Helioseismology shows that the Sun's convective envelope rotates differentially - the equator completing a rotation in about 25 days and the polar regions in about 34 days - while the radiative interior below the base of the convection zone (near 0.7R⊙0.7R_\odot) rotates almost as a solid body, with the transition confined to a thin shear layer, the tachocline, only about 4% of the solar radius thick. Spiegel and Zahn showed in 1992 that the envelope's differential rotation should burrow into the stably stratified interior over the Sun's lifetime through thermally mediated meridional circulation ("radiative spreading"), so an active confinement mechanism is required. Competing candidates have been proposed and partially modeled: strongly anisotropic turbulent viscosity; a weak fossil (primordial) poloidal magnetic field confined by penetrating meridional flow and enforcing uniform rotation through isorotation; skin-depth penetration of the cyclic dynamo magnetic field; self-excited non-axisymmetric dynamo action within the radiative interior; and angular-momentum transport by internal gravity waves. Recent global simulations (2021-2026) have for the first time produced self-consistently confined tachoclines in controlled parameter regimes, but none reach true solar diffusivities, and observations have not yet detected the interior magnetic fields or deep flows that would discriminate among the scenarios. The problem is to identify which mechanism, or combination, actually confines the tachocline in the Sun.

Background

The Sun's interior divides at radius r≈0.713R⊙r\approx0.713R_\odot into an outer convection zone, occupying the outer 30% by radius, in which energy is carried by turbulent convection, and a radiative interior below, in which energy is carried by radiation and the stratification is stable. Helioseismology - the inversion of the Sun's global acoustic oscillation frequencies into profiles of internal sound speed and rotation - shows that the convective envelope rotates differentially, with the equator rotating roughly 30% faster in angular velocity than the polar regions, whereas the radiative interior rotates nearly uniformly down to the limits of inversion (about 0.2R⊙0.2R_\odot). The transition between the two regimes is confined to the tachocline, a shear layer only about 0.04R⊙0.04R_\odot thick centered near the base of the convection zone.

A radially confined shear imposed at the top of a stably stratified radiative zone cannot simply stay put. The differential rotation drives a weak meridional circulation (a latitudinal flow with a compensating return flow) that burrows into the interior, and because thermal diffusion slowly erases the horizontal temperature contrasts that would otherwise shut the circulation down, the shear spreads inward on the Eddington-Sweet timescale, which for the Sun is shorter than its lifetime (Spiegel & Zahn 1992). Left unopposed, this "radiative spreading" would by now have spun much of the radiative interior into differential rotation, contradicting the helioseismic inversions. Spiegel and Zahn therefore proposed that turbulence in the tachocline transports angular momentum strongly anisotropically - with the horizontal (latitudinal) turbulent viscosity far exceeding the radial one - so that horizontal mixing keeps the burrowing layer nearly uniformly rotating and halts the spread. This is the hydrodynamic "fast tachocline"; its viability hinges on whether any real turbulence source produces the required anisotropy.

The main alternative is magnetic. A fossil field - a magnetic field trapped in the radiative interior since the Sun's pre-main-sequence contraction, slowly decaying by ohmic diffusion on timescales comparable to the solar age - threads the interior. By Ferraro's law of isorotation, a steady axisymmetric poloidal field cannot coexist in equilibrium with rotation that varies along its field lines, so such a field tends to enforce solid-body rotation (Gough & McIntyre 1998). But the field itself must be confined near the tachocline: if it connects freely to the differentially rotating envelope, it drags the interior into differential rotation instead. Gough and McIntyre proposed that downwelling branches of the meridional circulation just beneath the convection zone hold the field down against diffusion. Kitchatinov & Rüdiger (2006) quantified a related scenario in which a shallowly penetrating (at most about 1000 km) equatorward flow of roughly 10 m/s at the base of the convection zone confines the longest-lived dipolar decay mode of a fossil field, but only if the tachocline is hydrodynamically quiet, with an effective magnetic diffusivity no more than about 108 cm2 s−110^8\,\mathrm{cm^2\,s^{-1}} - a condition not independently established. A second magnetic route dispenses with the fossil field: the convection-zone dynamo's oscillating field penetrates a skin depth - the exponential decay depth of an oscillating magnetic field, scaling as (η/ω)1/2(\eta/\omega)^{1/2} for diffusivity η\eta and angular frequency ω\omega - into the upper radiative zone, and the resulting Maxwell stresses resist the spread (Forgács-Dajka & Petrovay 2001; Barnabé et al. 2017). Matilsky et al. (2022) demonstrated a stronger version in a global simulation: a self-excited, non-axisymmetric, quasi-cyclic dynamo imprints its field downward diffusively, the field is locally amplified throughout the radiative interior by horizontal motions driven by equatorial Rossby waves (large-scale vortical waves restored by the Coriolis force), and the resulting magnetic torques confine the tachocline.

Internal gravity waves - waves in a stably stratified fluid restored by buoyancy, excited by convection pounding on the base of the envelope - can also transport angular momentum and brake differential rotation (Charbonnel & Talon 2005; Rogers et al. 2008), and remain a live alternative or complement. Observationally, helioseismology pins down the present rotation profile, the tachocline thickness and its latitude dependence, and (through the surface lithium abundance) the depth of mixing permitted below the tachocline; but it constrains interior magnetic fields only weakly (existing bounds sit far above the roughly gauss-to-kilogauss amplitudes these models invoke), and the 1-10 m/s deep meridional flows required by the penetration-confinement scenario have not been detected.

Each scenario now has quantitative, partially successful realizations, reviewed in Garaud (2007) and in the introductions of the works below. Laminar axisymmetric solutions of the Gough-McIntyre model, with a confined fossil field and a uniformly rotating interior, exist numerically (Acevedo-Arreguín et al. 2013; Wood & Brummell 2018), but three-dimensional simulations of fossil fields show the fields tend to escape through the tachocline and couple the interior to envelope differential rotation unless the parameter regime is chosen carefully (Brun & Zahn 2006; Wood et al. 2011; Strugarek et al. 2011), and the stability of the confined states to non-axisymmetric magnetic instabilities is unproven. The quasi-linear theory of Leprovost & Kim (2006) supplies eddy viscosities anisotropic enough to satisfy the Spiegel-Zahn confinement criterion under idealized forcing, while acknowledging that shear-driven turbulence alone does not explain the deep interior's uniform rotation. On the simulation side, implicit large-eddy simulations - in which the numerical scheme's built-in dissipation plays the role of the unresolved subgrid turbulence - can produce a uniformly rotating stable lower layer and a tachocline without any explicit confining physics (Guerrero et al. 2012), a hint that stable stratification plus effective diffusion can self-organize the layer, though the result is strongly resolution-dependent and does not identify a solar mechanism. Most recently, a series of global dynamo simulations (2021-2026) achieved statistically steady confined tachoclines by dynamo-generated magnetic stresses, first in the viscous-spreading regime and then in the radiative-spreading regime relevant to the Sun; these still run at enhanced diffusivities and accelerated rotation. What actually operates in the Sun is undecided.

Problem Statement

Determine the physical mechanism - or combination of mechanisms - that confines the solar tachocline to its observed thinness (a few percent of the solar radius, centered near the base of the convection zone) and maintains the near-uniform rotation of the radiative interior against radiative spreading. A complete answer requires all three of the following parts together, and naming or ranking a candidate mechanism without the quantitative demonstration in (ii) is not sufficient: (i) identify which candidate process supplies the dominant confining angular-momentum transport in the actual Sun - anisotropic turbulent viscosity (in the Spiegel-Zahn sense or its modern reformulation), a fossil interior magnetic field confined by penetrating meridional circulation, skin-depth penetration of cyclic dynamo fields including their non-axisymmetric modes, self-excited dynamo action in the radiative interior, internal gravity-wave torques, or a combination; (ii) demonstrate quantitatively that the identified mechanism works at solar values of rotation rate, stratification, and microphysical diffusivities - not merely in simulations with enhanced diffusivities or accelerated rotation - including a quantitative (analytic or numerical) estimate of the amplitudes of the fields, flows, or wave fluxes it requires and of the resulting angular-momentum transport; and (iii) establish consistency with the helioseismic rotation profile and tachocline geometry, with observational bounds or detections of interior magnetic fields and deep meridional flow, and with constraints on mixing below the tachocline such as the solar lithium abundance.

References

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