Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus

Abstract

A series of laboratory experiments is presented investigating regular and chaotic baroclinic waves in a high–Prandtl number fluid contained in a rotating vessel and subjected to a horizontal temperature gradient. The study focuses on nonlinear aspects of mixed–mode states at moderate values of the forcing parameters within the regular wave regime. Frequency entrainment and phase locking of resonant triads and sidebands were found to be widespread. Cases were analysed in phase space reconstructions through a singular value decomposition of multi–variate time series. Four forms of mixed–mode states were found, each in well–defined regions of parameter space: (1) a nonlinear interference vacillation associated with strong phase locking through higher harmonics; (2) a modulated amplitude vacillation showing strong phase coherence in triads involving the long wave; (3) an intermittent bursting of secondary modes; (4) an attractor switching flow, where the dominant wave number switched at irregular intervals between two possible wave numbers. Many of the mixed–mode states are suggested to arise from homoclinic bifurcations, whereas no secondary Hopf bifurcations were found. One of the postulated homoclinic bifurcations was consistent with a bifurcation through intermittency. The bifurcation sequences, however, were strongly affected by phase locking between different wave number components and frequency locking between drift and modulation frequencies. When all free frequencies were locked, the flow reduced to a limit cycle which subsequently became unstable through an incomplete period–doubling cascade. The only observed case of torus–doubling was also associated with strong phase locking. Most of the observed regimes were consistent with low–dimensional dynamics involving a limited number of domain–filling modes, which can be represented in phase space reconstructions and characterized by invariants such as attractor dimensions and the Lyapunov exponents. Some flows associated with a weak structural vacillation, however, were not consistent with low–dimensional dynamics. It appeared rather that they were the result of spatially localized instabilities consistent with high–dimensional dynamics, which can be parametrized as stochastic dynamics.