The influence of a semi-infinite atmosphere on solar oscillations

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1 Jurnal f Physics: Cnference Series OPEN ACCESS The influence f a semi-infinite atmsphere n slar scillatins T cite this article: Ángel De Andrea Gnzález 014 J. Phys.: Cnf. Ser View the article nline fr updates and enhancements. This cntent was dwnladed frm IP address n 09/10/018 at 19:5

2 4th Internatinal Wrkshp & Summer Schl n Plasma Physics 010 Jurnal f Physics: Cnference Series 516 (014) di: / /516/1/01015 The influence f a semi-infinite atmsphere n slar scillatins Ángel De Andrea Gnzález Departament de Física. Escuela Plitécnica Superir. Uniersidad Carls III de Madrid. A. Uniersidad, Leganés. Spain. aandrea@fis.uc3m.es Abstract. The influence f a semi-infinitie atmsphere n slar inestigatins is inestigated using a mdel in which the crna is represented by a graitatinally stratified fluid. The slar crna can be mdeled as a semi-infinitie regin f plasma that ccupies the space abe the xy- plane in Cartesian crdinates with the z-axis taken alng the graitatinal acceleratin r r g = gu z. This assumptin is reasnable, as the plasma density f the atmsphere is much lwer than the density f the phtsphere. S, we cnsider the phstsphere as a slid and immbile bundary fr the atmsphere. The standard mathematical prcedures in eliseismlgy field are based n nrmal mde apprach fr arius slar mdels. The regin is assumed quasi-isthermal and withut magnetic fields. In this wrk, in rder t shw hw the mdes appear in the respnse t an initial perturbatin, we cnsider the initial alue prblem (IPV). The p-mdes and g-mdes pssess nly cntinuus spectra as ppsed t discrete spectrum like that preius inestigatrs fund fr this prblem- and the slutin t the initial alue prblem is btained thrugh an apprpiate Green s functin. 1. Intrductin Seeral authrs hae purpsed slutins f the initial alue prblem thrugh the use f a Laplace Transfrm in time [5-6]. The principal finding f the initial alue prblem apprach is that, in additin t the discrete eigenalues linked t the nrmal mdes, there exists a cntinuus spectrum f eigenalues. Thus, the mdal apprach cannt pride a cmplete slutin. We use the Cartesian crdinates with the z-axis pinting twards the interir f the Sun and parallel t the cnstant graity g r. As t the applicability f the Cartesian gemetry in ur analysis, it is alid fr perturbatins f a sufficiently small waelength when the effects f stellar sphericity are unimprtant. This then places the centre f the Sun int z /. Our mdel assumes an isthermal atmsphere at z< 0 abe the slar surface. The medium is cnsidered a perfect gas. In this paper we exclude macrscpic mtins and magnetic fields [1-3]. Cntent frm this wrk may be used under the terms f the Creatie Cmmns Attributin 3.0 licence. Any further distributin f this wrk must maintain attributin t the authr(s) and the title f the wrk, jurnal citatin and DOI. Published under licence by Ltd 1

3 4th Internatinal Wrkshp & Summer Schl n Plasma Physics 010 Jurnal f Physics: Cnference Series 516 (014) di: / /516/1/ Linear stability analysis and the initial prblem alue We start frm the standard set f hydrdynamic equatins describing the dynamics prcesses in a fully inized hydrgen plasma in presence f graity with cnstant acceleratin [4]. ρ r r + ( ρ ) = 0 r r r r r r ρ + ρ ( ρ ) = p ρ g ρ r r p ρ r r + p = γ + ρ ρ Cnsidering the displacement ectr r ξ ξ, where 1 =, linearizatin leads t: t ξ r r r r rr ρ ( γ ξ ) ρ ( ξ ) ρ ξ p + g g = 0 (1) () (3) (4).1. Results and discussin We define the Laplace transfrm ξ ( ) z z, ω in the cmplex -ω plane: 1 + i + σ iωt ξz ( z, ω) = ξz ( z, ω) e dω π (5) i + σ Then, we cnsider the initial alue prblem and intrduce the Laplace transfrm. S, we btain the fllwing ODE: being: N ( γ 1) s 4 d ξz d ξz ω k sω + k s N + λ + ξ z = f ( z, ω, k, F1, F ) (6) dz dz ω g = the Brunt-Väisäläa frequency; s 1 λ = ; F 1 and F are initial cnditins: ξ z F1 = ξ z ( z, t ) + ; F t= 0 = z ( z, t) + t= 0 (7) We can cnsider the Green s functin fr the frmer ODE, chsing the bundary cnditins G(, ω, z ) = 0 and G( 0, ω, z ) = 0 :

4 4th Internatinal Wrkshp & Summer Schl n Plasma Physics 010 Jurnal f Physics: Cnference Series 516 (014) di: / /516/1/01015 (, ω, z ) G z z( 1 + α ) Ae z < -z = Be + Ce -z < z < 0 z( 1 + α ) z( 1 α ) (8) On the ther hand: ( γ ) + ω ( ω sc ) 4 g k 1 k α = 1 (9) ω sc In accrd with the bundary cnditin ( ω ) Re(α ) > 0 fr Re (ω ) > σ. S, the slutin is: G,, z = 0, the square rt are defined s that ξ z ω = G z ω z f z ω k F F dz (10) (, ) (,, ) (,,,, ) z 1 0 On the ther hand, the cntinuity f G( z z ) at z = z, yields: (, ω, z ) G z, ω, at z = z, and the jump cnditin n z+ z z α + zα zα 1 e e - 1 z < -z α = ( z+ z )(1 + α ) zα 1 e e - 1 -z < z < 0 α ( ω z ) dg z,, dz (11) Expanding Green s functin abut α =0, this functin des nt turn ut t be an een functin f α. S, the Green s functin has branch pints at ω = γ b (assciated with cntinuus spectra fr p-mdes and g-mdes). S, the infinite thickness causes the degenerate g-mdes and p-mdes t be distended int a cntinuus spectrum. The fllwing figure 1 shws branch pints and dmains fr p-mdes cntinuus spectra and g- 6 mdes cntinuus spectra. The temperature fr slar crna is 10 K. Graitatinal acceleratin is assumed cnstant within the dmains f studied harmnic perturbatins and we take g= 74 m / s as it is n the slar surface. 3

5 4th Internatinal Wrkshp & Summer Schl n Plasma Physics 010 Jurnal f Physics: Cnference Series 516 (014) di: / /516/1/01015 Figure 1.Slid lines shw branch pints (frecuency cut-ff).the shaded areas shw dmains fr p-mdes cntinuus spectra (tp) and g-mdes cntinuus spectra (bttm). Discrete mdes d nt exist. 3. Cnclusin Oscillatins in slar crna is analyzed using initial alue prblem (IPV), s that a set f cntinuum p- mdes and g-mdes due t branch cuts in the cmplex plane, nt treated explicitly in the literature, appears. On the ther hand, discrete mdes d nt exist. The main cnsequence f a semi-infinitie atmsphere is a set f cntinuum p-mdes and g-mdes. 4. References [1] Vanlmmel E and Cadez V M 1998 Slar Physics [] Schmitz F and Steffens S 1999 Astrn. Astrphys [3] Cadez V M and Janic G 008 Publ. Astrn. Obs. Belgrade [4] Gedbled and Peds S 004 Principles f Magnethydrdynamics (Cambridge: Uniersity Press) [5] Ott E and Russell D A 1978 Physical Reiew Letters [6] Russell D A and Ott E 1979 Jurnal f Gephysical Research