Critical Self-Tuning: the Example of Zero Prandtl Number Convection

(Received 2 September 1994, revised 6 June 1995, accepted 20 February 1996)

Abstract
We present a new type of bifurcation scenario where nonlinear saturation of a stationary instability takes place only because of the competition with an oscillatory one. This is shown on the example of convection at zero Prandtl number between stress-free boundaries. We show with direct numerical simulations that time-dependent wavy rolls are generated at the onset of convection. Using a Galerkin model, we analyze the nonlinear interactions between rolls and waves and find that they maintain the system in the vicinity of the oscillatory instability onset, thus preventing the blow-up of the growing nonlinear roll solution. An interesting feature of this type of dynamics is that the system is self-tuned in the vicinity of a transition point.