D. Olvera

We discuss how searching for finite-amplitude disturbances of a given energy that maximize their subsequent energy growth after a certain later time $T$ can be used to probe the phase space around a reference state and ultimately to find other nearby solutions. The procedure relies on the fact that of all the initial disturbances on a constant-energy hypersphere, the optimization procedure will naturally select the one that lies closest to the stable manifold of a nearby solution in phase space if $T$ is large enough. Then, when in its subsequent evolution the optimal disturbance transiently approaches the new solution, a flow state at this point can be used as an initial guess to converge the solution to machine precision. We illustrate this approach in plane Couette flow by rediscovering the spanwise-localized “snake” solutions of Schneider et al. [ Phys. Rev. Lett. 104, 104501 (2010)], probing phase space at very low Reynolds numbers (less than $127.7$) where the constant-shear solution is believed to be the global attractor and examining how the edge between laminar and turbulent flow evolves when stable stratification eliminates the turbulence. We also show that the steady snake solution smoothly delocalizes as unstable stratification is gradually turned on until it connects (via an intermediary global three-dimensional solution) to two-dimensional Rayleigh-Bénard roll solutions.

The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds–Richardson number ( $Re$ – $Ri_{b}$ ) space for a fluid of unit Prandtl number $(Pr=1)$ using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge tracking – EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (J. Fluid Mech., vol. 745, 2014, pp. 25–61) – and found to connect with two-dimensional convective roll solutions when tracked to negative $Ri_{b}$ (the Rayleigh–Bénard problem with shear). Both these states and Nagata’s (J. Fluid Mech., vol. 217, 1990, pp. 519–527) original exact solution feel the presence of stable stratification when $Ri_{b}=O(Re^{-2})$ or equivalently when the Rayleigh number $Ra:=-Ri_{b}Re^{2}Pr=O(1)$ . This is confirmed via a stratified extension of the vortex wave interaction theory of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at $Ri_{b}=O(Re^{-2/3})$ . This corresponds to a stratified version of the boundary region equations regime of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85). Increasing the stratification further appears to lead to a third, ultimate regime where $Ri_{b}=O(1)$ in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar–turbulent boundary in the ( $Re$ – $Ri_{b}$ ) plane are briefly discussed.