Angular momentum transport by the GSF instability: non-linear simulations at the equator

Abstract

We present an investigation into the nonlinear evolution of the Goldreich-Schubert-Fricke (GSF) instability using axisymmetric and three-dimensional simulations near the equator of a differentially rotating radiation zone. This instability may provide an important contribution to angular momentum transport in stars and planets. We adopt a local Boussinesq Cartesian shearing box model, which represents a small patch of a differentially rotating stellar radiation zone. Complementary simulations are also performed with stress-free, impenetrable boundaries in the local radial direction. The linear and nonlinear evolution of the equatorial axisymmetric instability is formally equivalent to the salt fingering instability. This is no longer the case in 3D, but we find that the instability behaves nonlinearly in a similar way to salt fingering. Axisymmetric simulations -- and those in 3D with short dimensions along the local azimuthal direction -- quickly develop strong jets along the rotation axis, which inhibit the instability and lead to predator-prey-like temporal dynamics. In 3D, the instability initially produces homogeneous turbulence and enhanced momentum transport, though in some cases jets form on a much longer timescale. We propose and validate numerically a simple theory for nonlinear saturation of the GSF instability and its resulting angular momentum transport. This theory is straightforward to implement in stellar evolution codes incorporating rotation. We estimate that the GSF instability could contribute towards explaining the missing angular momentum transport required in red giant stars, and play a role in the long-term evolution of the solar tachocline.