Disruption of SSP/VWI states by a stable stratification

Tom Eaves (Lead / Corresponding author), Colm-cille P. Caulfield

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)
98 Downloads (Pure)


We identify ‘minimal seeds’ for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density Ec that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation 2H, moving with relative velocity 2ΔU, across which a constant density difference 2Δρ from a reference density ρr is maintained. To find minimal seeds, we use the ‘direct-adjoint-looping’ (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system’s ‘edge state’. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883–900), or equivalently finite Reynolds number solutions of the vortex–wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers RiB = gΔρH/(ρrΔU2) = O(Re-1), and exhibit chaotic motion for larger RiB. We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when RiB = O(Re-2).
Original languageEnglish
Pages (from-to)548-564
Number of pages17
JournalJournal of Fluid Mechanics
Early online date6 Nov 2015
Publication statusPublished - 10 Dec 2015


  • Nonlinear instability
  • transition to turbulence
  • variational methods


Dive into the research topics of 'Disruption of SSP/VWI states by a stable stratification'. Together they form a unique fingerprint.

Cite this