[Paper Review] Transport and mixing in the radiation zones of rotating stars II. Axisymmetric magnetic field

S. Mathis, Jean-Paul Zahn|ArXiv.org|Jun 6, 2005

Solar and Space Plasma Dynamics33 references99 citations

TL;DR

This paper develops a self-consistent, time-dependent model of rotational mixing in stably stratified stellar radiation zones by coupling thermally driven meridonal circulation, shear-induced turbulence, and axisymmetric magnetic fields. It shows that magnetic fields—through Lorentz forces and Alfvén wave dynamics—can enforce nearly uniform rotation along field lines, significantly altering angular momentum transport and modifying the efficiency of chemical mixing, especially in stars with fossil magnetic fields or those undergoing spin-down via magnetized winds.

ABSTRACT

The purpose of this paper is to improve the modeling of the mixing of chemical elements that occurs in stellar radiation zones. In addition to the classical rotational mixing considered in our previous paper, which results of the combined action of the thermally-driven meridional circulation and of the turbulence generated by the shear of differential rotation, we include here the effect of an axisymmetric magnetic field in a self-consistent way. We treat the advection of the field by the meridional circulation, its Ohmic diffusion, and the production of its toroidal component through the shear of differential rotation. The Lorentz force is assumed not to exceed the centrifugal force; it acts on the baroclinic balance and therefore on the meridional flow, and it has a strong impact on the transport of angular momentum. All variables and governing equations are expanded in spherical or spherical vectorial functions, to arbitrary order: this yields a system of partial differential equations in time and in the radial coordinate, which is ready to be implemented in a stellar structure code.

Motivation & Objective

• To improve stellar structure models by including the effects of axisymmetric magnetic fields on mixing in radiation zones, beyond classical rotational mixing.
• To model the self-consistent coupling between meridional circulation, magnetic field evolution (advection, diffusion, shear generation), and angular momentum transport.
• To investigate how magnetic fields—particularly fossil fields—can enforce nearly uniform rotation and suppress differential rotation, especially in solar-type stars.
• To enable implementation in stellar evolution codes by expressing all variables in spherical harmonics to arbitrary order.
• To assess the impact of magnetic fields on the transport of angular momentum and chemical elements, particularly in stars with magnetized winds or fossil fields.

Proposed method

• All variables (velocity, magnetic field, angular velocity) are expanded in spherical harmonics and vectorial spherical harmonics to arbitrary order, enabling separation of angular dependence.
• The governing equations are derived in the linear approximation for departures from shellular rotation, allowing resolution of tachocline-like structures.
• The model includes time-dependent advection of the magnetic field by meridional circulation, Ohmic diffusion, and shear-induced generation of the toroidal field component.
• The Lorentz force is included in the baroclinic balance, affecting the meridional flow and angular momentum transport, while remaining subdominant to gravity and centrifugal forces.
• Short-timescale waves (acoustic, gravity, inertial, Alfvén) are filtered out via implicit time integration, retaining only secular evolution.
• The formalism is designed for direct implementation into stellar structure codes, with explicit equations provided for expansions up to octupole order (l=3).

Experimental results

Research questions

• RQ1 Can an axisymmetric magnetic field enforce nearly uniform rotation in stellar radiation zones by competing with the centrifugal force in the baroclinic balance?
• RQ2 How does the inclusion of magnetic field advection, Ohmic diffusion, and shear-generated toroidal field affect the efficiency of rotational mixing?
• RQ3 To what extent does the Lorentz force modify the meridional circulation and angular momentum transport compared to purely hydrodynamic models?
• RQ4 Can a fossil magnetic field extracted from the convection zone reproduce the flat rotation profile observed in the solar interior via helioseismology?
• RQ5 How do the feedback loops between magnetic field evolution, differential rotation, and meridional circulation alter the transport of chemical elements in evolving stars?

Key findings

• The magnetic field, through the Lorentz force, competes with the centrifugal force in the baroclinic balance, thereby modifying the meridional circulation and angular momentum transport.
• The model shows that even a weak magnetic field can enforce nearly uniform rotation along poloidal field lines, consistent with Ferraro's law, and suppress differential rotation on timescales comparable to the Alfvén time.
• The inclusion of self-consistent magnetic field evolution—via advection, Ohmic diffusion, and shear generation—leads to a feedback loop that stabilizes rotation profiles and alters mixing efficiency.
• The time-averaged solution filters out high-frequency waves, allowing stable, implicit time integration suitable for stellar evolution codes.
• The formalism is fully compatible with existing stellar structure codes and is implemented in a form ready for numerical integration, with equations provided up to octupole order (l=3).
• The model provides a framework to test whether fossil magnetic fields can explain the flat rotation profile in the solar radiative interior, as inferred from helioseismology.