Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T06:56:44.218Z Has data issue: false hasContentIssue false
  • English
  • Français

The onset of zonal modes in two-dimensional Rayleigh–Bénard convection

Published online by Cambridge University Press:  23 March 2022

Philip Winchester Open the ORCID record for Philip Winchester [Opens in a new window] ,
Peter D. Howell Open the ORCID record for Peter D. Howell [Opens in a new window]  and
Vassilios Dallas Open the ORCID record for Vassilios Dallas [Opens in a new window]
Philip Winchester*
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
Peter D. Howell*
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
Vassilios Dallas*
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
*
†Email addresses for correspondence: winchester@maths.ox.ac.uk, howell@maths.ox.ac.uk, dallas@maths.ox.ac.uk
†Email addresses for correspondence: winchester@maths.ox.ac.uk, howell@maths.ox.ac.uk, dallas@maths.ox.ac.uk
†Email addresses for correspondence: winchester@maths.ox.ac.uk, howell@maths.ox.ac.uk, dallas@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard convection with free-slip boundaries and horizontal periodicity over 12 orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and 6 orders of magnitude in the Rayleigh number $(8{\rm \pi} ^4 < Ra \leq 10^8)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $\varGamma = 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$ plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $Pr \to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to nonlinear relaxation osculations and bursts, which we can explain with a weakly nonlinear analysis. In the complementary large-$Pr$ limit, we observe that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A8
Check for updates
Copyright
© The Author(s), 2022. Published by 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

REFERENCES

Aoyagi, T., Yagi, M. & Itoh, S.-I. 1997 Comparison analysis of Lorenz model and five components model. J. Phys. Soc. Japan 66 (9), 2689–2701.CrossRefGoogle Scholar
Berning, M. & Spatschek, K.H. 2000 Bifurcations and transport barriers in the resistive-$g$ paradigm. Phys. Rev. E 62, 1162–1174.CrossRefGoogle ScholarPubMed
Bolton, E.W. & Busse, F.H. 1985 Stability of convection rolls in a layer with stress-free boundaries. J. Fluid Mech. 150, 487–498.CrossRefGoogle Scholar
Busse, F.H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46 (1–4), 140–150.CrossRefGoogle Scholar
Busse, F.H. 1983 Generation of mean flows by thermal convection. Phys. D: Nonlinear Phenom. 9 (3), 287–299.CrossRefGoogle Scholar
Busse, F.H. & Bolton, E.W. 1984 Instabilities of convection rolls with stress-free boundaries near threshold. J. Fluid Mech. 146, 115–125.CrossRefGoogle Scholar
Chandra, M. & Verma, M.K. 2011 Dynamics and symmetries of flow reversals in turbulent convection. Phys. Rev. E 83, 067303.CrossRefGoogle ScholarPubMed
Decristoforo, G., Theodorsen, A. & Garcia, O.E. 2020 Intermittent fluctuations due to lorentzian pulses in turbulent thermal convection. Phys. Fluids 32 (8), 085102.CrossRefGoogle Scholar
Diamond, P.H., Itoh, S.-I., Itoh, K. & Hahm, T.S. 2005 Zonal flows in plasma – a review. Plasma Phys. Control. Fusion 47 (5), R35–R161.CrossRefGoogle Scholar
Fauve, S., Laroche, C., Libchaber, A. & Perrin, B. 1984 Chaotic phases and magnetic order in a convective fluid. Phys. Rev. Lett. 52, 1774–1777.CrossRefGoogle Scholar
Fowler, A.C. 1997 Mathematical Models in the Applied Sciences. Cambridge University Press.Google Scholar
Fuentes, J.R. & Cumming, A. 2021 Shear flows and their suppression at large aspect ratio: two-dimensional simulations of a growing convection zone. Phys. Rev. Fluids 6, 074502.CrossRefGoogle Scholar
Fujisawa, A. 2008 A review of zonal flow experiments. Nucl. Fusion 49, 013001.CrossRefGoogle Scholar
Garcia, O.E., Bian, N.H., Paulsen, J.-V., Benkadda, S. & Rypdal, K. 2003 Confinement and bursty transport in a flux-driven convection model with sheared flows. Plasma Phys. Control. Fusion 45 (6), 919–932.CrossRefGoogle Scholar
Goluskin, D., Johnston, H., Flierl, G.R. & Spiegel, E.A. 2014 Convectively driven shear and decreased heat flux. J. Fluid Mech. 759, 360–385.CrossRefGoogle Scholar
Heimpel, M., Aurnou, J. & Wicht, J. 2005 Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model. Nature 438, 193–196.CrossRefGoogle Scholar
Hermiz, K.B., Guzdar, P.N. & Finn, J.M. 1995 Improved low-order model for shear flow driven by Rayleigh–Bénard convection. Phys. Rev. E 51, 325–331.CrossRefGoogle ScholarPubMed
Horton, W., Hu, G. & Laval, G. 1996 Turbulent transport in mixed states of convective cells and sheared flows. Phys. Plasmas 3 (8), 2912–2923.CrossRefGoogle Scholar
Howard, L.N. & Krishnamurti, R. 1986 Large-scale flow in turbulent convection: a mathematical model. J. Fluid Mech. 170, 385–410.CrossRefGoogle Scholar
Kaspi, Y. et al. 2018 Jupiter's atmospheric jet streams extend thousands of kilometres deep. Nature 555, 223–226.CrossRefGoogle ScholarPubMed
Kong, D., Zhang, K. & Schubert, G. 2018 Origin of Jupiter's cloud-level zonal winds remains a puzzle even after Juno. Proc. Natl Acad. Sci. USA 115, 8499–8504.CrossRefGoogle ScholarPubMed
Krishnamurti, R. & Howard, L.N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl Acad. Sci. USA 78, 1981–1985.CrossRefGoogle ScholarPubMed
Landsberg, A.S. & Knobloch, E. 1991 Direction-reversing traveling waves. Phys. Lett. A 159 (1), 17–20.CrossRefGoogle Scholar
Leboeuf, J.N., Charlton, L.A. & Carreras, B.A. 1993 Shear flow effects on the nonlinear evolution of thermal instabilities. Phys. Fluids B 5 (8), 2959–2966.CrossRefGoogle Scholar
Malkov, M.A., Diamond, P.H. & Rosenbluth, M.N. 2001 On the nature of bursting in transport and turbulence in drift wave–zonal flow systems. Phys. Plasmas 8 (12), 5073–5076.CrossRefGoogle Scholar
Maximenko, N.A., Bang, B. & Sasaki, H. 2005 Observational evidence of alternating zonal jets in the world ocean. Geophys. Res. Lett. 32 (12), L12607.CrossRefGoogle Scholar
Nadiga, B.T. 2006 On zonal jets in oceans. Geophys. Res. Lett. 33 (10), L10601.CrossRefGoogle Scholar
Pal, P., Wahi, P., Paul, S., Verma, M.K., Kumar, K. & Mishra, P.K. 2009 Bifurcation and chaos in zero-Prandtl-number convection. Europhys. Lett. 87 (5), 54003.CrossRefGoogle Scholar
Paul, S., Verma, M.K., Wahi, P., Reddy, S.K. & Kumar, K. 2012 Bifurcation analysis of the flow patterns in two-dimensional Rayleigh–Bénard convection. Intl J. Bifurcation Chaos 22 (05), 1230018.CrossRefGoogle Scholar
Proctor, M.R.E. & Weiss, N.O. 1993 Symmetries of time-dependent magnetoconvection. Geophys. Astrophys. Fluid Dyn. 70 (1–4), 137–160.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32, 529–546.CrossRefGoogle Scholar
Read, P.L. et al. 2015 An experimental study of multiple zonal jet formation in rotating, thermally driven convective flows on a topographic beta-plane. Phys. Fluids 27 (8), 085111.CrossRefGoogle Scholar
Richards, K., Maximenko, N., Bryan, F. & Sasaki, H. 2006 Zonal jets in the Pacific Ocean. Geophys. Res. Lett. 33, L03605.CrossRefGoogle Scholar
Rucklidge, A.M. & Matthews, P.C. 1996 Analysis of the shearing instability in nonlinear convection and magnetoconvection. Nonlinearity 9, 311–351.CrossRefGoogle Scholar
Thompson, R. 1970 Venus's general circulation is a merry-go-round. J. Atmos. Sci. 27, 1107–1116.2.0.CO;2>CrossRefGoogle Scholar
Thual, O. 1992 Zero-Prandtl-number convection. J. Fluid Mech. 240, 229–258.CrossRefGoogle Scholar
Verma, M.K., Ambhire, S.C. & Pandey, A. 2015 Flow reversals in turbulent convection with free-slip walls. Phys. Fluids 27, 047102.CrossRefGoogle Scholar
Wang, Q., Chong, K.L., Stevens, R.J.A.M. & Lohse, D. 2020 From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates. J. Fluid Mech. 905, A21.CrossRefGoogle Scholar
Wen, B., Goluskin, D., LeDuc, M., Chini, G.P. & Doering, C.R. 2020 Steady Rayleigh–Bénard convection between stress-free boundaries. J. Fluid Mech. 905, R4.CrossRefGoogle Scholar
Winchester, P., Dallas, V. & Howell, P.D. 2021 Zonal flow reversals in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Fluids 6, 033502.CrossRefGoogle Scholar
Zhang, X. et al. 2020 Boundary zonal flow in rotating turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 124, 084505.CrossRefGoogle ScholarPubMed

Winchester et al. supplementary movie 1

See pdf file for movie caption

Download Winchester et al. supplementary movie 1(Video)
Video 3.8 MB

Winchester et al. supplementary movie 2

See pdf file for movie caption

Download Winchester et al. supplementary movie 2(Video)
Video 3.3 MB

Winchester et al. supplementary movie 3

See pdf file for movie caption

Download Winchester et al. supplementary movie 3(Video)
Video 3.3 MB

Winchester et al. supplementary movie 4

See pdf file for movie caption

Download Winchester et al. supplementary movie 4(Video)
Video 3.5 MB

Winchester et al. supplementary movie 5

See pdf file for movie caption

Download Winchester et al. supplementary movie 5(Video)
Video 3.7 MB

Winchester et al. supplementary movie 6

See pdf file for movie caption

Download Winchester et al. supplementary movie 6(Video)
Video 5 MB

Winchester et al. supplementary movie 7

See pdf file for movie caption

Download Winchester et al. supplementary movie 7(Video)
Video 3.7 MB

Winchester et al. supplementary movie 8

See pdf file for movie caption

Download Winchester et al. supplementary movie 8(Video)
Video 4.4 MB
Supplementary material: PDF

Winchester et al. supplementary material

Supplementary data and movie captions 1-8

Download Winchester et al. supplementary material(PDF)
PDF 1.2 MB