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Onset of Küppers–Lortz-like dynamics in finite rotating thermal convection

Published online by Cambridge University Press:  28 January 2010

A. RUBIO ,
J. M. LOPEZ  and
F. MARQUES
A. RUBIO
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe AZ 85287, USA
J. M. LOPEZ*
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe AZ 85287, USA
F. MARQUES
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
*
†Email address for correspondence: [email protected]
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Abstract

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

JFM classification

Convection: Convection Geophysical and Geological Flows: Rotating flows Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 337 - 357
Copyright
Copyright © Cambridge University Press 2010

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References

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Rubio et al. supplementary movie

Movie 1. This movie corresponds to figure 5(a) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2372, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 76,500 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 858.6 KB

Rubio et al. supplementary movie

Movie 1. This movie corresponds to figure 5(a) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2372, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 76,500 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 667.4 KB

Rubio et al. supplementary movie

Movie 2. This movie corresponds to figure 5(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2373, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 95,400 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 1.2 MB

Rubio et al. supplementary movie

Movie 2. This movie corresponds to figure 5(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2373, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 95,400 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 940.8 KB

Rubio et al. supplementary movie

Movie 3. This movie corresponds to figure 5(c) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2374, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 130,050 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 3. This movie corresponds to figure 5(c) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2374, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 130,050 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2 MB

Rubio et al. supplementary movie

Movie 4. This movie corresponds to figure 13(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2390, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 4. This movie corresponds to figure 13(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2390, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 5. This movie corresponds to figure 13(c) in the paper over a time span indicated by box C in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2400, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 3.9 MB

Rubio et al. supplementary movie

Movie 5. This movie corresponds to figure 13(c) in the paper over a time span indicated by box C in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2400, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.6 MB

Rubio et al. supplementary movie

Movie 6. This movie corresponds to figure 13(d) in the paper over a time span indicated by box A in figure 14. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.5 MB

Rubio et al. supplementary movie

Movie 6. This movie corresponds to figure 13(d) in the paper over a time span indicated by box A in figure 14. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB

Rubio et al. supplementary movie

Movie 7. This movie corresponds to figure 13(d) in the paper over a time span indicated by box B in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.6 MB

Rubio et al. supplementary movie

Movie 7. This movie corresponds to figure 13(d) in the paper over a time span indicated by box B in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB

Rubio et al. supplementary movie

Movie 8. This movie corresponds to figure 5(f) in the paper over a time span indicated by box A in figure 7. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=0. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.2 MB

Rubio et al. supplementary movie

Movie 8. This movie corresponds to figure 5(f) in the paper over a time span indicated by box A in figure 7. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=0. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB