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Transitions to chaos in two-dimensional double-diffusive convection. (English) Zbl 0612.76047

(Authors’ summary.) The partial differential equations governing two- dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of periodic-doubling bifurcations.
Overstability occurs if the ratio of the solutal to the thermal diffusivity \(\tau <1\) and the solutal Rayleigh number \(R_ S\) is sufficiently large. Solutions have been obtained for two representative values of \(\tau\). For \(\tau =0.316\), \(R_ S=10^ 4\), symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number \(R_ T\) is increased. At higher values of \(T_ T\), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As \(R_ T\) is further increased more period-doubling cascades, follwed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist.
Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for \(\tau =0.1\) by varying \(R_ T\) for several different values of \(R_ S\). When \(R_ S\) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As \(R_ S\) is increased, chaos appears in the neighbourhood of these heteroclinic orbits. At higher values of \(R_ S\), chaos is found for a broader range in \(R_ T\). A truncated fifth-order model suggests that the appearance of chaos is associated with heteroclinic bifurcations.
Reviewer: H.S.Takhar

MSC:

76E15 Absolute and convective instability and stability in hydrodynamic stability
76F99 Turbulence
76R99 Diffusion and convection
76M99 Basic methods in fluid mechanics
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