The 1: 2 Mode Interaction in Rayleigh-bÉnard convection with and without Boussinesq Symmetry

Nonlinear two-dimensional Rayleigh–Benard convection with periodic boundary conditions in the horizontal is studied for spatial periods near the 1:2 steady state mode interaction. The boundary conditions at the bottom are no-slip, and convection is driven by a fixed imposed temperature difference across the layer. Homotopic continuation is used to continue the boundary conditions at the top from no-slip (β=0) to stress-free (β=1). When β=0 and non-Boussinesq effects are absent the system has midplane reflection symmetry and the 1:2 resonance is weak. When β=1 this symmetry is strongly broken and the resonance is strong. The transition between these two cases is explored for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

[1]  E. Knobloch,et al.  Resonant mode interactions in Rayleigh-Benard convection , 1998 .

[2]  M. Proctor,et al.  Strong spatial resonance and travelling waves in benard convection , 1987 .

[3]  Edgar Knobloch,et al.  Robust heteroclinic Cycles in Two-Dimensional Rayleigh-bÉnard convection without Boussinesq Symmetry , 2002, Int. J. Bifurc. Chaos.

[4]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[5]  E. Knobloch,et al.  Three-dimensional Marangoni-Benard flows in square and nearly square containers , 2001 .

[6]  John Guckenheimer,et al.  Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry , 1988 .

[7]  D. Armbruster.  0(2)-symmetric bifurcation theory for convection rolls , 1987 .

[8]  Mark R. Proctor,et al.  The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance , 1988, Journal of Fluid Mechanics.

[9]  David G. Schaeffer,et al.  Qualitative analysis of a model for boundary effects in the Taylor problem , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  D. Moore,et al.  Asymmetric oscillations in thermosolutal convection , 1991, Journal of Fluid Mechanics.

[11]  Hopf Bifurcation with O(2) Symmetry , 1988 .

[12]  S. Cox.  Mode interaction in Rayleigh-Be´nard convection , 1996 .

[13]  E. Knobloch,et al.  On degenerate Hopf bifurcation with broken O(2) symmetry , 1988 .

[14]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[15]  Minimal model of binary fluid convection. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[16]  E. Knobloch,et al.  Symmetry and Symmetry-Breaking Bifurcations in Fluid Dynamics , 1991 .

[17]  Edgar Knobloch,et al.  The 1: 2 Mode Interaction in Rayleigh-BéNard convection with Weakly Broken midplane Symmetry , 2001, Int. J. Bifurc. Chaos.

[18]  Martin Golubitsky,et al.  Symmetries and pattern selection in Rayleigh-Bénard convection , 1984 .

[19]  Busse,et al.  Hexagonal convection cells under conditions of vertical symmetry. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  P. Metzener,et al.  Strong resonance in two‐dimensional non‐Boussinesq convection , 1994 .

[21]  Edgar Knobloch,et al.  New type of complex dynamics in the 1:2 spatial resonance , 2001 .

[22]  Friedrich H. Busse,et al.  Nonlinear evolution of spatio-temporal structures in dissipative continuous systems , 1990 .

[23]  E. Knobloch.  SYMMETRY AND INSTABILITY IN ROTATING HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC FLOWS , 1996 .

[24]  Subharmonic and asymmetric convection rolls , 1986 .

[25]  V. S. Gils.  Hopf bifurcation and symmetry: travelling and standing waves on the circle , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  Edgar Knobloch,et al.  The double Hopf bifurcation with 2:1 resonance , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  P. Matthews,et al.  Compressible magnetoconvection in oblique fields: linearized theory and simple nonlinear models , 1992, Journal of Fluid Mechanics.

[28]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[29]  Friedrich H. Busse,et al.  The stability of finite amplitude cellular convection and its relation to an extremum principle , 1967, Journal of Fluid Mechanics.