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Abstract Differentially rotating stars and planets transport angular momentum (AM) internally due to turbulence at rates that have long been a challenge to predict reliably. We develop a self-consistent saturation theory, using a statistical closure approximation, for hydrodynamic turbulence driven by the axisymmetric Goldreich–Schubert–Fricke instability at the stellar equator with radial differential rotation. This instability arises when fast thermal diffusion eliminates the stabilizing effects of buoyancy forces in a system where a stabilizing entropy gradient dominates over the destabilizing AM gradient. Our turbulence closure invokes a dominant three-wave coupling between pairs of linearly unstable eigenmodes and a near-zero frequency, viscously damped eigenmode that features latitudinal jets. We derive turbulent transport rates of momentum and heat and provide them in analytic forms. Such formulae, free of tunable model parameters, are tested against direct numerical simulations; the comparison shows good agreement. They improve upon prior quasi-linear or “parasitic saturation” models containing a free parameter. Given model correspondences, we also extend this theory to heat and compositional transport for axisymmetric thermohaline-instability-driven turbulence in certain regimes.
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This content will become publicly available on May 21, 2026
Toward a Self-consistent Hydrodynamical Model of the Solar Tachocline
The solar tachocline is a thin internal boundary layer in the Sun located between the differentially rotating convection zone and the uniformly rotating region of the radiative interior beneath. E. A. Spiegel & J. P. Zahn proposed the first hydrodynamical model, which here we call SZ92, arguing that the tachocline is essentially in a steady state of thermal-wind balance, angular-momentum balance, and thermal equilibrium. Angular momentum transport in their model is assumed to be dominated by strongly anisotropic turbulence, which is primarily horizontal, owing to the strong stable stratification of the radiative interior. By contrast, the heat transport is assumed to be dominated by a predominantly vertical diffusive heat flux, owing to the thinness of the tachocline. In this paper, we demonstrate that these assumptions are not consistent with the new model of stratified turbulence recently proposed by G. P. Chini et al. and K. Shah et al., which has been numerically validated by P. Garaud et al. We then propose a simple self-consistent alternative to the SZ92 model—namely, a scenario wherein angular momentum and heat transport are both dominated by horizontal turbulent diffusion. The thickness of the tachocline in the new model scales as Ω/N, where Ω is the mean angular velocity of the Sun and N is the characteristic buoyancy frequency in the tachocline region. We discuss other properties of the model and show that it has several desirable features but does not resolve some of the other well-known problems of the SZ92 model.
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- PAR ID:
- 10607874
- Publisher / Repository:
- American Astronomical Society
- Date Published:
- Journal Name:
- The Astrophysical Journal
- Volume:
- 985
- Issue:
- 2
- ISSN:
- 0004-637X
- Page Range / eLocation ID:
- 151
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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