Three-dimensional shear-flow instability saturation via stable modes

Abstract

Turbulence in three dimensions (3D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here that the shear-flow-instability-supplied 3D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, and such inverse transfer is more efficient for turbulent fluctuations in 3D than in two dimensions (2D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in 3D than in 2D (≈70% vs ≈50%). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of 2D shear flow. The more widely occurring 3D turbulence is thus inherently different from the commonly studied 2D turbulence, despite both saturating via stable modes.

Predicting the Slowing of Stellar Differential Rotation by Instability-driven Turbulence

“…For example, recent simulations of fully 3D spherical shell nonrotating fingering convection have exhibited strong, large-scale jets (Tassin et al 2023). Such coherent jets are ubiquitous in various settings, such as in geo-and astrophysical observations, numerical simulations of 3D shear-flow-instability-driven turbulence (Tripathi et al 2023b), and in laboratory fusion plasmas (Terry 2019). The examples show that largescale flows can emerge even in global geometry.…”

“…To expedite analytic calculations, one may solve for the adjoint solutions Y j of L, which are the eigenvectors of L † . Such adjoint solutions Y j form a biorthogonal basis with the eigenvectors X j of L (Fraser et al 2021;Tripathi et al 2022aTripathi et al , 2023aTripathi et al , 2023b. That is,…”

“…Equation (2) shows that on the (k x , k z )-plane, the growth rate exhibits strong anisotropy: fluctuations with k x = 0 ("elevator modes") grow the fastest, whereas those with k z = 0 are linearly stable. This observation is critical for the nonlinear saturation of the GSF instability, because the anisotropy of the linear physics, in particular the k z = 0 fluctuation, can impose its anisotropy on the nonlinear energy transfer, which is otherwise isotropic; such consequential effects have been found in various systems such as 3D Kelvin-Helmholtz instability (Tripathi et al 2023b), rotating (Waleffe 1993;Smith & Waleffe 1999) and stably stratified turbulence (Riley & Lelong 2000), and turbulence with an external magnetic field in astrophysical (Ng & Bhattacharjee 1996;Du et al 2023) and fusion plasmas (Biskamp & Zeiler 1995;Terry 2004). Using the complete basis provided by the eigenvectors of the linear operator L, we can decompose the arbitrary incompressible fluctuation…”