Exploring hydrodynamical stellar tachoclines
Abstract
Stellar tachoclines are thin transition regions located between theradiative core and the convective envelope of solar-type stars. As these tworegions generally rotate differently and differentially for the convective region,tachoclines are naturally the seat of strong shear and turbulence. Therefore,they link the dynamics of both regions and, as such, understanding thedynamics of the transport and mixing inside the tachocline would shed light onhow the dynamical processes of the convection zone might affect the seculartransport inside the radiative zone. In particular, we expect the evolution ofthe differential rotation in the convection zone to affect the dynamics of thetachocline. Indeed, as solar-type stars are braked on the Main Sequence, thedifferential rotation in the convection zone is expected to evolve from acylindrical rapidly-rotating regime (columns of varying velocities, aligned withthe rotation axis) to a conical solar-like regime (with an equatorial accelerationas in the case of the Sun) and finally to a conical anti-solar-like regime (with apolar acceleration). As of today, stellar evolutionary codes only consider thesolar conical regime to study the dynamics of stellar tachoclines throughout theevolution of stars. We discuss different possibilities to model hydrodynamicaltachoclines and we show that Mathis & Zahn 2004’s formalism is able to treatcoherently hydrodynamical stellar tachoclines when taken in the thin layerapproximation. We use it to model the differential rotation, meridionalcirculation and mixing coefficients inside the tachocline in order to examine theeffect of the different rotation regimes on the transport.
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TASC9/KASC16 – Camille Moisset Exploring hydrodynamical stellar tachoclines Camille Moisset, Stéphane Mathis, Louis Amard 1
Introducing the solar tachocline García et al. 2007 • First observed for the Sun, using helioseismology (Brown et al. 1989, Kosovitchev et al. 1997). • First hydrodynamical model by Spiegel & Zahn (1992). - Chemical elements mixing at the interface CZ: influence the surface abundances (Brun et al.1999). - Bulk turbulent convective motions and plumes excite waves inside the RZ (Press 1981, Pinçon et al. 2016). - Candidate for the generation and maintenance of the dynamo magnetic field and magnetic cycle in the Sun. Necessary region to take into account when modelling the evolution of stars and the coupling between their radiative interior and convective envelope. •Thin layer (below ). •Strong latitudinal and radial large-scale shear. •Key region to understand the transport of angular momentum and the mixing of chemical elements. <latexit sha1_base64="TPVTtIxF6y3mBYHKjUiWYAJJ9lc=">AAAB+HicbVDLSsNAFJ34rPXRqEs3g6XgQkoiWl0W3bisYh/QhDCZTNqhk5kwMxFq6Je4caGIWz/FnX/jtM1CWw9cOJxzL/feE6aMKu0439bK6tr6xmZpq7y9s7tXsfcPOkpkEpM2FkzIXogUYZSTtqaakV4qCUpCRrrh6Gbqdx+JVFTwBz1OiZ+gAacxxUgbKbArF14Neqf3Qe6JSOhJYFedujMDXCZuQaqgQCuwv7xI4CwhXGOGlOq7Tqr9HElNMSOTspcpkiI8QgPSN5SjhCg/nx0+gTWjRDAW0hTXcKb+nshRotQ4CU1ngvRQLXpT8T+vn+n4ys8pTzNNOJ4vijMGtYDTFGBEJcGajQ1BWFJzK8RDJBHWJquyCcFdfHmZdM7qbqPeuDuvNq+LOErgCByDE+CCS9AEt6AF2gCDDDyDV/BmPVkv1rv1MW9dsYqZQ/AH1ucPiSuSZQ==</latexit> 5% R→ •Prime importance in the interaction between the radiative and convective regions: Thompson et al. 2003 2 TASC9/KASC16 – Camille Moisset
3 Why is the solar tachocline so thin? Strugarek et al. 2023 Many possible candidates Complicated dynamics •Radiative spreading of the tachocline: it should be around 30% of the solar radius today (Brun & Zahn 2006, Strugarek et al 2023). No consensus has been reached yet Need a confinement mechanism for the tachocline. TASC9/KASC16 – Camille Moisset
4 •Hydrodynamical tachoclines: •Magnetic tachoclines: Radiative spreading countered by strong anisotropic turbulence (Spiegel & Zahn 1992, Garaud et al. 2025). Two families of models •Confinement with a fossil magnetic field in the radiative zone (Gough & McIntyre 1998, Garaud 2001). •Confinement with a dynamo field in the convective zone (Barnabé et al 2017, Matilsky et al. 2024). Why is the solar tachocline so thin? Strugarek et al. 2023 Need a confinement mechanism for the tachocline. TASC9/KASC16 – Camille Moisset
5 •Hydrodynamical tachoclines: •Magnetic tachoclines: Radiative spreading countered by strong anisotropic turbulence (Spiegel & Zahn 1992, Garaud et al. 2025). Two families of models •Confinement with a fossil magnetic field in the radiative zone (Gough & McIntyre 1998, Garaud 2001). •Confinement with a dynamo field in the convective zone (Barnabé et al 2017, Matilsky et al. 2024). What about the magnetic field? How to effectively confine the tachocline? Why is the solar tachocline so thin? Strugarek et al. 2023 Need a confinement mechanism for the tachocline. TASC9/KASC16 – Camille Moisset
Strugarek et al. 2023 6 •Hydrodynamical tachoclines: •Magnetic tachoclines: Radiative spreading countered by strong anisotropic turbulence (Spiegel & Zahn 1992, Garaud et al. 2025). Two families of models •Confinement with a fossil magnetic field in the radiative zone (Gough & McIntyre 1998, Garaud 2001). •Confinement with a dynamo field in the convective zone (Barnabé et al 2017, Matilsky et al. 2024). Why is the solar tachocline so thin? Need a confinement mechanism for the tachocline. TASC9/KASC16 – Camille Moisset
Noraz et al. 2024 The rotational evolution of solar-type star •The rotation in the convection zone evolves during the Main Sequence: cylindrical profile, conical solar-like (fast equator, slow poles) and conical anti-solar-like (slow equator, fast poles). •The key physical parameter is the fluid Rossby number (Brun et al. 2017, 2022, Noraz et al. 2024): <latexit sha1_base64="Lzdc+PYTz3iAOIsUznGjx3XzAko=">AAACDHicbVC7SgNBFJ2NrxhfUUubxSDEJu4GiTZC0MbOKOYB2RBmJzfJkNmdZeauEJZ8gI2/YmOhiK0fYOffOHkUmnhg4HDOudy5x48E1+g431ZqaXlldS29ntnY3Nreye7u1bSMFYMqk0Kqhk81CB5CFTkKaEQKaOALqPuDq7FffwCluQzvcRhBK6C9kHc5o2ikdjZ3Jy885KIDiScD6NHRSb7o3YxZO/E0UjU6Nimn4ExgLxJ3RnJkhko7++V1JIsDCJEJqnXTdSJsJVQhZwJGGS/WEFE2oD1oGhrSAHQrmRwzso+M0rG7UpkXoj1Rf08kNNB6GPgmGVDs63lvLP7nNWPsnrcSHkYxQsimi7qxsFHa42bsDlfAUAwNoUxx81eb9amiDE1/GVOCO3/yIqkVC26pULo9zZUvZ3WkyQE5JHnikjNSJtekQqqEkUfyTF7Jm/VkvVjv1sc0mrJmM/vkD6zPH3JYm0E=</latexit> Ro =˜ω/(2!ω) Evolution of differential rotation in the convective zone: not included in stellar evolutionary codes.7 <latexit sha1_base64="9NRiOKZLEoCTRgye63NdB6w1TyU=">AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiYi1WXRjcta7AOaECaTSTt0MhNmJkIJ9VfcuFDErR/izr9x2mahrQcuHM65l3vvCVNGlXacb6u0tr6xuVXeruzs7u0f2IdHXSUyiUkHCyZkP0SKMMpJR1PNSD+VBCUhI71wfDvze49EKir4g56kxE/QkNOYYqSNFNjVtjhviyD3RCT01FM0gW5g15y6MwdcJW5BaqBAK7C/vEjgLCFcY4aUGrhOqv0cSU0xI9OKlymSIjxGQzIwlKOEKD+fHz+Fp0aJYCykKa7hXP09kaNEqUkSms4E6ZFa9mbif94g0/G1n1OeZppwvFgUZwxqAWdJwIhKgjWbGIKwpOZWiEdIIqxNXhUTgrv88irpXtTdRr1xf1lr3hRxlMExOAFnwAVXoAnuQAt0AAYT8AxewZv1ZL1Y79bHorVkFTNV8AfW5w87H5SE</latexit> Ro/Ro→→1 <latexit sha1_base64="zSEukxLbJ9FV0DTkaBAyPFZsSkI=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXMVEtLosunFZi31AE8JkMmmHTjJhZiKU2IW/4saFIm79DXf+jdM2C209cOFwzr3ce0+QMiqVbX8bpaXlldW18nplY3Nre8fc3WtLnglMWpgzLroBkoTRhLQUVYx0U0FQHDDSCYY3E7/zQISkPLlXo5R4MeonNKIYKS355kGTnza5n7s85GrsShpD23IufLNqW/YUcJE4BamCAg3f/HJDjrOYJAozJGXPsVPl5UgoihkZV9xMkhThIeqTnqYJion08un9Y3islRBGXOhKFJyqvydyFEs5igPdGSM1kPPeRPzP62UquvJymqSZIgmeLYoyBhWHkzBgSAXBio00QVhQfSvEAyQQVjqyig7BmX95kbTPLKdm1e7Oq/XrIo4yOARH4AQ44BLUwS1ogBbA4BE8g1fwZjwZL8a78TFrLRnFzD74A+PzB58vlTU=</latexit> Ro/Ro→→0.15 TASC9/KASC16 – Camille Moisset
Evolution of stellar parameters Along with the differential rotation in the convective region, several other parameters change during the evolution of a star: • Radius of the star and thickness of the convective region. • Rotation, stratification and turbulent transport. • Latitudinal shear at the top of the tachocline (Brun et al. 2022). All these parameters influence the dynamics of the tachocline over evolutionary timescales. <latexit sha1_base64="pl4sLM8TNk1kDGVBrZksUKPgKnY=">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</latexit> ω=!! N"1/2!εv ϑh"1/4 <latexit sha1_base64="BYbHGt9JSPWL0WJJYeNoA1dFN1A=">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</latexit> STAREVOL 1M→model Radiative zone Convective zone 8 TASC9/KASC16 – Camille Moisset
ØFormalism Mathis & Zahn 2004 + thin layer approximation: recover Spiegel & Zahn’s model for the solar tachocline. Hydrodynamical tachocline for stellar evolutionary codes Forcing from the convective zone Rotation in the tachocline ØFormalism allows the extension of the tachocline’s model for any profile of rotation in the convective zone. ØPrescriptions for the differential rotation, meridional circulation and effective turbulent diffusion in the tachocline. ØAllows for a direct implementation in stellar evolutionary codes (1D and 2D). Coherent results for the rotation profile in the tachocline for the solar case. 9 TASC9/KASC16 – Camille Moisset
Conclusion & perpectives New model for hydrodynamical tachoclines from Mathis & Zahn’s 2004 formalism that allows: •To compute the dynamics in both the bulk of the radiative zone and the tachocline. •To include the long term evolution of the convective region with any profile of differential rotation. •To be directly implemented in 1D and 2D stellar evolutionary codes. Complementary model incoming, based on a dominant transport of heat ensured by the horizontal radiative diffusion (Garaud et al. 2025). •To prescribe the differential rotation, meridional circulation and effective turbulent diffusion of the elements in the tachocline. Stay tuned! TASC9/KASC16 – Camille Moisset