The combined effects of vertical and horizontal shear instabilities

TL;DR

This study uses direct numerical simulations to analyze how combined vertical and horizontal shear instabilities influence turbulence in stellar radiative zones, revealing that vertical shear generally does not affect turbulence unless its Richardson number is below one.

Contribution

It provides new insights into the nonlinear saturation of shear instabilities with combined shear directions, showing vertical shear's limited impact in most stellar conditions.

Findings

Vertical shear has negligible effect on turbulence unless Richardson number < 1.

Horizontal shear dominates the turbulence dynamics in most stellar regimes.

Vertical shear can be ignored in stellar mixing models for regions with high Richardson number.

Abstract

Shear instabilities can be the source of significant amounts of turbulent mixing in stellar radiative zones. Past attempts at modeling their effects (either theoretically or using numerical simulations) have focused on idealized geometries where the shear is either purely vertical or purely horizontal. In stars, however, the shear can have arbitrary directions with respect to gravity. In this work, we use direct numerical simulations to investigate the nonlinear saturation of shear instabilities in a stably stratified fluid, where the shear is sinusoidal in the horizontal direction, and either constant or sinusoidal in the vertical direction. We find that, in the parameter regime studied here (non-diffusive, fully turbulent flow), the mean vertical shear does not play any role in controlling the dynamics of the resulting turbulence unless its Richardson number is smaller than one (approximately). As most stellar radiative regions have a Richardson number much greater than one, our result implies that the vertical shear can essentially be ignored in the computation of the vertical mixing coefficient associated with shear instabilities for the purpose of stellar evolution calculations, even when it is much larger than the horizontal shear (as in the solar tachocline, for instance).