Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability

Cyclic forcing on many timescales is believed to have a significant effect on various quasi-periodic, geophysical phenomena such as El Niño, the Quasi-Biennial Oscillation, and glacial cycles. This variability has been investigated by numerous previous workers, in models ranging from simple energy balance constructions to full general circulation models. We present a numerical study in which periodic forcing is applied to a highly idealised, two-layer, quasi-geostrophic model on a β-plane. The bifurcation structure and (unforced) behaviour of this particular model has been extensively examined by Lovegrove et al. (2001) and Lovegrove et al. (2002). We identify from their work three distinct regimes on which we perform our investigations: a steady, travelling wave regime, a quasi-periodic, modulated wave regime and a chaotic regime. In the travelling wave regime a nonlinear resonance is found. In the periodic regime, Arnol’d tongues, frequency locking and a Devil’s staircase is seen for small amplitudes of forcing. As the forcing is increased the Arnol’d tongues undergo a period doubling route to chaos, and for larger forcings still, the parameter space we explored is dominated by either period 1 behaviour or chaotic behaviour. In the chaotic regime we extract unstable periodic orbits (UPOs) and add the periodic forcing at periods corresponding to integer multiples of the UPO periods. We find regions of synchronization, similar to Arnol’d tongue behaviour but more skewed and centred approximately on these periods. The regions where chaos suppression took place are smaller than the synchronization regions, and are contained within them. Correspondence to: P. L. Read ([email protected])