physical Sciences, The University of Wales, Aberystwyth, Penglais Campus, Aberystwyth, Ceredigion, SY23 3BZ, Wales, UK e-mail: b.pinter@aber.ac.uk 2 Solar Physics and Space Plasma Research Centre (SP 2 RC), Department of Applied Mathematics, University of She?eld, Hicks Building, Hounsfield Road, She?eld, S3 7RH, England, UK e-mail: robertus@sheffield.ac.uk 3 Centre for Plasma-Astrophysics, Departement Wiskunde, Faculteit Wetenschappen, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium e-mail: Marcel.Goossens@wis.kuleuven.ac.be Received 9 July 2006/Accepted 29 January 2007 ABSTRACT The coupling of solar global acoustic oscillations to a magnetised solar atmosphere is studied here. The solar interior ? atmosphere interface is modelled by a non-magnetic polytrope interior overlayed by a planar atmosphere embedded in non-uniform horizontal atmospheric magnetic field. Pint?r & Goossens (1999, A&A, 347, 321) showed that parallel propagating acoustic waves can couple resonantly to local magnetohydrodynamic (MHD) slow continuum modes only. In general, global acoustic modes can, however, propagate in arbitrary directions with respect to local atmospheric fields giving rise to an additional e?cient coupling mechanism that has consequences on mode damping and atmospheric energetics. In this paper we study obliquely propagating global modes that can couple also to local MHD Alfv?n continuum modes. The atmospheric magnetic e?ects on global mode frequencies are still much of a debate. In particular, the resulting frequency shifts and damping rates of global modes caused by the resonant interaction with both local Alfv?n and slow waves are investigated. We found the coupling of global f and p modes and the Lamb mode, that penetrate into the magnetic solar atmosphere, will strongly depend on the direction of propagation with respect to the solar atmospheric magnetic field. These frequency shifts, as a function of the propagation direction, give us a further elegant tool and refinement method of local helioseismology techniques. Finally we briefly discuss the importance of studying obliquely propagating waves and discuss the results in the context of possible helioseismic observations. Key words. Sun: helioseismology ? Sun: oscillations ? Sun: atmosphere ? Sun: chromosphere ? Sun: magnetic fields ? magnetohydrodynamics (MHD) 1. Introduction Observations reveal complex structures of the solar atmospheric magnetic field. Beneath the solar surface the magnetic field can be described by confined, vertical thin flux tubes. When these flux tubes break through the photosphere, it is observed that the magnetic field lines incline in most cases from the verti- cal direction. They fan out and create a local magnetic canopy, i.e. structures with horizontal magnetic field, throughout the chromosphere. From the solar atmospheric magnetic structures we are inter- ested in global (or coherent) structures that exist for a relatively long time, at least when compared to the life time of the global oscillations. In practical terms this means a few hours. Title & Schrijver (1998) argue that the life time of the lower solar atmo- spheric magnetic carpet satisfies this condition. Although esti- mates are changing as instrumentation improves, it is now gen- erally believed that the replacement time of the magnetic carpet is of the order of 10?14 h; well above the life time of the global oscillations, that is around 7?10 periods. When considering the star Appendix A is only available in electronic form at http://www.aanda.org global p-modes, their life time is around 2000 to 3000 s. This observation allows us to work in a framework where the time- dependent variation of the background magnetic carpet can be neglected. If the picture of static magnetic carpet holds, this is even more true for global chromospheric and coronal background fields on the time scales of the life time of acoustic global os- cillations allowing an investigation of the coupling of these os- cillations to the solar atmosphere in stationary state. Before we embark on the analysis, let us briefly recall some evidence that may directly or indirectly indicate the mechanism of such cou- pling. Of course, there is no direct observational proof of the actual resonant layer since that would require resolution of the order of few hundreds of meters in the solar plasma. However, there are studies indicating that oscillations found in the lower solar atmosphere, also called the lower boundary layer, or even in the low corona show strong correlation with the periodic mo- tions that characterize the photosphere (Erd?lyi 2006). The existence of oscillations within the atmospheric mag- netic structures is clear from observations. The presence of os- cillations in sunspots was already known (e.g. Bogdan 2000) before the generation of high angular and temporal resolution Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20041632 378 B. Pint?r et al.: Obliquely propagating global solar oscillations. II. measurements. Images from SOHO and TRACE prove unques- tionably that di?erent kinds of oscillations propagate from the solar interior through the chromosphere, and reach even the corona (e.g., Schrijver et al. 2002; Aschwanden et al. 2002; and Roberts 2002). De Pontieu et al. (2003) found correlation between lower transition region oscillations and photospheric p-mode oscilla- tions. De Pontieu et al. (2004) showed how global modes leak into the atmosphere along vertical thin flux tubes and can drive periodic spicules along inclined thin flux tubes. This leakage can even cause loop oscillations if these photospheric motions reach the lower corona, as show by De Pontieu et al. (2005) and De Pontieu & Erd?lyi (2006). Last but not least, Rutten & Krijger (2003) have studied low- frequency brightness modulation of internetwork regions in the low chromosphere using image sequences from TRACE, and have shown that even atmospheric gravity waves are at present in the low chromosphere. The examples will improve with the launch of Solar-B giving a timely aspect to questions like: what are the physical details of the coupling of global oscillations to the lower boundary layer? What is the coupling mechanism? Could resonant absorption, an already popular mechanism for plasma heating and damping of loop oscillations, play a role in the coupling? What could be the manifestation of resonant cou- pling of global modes? In this paper we try to demonstrate the answers to these questions in a simple theoretical model. A number of solar models were put forward to investigate the features of global oscillations in the solar atmosphere. An ap- proximation for the inhomogeneous solar interior and atmo- sphere is a two-layer model, where the lower layer represents the magnetic field-free sub-photospheric region while the upper one is for the atmosphere embedded in a horizontal (canopy) mag- netic field with constant Alfv?n speed. Miles et al. (1992) used that model with isothermal layers to study magnetoacoustic- gravity surface waves. Miles & Roberts (1992) carried out an- other follow-up study, in which the upper layer has a uniform magnetic field. Campbell & Roberts (1989) studied the e?ects of an atmospheric magnetic field on global solar oscillations, sim- ilarly in a two-layer model, with constant Alfv?n speed in the layer above the interface. They considered the solar interior as a polytrope, instead of an isothermal medium. Evans & Roberts (1990) removed the assumption of constant Alfv?n speed and considered a uniform chromospheric magnetic field instead. Vanlommel et al. (2002) also studied the e?ects of a uniform, horizontal coronal magnetic field on global oscillations. They focused on coupling of global modes (which propagate paral- lel to the magnetic field lines) to local slow MHD oscillations, and used an equilibrium model somewhat more realistic than in Campbell & Roberts (1989) and Evans & Roberts (1990). Magnetic e?ects on obliquely propagating f and p modes were investigated by Jain & Roberts (1994) in the two-layer model, with constant Alfv?n speed in the upper layer. Jain & Roberts came to the conclusion that the positive frequency shift of the f mode is reduced as?is increased from zero to 90 ? .They explained this result as being due to the magnetic tension, which is reduced when the angle of the wave vector with respect to the magnetic field becomes larger. They also found that a pos- itive shift in p-mode frequencies is reduced by an increase in the propagation angle. The e?ect is smaller for p modes with higher radial order since their frequency shifts are thought to depend strongly on temperature and magnetic pressure in the chromosphere, and so the orientation of the wave vector is less important. A more advanced three-layer model with an intermediate zone, where the magnetic field, together with the Alfv?n speed, varies continuously from zero was introduced by Tirry et al. (1998). The importance of the Alfv?n continuum is that global modes may interact resonantly to local Alfv?n oscillations at the height were their frequency matches the frequency of the global mode, hence the model can be used to investigate the e?ects of resonant coupling between global modes and local magnetohy- drodynamic (MHD) oscillations. Tirry et al. (1998) gives a thor- ough mathematical description of the three-layer model, and out- lines the basic idea of resonant coupling. However, the physical results presented in Tirry et al. (1998) are far from complete, details were left for subsequent studies. Pint?r & Goossens (1999), hereafter Paper I, exposed the case of parallel propagation in a follow-up study to Tirry et al. (1998). Di?erent types of oscillation modes have been deter- mined for a wide range of the magnetic field strength and for dif- ferent degrees of the spherical harmonic. The emphasis is on the possible coupling of global solar oscillation modes to localized continuum eigenmodes of the magnetic atmosphere. For propa- gation parallel to the magnetic field, the global oscillation modes can couple only to slow continuum modes and this has been found to occur for a rather large range of parameters. Damping of parallel global oscillation modes due to resonant absorption and frequency shifts of global modes due to the magnetic field have been examined. The model in the present paper is an enhancement of models that have been used by those referred to previously (Campbell & Roberts 1989 to Pint?r & Goossens 1999). We consider the phenomena of resonant coupling of solar global oscillations to the inhomogeneous solar atmosphere for oblique propagation. Figure 1 shows the equilibrium profiles for the plasma den- sity, pressure, sound speed and Alfv?n speed for some typical values. Notice that the number density, n 0 (z) is plotted in Fig. 1a, though in the calculations the mass density,? 0 (z) ? m p n 0 (z)is used, where m p is the average mass of the plasma particles. The atmospheric scale height, H co ,inFig.1aismuchlarger than the range of the atmosphere shown. This is why the rate of the exponential decay of n 0 (z)andp 0 (z) in the upper layer is hardly recogniseble. Although the model is an e?ective tool to describe crucial solar phenomena, we have to emphasize that the current ap- proach does not claim to be a perfect representation of the highly complex and dynamic Sun. The assumption of a steady uni- directional horizontal atmospheric magnetic field is obviously a crude representation of the three-dimensional magnetic struc- tures of the real solar chromosphere and corona. Magnetic fluxes are continuously emerging at the solar surface and expanding into the atmosphere. Consequently, the orientation of the atmo- spheric magnetic field changes temporarily and spatially. It is our aim to further develop the present model by considering stochas- tic magnetic fields, similar to that in Erd?lyi et al. (2004a,b, 2005). We describe the model in Sect. 2. In Sect. 3 the structure of the frequency spectrum and the global oscillation modes to- gether with spatial solutions are derived. Section 4 is devoted to a detailed investigation of the obtained atmospheric e?ects on the f-andp-modes. Section 5 is for the summary and discussion of the results. 2. The model The main characteristics of the model are presented in this sec- tion. First, a basic description is given, explaining the main B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 379 Fig.1. Illustration of the equilibrium of a) plasma density and pressure and b) sound and Alfv?n speeds. (The Alfv?n speed profile is given for B L ?90 G. The scale heights shown are defined in Eqs. (11) and (12)). features of the model, its relevance to the Sun together with its limitations. Finally, the dispersion relation between the wave number, k, and the frequency,?, of the eigenoscillations is de- rived. Model description. A plane-parallel, three-layer model in Cartesian coordinate system is used here, similar to the one in Paper I. While Paper I investigated the case of parallel propa- gation, the present study focuses on the results for the case of non-parallel propagation in detail. The static equilibrium model is derived from the ideal mag- netohydrodynamic (MHD) equations, which are the continuity equation, the equation of momentum, the equation of energy and the induction equation: ?? ?t +?.(?u)=0, (1) ? bracketleftBigg ? ?t +u.? bracketrightBigg u=??p+ 1 ? (??B)?B+?g, (2) bracketleftBigg ? ?t +u.? bracketrightBigg p? ?p ? bracketleftBigg ? ?t +u.? bracketrightBigg ?=0, (3) ?B ?t =??(u?B), ?.B=0. (4) The adiabatic index (or ratio of specific heats) is taken?= 5/3 throughout the paper. The three layers are the semi-infinite solar interior (z< 0) and atmosphere (z>L) ? which is basically the corona ? with a transitional layer between them (0 ? z ? L). Note that this transitional layer is not taken just for the transition region of the Sun. The equilibrium quantities (temperature, T 0 , density, ? 0 , gas pressure, p 0 , and magnetic induction, B 0 )are inhomogeneous and vary continuously in the z-direction, which is oriented towards the solar centre. (The 0 index refers to the state of equilibrium.) The interior is a polytrope (i.e., p(z)?? ? (z)), where?=5/3 is the ratio of specific heats or the adiabatic index, with an equi- librium temperature decreasing from central to surface regions. The top of the interior is the photosphere (z= 0). The temper- ature, T 0 (z), increases linearly in the intermediate transitional layer from its photospheric minimum to its maximum, which is the temperature of the isothermal corona. The plasma density, ? 0 (z), decreases throughout the three layer of the model with di?erent steepness. Its decay is exponential in the corona. A horizontal magnetic field, B 0 (z), is considered in the at- mosphere (z>0) representing a canopy-like structure. From the non-magnetic photosphere, the field strength increases sharply in the transitional layer to its top in a way that the square of the Alfv?n speed, v 2 A (z) ? B 2 0 (z)/(?? 0 (z)), increases linearly. Above that, the magnetic field strength decreases exponentially together with the plasma density so that the Alfv?n speed re- mains constant in the corona. Such an equilibrium will have all the necessary ingredients to investigate the mechanism of reso- nant coupling of solar global oscillations and will also be treat- able analytically. The equilibrium profile of the plasma pressure, p 0 (z), can be derived from the temperature, density and magnetic induction by assuming pressure equilibrium and that the plasma obeys the perfect gas law. The equilibrium density, plasma pressure and Alfv?n speed square, hence, have the following profiles, respectively: ? 0 (z)= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ph parenleftBigg 1? z H in parenrightBigg 1/(??1) , z?0, ? ph parenleftBigg 1+ z H tr2 parenrightBigg ? , 0?z?L, ? L exp parenleftBigg ? z?L H co parenrightBigg , L?z, (5) p 0 (z)= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? p ph parenleftBigg 1? z H in parenrightBigg ?/(??1) , z?0, p ph parenleftBigg 1+ z H tr1 parenrightBigg parenleftBigg 1+ z H tr2 parenrightBigg ? , 0?z?L, p L exp parenleftBigg ? z?L H co parenrightBigg , L?z, (6) 380 B. Pint?r et al.: Obliquely propagating global solar oscillations. II. Fig.2. Sketch of the three-layer model. v 2 s (z)= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? v 2 s,ph parenleftBigg 1? z H in parenrightBigg , z?0, v 2 s,ph parenleftBigg 1+ z H tr1 parenrightBigg , 0?z?L, v 2 s,co , L?z, (7) v 2 A (z)= ? ? ? ? ? ? ? ? ? ? ? 0 ,z?0, v 2 A,co z L , 0?z?L, v 2 A,co , L?z, (8) with p ph = 1 ? ? ph v 2 s,ph ,v 2 s,co = T co T ph v 2 s,ph (9) ? L =? 0 (z=L), p L = p 0 (z=L) (10) H in = v 2 s,ph (??1)g , H co = 1+? ? v 2 s,co ?g (11) H tr1 = v 2 s,ph L v 2 s,co ?v 2 s,ph ,H tr2 = 2v 2 s,ph L 2v 2 s,co ?2v 2 s,ph +?v 2 A,co (12) ?=1+ 2?gL 2v 2 s,co ?2v 2 s,ph +?v 2 A,co ,?= 2v 2 s,co ?v 2 A,co ? (13) The indices ph, L and co refer to equilibrium quantities taken at the photosphere (z = 0), at the top of the transitional layer (z=L) and in the corona (z?L), respectively. The equilibrium profiles, Eqs. (5) to (8), can be derived from the vertical component of the equation of motion, Eq. (2) (which takes the form dP 0 (z)/dz =?g? 0 (z) for the equilibrium quantities) and from the definition of the sound speed square, v 2 s (z)??p 0 (z)/? 0 (z). Here P 0 is the total pressure of the equilib- rium: P 0 (z)? p 0 (z)+B 2 0 (z)/2?. Although the model is simple, it mimics a fairly realistic ba- sic solar structure. The atmosphere features a sudden density and pressure decrease in the transitional layer. The temperature pro- file in the model reflects the observed transitional profile in the sense that the interior becomes cooler towards the photosphere, above which the corona becomes hot again at a certain height. Most importantly, the magnetic profile represents a canopy mag- netic field above the observed magnetic carpet, where the con- densed magnetic flux tubes ? penetrating from the photosphere into the atmosphere ? fan out and the magnetic field lines form horizontal, canopy structures. The solar plasma is stratified in the model by constant gravity,g=?ge z , withg= 274 m/s 2 ,using its observed photospheric value. Because the global waves are expected to be concentrated right beneath the photosphere, they are most a?ected by the photospheric values of the equilibrium quantities. Governing equations. The disturbances in the plasma are de- scribed by the linearized MHD equations, derived from Eqs. (4). Ohmic heating is included via resistivity. Dissipation has to be taken into account in regions only where steep gradients occur, i.e., in the vicinity of the resonant heights. Elsewhere the ideal MHD equations give an accurate description. The perturbed quantities take their Fourier-transformed form f 1 (x,y,z,t)= f(z;?,k x ,k y )e i(k x x+k y y??t) . (14) The aim is to obtain a dispersion relation, f(?,k x ,k y ) = 0. The linear Fourier transformed ideal MHD equations can be reduced to two ordinary di?erential equations of the first order for the vertical component of the Lagrangian displacement, ? z (z)and for the Eulerian perturbation of total pressure, P(z): D d? z dz =C 1 ? z ?C 2 P, D dP dz =C 3 ? z ?C 1 P. (15) The total pressure is the sum of the plasma pressure and the mag- netic pressure, P(z) ? p(z)+B 2 (z)/(2?). The coe?cient func- tions D, C 1 , C 2 and C 3 in (15) are given by D(z)=? 0 (v 2 s +v 2 A )(? 2 ?? 2 c )(? 2 ?? 2 A ), C 1 (z)=g? 0 ? 2 (? 2 ?? 2 A ), C 2 (z)=? 4 ?k 2 (v 2 s +v 2 A )(? 2 ?? 2 c ), C 3 (z)= parenleftBigg ? 0 (? 2 ?? 2 A )+g d? 0 dz parenrightBigg D+g 2 ? 2 0 (? 2 ?? 2 A ) 2 . (16) The two components of the horizontal wave vector, k,arek x = k cos?and k y =k sin?. The angle?measures the inclination of the propagation direction to the magnetic field lines (see Fig. 3). The x axis in the model is taken to be parallel to the magnetic field lines, i.e. cos?? k.B kB ? (17) The local (i.e. z-dependent) Alfv?n frequency? A (z) and the lo- cal slow or cusp frequency? c (z) play a fundamental role in the coupling of global solar oscillations to continuum oscillations. The local sound frequency? s (z) is, indirectly, also a key param- eter for the frequency spectrum. The squares of these frequencies are given by ? 2 s (z)=(k 2 x +k 2 y )v 2 s (z), ? 2 A (z)=k 2 x v 2 A (z), ? 2 c (z)=k 2 x v 2 c (z), (18) where the cusp speed square is given by v 2 c (z)? v 2 s (z)v 2 A (z) v 2 s (z)+v 2 A (z) ? Equations (15) describe how linear oscillations of a one- dimensional, inhomogeneous magnetic plasma are governed. B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 381 Fig.3. The idea of oblique propagation. The three layers of the model (interior, transitional layer and corona) are joint by the physical requirements that the Lagrangian displacement,? z (z), and the Lagrangian perturbation of total pressure, P(z)+g? 0 (z)? z (z), must be continuous functions of height. Equations (15) with these boundary conditions and boundary conditions at??define an eigenvalue problem for the global frequency?. The boundary conditions far away from the transitional layer are that the kinetic and magnetic energy of the eigenoscillations diminish to zero for z???(toward the solar interior) and for z ??(toward the outer corona). The aim of this study is to understand how the frequency spectrum of linear eigenoscillations is changed when the direction of propagation, ?,varies. 3. Solutions to the governing equations Interior. The general solutions of Eqs. (15) for? z (z)andP(z)in the polytrope interior are a product of an exponential term and a linear combination of the Kummer functions M and U. (About Kummer functions, see, e.g., Abramovitz & Stegun 1965). The integrational coe?cient of the terms containing function M has to be zero due to the boundary condition that the kinetic energy density, E kin ?? 0 (z)? 2 ? 2 z (z)/2, of the perturbation has a finite asymptotic value as z tends to??. The solutions remain free up to a constant factor, as the linear wave theory does not predict the size of the perturbations. Transitional layer. Once having the values ? z (z = 0) and P(z = 0), from the internal solutions, a numerical integration of Eqs. (15) can be carried out from z= 0toL, using the equi- librium profile of the transitional layer, in case of no resonance (?>? A ). If resonance occurs in the transitional layer, the numer- ical integration evaluates? z (z)andP(z) at the lower edge of the resonant layer; there the connection formulae are used to obtain ? z (z)andP(z) at the top of the resonant layer, which are then the initial values for further numerical integration, up to z = L.If the global frequency takes place not only in the Afv?n but also in the slow continuum, both sets of connection formulae have to be used: around the slow resonant position, z C , and then around the Alfv?n resonant position, z A . In this latter case the evalua- tion of ? z (z = L)andP(z = L) consists of numerical integra- tions along three sections in the transitional layer (first, below the Alfv?n resonant height, then between the Alfv?n and slow resonant heights, finally, above the slow resonant height) and the use of two connection formulae (first, across the Alfv?n resonant height, then across the slow resonant height). Corona. In order to properly investigate the cut-o?frequencies (see the following paragraph), it is useful to write down explic- itly the coronal solutions for? z (z)andP(z). The general coronal solution is ? z (z)=A 0 exp parenleftBiggparenleftBigg 1 2H ?? parenrightBigg z parenrightBigg +A 1 exp parenleftBiggparenleftBigg 1 2H +? parenrightBigg z parenrightBigg , P(z)= D C 2 ? c ? bracketleftBiggparenleftBigg C 1 D ? 1 2H +? parenrightBigg A 0 exp parenleftBigg ? parenleftBigg 1 2H +? parenrightBigg z parenrightBigg + parenleftBigg C 1 D ? 1 2H ?? parenrightBigg A 1 exp parenleftBiggparenleftBigg ? 1 2H +? parenrightBigg z parenrightBiggbracketrightBigg , (19) whereA 0 andA 1 are integration constants,? c is the equilibrium density at the top of the transitional layer (at z = L)andthe parameter?is defined by its square as ? 2 (z)= parenleftBigg C 1 D ? 1 2H parenrightBigg 2 ? C 2 C 3 D 2 ? (20) The boundary condition lim z?? E kin (z)=0 is satisfied for positive values of? 2 and forA 1 =0 in Eq. (19). Characteristic frequencies. The structure of the inhomoge- neous model indicates that the frequency spectrum of the eigenoscillations is divided into di?erent regions by character- istic frequencies. Although such divisions of the solar eigen- spectrum have not been observed clearly yet, it is necessary to analyse the e?ects of the characteristic frequencies arising in the present structured model. It follows from the definitions of the coe?cients D and C 1 ?C 3 , Eqs. (16), that ? 2 can be written as a cubic poly- nomial divided by a quadratic polynomial in? 2 : ? 2 =? (? 2 ?? 2 I )(? 2 ?? 2 II )(? 2 ?? 2 III ) (v 2 s +v 2 A )(? 2 ?? 2 A )(? 2 ?? 2 c ) ? (21) Here? 2 I ,? 2 II and? 2 III denote the three yet unknown roots of the cubic polynomial in? 2 and play the role of cut-o?frequencies. The parameter? 2 changes sign when? matches one of the cut-o?frequencies or one of the two other characteristic frequen- cies, ? A or? c . The importance of the sign of? 2 is that pertur- bations in the corona (see Eqs. (19)) propagate for? 2 <0 (leaky modes) and are evanescent for? 2 >0 (eigenmodes). The frequency spectrum in an equilibrium which is free of a magnetic field is characterized only by two cut-o?frequencies, ? I and? II . They can be expressed as ? 2 I | B=0 = v 2 s 2 parenleftBigg k 2 + 1 4H 2 parenrightBigg parenleftBig 1? ? 1?? parenrightBig , ? 2 II | B=0 = v 2 s 2 parenleftBigg k 2 + 1 4H 2 parenrightBigg parenleftBig 1+ ? 1?? parenrightBig , (22) 382 B. Pint?r et al.: Obliquely propagating global solar oscillations. II. where ?= 64k 2 H 2 (v 2 s ?gH)gH v 4 s (1+4k 2 H 2 ) 2 , (23) The sound speed,v s , and the isothermal density scale height, H? v 2 s /?g, are all constant in the corona. Notice that ?, defined in Eq. (23), is always positive, as ?(=5/3) > 1. It follows from this that it never happens in the non-magnetic model that only the upper cut-o? frequency ex- ists while the lower cut-o? frequency does not exist (i.e., if ? 2 II (B0)>0then? 2 I (B0)>0 too). Equation (20) reduces to a quadratic equation for ? 2 for an equilibrium with a magnetic field (B nequal 0) and for parallel propagation (k y = 0). The three frequencies characterising the eigenspectrum are now? I ,? II and? c . The two cut-o?frequen- cies can be expressed analytically as ? 2 I vextendsingle vextendsingle vextendsingle k y =0 = 1 2 parenleftBigg k 2 x + 1 4H 2 co parenrightBigg (v 2 s +v 2 A ) parenleftBig 1? ? 1?? parenrightBig , ? 2 II vextendsingle vextendsingle vextendsingle k y =0 = 1 2 parenleftBigg k 2 x + 1 4H 2 co parenrightBigg (v 2 s +v 2 A ) parenleftBig 1+ ? 1?? parenrightBig , (24) where ?= 4k 2 x parenleftBiggparenleftBigg k 2 x + 1 4H 2 co parenrightBigg v 2 s v 2 A + parenleftBigg ?? 1+? ?1 parenrightBigg g 2 parenrightBigg parenleftBigg k 2 x + 1 4H 2 co parenrightBigg 2 (v 2 s +v 2 A ) 2 ? (25) The coronal magnetic density scale-height, H co modified by the presence of the magnetic canopy, is defined in Eq. (11). The Alfv?n speed, the sound speed, the plasma-?and hence H co are all constant in the corona. The Alfv?n and slow frequencies tend to zero while the sound frequency becomes? s =k y v s for k x ? 0. There are only two cut-o?frequencies for k x =0. Their asymptotic values are ? 2 I vextendsingle vextendsingle vextendsingle k x =0 = 1 2 (k 2 y + 1 4H 2 co )(v 2 s +v 2 A ) parenleftBig 1? ? 1?? parenrightBig , ? 2 II vextendsingle vextendsingle vextendsingle k x =0 = 1 2 (k 2 y + 1 4H 2 co )(v 2 s +v 2 A ) parenleftBig 1+ ? 1?? parenrightBig , (26) where in this limit ?= (??1)??1 1+? k 2 y parenleftBigg k 2 y + 1 4H 2 co parenrightBigg 2 g 2 (v 2 A +v 2 s ) 2 ? (27) The transitional layer between the photosphere and the corona introduces an Alfv?n continuum and a slow continuum in the fre- quency spectrum of oscillations. These two frequency continua are [min(? A (z)), max(? A (z))] (i.e. [0, ? A ]) and [min(? c (z)), max(? c (z))] (i.e. [0,? c ]), respectively. A global mode that has a matching frequency in the slow and/or Alfv?n continuum cou- ples resonantly to a local slow mode and/or a local Alfv?n mode, respectively. The coupling also transforms the eigenmode into a damped mode. The damping rate is the non-zero imaginary part of the eigenfrequency ? as investigated in detail in the Appendix. Global oscillation modes propagating parallel to the atmo- spheric magnetic field lines (?= 0) can interact only with local slow continuum modes. However, obliquely propagating global oscillations (? nequal 0) can be coupled resonantly also to local Alfv?n continuum modes. Although the frequencies are expressed in terms of ? in the derivations, henceforth the results are presented in terms of ?(??/2?), which is more commonly used when quoting obser- vations. The imaginary part, Im?, which represents the damping rate of the mode, is replaced in the discussions by the, more fa- miliar, line width of the modes,?(??2Im?), for similar reason. The characteristic frequencies are labeled in the plots of the solu- tions of the eigenvalue problem with I, II, III, A and C, referring to the frequencies of? I ,? II ,? III ,? A and? C , respectively. It was discussed earlier that? 2 , defined in Eq. (20), has a pos- itive value for eigenmodes with a frequency right below? II .The parameter? 2 is complex in the Alfv?n continuum and in the slow continuum. The regions with positive real part of? 2 inthe Alfv?n and slow continua are for (damped) eigenmodes, while those with negative real part of? 2 inthe Alfv?n and slow continua are for leaky modes. Input parameters. The focus is on the e?ects introduced by the atmospheric magnetic field strength and the angle of prop- agation on the frequency spectrum. Hence the thickness of the transitional layer is fixed at L = 2 Mm throughout the numer- ical analysis, representing a transition between the photosphere and corona. Also, the temperature increase through the transi- tional layer is fixed at T co /T ph = 200. The sound speed is taken at the photosphere v s,ph = 7.6kms ?1 , which corresponds to T ph ? 4170 K photospheric temperature minimum. These im- ply that v s,co ? 108 km s ?1 and T co ? 834 000 K. The plasma density at the photosphere is fixed at ? ph = 0.17 g m ?3 .The avearage molar mass of the plasma particles is approximated in the model as M p (?N A m p ) = 1.3gmol ?1 . From this, the pho- tospheric plasma pressure is p ph ? 8320 N m ?2 . The plasma density,? L ?? 0 (z = L), and pressure, p L ? p 0 (z = L), at the top of the transitional layer are a function of the magnetic field strength B L , as given by Eqs. (5) and (6). The dependence of the mode frequencies on the harmonic degree, l, is basically parabolic,?? ? l, as also obtained in many other early studies, e.g. in Campbell & Roberts (1989). (The re- lation between the wave number, k, and the harmonic degree, l, of an oscillation mode is k? ? l(l+1)/R circledot .) The only restriction to the possible choices of l in the present planar geometry comes from the requirement that the horizontal wavelength of the per- turbations has to be small compared to the solar radius, which is R circledot = 696 Mm in the model. It can be easily shown that this condition is fulfilled for l > 6. Since at present we do not wish to study the l dependence of the results, the harmonic degree is fixed at 100 throughout the paper. Structure of spectrum. First, we study the properties of charac- teristic frequencies, that define and border the di?erent regions of the frequency spectrum. The propagation angle is varied be- tween 0 and 90 ? in Figs. 4a,b for l = 100, B L = 10and50G, respectively. B L ? B(z = L) is the magnetic field strength at the top of the transitional layer, characterizing the overall atmo- spheric magnetic field strength. The values 10 and 50 G for B L are chosen to represent weak atmospheric magnetic fields. The propagation angle,?, can obviously take values also be- tween 0 and?90 ? , but all the mathematical expressions related to the frequency spectra are even functions of ?, and so solu- tions for characteristic frequencies of eigenfrequencies for any B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 383 Fig.4. Frequency spectrum with varying ? for l = 100 and for weak atmospheric magnetic fields: a) B L =10 G and b) B L =50 G. value of?90 ? ???0 are equal to the solution for??,whichis between 0 and 90 ? . The characteristic Alfv?n frequency,? A , and the third cut- o? frequency, ? III , are distinct for ? nequal 0, and ? A , ? c and ? III decrease to zero as ? ? 90 ? . The di?erences between ? III , ? c and ? A in Fig. 4a are so small that the frequencies cannot be easily distinguished. A small region of the spectrum, around?= 50 ? , is enlarged and inserted in Fig. 4a to show the thin (<1?Hz) layer between? A and? c (which is part of the Alfv?n continuum), and between ? c and ? III (which is for leaky modes). The lower cut-o?frequencies,? I and? III , decrease while the upper cut-o? frequency,? II , increases with increasing?. Next, let us embark on how the characteristic frequencies di- vide the frequency spectrum into di?erent regions for weak mag- netic field. Eigenmodes have real frequencies between? I and? II . The thin region max(? III ,? c )? II , ? A 1) to the strong field region (?<1). For a weak field the lower curve belongs to? III and the upper one is ? I , while for a strong field the upper curve repre- sents? III and the lower one is for? I . The boundary between the two regions,?= 1, corresponds to B L ? 83 G. This change of character is an example of coupling or avoided crossing between characteristic frequencies. 4. Global oscillations The inhomogeneous (e.g., three-layer) structure of the solar atmosphere introduces characteristic frequencies. It has been shown in Sect. 3 how strongly an atmospheric magnetic field af- fects them. Here we turn our attention to the global oscillations themselves, and investigate how their frequencies are influenced by the magnetic solar atmosphere. 4.1. Frequencies of eigenmodes in weak atmospheric magnetic field For a weak atmospheric magnetic field, the slow and Alfv?n con- tinua are confined at the low-frequency region of the spectrum (Fig. 4). Consequently, the eigenmodes, of which the frequen- cies are higher, are not coupled resonantly to either Alfv?n or slow continuum modes. Figure 4a is the frequency spectrum for a weak, B L = 10 G, field. The eigenmode with the lowest fre- quency in the spectrum is the fundamental mode (or f mode). Above that the frequencies of the first two pressure modes (or p modes) can be seen. Besides the f and p modes, the a mode (also known as Lamb mode) also appears as an eigenmode, with highest frequencies. The a mode is an acoustic mode, first time found by Lamb (1932). In a non-magnetic medium, the Lamb mode frequency is near the sound frequency. The Lamb mode usually appears as an eigenmode in isothermal models. The ex- istence of this mode in the present model is most likely due to the isothermal corona. The Lamb mode can also arise in a model in which the magnetic field of then isothermal corona is uniform, i.e.v A,co =cst.is replaced by B co =cst.(Pint?r 1999). For more details on this acoustic mode see also Paper I or, e.g., Vanlommel & ? Cade? (1998). The frequency of the a mode is right below the upper cut-o?frequency for all values of?. According to this, the a mode oscillation can propagate parallel as well as obliquely to the magnetic field lines. All the four modes (f, p 1 , p 2 and a) in Fig. 4a are hardly influenced by the weak magnetic field, the shifts of the mode frequencies are of the order of?Hz. More pronounced magnetic e?ects can be seen in Fig. 4b, where B L =50 G. First of all, there are two examples of charac- ter change between eigenoscillation modes, similar to that seen between the two lower cut-o?frequencies, discussed in the pre- vious subsection and visualised in Fig. 6. One of them happens between the a and p 3 modes. For increasing?,thea-mode fre- quency moves away from? II at?? 20 ? ,andthea mode trans- forms into the p 3 mode. The p 3 mode comes to existence with increasing propagation angle at?? 28 ? , due to the fact that the upper cut-o?frequency is increased by the increase of?. The fre- quency of p 3 starts to increase sharply at?? 30 ? , showing that the p 3 mode transforms into the a mode, of which the frequency stays near below? II for further increasing?. The p modes of radial order one and two exist between ? I and ? II in the whole range of? without any gap. The variation of their frequencies with?is not large enough to see in Fig. 4b, as it is still of the order of few tens of nHz, while the scale here is?mHz. The fundamental mode (f mode) is also present in the spec- trum. The frequency of the f mode falls into the Alfv?n and slow continua for small?, i.e., the mode is resonantly coupled to an Alfv?n mode and to a slow mode. For a non-zero ?, the third cut-o? frequency is no longer equal to the Alfv?n frequency, as? III ?? A for?=0, and eigen- modes may occur also with frequencies between? A and? III .We found a new mode of which the real frequency is within this re- gion, and exists for?>0 only, i.e. it does not propagate along the magnetic field lines, only obliquely. We call this new mode b mode. Being in the Alfv?n continuum, the b mode is coupled resonantly to a local Alfv?n oscillation. The b-mode frequency is always just above the third cut-o?frequency such as the a-mode frequency is always right below the upper cut-o?frequency. The apparent similarities between the a and b modes allow us to con- sider the b mode another Lamb mode, that exists together with the a mode, but is coupled to Alfv?n modes, and propagates only obliquely to the magnetic field lines. We refer to this mode as the magnetic Lamb mode. The f-mode frequency occurs also between ? A and ? III for propagation angles 26 ? 40 ? , above the slow and Alfv?n continua. 4.2. Frequencies of eigenmodes in strong atmospheric magnetic field For strong atmospheric magnetic fields the increase of the upper cut-o?frequency allows the presence of more p mode frequen- cies within the Alfv?n and slow continua. Ten of them is plot- ted in Fig. 5a. The frequency spectrum is shown as a function of?, as in Figs. 4a,b, but here with B L fixed at 100 G. The third cut-o?frequency is higher then? A for?>0and? A decreases with increasing?slightly faster then? III . The gap between them, which is for leaky mode frequencies, increases with increasing?. Hence, the gap in the p-mode frequencies due to the leaky region between? A and? III is more and more recognisable for p modes of lower order. An important consequence of this phenomenon B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 385 could be the direct leakage of solar oscillations into the magnetic solar atmosphere (see e.g. De Pontieu et al. 2005; De Pontieu & Erd?lyi 2006). The magnetic Lamb mode can be found also for strong mag- netic fields. Its frequency is near the lowest cut-o? frequency, which is? I ,as? III exceeds? I for strong magnetic fields. As the b mode frequency decreases together with? I , the mode changes character with the p 2 , p 1 and f modes with increasing ?.The avoided crossings of the b and f modes are enlarged in the inserted panel in Fig. 5a. The frequencies of both the b and f modes can be found above the slow and Alfv?n continua for large propagation angles. The large region of slow continuum contains the frequencies of the p 1 , f and b modes until a certain propagation angle. Panel b in Fig. 5 is the frequency spectrum with the fre- quencies of the f, b and p modes for B L = 120 G, a mag- netic field strength possibly more characteristic for active re- gions. The p-mode frequencies are displayed up to the p 8 mode. The slow continuum is not increased much compared to that for B L = 100 G, contrary to the Alfv?n continuum, which al- most fills now the entire spectrum for the given frequency range (0 ??? 5 mHz). The gaps in the eigenmodes due to the leaky regions between ? I and ? c and between ? A and ? III are hardly noticeable. 4.3. Frequency shift and line-width variation due to varying propagation angle Although the frequency spectra for di?erent atmospheric mag- netic field strengths show remarkable di?erences, only those due to the varying characteristic frequencies are recogniseable, as seen in Figs. 4 and 5. Frequency shifts of eigenmodes caused by the varying magnetic field strength and propagation angle are smaller than the order of mHz. However, they are of high impor- tance from observational point of view. The frequency shift related to the frequencies for parallel propagation,??(?; B L ) ??(?; B L )??(?= 0; B L ), and the line width, ?(?; B L ), of the f, p 1 , p 2 , p 3 and p 4 modes are given for propagation angle 0 ??? 90 ? in Figs. 7a,b, respectively, for B L = 100 G. The f, p 1 and p 2 modes have a gap in the Alfv?n continuum. Notice that here the frequency shift caused only by vary- ing ? is studied, because frequency shifts due to varying mag- netic fields have already been investigated in Paper I. The frequency shifts of the f, p 1 and p 2 global modes due to non-zero propagation angle are basically negative and their mod- ulus is less than 0.2?Hz. The frequencies of the three modes are in the slow continuum and also in the Alfv?n continuum (which is the frequency region coloured in green in Fig. 5a) for propa- gation angles smaller than 59.4 ? , 38.8 ? and 13.0 ? , respectively. The modes, hence, are coupled resonantly to a local slow mode and to a local Alfv?n mode in the atmosphere. The damping rate of the global modes caused by resonant absorption can be mea- sured by the non-zero line width of the modes (which is shown in the lower panel, in Fig. 7b). The three modes modes do not propagate in certain directions, as their frequency is in a region for leaky modes for an interval of?.Thep 2 mode terminates at ??13 ? ,thep 1 mode at??39 ? and the f mode at??59 ? .The lack of the three modes appears also in Fig. 5a as gaps in the ??(?) graphs. The frequencies of the f, p 1 and p 2 modes reap- pear for larger values of?. With a frequency right above? I ,the magnetic Lamb mode changes character first with p 2 (at??27 ? ) then with p 1 (at??43 ? ) and finally with the f mode (at??60 ? ) (see also Fig. 5a). The influence of the magnetic Lamb mode on Fig.7. a) Frequency shift and b) line width of the f and first four p modes as functions of propagation angle, ?,forl = 100 and B L =100 G. ? The actual value for the line width of the f mode can be obtained by multiplying the values shown in panel b) by a factor of ten. The frequency shifts in panel a) and line widths in panel b) are plotted as dashed lines for which?the f, p 1 or p 2 mode shows the characteris- tics of the Lamb mode. the f, p 1 and p 2 mode frequencies around the given propagation angles appears as a sharp increase in the frequency shift with decreasing?. Hence, positive frequency shifts occur only due to the magnetic Lamb mode. The frequency shifts e?ected by the interaction with the magnetic Lamb mode is plotted with dashed lines in Fig. 7a. The frequencies of p 3 and p 4 , however, are shifted further from the values obtained for parallel propagation, also to the negative direction (i.e.??(?)<0). The negative frequency shifts take their maximal values at?= 90 ? , which shift is of the order of?Hz. The f, p 1 and p 2 modes are coupled resonantly to a local slow and a local Alfv?n mode for the intervals of ? given in the latter paragraph. The largest line width can be found for the f mode, for 0