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Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.
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We investigate the instability and nonlinear saturation of temperature-stratified Taylor-Couette flows in a finite height cylindrical gap and calculate angular-momentum transport in the nonlinear regime. The model is based on an incompressible fluid in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients itself, the differential rotation as well as the stratification due to the temperature gradient, are stable, together the system becomes subject of the stratorotational instability and nonaxisymmetric flow pattern evolve. This flow configuration transports angular momentum outwards and will therefor be relevant for astrophysical applications. The belonging viscosity $alpha$ coefficient is of the order of unity if the results are adapted to the size of an accretion disc. The strength of the stratification, the fluids Prandtl number and the boundary conditions applied in the simulations are well-suited too for a laboratory experiment using water and a small temperature gradient below five Kelvin. With such a rather easy realizable set-up the SRI and its angular momentum transport could be measured in an experiment.
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