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Statistical state dynamics modes and equilibria underlie the structure and mechanism of wide channel Couette turbulence

Wide channel Couette (WCC) turbulence is striking in being dominated by a large-scale spanwise periodic structure composed of streamwise streaks and associated roll superstructures. This apparent equilibrium is shown in this work to correspond to a fixed point solution of the Navier-Stokes equations expressed in the statistical state dynamics (SSD) framework. Moreover, this fixed point solution is found to be rank-three, consisting of one analytically identified roll-streak structure (RSS) constituting the first cumulant of the SSD, and two analytically determined eigenmodes supporting the second cumulant. This minimal representation captures both the structure and the dynamics of WCC turbulence, while the remaining spectral components contribute negligibly to the equilibrium dynamics. Turbulent Couette flows other than WCC can be understood to be limit cycle and chaotic extensions supported by the same underlying mechanism as the fixed point WCC turbulence except for the two eigenmodes supporting WCC turbulence being replaced by two Floquet modes and two Lyapunov vectors, respectively. These results provide an analytic solution for turbulence in Couette flow.