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A More Realistic Representation of Overshoot at the Base of the Solar Convective Envelope As Seen by Helioseismology

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A More Realistic Representation of Overshoot at the Base of the Solar Convective Envelope As Seen by Helioseismology Mon. Not. R. Astron. Soc. 414, 1158–1174 (2011) doi:10.1111/j.1365-2966.2011.18460.x A more realistic representation of overshoot at the base of the solar convective envelope as seen by helioseismology J. Christensen-Dalsgaard,1,2 M. J. P. F. G. Monteiro,3,4 M. Rempel2 and M. J. Thompson2,5 1Danish AsteroSeismology Centre, and Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark 2High Altitude Observatory, NCAR†, Boulder, CO 80307-3000, USA 3Centro de Astrof´ısica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 4Departamento de F´ısica e Astronomia, Faculdade de Cienciasˆ da Universidade do Porto, Portugal 5School of Mathematics & Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH Accepted 2011 February 1. Received 2011 January 31; in original form 2010 October 28 ABSTRACT The stratification near the base of the Sun’s convective envelope is governed by processes of convective overshooting and element diffusion, and the region is widely believed to play a key role in the solar dynamo. The stratification in that region gives rise to a characteristic signal in the frequencies of solar p modes, which has been used to determine the depth of the solar convection zone and to investigate the extent of convective overshoot. Previous helioseismic investigations have shown that the Sun’s spherically symmetric stratification in this region is smoother than that in a standard solar model without overshooting, and have ruled out simple models incorporating overshooting, which extend the region of adiabatic stratification and have a more-or-less abrupt transition to subadiabatic stratification at the edge of the overshoot region. In this paper we consider physically motivated models which have a smooth transition in stratification bridging the region from the lower convection zone to the radiative interior beneath. We find that such a model is in better agreement with the helioseismic data than a standard solar model. Key words: asteroseismology – convection – Sun: helioseismology – Sun: interior – stars: interiors. age estimates of stars, and so improved constraints on theories of 1 INTRODUCTION overshooting obtained from a study of how overshooting works An understanding of the overshoot region at the bottom of the Sun’s in the solar case can be important for understanding stars more convective envelope is important for a number of reasons. The widely. overshoot region approximately coincides with the solar tachocline, Helioseismology provides a means of probing directly the con- a region of rotational shear which is generally believed to play a key ditions inside the Sun, because the frequencies of resonant modes role in the solar dynamo: overshooting is likely to be important for set up by acoustic waves propagating in the solar interior depend in helping to store the magnetic flux below the convection zone during particular on the adiabatic sound speed c which is given by the solar cycle. Bulk motion in the overshoot region also affects the p k T 2 = 1 1 B thermal stratification and it may contribute to significant mixing of c . (1) ρ muμ chemical elements, for example transporting fragile elements such Here, is the logarithmic derivative of pressure p with respect to as lithium to hotter regions where they are destroyed more easily 1 density ρ at constant specific entropy, T is temperature, μ is the than in the convection zone. More generally, convective overshoot mean molecular weight, k is Boltzmann’s constant and m is the in stars (particularly those with convective cores) is likely to be B u atomic mass unit. Hence, the sound-speed gradient with respect to an important and as yet imperfectly understood process affecting depth depends on the temperature gradient, which itself depends on the mechanism by which heat is transported. The transition between E-mail: [email protected] (JCD); [email protected] (MJPFGM); fully radiative heat transport beneath the convection zone and con- [email protected] (MR); [email protected] (MJT) vective heat transport within the convection zone is manifested in †The National Center for Atmospheric Research is operated by the Uni- the temperature gradient and hence too in the sound-speed gradient. versity Corporation for Atmospheric Research under sponsorship of the If the transition in sound-speed gradient takes place over a distance National Science Foundation. that is small compared with the vertical wavelength of the acoustic C 2011 The Authors Monthly Notices of the Royal Astronomical Society C 2011 RAS Overshoot in the Sun 1159 waves near the base of the convection zone, then the transition ap- When the observed and model frequencies are compared, it is pears to the waves to be more-or-less sharp and this gives rise to an found that the amplitude of δωp in the Sun is comparable with or oscillatory signal in the mode frequencies ω: the form of the signal smaller than that in models without overshooting, implying that gives information about the location and nature of this ‘acoustic the amount of overshooting of the kind predicted by these mixing- glitch’ (Gough 2002a). length models is very small. Monteiro et al. (1994) and Christensen- Monteiro, Christensen-Dalsgaard & Thompson (1994) repre- Dalsgaard et al. (1995) concluded that the extent of any such over- sented the effect of the base of the convection zone in terms of shoot at the base of the convection zone was less than one tenth an additive contribution δωp to the frequencies, relative to those of of a pressure scaleheight. A similar limit was found by Basu et al. a corresponding model in which the transition had been smoothed (1994), while Roxburgh & Vorontsov (1994) obtained a somewhat out. For low-degree modes (see Monteiro, Christensen-Dalsgaard weaker limit. Basu & Antia (1994) noted that the composition gra- & Thompson 2000) a break in the first derivative of the sound speed dient caused by the inclusion of helium settling produced a sharper gives rise to a contribution of the form transition in the sound-speed gradient, even in models without over- shoot, and hence a larger oscillatory signal. From these analyses, δω = A(ω)cos(2ωτ¯ + 2φ ) , (2) p d 0 it would appear that the transition in sound-speed gradient at the while a break in the second derivative gives rise to a similar signal base of the solar convection zone is if anything smoother than in the but with a sine term instead of a cosine, and with a different fre- non-overshoot models. A caveat is that what we purport to measure quency dependence of A(ω). Here,τ ¯d is essentially the value of τ at in the above studies is the spherically symmetric component of the the location of the acoustic glitch, where structure: departures from sphericity, such as a latitudinal depen- dence to the shape of the base of the convection zone, could make R dr τ = (3) the transition appear smoother than it is locally. The helioseismic r c evidence, however, is that the location of the base of the convection is the acoustic depth beneath the surface, r being the corresponding zone is independent of latitude (Monteiro & Thompson 1998; Basu distance to the centre and R the surface radius of the Sun. Also, φ0 & Antia 2001). Changes of the base of the convection zone on time- is a phase introduced by the reflection of the mode at the turning scales shorter than the observation interval could also have a similar points, depending in particular on the near-surface structure. The effect by introducing a time-averaging effect on the mode frequen- function A(ω) is an amplitude which depends on the sharpness and cies that would mimic a smoother transition. The importance of this nature of the convection-zone base: the smoother the transition, the effect is difficult to estimate, however, and depends strongly on the smaller in general will be the amplitude. However, if moderate- 3D nature of convection at the base of the envelope. degree data are used, as in the case of Sun where we have accurate Overshoot has been addressed in the last two decades by a va- data for modes whose degree is above 3, the above expression needs riety of 2D and 3D numerical simulations (Roxburgh & Simmons to include additional terms, both in the amplitude (Monteiro et al. 1993; Hurlburt et al. 1994; Singh, Roxburgh & Chan 1995, 1998; 1994) and in the argument of the signal (Christensen-Dalsgaard, Saikia et al. 2000; Brummell, Clune & Toomre 2002; Rogers & Monteiro & Thompson 1995), to account for the first-order effect Glatzmaier 2005a,b). Whilst the non-local mixing-length models of the mode degree on the signal. have clearly predicted an adiabatic overshoot region of a sizeable Overshoot at the base of the solar convection has traditionally fraction of a pressure scaleheight and a rather sharp transition to the been modelled using non-local mixing-length theory (e.g., Zahn radiative zone beneath, the numerical simulations show a greater 1991). Such models mostly predict an overshoot region that is nearly variety of possible behaviours. The work by Brummell et al. (2002) adiabatically stratified; in terms of the logarithmic temperature gra- is currently one of the best resolved and most turbulent investiga- −6 dient ∇=dlnT/dlnp one finds that δ ≡∇−∇ad ∼−10 ,where tions: it shows strongly subadiabatic overshoot with very smooth ∇ad = (∂ ln T/∂ ln p)s is the adiabatic temperature gradient, s be- transition towards the radiative temperature gradient. Most of the ing specific entropy. The depth of the overshoot region is typically earlier 2D and 3D simulation were more in the laminar regime and between 0.2Hp and 0.4Hp,whereHp is the pressure scaleheight at found, depending on their parameters (mainly the stiffness of the the base of the convection zone, with a very steep transition towards subadiabatic layer), both nearly adiabatic overshoot and extended the radiative temperature gradient.
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