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

Abstract

We 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 dynamicein 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.

Noise-induced statistically stable oscillations in a deterministically divergent nonlinear dynamical system

“…Note that the system we consider serves only as a prototype of this behavior and it is not derived from the convection problem. On the other hand, a 3D model with similar behavior was systematically derived for a homogeneous thermohaline convection problem directly from the partial differential equations [10]. Here we aim to capture the central qualitative features in a two dimensional model.…”

Predictability of noise-controlled dynamics

“…The phenomenon has been reported in a one-dimensional map [5], in repeated passage through a saddle-node bifurcation [13], and in ordinary differential equations describing the resonant interaction of wave modes [6,10], the intermittent destabilization of convection by shear [7,11], pulsating laser oscillations [9] and plane Poiseuille flow [14]. Numerical evidence for similar behavior has been found in a set of partial differential equations describing the shear instability of thermohaline convection [8].…”

“…Successive maxima are independent, depending only the realization of the noise in the slow phase leading up to it. This is the key to the simplification of the dynamics produced by noise [6,7,10,11]. We shall calculate the density of maxima of |a a a t (0)| as follows.…”

“…2 illustrates the increasingly complicated behavior of solutions as a function of α. A description in terms of a map of successive turning points of a a a t (0) can be constructed, where the fast phase is also analysed [6,7,10,11]. In the simplest approximation [6,7,10,11], the fast phase begins at a a a max , the maximum of a a a t (0), and ends at a a a t (0) = a a a min where a a a max + a a a min = κ.…”

“…In noise-controlled dynamics [5][6][7][8][9][10][11][12], timescales and qualitative behaviors of solutions are controlled by tiny amounts of additive noise. The phenomenon has been reported in a one-dimensional map [5], in repeated passage through a saddle-node bifurcation [13], and in ordinary differential equations describing the resonant interaction of wave modes [6,10], the intermittent destabilization of convection by shear [7,11], pulsating laser oscillations [9] and plane Poiseuille flow [14].…”