Statistical State Dynamics Eigenmodes and Equilibria Form the Structural Basis of Couette Turbulence

Brian F. Farrell; Petros J. Ioannou

Statistical State Dynamics Eigenmodes and Equilibria Form the Structural Basis of Couette Turbulence

Fluid Dynamics 2026-07-17 v1

Authors: Brian F. Farrell , Petros J. Ioannou

Abstract

Wide channel Couette (WCC) turbulence consists primarily of steady roll streak structures (RSS) maintained by a self-sustaining process, yet the Navier Stokes equations in velocity variables admit no linear RSS instability or stable equilibrium, leaving the WCC turbulent state's analytical basis obscure. We show that in a second order statistical state dynamics (SSD) a fixed-point state of Couette turbulence arises from a modal RSS instability and equilibrates as an exact, attracting RSS at spanwise wavenumber 3. This fixed point comprises a rank-1 streamwise mean flow and a conjugate pair of neutral eigenmodes, regularized by roll advection rather than viscosity, forming a rank-2 fluctuation covariance. WCC turbulence arises as a spanwise tiling by this fixed-point RSS unit cell.

Keywords

statistical state dynamics Couette turbulence roll-streak structures eigenmodes fixed point self-sustaining process

Cite

@article{arxiv.2607.15842,
  title  = {Statistical State Dynamics Eigenmodes and Equilibria Form the Structural Basis of Couette Turbulence},
  author = {Brian F. Farrell and Petros J. Ioannou},
  journal= {arXiv preprint arXiv:2607.15842},
  year   = {2026}
}

Comments

6 pages, 5 figures, submitted to PRL