Exact coherent structures in stably stratified plane Couette flow

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 D 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 DO.Re..2/or equivalently when the Rayleigh number Ra VD ..Ri_bRe₂Pr D 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 D 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 D 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.