Oscillations in double-diffusive convection

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently smell and the solutal Rayleigh number, R,, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, R,, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of R, the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of Rs a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of R,. The chaotic solutions persist as R, is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that perioddoubling, followed by the appearance of a strange attractor, is a characteristic feature o€ double-diffusive convection.

[1]  D. Sattinger.  Topics in stability and bifurcation theory , 1973 .

[2]  W. Siegmann,et al.  A Nonlinear Model for Double-Diffusive Convection , 1975 .

[3]  Akira Itō,et al.  Successive Subharmonic Bifurcations and Chaos in a Nonlinear Mathieu Equation , 1979 .

[4]  Edward A. Spiegel,et al.  Ordinary differential equations with strange attractors , 1980 .

[5]  Hermann Haken,et al.  Analogy between higher instabilities in fluids and lasers , 1975 .

[6]  H. Huppert.  Transitions in double-diffusive convection , 1976, Nature.

[7]  Paul C. Martin,et al.  Transition to turbulence in a statically stressed fluid system , 1975 .

[8]  Transition between turbulent and periodic states in the Lorenz model , 1978 .

[9]  H. Poincaré.  Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation , 1885, Bulletin astronomique.

[10]  D. W. Moore,et al.  APERIODIC BEHAVIOUR OF A NON-LINEAR OSCILLATOR , 1971 .

[11]  S. Rosseland.  Astronomy and Cosmogony , 1928, Nature.

[12]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[13]  Edgar Knobloch,et al.  Nonlinear periodic convection in double-diffusive systems , 1981 .

[14]  E. Spiegel.  Convection in Stars II. Special Effects , 1972 .

[15]  Alain Arneodo,et al.  Transition to stochasticity for a class of forced oscillators , 1979 .

[16]  K. Robbins,et al.  Periodic solutions and bifurcation structure at high R in the Lorenz model , 1979 .

[17]  H. Huppert,et al.  Nonlinear double-diffusive convection , 1976, Journal of Fluid Mechanics.

[18]  Mitchell J. Feigenbaum,et al.  The onset spectrum of turbulence , 1979 .

[19]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[20]  Chaos and limit cycles in the Lorenz model , 1978 .

[21]  W. Siegmann,et al.  Nonlinear dynamic theory for a double-diffusive convection model , 1977 .

[22]  A. E. Gill,et al.  On thermohaline convection with linear gradients , 1969, Journal of Fluid Mechanics.

[23]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[24]  George Veronis,et al.  Effect of a stabilizing gradient of solute on thermal convection , 1968, Journal of Fluid Mechanics.

[25]  M. Stern.  The "Salt-Fountain" and Thermohaline Convection , 1960 .

[26]  Carlo Boldrighini,et al.  A five-dimensional truncation of the plane incompressible Navier-Stokes equations , 1979 .

[27]  D. Moore,et al.  Two-dimensional Rayleigh-Benard convection , 1973, Journal of Fluid Mechanics.

[28]  Y. Pomeau.  Turbulence : Determinism and chaos , 1977 .

[29]  I. Shimada,et al.  The Iterative Transition Phenomenon between Periodic and Turbulent States in a Dissipative Dynamical System , 1978 .

[30]  E. Knobloch,et al.  Oscillatory and steady convection in a magnetic field , 1981, Journal of Fluid Mechanics.

[31]  K. Robbins,et al.  A new approach to subcritical instability and turbulent transitions in a simple dynamo , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.

[32]  James H. Curry,et al.  A generalized Lorenz system , 1978 .

[33]  Claudio Tebaldi,et al.  Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations , 1979 .

[34]  Valter Franceschini,et al.  A Feigenbaum sequence of bifurcations in the Lorenz model , 1980 .