[Paper Review] Transport and mixing in the radiation zones of rotating stars: I-Hydrodynamical processes

S. Mathis, Jean-Paul Zahn|ArXiv.org|Jun 18, 2004

Stellar, planetary, and galactic studiesPhysics and Astronomy20 references104 citations

TL;DR

This paper improves modeling of rotational mixing in stellar radiation zones by extending hydrodynamical treatments to higher-order spherical harmonics and retaining full time derivatives, enabling accurate simulation of tachocline circulation and rapid evolutionary phases. It resolves limitations in prior models by capturing latitudinal differential rotation and non-stationary dynamics, particularly for massive stars with strong shear-driven turbulence and meridional circulation.

ABSTRACT

The purpose of this paper is to improve the modelization of the rotational mixing which occurs in stellar radiation zones, through the combined action of the thermally driven meridional circulation and of the turbulence generated by the shear of differential rotation. The turbulence is assumed to be anisotropic, due to the stratification, with stronger transport in the horizontal directions than in the vertical. The main difference with the former treatments by Zahn (1992) and Maeder & Zahn (1998) is that we expand here the departures from spherical symmetry to higher order, and include explicitly the differential rotation in latitude, to first order. This allows us to treat simultaneously the bulk of a radiation zone and its tachocline(s). Moreover, we take fully into account the non-stationarity of the problem, which will enable us to tackle the rapid phases of evolution. The system of partial differential equations, which govern the transport of angular momentum, heat and chemical elements, is written in a form which makes it ready to implement in a stellar evolution code. Here the effect of a magnetic field is deliberately ignored; it will be included in forthcoming papers.

Motivation & Objective

• To improve the modeling of rotational mixing in stellar radiation zones by extending the treatment of departures from spherical symmetry beyond the standard 1.5D approximation.
• To explicitly include the effects of latitudinal differential rotation in the linear approximation, enabling treatment of tachocline circulation.
• To retain all time derivatives (except for dynamical relaxation) to better resolve rapid evolutionary phases with steep gradients in rotation and composition.
• To provide a system of partial differential equations suitable for implementation in stellar evolution codes, with a focus on angular momentum, heat, and chemical element transport.
• To lay the groundwork for future inclusion of magnetic fields in rotational mixing processes, which are currently neglected.

Proposed method

• Expanding all hydrodynamical variables in spherical harmonics to higher order (beyond the standard 1.5D approach), allowing for explicit treatment of latitudinal variations.
• Formulating the system of equations using a relaxation approach for temperature fluctuations (Ψₗ), which stabilizes numerical solutions near steep composition and rotation gradients.
• Modeling turbulent transport as anisotropic (νᵥ ≪ νₕ, Dᵥ ≪ Dₕ) due to stable stratification, favoring horizontal over vertical mixing.
• Including the meridional circulation driven by thermal imbalance (from von Zeipel’s work) and its feedback on rotation via angular momentum transport.
• Applying boundary conditions that ensure continuity with a potential field at the stellar surface (r = R), including vanishing radial derivative for Ψₗ and a mixed condition for Λₗ.
• Using a linearized, axisymmetric formulation to describe the interaction between differential rotation (via Ψₗ) and composition inhomogeneities (via Λₗ), avoiding non-linear terms that complicate numerical treatment.

Experimental results

Research questions

• RQ1 How can the treatment of meridional circulation and shear-driven turbulence in radiation zones be improved to capture tachocline dynamics accurately?
• RQ2 What is the impact of including higher-order spherical harmonic terms on the modeling of latitudinal differential rotation and circulation?
• RQ3 How does retaining full time derivatives enhance the simulation of rapid evolutionary phases with strong gradients in rotation and composition?
• RQ4 In what way does the relaxation formulation of the heat equation improve numerical stability near steep gradients in stellar interiors?
• RQ5 How can the system of equations be structured to be directly implementable in stellar evolution codes while preserving physical consistency?

Key findings

• The model successfully captures the octupolar circulation in tachoclines induced by differential rotation between convection zones and radiation zones, which was previously treated ad hoc.
• By retaining full time derivatives (except for dynamical relaxation), the model enables accurate simulation of rapid evolutionary phases where gradients in angular velocity and composition steepen significantly.
• The relaxation formulation of the temperature fluctuation Ψₗ reduces numerical sensitivity to steep composition gradients, improving stability in stellar evolution codes.
• The inclusion of higher-order spherical harmonics allows for a more accurate representation of latitudinal variations in rotation and circulation, moving beyond the 1.5D shellular approximation.
• The model provides a self-consistent framework for angular momentum, heat, and chemical transport that is suitable for integration into stellar evolution codes.
• The framework is explicitly designed to allow future inclusion of magnetic fields, which are identified as essential for explaining the flat rotation profile in solar-type stars, contrary to the model’s current prediction of a rapidly spinning core.