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A low-order model of the shear instability of convection: chaos and the effect of noise

The authors consider a third-order system of ordinary differential equations aimed at modelling the shear instability of tall thin cells as found, for example, in thermohaline convection. The nonlinear interactions between the zeroth and first harmonics of the convective cells (A and B) fuel the growth of a horizontal shear (C) which, in turn, becomes of sufficient magnitude to destabilise the convective motions. The most interesting feature of the model is that in certain parameter regimes a small amount of random noise has a tremendous influence on the dynamics-in others the noise is unimportant and the system displays many typical features of low-order dynamical systems, such as sequences of period-doubling bifurcations. When the decay rates of B and C are comparable, analytic solutions can be found for both regimes; for the former by solving the Fokker-Planck equation describing the probability distribution of A, B and C, for the latter by reducing the equations to a simple one-dimensional map. These analytic results are a considerable aid in understanding the numerical solutions of the equations for general parameter values.