Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T06:03:58.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillations in double-diffusive convection

Published online by Cambridge University Press:  20 April 2006

L. N. Da Costa ,
E. Knobloch  and
N. O. Weiss
L. N. Da Costa
Affiliation:
Department of Physics, Harvard University, Cambridge, Massachusetts
E. Knobloch
Affiliation:
Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts
N. O. Weiss
Affiliation:
Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts Permanent address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.
Get access Rights & Permissions [Opens in a new window]

Abstract

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 small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, 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 RS 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 RT. The chaotic solutions persist as RT 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 period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 109 , August 1981 , pp. 25 - 43
Copyright
© 1981 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.Google Scholar
Baker, N. H., Moore, D. W. & Spiegel, E. A. 1971 Aperiodie behaviour of a nonlinear oscillator. Quart. J. Mech. Appl. Math. 24, 391–422.Google Scholar
Boldrighini, C. & Franceschini, V. 1979 A five-dimensional truncation of the plane incompressible Navier-Stokes equations. Commun. Math. Phys. 64, 159–170.Google Scholar
Coullet, P., Tresser, C. & Arnéodo, A. 1979 Transition to stochasticity for a class of forced oscillators. Phys. Lett. A 72, 268–270.Google Scholar
Curry, J. H. 1978 A generalized Lorenz system. Commun. Math. Phys. 60, 193–204.Google Scholar
Feigenbaum, M. J. 1978 Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52.Google Scholar
Feigenbaum, M. J. 1979 The onset spectrum of turbulence. Phys. Lett. A 74, 375–378.Google Scholar
Franceschini, V. 1980 A Feigenbaum sequence of bifurcations in the Lorenz model. J. Stat. Phys. 22, 397–407.Google Scholar
Franceschini, V. & Tebaldi, C. 1979 Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations. J. Stat. Phys. 21, 707–726.Google Scholar
Haken, H. 1975 Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53, 77–78.Google Scholar
Hopf, E. 1942 Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math. Phys. Sächs. Akad. Wiss. 94, 1–22 (transl. in Marsden & McCracken 1976, pp. 163–193).Google Scholar
Huppert, H. E. 1976 Transitions in double-diffusive convection. Nature 263, 20–22.Google Scholar
Huppert, H. E. 1977 Thermosolutal convection. In Problems of Stellar Convection (ed. E. A. Spiegel & J. P. Zahn), Lecture Notes in Physics, vol. 71, pp. 239–254. Springer.
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Ito, A. 1979 Successive subharmonic bifurcations and chaos in a nonlinear Mathieu equation. Prog. Theor. Phys. 61, 815–824.Google Scholar
Jeans, J. H. 1928 Astronomy and Cosmogony, pp. 179–186. Cambridge University Press.
Knobloch, E. & Proctor, M. R. E. 1981 Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar
Knobloch, E., Weiss, N. O. & Da Costa, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. (in press).
Lorenz, E. N. 1963 Deterministic nonperiodie flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Mclaughlin, J. B. & Martin, P. C. 1975 Transition to turbulence in a statically stressed fluid system. Phys. Rev. A 12, 186–203.Google Scholar
Malkus, W. V. R. 1972 Non-periodic convection at high and low Prandtl number. Mém. Soc. Roy. Sci. Liège 4, 125–128.Google Scholar
Marsden, J. E. & McCracken, M. 1976 The Hopf Bifurcation and its Applications. Springer.
Marzec, C. J. & Spiegel, E. A. 1980 Ordinary differential equations with strange attractors. SIAM J. Appl. Math. 38, 403–421.Google Scholar
May, R. M. 1976 Simple mathematical models with very complicated dynamics. Nature 261, 459–467.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh — Bénard convection. J. Fluid Mech. 58, 289–312.Google Scholar
Morioka, N. & Shimizu, T. 1978 Transition between turbulent and periodie states in the Lorenz model. Phys. Lett. A 66, 447–449.Google Scholar
Poincaré, H. 1885 Sur l’équilibre d'une masse fluide animée d'un mouvement de rotation. Acta Math. 7, 259–380.Google Scholar
Pomeau, Y. 1977 Turbulence: determinism and chaos, in Problems of Stellar Convection (ed. E. A. Spiegel & J. P. Zahn). Lecture Notes in Physics, vol. 71, pp. 337–348. Springer.
Robbins, K. A. 1977 A new approach to suberitical instability and turbulent transitions in a simple dynamo. Math. Proc. Camb. Phil. Soc. 82, 309–325.Google Scholar
Robbins, K. A. 1978 Periodie solutions and bifureation structure at high R in the Lorenz model. SIAM J. Appl. Math. 36, 457–472.Google Scholar
Rubenfeld, L. A. & Siegmann, W. L. 1977 Nonlinear dynamic theory for a double-diffusive convection model. SIAM J. Appl. Math. 32, 871–894.Google Scholar
Sattinger, D. H. 1973 Topics in stability and bifurcation theory. Lecture Notes in Mathematics, vol. 309, Springer.
Schechter, R. S., Velarde, M. G. & Platten, J. K. 1974 The two component Bénard problem. In Advances in Chemical Physics (eds. I. Prigogine and S. A. Rice), vol. 26, pp. 265–301. Interscience.
Shimada, I. & Nagashima, T. 1978 The iterative transition phenomenon between periodic and turbulent states in a dissipative dynamical system. Prog. Theor. Phys. 59, 1033–1036.Google Scholar
Shimizu, T. & Morioka, S. 1978 Chaos and limit cycles in the Lorenz model. Physics Lett. A 66, 182–184.Google Scholar
Siegmann, W. L. & Rubenfeld, L. A. 1975 A nonlinear model for double-diffusive convection. SIAM J. Appl. Math. 29, 540–557.Google Scholar
Spiegel, E. A. 1972 Convection in stars. II. Special effects. Ann. Rev. Astron. Astrophys. 10, 261–304.Google Scholar
Stern, M. E. 1960 The ‘salt-fountain’ and thermohaline convection. Tellus 12, 172–175.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Veronis, G. 1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1–17.Google Scholar
Veronis, G. 1968 Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.Google Scholar