Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-08-04T13:06:58.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusion process produced by random internal waves

Published online by Cambridge University Press:  26 April 2006

Evry Schatzman
Evry Schatzman
Affiliation:
Observatoire de Paris, Section de Meudon, DASGAL, CNRS URA 335, GDR 131, 92195 Meudon, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of the paper is to present a new transport process which is likely to have great importance for understanding the internal constitution of the stars.

In order to set the problem in context, we first give a short presentation of the physical properties of the Sun and stars, described usually under the names Standard Solar Model or Standard Stellar Models (SSM). Next we show that an important shortcoming of SSM is that they do not explain the age dependence of the lithium deficiency of stars of known age: stars of galactic clusters and the Sun. It was suggested a long time ago that the presence of a macroscopic diffusion process in the radiative zone should be assumed, below the surface convective zone of solar-like stars. It is then possible for the lithium present in the convective zone to be carried to the thermonuclear burning level below the convective zone. The first assumption was that differential rotation generates turbulence and therefore that a turbulent diffusion process takes place. However, this model predicts a lithium abundance which is strongly rotation dependent, contrary to the observations. Furthermore, as the diffusion coefficient is large all over the radiative zone, it prevents the possibility of gravitational separation by diffusion and consequently leads to the impossibility of explaining the difference in helium abundance between the surface and the centre of the Sun. The consequence is obviously that we need to take into account another physical process.

Stars having a mass M < 1.3M[odot ] have a convective zone which begins close to the stellar surface and extends down to a depth which is an appreciable fraction of the stellar radius. In the convective zone, strong stochastic motions carry, at least partially, heat transfer. These motions do not vanish at the lower boundary and generate internal waves into the radiative zone. These random internal waves are at the origin of a diffusion process which can be considered as responsible for the diffusive transport of lithium down to the lithium burning level. This is certainly not the only physical process responsible for lithium deficiency in main sequence stars, but its properties open the way to a completely consistent analysis of lithium deficiency.

The model of generation of gravity waves is based on a model of heat transport in the convective zone by diving plumes. The horizontal component of the turbulent motion at the boundary of the convective zone is assumed to generate the horizontal motion of internal waves. The result is a large horizontal component of the diffusion coefficient, which produces in a short time an horizontally uniform chemical composition. It is known that gravity waves, in the absence of any dissipative process, cannot generate vertical mixing. Therefore, the vertical component of the diffusion coefficient is entirely dependent on radiative damping. It decreases quickly in the radiative zone, but is large enough to be responsible for lithium burning.

Owing to the radial dependence of velocity amplitude, the diffusion coefficient increases when approaching the stellar centre. However, very close to the centre, nonlinear dissipative and radiative damping of internal waves become large and the diffusion coefficient vanishes at the very centre.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 322 , 10 September 1996 , pp. 355 - 382
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aller, L. H. & Chapman, S. 1960 Diffusion in the Sun. Astrophys. J. 132, 466–472.Google Scholar
Asaeda, T. & Imberger, J. 1993 Structure of bubble plumes in linearly stratified environments. J. Fluid Mech. 249, 35–47.Google Scholar
Baglin, A. & Lebreton, Y. 1990 Surface abundances of light elements as diagnostics of transport processes in the Sun and in solar-type stars. In Inside the Sun (ed. G. Berthomieu & M. Cribier), pp. 437–448.
Baglin, A., Morel, M. & Schatzman, E. 1985 Stellar evolution with turbulent diffusion mixing. V. Lithium abundance in the lower main sequence. Astron. Astrophys. 149, 309–314.Google Scholar
Bahcall, J. N. & Kumar, P. 1993 G-modes and the solar neutrinos problem. Astrophys. J. Lett. 409, 473–476.Google Scholar
Balachandran, S. 1993 Observable effect of rotation on stellar abundance. IAU Colloquium 137: Inside the Stars. Astron. Soc. Pac. Conf. Ser., Vol. 40 (ed. W. W. Weiss & A. Baglin), pp. 333–346.
Balachandran, S. 1994 Lithiium on the main sequence. 8th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun (ed. J. P. Caillaut). Astron. Soc. Pac. Conf. Ser., Vol. 64, pp. 234–243.
Batchelor, G. K. 1952 Diffusion in a field of homogeneous turbulence. II. The relative motion of particles.. Proc. R. Soc. Lond. A 213, 345–362.Google Scholar
Berthomieu, G., Morel, P., Provost, J. & Zahn, J. P. 1993 Seismological constraints on convective penetration in the Sun. Inside the Stars, IAU Colloquium 137 (ed. W. W. Weiss & A. Baglin). ASP Conf. Ser., Vol. 40, pp. 60–62.
Bodenheimer, P. 1965 Studies in stellar evolution. II. Lithium depletion during the pre-main- sequence contraction. Astrophys. J. 142, 451–461.Google Scholar
Bouvier, J. 1994 The rotational evolution of low-mass stars. In 8th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun (ed. J. P. Caillaut). Astron. Soc. Pac. Conf. Ser., Vol. 64, pp. 151–162.
Bretherton, F. P. 1969 On the mean motion induced by.internal gravity waves. J. Fluid Mech. 36, 785–803.Google Scholar
Carruthers, D. L. & Hunt, J. C. R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer. J. Fluid Mech. 165, 475–501.Google Scholar
Cattaneo, F. & Malagoli, A. 1992 Seventh Workshop on Cool Stars, Stellar Systems and the Sun (ed. M. S. Gianpapa & J. A. Bookbinde). Astron. Soc. Pac. Conf. Ser., Vol. 26, p. 139.
Chandrasekhar, S. 1943 Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89.Google Scholar
Chajrbonnel, C. & Vauclair, S. 1993 New constraints on the rotation-induced mixing in stars, from lithium observations in main sequence F-type stars and subgiants. In IAU Colloquium 137: Inside the Stars (ed. W. W. Weiss & A. Baglin). Astron. Soc. Pac. Conf. Ser., Vol. 64, pp. 284–286.
Christansen-Dalsgaard, J., Proffitt, C. R. & Thomson, M. J. 1993 Effects of diffusion on solar models and their oscillations frequencies. Astrophys. J. 403, L75–L78.Google Scholar
Cowling, T. G. 1941 The non-radial oscillations of tropic stars. Mon. Not. R. Astron. Soc. 101, 367–375.Google Scholar
D'Antona, F. & Mazzitelli, I. 1984 Lithium depletion in stars. Pre-main sequence burning and extra mixing. Astron. Astrophys. 138, 431–442.Google Scholar
D'Antona, F. & Mazzitelli, I. 1994 New pre-main sequence tracks for M 2.5 M as test of opacities and convection models. Astrophys. J. Suppl. 90, 467–500.Google Scholar
Frisch, U. 1987 Lectures on turbulence and lattice gas dynamics. In Summer School on Turbulence, NCAR, Boulder, Colorado, June 1987.
Garcia-Lopez, R. J. & Spruit, H. C. 1991. Li depletion in F stars by internal gravity waves Astrophys. J. 337, 268Google Scholar
Goldreich, P. & Nicholson, P. 1989O Tides In Rotating Fluids. Astrophys. J. 342, 1075–1078.Google Scholar
Goldreich, P. & Nicholson, P. 19896 Tidal friction in early-type stars. Astrophys. J. 342, 1079–1084.Google Scholar
Gough, D. O. 1991 Seismic constraints on the solar neutrino proble. Ann. N. Y. Acad. Sci. 647, 199–217.Google Scholar
Grimshaw, R. 1984 Wave action and wave mean flow interaction, with applications to stratified shear-flow. Ann. Rev. Fluid Mech. 16, 11–44.Google Scholar
Herbig, G. H. 1965 Lithium abundances in F5-G8 stars. Astrophys. J. 141, 588.Google Scholar
Hurlburt, N. E., Toomre, J. & Massaguer, J. M. 1986 Non-linear compressible convection penetrating into stable layers and producing internal gravity waves. Astrophys. J. 311, 563–577.Google Scholar
Knobloch, E. 1977 The diffusion of scalar and vector field by homogeneous stationary turbulence. J. Fluid Mech. 83, 129–140.Google Scholar
Knobloch, E. 1991 Towards a Theory of Wave Transport. Lecture Notes in Helioseismology, Vol. 388, pp. 241–252.
Kurucz, R. L. 1991 New opacity calculations in stellar atmosphere beyond classical models. Proc. NATO Advanced Research Workshop on Stellar Atmosphseres: Beyond Classical Models, Trieste (ed. L. Orivellari, I. Herberry & D. G. Hummer), pp. 441–448. Kluwer.
Larson, M. & JÖnssen, L. 1994 Mixing in a two-layer stably stratified fluid by a turbulent jet. J. Hydraul. Res. 32, 271.Google Scholar
Larson, M. & JÖnssen, L. 1995 A model of jet-plume interaction in a stably stratified ambient. Communication at Euromech. meeting, Sept. 1995.
Ledoux, P. & Walraven, Th. 1958 Variable stars. In Handbuch der Physik, LI (ed. S. Flügge), pp. 353–601. Springer.
Lesieur, M. 1993 Turbulence in Fluids, pp. 1–412. Kluwer.
List, E. J. 1982 Turbulent jets and plumes. Ann. Rev. Fluid Mech. 14, 189–212.Google Scholar
Lo, Y. C. 1996 Plumes. Astron. Astrophys. (to be submitted).
Martin, E. L. & Rebolo, R. 1993 EK Cephei B: a test object for pre-ZAMS models of solar type stars. Astron. Astrophys. 274, 274–278.Google Scholar
McIntyre, M. E. 1973 Mean motion and impulse of a guided internal gravity wave packet. J. Fluid Mech. 60, 801–811.Google Scholar
Michaud, G. 1970 Diffusion process in peculiar A stars. Astrophys. J. 160, 641–658.Google Scholar
Michaud, G. & Charbonneau, P. 1991 The Lithium abundance in stars. Space Sci. Rev. 57, 1–5.Google Scholar
Michaud, G. & Profitt, C. R. 1993 Particle transport process. Inside the Stars, IAU Colloquium 137 (ed. W. W. Weiss & A. Baglin). Astron. Soc. Pac. Conf. Ser. Vol. 40, pp. 246–259.
Montalbán, J. 1994 Mixing by internal waves. I. Lithium depletion in the Sun. Astron. Astrophys. 281, 421–432.Google Scholar
Montalbán, J. 1994 The generation of internal waves, IAU Colloquium 137, Inside the Stars (ed. W. W. Weiss & A. Baglin). Astron. Soc. Pac. Conf. Ser. Vol. 40, pp. 278–280.
Montalbán, J. & Schatzman, E. 1994. Mixing By Internal Waves. Inside The Stars, Iau Colloquium 137 (ed. W. W. Weiss & A. Baglin). Astron. Soc. Pac. Conf. Ser. Vol. 40, pp. 281–283.
Montalbán, J. & Schatzman, E. 1996 Mixing by internal waves. II. Li and Be depletion rate in low mass main-sequence. Astron. Astrophys. 305, 513–518.Google Scholar
Morton, G. L. Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitationnal convection from maintained and instantaneous sources.. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Pasquini, L., Edvardsson, R., Liu, Q. & Pallavicini, R. 1994a Lithium Abundances In Near-By Solar Like Stars. 8th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun (ed. J. P. Caillaut). Astron. Soc. Pac. Conf Ser. Vol. 64, pp. 294–296.
Pasquini, L., Liu, Q. & Pallavicini, R. 1994b Lithium abundances in nearby solar-like stars. Astron. Astrophys. 287, 191–205.
Perez Hernández, F. & Christansen-Dalsgaard, J. 1994 The phase function for stellar acuostic oscillations. III. The solar case. Mon. Not. R. Astron. Soc. 269, 475–492.Google Scholar
Press, W. H. 1981 Radiative and other effects from internal waves in solar and stellar interior. Astrophys. J. 245, 286–303.Google Scholar
Press, W. H. & Ribicky, G. B. 1981 Enhancement of passive diffusion and suppression of heat flux in a fluid with time varying shear. Astrophys. J. 248, 751–766.Google Scholar
Randich, S. & Pallavicini, R. 1991 Lithium abundance and chromospheric activity. Mem. Soc. Astron. Ital. 62, 75–90.Google Scholar
Rayleigh, Lord 1880 On The Resultant Of A Large Number Of Vibrations Of The Same Pitch And Of Arbitrary Phase. Phil. Mag. 10, 73–78 (In Scientific Papers, Vol. I, P. 491. Cambridge University Press, 1899.)
Rayleigh, Lord 1899 On James Bernouilli'S Theorem In Probabilities. Phil. Mag. 47, 246–251 (In Scientific Papers, Vol. Iv P. 370. Cambridge University Press, 1903).
Rieutord, M. & Zahn, J.-R. 1995 Turbulent plumes in stellar convective envelopes. Astron. Astrophys. 296, 127–138.Google Scholar
Rujula, A. De & Glashow, S. 1992 CERN-Th 6608/92, preprint
Schatzman, E. 1969 Gravitationnal separation of the elements and turbulent transport. Astron. Astrophy. 3, 331–346.Google Scholar
Schatzman, E. 1991a Lithium, Rotation And Age. In Angular Momentum Evoluton of Young Stars (ed. S. Catalono & J. R. Stauffers), pp. 223–242. Kluwer.
Schatzman, E. 1991b Mixing in the radiative zone. In Solar Interior and Atmosphere (ed. S. Catalano & M. S. Matthews), pp. 192–222. University of Arizona Press
Schatzman, E. 1991c On lithium depletion: a possible discrimination of the physical processes. Mem. Soc. Astron. Ital. 62, 111–130.Google Scholar
Schatzman, E. 1993a Transport Of Angular Momentum And Diffusion By The Action Of Internal Waves. Astron. Astrophys. 279, 431–446.Google Scholar
Schatzman, E. 1993b Filtering of gravity waves. Astron. Astrophys. 271, L29–L30.Google Scholar
Schatzman, E. 1995 Solar neutrinos and transport processes. In Physical Processes in Astrophysics (ed. I. W. Roxburgh & J. L. Masnou). Lecture Notes in Physics, vol. 458, pp. 171–184. Springer.
Schatzman, E. & Baglin, A. 1991 On the physcs of lithium depletion. Astron. Astrophys. 125–133.Google Scholar
Schatzman, E. & Maeder, A. 1981 Solar neutrinos and turbulent diffusion. Nature 290, 683–686.Google Scholar
Schatzman, E., Maeder, A., Angrand, F. & Glowinski, R. 1991 Stellar evolution with turbulent diffusion mixing. II. The solar model and the neutrino problem. Astron. Astrophys. 96, 1–16.Google Scholar
Schatzman, E. & Montalban, J. 1995 Transport processes due to internal waves. In Solar Modeling (ed. A. B. Balantekin & J. N. Bahcall), pp. 221–231. World Scientific.
Soderblom, D. R. 1991 The study of lithium in stars like the Sun. Mem. Soc. Astron. Ital. 62, 33–42.Google Scholar
Soderblom, D. R., Jones, B. F., Balachandran, S., Stauffer, J. R., Duncan, D. K., Fedele, S. & Hudon, J. D. 1993B The Evolution Of The Lithium Abundance Of Solar-Type Stars. III. The Pleiades. Astron. J. 106, 1059–1079.Google Scholar
Soderblom, D. R., Oey, M. S., Johnson, D. R. H. & Stones, R. P. S. 1990 The evolution of the lithium abundance of solar-type stars. I. The Hyades and Coma Beren ices clusters. Astron. J. 90, 595–607.Google Scholar
Soderblom, D. R., Pilachowski, C. A., Fedele, S. B. & Jones, B. F. 1993A The Evolution Of The Lithium Abundance Of Solar-Type Stars. Ii. The Ursae Major Group. Astron. J. 105, 2299–2307.Google Scholar
Steepen, M. 1993 The depth of the solar convective zone inferred from hydrodynamical models of the surface layers. In Inside the Stars, IAU colloquium 137 (ed. W. W. Weiss & A. Baglin). Astron. Soc. Pac. Conf. Ser. vol. 40, pp. 300–303.
Steffen, M. & Freytag, B. 1991 Hydronynamics of the solar photosphere: model of calculations and spectroscopic observations. Rev. Mod. Astron. 4, 43–60.Google Scholar
Stull, R. B. 1976 Internal grvity waves generated by penetrative convection. J. Atmos. Sci. 33, 1279–86.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. R. Soc. Lond. A 158, 196–212.Google Scholar
Townsend, A. A. 1965 Internal waves produced by a convective layer. J. Fluid Mech. 24, 307–319.Google Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption and its application to geophysical flows. J. Fluid Mech. 173, 451–471.Google Scholar
Vauclair, G. & Vauclair, S. 1982 Elements segregation in Stellar outer layers. Ann. Rev. Astron. Astrophys. 20, 37–00.Google Scholar
Zahn, J.-P. 1983 Instability and mixing processes in upper main-sequence. In Astrophysical Processes in Upper Main-Sequence Stars (ed. B. Hauck & A. Maeder), pp. 253–329. Publ. Observatoire de Geneve.
Zahn, J.-P. 1991 Convective penetration in stellar interiors. Astron. Astrophys. 252, 179–188.Google Scholar
Zahn, J.-P. 1992 Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115–132.Google Scholar