Secondary bifurcations of stationary flows

  • Rita Meyer-Spasche
  • John H. Bolstad
  • Frank Pohl
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 549)


Numerical investigations of stationary Taylor vortex flows with periodic boundary conditions led Meyer-Spasche/Wagner [13] to speculate that there is a curve of loci of secondary bifurcations in the (period, Reynolds number)-plane which connects two double points, i.e. two intersection points of neutral curves of flows with different numbers of vortices (n-vortex flows, n = 2, 4, and a double vortex flow). Lortz et al. [12] proved analytically the existence of such a curve for a model problem derived from the equations of the Boussinesq approximation, and also the existence of a second such curve.

It remained unclear then if the results carry over to the full Boussinesq system and thus to the narrow-gap approximation, and how the results generalize to flows with other numbers of vortices. These questions were treated in [14]. In the present contribution we continue these investigations and present computations that justify speculations in [14] about the structure of other secondary interactions. We also present a curve of Hopf bifurcation points which was found in the course of computations discussed here, as a by-product.


Rayleigh Number Bifurcation Point Double Point Critical Curve Convection Roll 
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  1. 1.
    J.H. Bolstad, H.B. Keller: ‘Computation of anomalous modes in the Taylor experiment’. J. Comp. Phys. 69, 230–251 (1987)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    J.H. Bolstad: Following paths of symmetry-breaking bifurcation points. LLNL Tech. Rep. UCRL-99530, 1988; Int. J. Bif. Chaos 3 (1992) 559-576Google Scholar
  3. 3.
    J.H. Bolstad: Following paths of pitchfork bifurcation points. LLNL Tech. Rep. UCRL-JC-110570, 1992Google Scholar
  4. 4.
    J.H. Bolstad: Following paths of transcritical bifurcation points. LLNL Tech. Rep. UCRL-JC-113312, March, 1993.Google Scholar
  5. 5.
    F.H. Busse, A.C. Or: Subharmonic and asymmetric convection rolls. ZAMP 37, 608–623 (1986)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability. (Oxford University Press, London1961)zbMATHGoogle Scholar
  7. 7.
    J.H. Curry: A generalized Lorenz system. Commun. math. Phys. 60, 193–204 (1978)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    K. Gatermann, B. Werner: ‘Secondary Hopf bifurcation caused by steady-state steady-state mode interaction’. In: Pattern Formation: Symmetry Methods and Applications ed. by J. Chadam et al., Fields Institute Comm. 5, 1996Google Scholar
  9. 9.
    H.B. Keller: Numerical solution of bifurcation and nonlinear eigenvalue problems. 359–384 in Applications of Bifurcation Theory. P. Rabinowitz, ed., (Academic Press, New York 1977)Google Scholar
  10. 10.
    E. Knobloch, J. Guckenheimer: Convective transitions induced by a varying aspect ratio. Phys. Rev. A 27, 408–417 (1983)CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    E.N. Lorenz: ‘Deterministic nonperiodic flow’. J. Atmospheric Sci. 20, 130–141 (1963)CrossRefADSGoogle Scholar
  12. 12.
    D. Lortz, R. Meyer-Spasche, P. Petro.: A global analysis of secondary bifurcations in the Bénard problem and the relationship between the Bénard and Taylor problems. Methoden und Verfahren der mathematischen Physik 37 (Verlag Peter Lang, Frankfurt a.M., 1991) pp. 121–142Google Scholar
  13. 13.
    R. Meyer-Spasche, M. Wagner: The basic (n, 2n)-fold of steady axisymmetric Taylor vortex flows. In: The Physics of Structure Formation, ed. by W. Güttinger, G. Dangelmayr (Springer, Berlin, New York 1987) pp. 166–178Google Scholar
  14. 14.
    R. Meyer-Spasche: Pattern Formation in Viscous Flows, ISNM vol. 128, (Birkhäuser, Basel, Boston 1999)Google Scholar
  15. 15.
    L. A. Segel: The non-linear interaction of two disturbances in the thermal convection problem. J. Fluid Mech. 14, 97–114 (1962)zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    J.H. Wilkinson: Rounding Errors in Algebraic Processes, Her Magesty’s Stationery Office, 1963, Springer Verlag 1969, Dover Reprint 1994Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Rita Meyer-Spasche
    • 1
  • John H. Bolstad
    • 2
  • Frank Pohl
    • 1
  1. 1.MPI für PlasmaphysikEURATOM-AssociationGarchingGermany
  2. 2.Laurence Livermore National Laboratory, L-23U of CaliforniaLivermoreUSA

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