Blue Sky Catastrophe in Double-Diffusive Convection

Abstract

A global bifurcation of the blue sky catastrophe type has been found in a small Prandtl number binary mixture contained in a laterally heated cavity. The system has been studied numerically applying the tools of bifurcation theory. The catastrophe corresponds to the destruction of an orbit which, for a large range of Rayleigh numbers, is the only stable solution. This orbit is born in a global saddle-loop bifurcation and becomes chaotic in a period doubling cascade just before its disappearance at the blue sky catastrophe.PACS numbers: 47.27. Te, 47.20.Ky, 44.25.+f Bifurcation theory has long been a very helpful tool in the analysis of complex dynamics of nonlinear systems [1,2]. Whereas different devised scenarios have been found in theoretical models with a few variables, there is a growing interest both in relating real systems with that kind of models (e.g. projecting their dynamics to some relevant degrees of freedom [3]) and in directly analyzing the behavior of these systems in terms of dynamical systems theory (by studying them either experimentally or by realistic models). In this context a great deal of work has been devoted to convection in fluids. Qualitative changes in the dynamics of fluxes maintained out of equilibrium by imposed thermal gradients have provided examples of most of the known bifurcations, and have become a main subject in the area of nonlinear dynamics.In this letter we will show the occurrence of a blue sky catastrophe [BSC] in double diffusive convection. The BSC is a codimension-1 bifurcation that consists in the destruction of a stable periodic orbit as its length and period tend to infinity, while the cycle remains bounded and located at a finite distance from all the equilibrium solutions [1,4]. This destruction is caused by the collision with a non-hyperbolic cycle that appears at the bifurcation point. While approaching the bifurcation the orbit increasingly coils in the zone where the new cycle will appear, which originates the divergence in both period and length. In that point the original cycle becomes an orbit homoclinic to the new cycle. This type of bifurcation is relatively exotic, but can easily be found in slow-fast (i.e. singularly perturbed) systems with at least two fast variables [5].We are interested in double-diffusive fluxes that occur when convection is driven by simultaneous thermal and concentration gradients in a binary mixture [6]. Doublediffusive convection in cavities with imposed vertical gradients exhibits very rich dynamics, and has been used as a system to study pattern formation [7] and transition to chaos [8]. The case of horizontal gradients, which arises naturally in applications such as crystal growth [9] or oceanography [6], has received less attention. In this work we numerically study this latter configuration for a small Prandtl number binary mixture. We consider the case when thermal and solutal buoyancy forces exactly compensate each other, which allows the existence of a quiescent (conductive) state [10,11,12,13]. We have found th...

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Transitions between symmetric and nonsymmetric regimes in binary-mixture convection

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Natural convection in a horizontal cylinder with axial rotation

“…We expect that the complex bifurcation scenarios we have identified for moderate values of the Rayleigh number and slow rotation rates lead to complex spatiotemporal dynamics, as it occurred in other analogous two-dimensional laterally heated systems, with similar aspect ratios and comparable values of the Prandtl number, that we had analyzed in the past [8][9][10][11]. However, a complete study of this emerging nonlinear dynamics would merit further study and is beyond the scope of this paper.…”

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