Spirals and ribbons: frequency prediction from mean flows and a new heteroclinic orbit

A number of time-periodic flows, (e.g. the wake of a circular cylinder and the flow over an open cavity) have been found to have the RZIF property: a linear stability analysis carried out about the temporal mean (rather than the usual steady state) leads to an eigenvalue whose Real part is near Zero and whose Imaginary part is the nonlinear Frequency.  For 2D thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy RZIF 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. In addition, the ribbon branch is succeeded by a heteroclinic orbit anchored by unstable axisymmetric Taylor-vortex flows with excursions into the ribbon regime.