Double-diffusive mixing-length theory, semiconvection and massive star evolution
Open accessAbstract
Double-diffusive convection refers to mixing where the effects of thermal and composition gradients compete to determine the stability of a fluid. In addition to the familiar fast convective instability, such fluids exhibit the slow, direct salt finger instability and the slow, overstable semiconvective instability. Previous approaches to this subject usually have been based on linear stability analyses. We develop here the non-linear mixing-length theory (MLT) of double-diffusive convection, in analogy to the more familiar MLT for a fluid of homogeneous composition. We present approximate solutions for the mixing rate in the various regimes, and show that the familiar Schwarzschild and Ledoux stability criteria are good approximations to the precise criteria in stellar interiors. We have implemented the self-consistent computation of the temperature gradient and turbulent mixing rate in a stellar evolution code, and solved a diffusion equation to mix the composition at the appropriate rate. We have evolved 15- and 30-M⊙ stars from the zero-age main sequence to the end of core He-burning. Semiconvective mixing is fast enough to alter stellar composition profiles on relevant time-scales, but not so fast that instantaneous readjustment is appropriate.
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Mixing can occur if the gradient of the molecular weight is negative, even though the material is formally stable.
“The magenta region shows that although formally stable, mixing can occur if the gradient of the molecular weight is negative.”
From The Dawes Review 2: Nucleosynthesis and Stellar Yields of Low- and Intermediate-Mass Single Stars (2014) · cites this paper as Grossman & Taam (1996)
Convection is expected above the Ledoux criterion line.
“The Ledoux criterion is the blue line, with convection expected above this line.”
From The Dawes Review 2: Nucleosynthesis and Stellar Yields of Low- and Intermediate-Mass Single Stars (2014) · cites this paper as Grossman & Taam (1996)
Convection is expected to the right of the Schwarzschild criterion line.
“The left panel shows the Schwarzschild criterion as a red line, with convection expected to the right of the red line.”
From The Dawes Review 2: Nucleosynthesis and Stellar Yields of Low- and Intermediate-Mass Single Stars (2014) · cites this paper as Grossman & Taam (1996)
Semiconvection is the region where material is stable according to the Ledoux criterion but unstable according to the Schwarzschild criterion.
“The green region shows where the material is stable according to the Ledoux criterion but unstable according to the Schwarzschild criterion: this is semiconvection.”
From The Dawes Review 2: Nucleosynthesis and Stellar Yields of Low- and Intermediate-Mass Single Stars (2014) · cites this paper as Grossman & Taam (1996)
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Cites
Stellar Structure and Evolution
Kippenhahn, R., Weigert, A., & Weiß, A. (2012) · Astronomy and astrophysics library · doi:10.1007/978-3-642-30304-3 · Cited by 2,001 · see citing works
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- Stevenson, D.J. 1977, Proc. Astron. Soc. Australia, 3, 165
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MESA ISOCHRONES AND STELLAR TRACKS (MIST). I. SOLAR-SCALED MODELS
Choi, J., Dotter, A., Conroy, C., Cantiello, M., Paxton, B., & Johnson, B. D. (2016) · The Astrophysical Journal · doi:10.3847/0004-637x/823/2/102 · Cited by 2,922 · see citing works
“The importance of semiconvection on the evolution of massive stars has been studied for many decades, e.g., during the core helium burning phase (CHeB) (Stothers & Chin 1975; Langer et al. 1985; Grossman & Taam 1996).”
BackgroundThe Dawes Review 2: Nucleosynthesis and Stellar Yields of Low- and Intermediate-Mass Single Stars
Karakas, A. I., & Lattanzio, J. C. (2014) · Publications of the Astronomical Society of Australia · doi:10.1017/pasa.2014.21 · Cited by 735 · see citing works
“One rather nice way of viewing the various mixing mechanisms in stars is given in Figure 11, based on Figure 2 of Grossman & Taam (1996).”
MethodAlthaus, L. G., Serenelli, A. M., Panei, J. A., et al. (2005) · Astronomy and Astrophysics · doi:10.1051/0004-6361:20041965 · Cited by 241 · see citing works
“Our calculations made use the double-diffusive mixing length theory of convection for fluids with composition gradients (Grossman & Taam 1996).”
BackgroundMethodInfluentialModeling He-rich subdwarfs through the hot-flasher scenario
Bertolami, M. M. M., Althaus, L. G., Unglaub, K., et al. (2008) · Astronomy and Astrophysics · doi:10.1051/0004-6361:200810373 · Cited by 145 · see citing works
“To analyze the effect of chemical gradients during deep-mixing events, we performed some simulations of deep-mixing for a model of 0.4641M⊙ and Z=0.02 with the theory of convection developed by Grossman et al. (1993), in its local version presented by Grossman & Taam (1996)5.”
InfluentialConstraining mixing in massive stars in the Small Magellanic Cloud
Schootemeijer, A., Langer, N., Grin, N. J., et al. (2019) · Astronomy and Astrophysics · doi:10.1051/0004-6361/201935046 · Cited by 132 · see citing works
New evolutionary models for massive ZZ Ceti stars. I. First results for their pulsational properties
Althaus, L. G., Serenelli, A. M., Córsico, A. H., et al. (2003) · Astronomy and Astrophysics · doi:10.1051/0004-6361:20030472 · Cited by 108 · see citing works
“Treatment of convection: For this work, we included in LPCODE the extended mixing length theory of convection for fluids with composition gradients developed by Grossman et al. (1993) in its local approximation as given by Grossman & Taam (1996).”
MethodConstraining mixing processes in stellar cores using asteroseismology
Aguirre, V. S., Ballot, J., Serenelli, A. M., et al. (2011) · Astronomy and Astrophysics · doi:10.1051/0004-6361/201015847 · Cited by 90 · see citing works
“Throughout the years, several approaches have been proposed to deal with semiconvective regions when they appear in stellar models (Stevenson 1979; Langer et al. 1983; Spruit 1992; Grossman & Taam 1996).”
BackgroundConstraining mixing in massive stars in the Small Magellanic Cloud
Schootemeijer, A., Langer, N., Grin, N. J., et al. (2019) · Springer Link (Chiba Institute of Technology) · doi:10.1051/0004-6361/201935046/pdf · Cited by 88 · see citing works
New evolutionary calculations for the born again scenario
Bertolami, M. M. M., Althaus, L. G., Serenelli, A. M., et al. (2006) · Astronomy and Astrophysics · doi:10.1051/0004-6361:20053804 · Cited by 88 · see citing works
“We extend the scope of the paper by exploring the role played by the molecular weight gradient (∇µ) induced by proton burning by considering the double-diffusive MLT for fluids with composition gradients (Grossman & Taam 1996).”
MethodThe structure of the solar convective overshooting zone
Xiong, D. R., & Deng, L. (2001) · Monthly Notices of the Royal Astronomical Society · doi:10.1046/j.1365-8711.2001.04820.x · Cited by 58 · see citing works
“Although great advancements have been achieved in this field (Xiong 1979, 1981; Canuto 1993, 1997a; Grossman, Narayan & Arnett 1993; Grossman 1996; Grossman & Taam 1996), the phenomenological non-local mixinglength theory is still the most usual means for the treatment of convective overshooting in…”
BackgroundCanuto, V. M. (1999) · The Astrophysical Journal · doi:10.1086/307783 · Cited by 52 · see citing works
“The models of Xiong (1985a, 1985b, 1986) and Grossman & Taam (1996) have already been discussed in section ° 8, and we shall therefore comment only on the recent work of Umezu (1998).”
BackgroundInfluentialThermohaline mixing and the photospheric composition of low-mass giant stars
Wachlin, F. C., Bertolami, M. M. M., & Althaus, L. G. (2011) · Astronomy and Astrophysics · doi:10.1051/0004-6361/201117029 · Cited by 51 · see citing works
“In particular, we adopt the very recent prescription of Traxler et al. (2011) and the double diffusive mixing-length theory of Grossman & Taam (1996).”
BackgroundMethodInfluentialAguirre, V. S., Ballot, J., Serenelli, A. M., et al. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1102.0779 · Cited by 48 · see citing works
“Throughout the years, several approaches have been proposed to deal with semiconvective regions when they appear in stellar models (Stevenson 1979; Langer et al. 1983; Spruit 1992; Grossman & Taam 1996).”
BackgroundNew evolutionary models for massive ZZ Ceti stars
Córsico, A. H., Althaus, L. G., Montgomery, M. H., et al. (2004) · Astronomy and Astrophysics · doi:10.1051/0004-6361:20041101 · Cited by 41 · see citing works
“In particular, we considered the extended mixing length theory of convection for fluids with composition gradients developed by Grossman et al. (1993) in its local approximation as given by Grossman & Taam (1996).”
Pulsations of massive ZZ Ceti stars with carbon/oxygen and oxygen/neon cores
Córsico, A. H., García-Berro, E., Althaus, L. G., et al. (2004) · Astronomy and Astrophysics · doi:10.1051/0004-6361:20040416 · Cited by 39 · see citing works
“Finally, convection is treated according to the mixing length theory for fluids with composition gradients (Grossman & Taam 1996) that applies in the convective, semiconvective, and salt finger instability regimes.”
MAIN SEQUENCE EVOLUTION WITH LAYERED SEMICONVECTION
Moore, K., & Garaud, P. (2016) · The Astrophysical Journal · doi:10.3847/0004-637x/817/1/54 · Cited by 35 · see citing works
Fingering convection in red giants revisited
Wachlin, F. C., Vauclair, S. D., & Althaus, L. G. (2014) · Astronomy and Astrophysics · doi:10.1051/0004-6361/201424580 · Cited by 29 · see citing works
“In particular, and for the relevance of the present paper, th code includes a generalized version of the mixing length theory developed by Grossman et al. (1993, GNA hereafter) (see also Grossman & Taam 1996).”
BackgroundOvershooting and semiconvection: structural changes and asteroseismic signatures
Noels-Grötsch, A., Montalbán, J., Miglio, A., et al. (2009) · Astrophysics and Space Science · doi:10.1007/s10509-009-0203-7 · Cited by 28 · see citing works
“…∇ − ∇ad ∇L − ∇ (8) where α is an efficiency factor of the order of 0.1, cP is the specific heat at constant pressure, κr, the radiative conductivity, , the density, and ∇L, the Ledoux gradient (= ∇ad + β4−3β ∇μ, β is the ratio of gas pressure to total pressure) (see also Grossman and Taam 1996).”
InfluentialCórsico, A. H., Althaus, L. G., Montgomery, M. H., et al. (2004) · Americanae (AECID Library) · doi:10.48550/arxiv.astro-ph/0408236 · Cited by 22 · see citing works
“In particular, we considered the extended mixing length theory of convection for fluids with composition gradients developed by Grossman et al. (1993) in its local approximation as given by Grossman & Taam (1996).”
Turbulence in Astrophysical and Geophysical Flows
Canuto, V. M. (2008) · Lecture notes in physics · doi:10.1007/978-3-540-78961-1_4 · Cited by 17 · see citing works
Layer formation in double-diffusive convection over resting and moving heated plates
Zaussinger, F., & Kupka, F. (2019) · Theoretical and Computational Fluid Dynamics · doi:10.1007/s00162-019-00499-7 · Cited by 16 · see citing works
Semiconvection and Overshooting: Schwarzschild and Ledoux Criteria Revisited
Canuto, V. M. (2000) · The Astrophysical Journal · doi:10.1086/312654 · Cited by 16 · see citing works
“A similar result was obtained by Grossman & Taam (1996).”
MethodResultThe extent of the solar overshooting zone deduced from Li abundance
Xiong, D. R., & Deng, L. (2002) · Monthly Notices of the Royal Astronomical Society · doi:10.1046/j.1365-8711.2002.05767.x · Cited by 16 · see citing works
Layer formation in double-diffusive convection over resting and moving heated plates
Zaussinger, F., & Kupka, F. (2019) · MPG.PuRe (Max Planck Society) · Cited by 11 · see citing works
Trimble, V., & McFadden, L. (1998) · Publications of the Astronomical Society of the Pacific · doi:10.1086/316143 · Cited by 9 · see citing works
Stothers, R. B., & Chin, C.-W. (2000) · The Astrophysical Journal · doi:10.1086/309349 · Cited by 9 · see citing works
A hot white dwarf merger remnant revealed by an ultraviolet detection of carbon
Sahu, S., Bédard, A., Gaensicke, B., et al. (2025) · Nature Astronomy · doi:10.1038/s41550-025-02590-y · Cited by 5 · see citing works
“Still , the emergence of a staircase-like chemical profile is informative as it constitutes a well-known signature of semi-convection in one-dimensional stellar models 48,49 and even in three-dimensional hydrodynamical simulations 50 .”
Zaussinger, F., Kupka, F., & Muthsam, H. (2012) · Lecture notes in physics · doi:10.1007/978-3-642-33380-4_11 · Cited by 3 · see citing works
Convection and mixing in stars: theory versus observations
Chiosi, C. (2006) · Proceedings of the International Astronomical Union · doi:10.1017/s1743921307000506 · Cited by 2 · see citing works
Modeling He-rich subdwarfs through the hot-flasher scenario
Bertolami, M. M. M., Althaus, L. G., Unglaub, K., et al. (2009) · Journal of Physics Conference Series · doi:10.1088/1742-6596/172/1/012014 · Cited by 2 · see citing works
Hydrogen Flux in the Semiconvection Zone of the Massive Main Sequence Stars
Umezu, M. (1998) · Astrophysics and Space Science · doi:10.1023/a:1001743621424 · Cited by 2 · see citing works
On the local theory of stellar convection with μ-gradient using one-point correlation functions
Umezu, M. (2008) · Monthly Notices of the Royal Astronomical Society · doi:10.1111/j.1365-2966.2008.13964.x · Cited by 1 · see citing works
Theoretical modeling of convection I. Key physical processes
Canuto, V. M. (2006) · Proceedings of the International Astronomical Union · doi:10.1017/s1743921307000051 · Cited by 1 · see citing works
Stellar Physics and Starburst Evolution
Meynet, G. (1998) · Springer eBooks · doi:10.1007/978-3-662-29742-1_5
Causal Propagation of Heat and Thermohaline Instability in Quasi-static Phenomena
Aguirre-Guzmán, A., & Falcon, N. L. (2016) · Environmental science and engineering · doi:10.1007/978-3-319-27965-7_31
García-Berro, E., & Althaus, L. G. (2013) · Springer eBooks · doi:10.1007/978-94-007-5615-1_1
Overshooting and semiconvection: structural changes and asteroseismic signatures
Noels-Grötsch, A., Montalbán, J., Miglio, A., et al. (2009) · Springer eBooks · doi:10.1007/978-90-481-9198-7_38
- Authors
- Grossman, S. A., & Taam, R. E.
- Affiliations
- Northwestern University (US)
- Venue
- Monthly Notices of the Royal Astronomical Society
- Publisher
- Oxford University Press (OUP)
- Published
- 1996-12-11
- Volume
- 283
- Issue
- 4
- Pages
- 1165-1178
- ISSN
- 0035-8711 (ISSN-L)
- 1365-2966
- OpenAlex
- W1965316083
- Semantic Scholar
- 66f2791a431879d933b76abb64861458e64ae81f
- Cited by
- 38
- References
- 1
- Citations by year
- 2025: 1 · 2019: 4 · 2016: 3 · 2014: 2 · 2013: 2 · 2012: 1
- FWCI
- 1.44
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Monthly Notices of the Royal Astronomical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Stellar, planetary, and galactic studies
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Solar and Space Plasma Dynamics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Astrophysics and Star Formation Studies
- Keywords
- double-diffusive convection, mixing length theory, semiconvection, massive stars, stellar evolution, Ledoux criterion, core helium burning, stellar interiors
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