Spirals and ribbons in counter-rotating Taylor-Couette flow: Frequencies from mean flows and heteroclinic orbits

A number of time-periodic flows have been found to have a property called RZIF: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state), an eigenvalue is obtained whose real part is zero and whose imaginary part is the nonlinear frequency. For two-dimensional thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy the RZIF property and standing waves which do not. We have investigated this property numerically for counter-rotating Couette-Taylor flow, in which a Hopf bifurcation gives rise to branches of upwards and downwards traveling spirals and ribbons which are an equal superposition of the two. In the regime that we have studied, we find that both spirals and ribbons satisfy the RZIF property. As the outer Reynolds number is increased, the ribbon branch is succeeded by two types of heteroclinic orbits, both of which connect saddle states containing two axially stacked pairs of axisymmetric vortices. One heteroclinic orbit is nonaxisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric.