Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counter-rotating Taylor-Couette flow

Taylor-Couette flow in a small aspect-ratio wide-gap annulus in the counter-rotating regime is investigated by solving the full 3D Navier-Stokes equations. The system is invariant under rotations about the axis, reflection about the mid-plane, and time translations. A systematic investigation is presented, both in terms of the flow physics, the numerical simulations and from a dynamical systems perspective. The dynamics are primarily associated with the behavior of the jet of angular momentum that emerges from the inner cylinder boundary layer at about the mid-plane. The sequence of bifurcations as the rotation is increased consists of a Hopf bifurcation breaking the reflection symmetry and leading to an axisymmetric limit cycle associated to an invariant one-torus manifold with a spatio-temporal symmetry. This undergoes a Hopf bifurcation breaking axisymmetry, leading to quasi-periodic solutions evolving on a 2-torus that is only setwise symmetric due to precesion. These undergo a further Hopf bifurcation introducing a third incommensurate frequency leading to a 3-torus. On it,as the rotation is further increased, a SNIC bifurcation happens, destroying the 3-torus and leaving a pair of symmetrically related 2-tori states on which all symmetries of the system have been broken.