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Robust wall states in rapidly rotating Rayleigh–Bénard convection

Published online by Cambridge University Press:  11 May 2020

Benjamin Favier Open the ORCID record for Benjamin Favier [Opens in a new window]  and
Edgar Knobloch
Benjamin Favier*
Affiliation:
Aix-Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
Edgar Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA94720, USA
*
†Email address for correspondence: favier@irphe.univ-mrs.fr
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Abstract

We show, using direct numerical simulations with experimentally realizable boundary conditions, that wall modes in Rayleigh–Bénard convection in a rapidly rotating cylinder persist even very far from their linear onset. These nonlinear wall states survive in the presence of turbulence in the bulk and are robust with respect to changes in the shape of the boundary of the container. In this sense, these states behave much like the topologically protected states present in two-dimensional chiral systems even though rotating convection is a three-dimensional nonlinear driven dissipative system. We suggest that the robustness of this nonlinear state may provide an explanation for the strong zonal flows observed recently in experiments and simulations of rapidly rotating convection at high Rayleigh number.

JFM classification

Convection: Convection in cavities Geophysical and Geological Flows: Rotating flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 895 , 25 July 2020 , R1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Aurnou, J. M., Bertin, V., Grannan, A. M., Horn, S. & Vogt, T. 2018 Rotating thermal convection in liquid gallium: multi-modal flow, absent steady columns. J. Fluid Mech. 846, 846–876.CrossRefGoogle Scholar
Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. 2017 Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573.CrossRefGoogle ScholarPubMed
Busse, F. H. 1968 Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Clune, T. & Knobloch, E. 1993 Pattern selection in rotating convection with experimental boundary conditions. Phys. Rev. E 47, 2536–2550.Google ScholarPubMed
Dasbiswas, K., Mandadapu, K. K. & Vaikuntanathan, S. 2018 Topological localization in out-of-equilibrium dissipative systems. Proc. Natl Acad. Sci. USA 115, E9031–E9040.CrossRefGoogle ScholarPubMed
Delplace, P., Marston, J. B. & Venaille, A. 2017 Topological origin of equatorial waves. Science 358, 1075–1077.CrossRefGoogle ScholarPubMed
Ecke, R. E., Zhong, F. & Knobloch, E. 1992 Hopf bifurcation with broken reflection symmetry in rotating Rayleigh–Bénard convection. Eur. Phys. Lett. 19, 177–182.CrossRefGoogle Scholar
Favier, B., Silvers, L. J. & Proctor, M. R. E. 2014 Inverse cascade and symmetry breaking in rapidly rotating Boussinesq convection. Phys. Fluids 26, 096605.CrossRefGoogle Scholar
Fischer, P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comp. Phys. 133, 84–101.CrossRefGoogle Scholar
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a contained rotating stratified fluid. J. Fluid Mech. 114, 123–156.CrossRefGoogle Scholar
Früh, W.-G. & Read, P. L. 1999 Experiments on a barotropic rotating shear layer. Part 1. Instability and steady vortices. J. Fluid Mech. 383, 143–173.CrossRefGoogle Scholar
Goldstein, H., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.CrossRefGoogle Scholar
Guervilly, C., Hughes, D. W. & Jones, C. A. 2014 Large-scale vortices in rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 758, 407–435.CrossRefGoogle Scholar
Herrmann, J. & Busse, F. H. 1993 Asymptotic theory of wall-localized convection in a rotating fluid layer. J. Fluid Mech. 255, 183–194.CrossRefGoogle Scholar
Hide, R. & Titman, C. W. 1967 Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60.CrossRefGoogle Scholar
Horn, S. & Schmid, P. J. 2017 Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection. J. Fluid Mech. 831, 182–211.CrossRefGoogle Scholar
Janiaud, B., Pumir, A., Bensimon, D., Croquette, V., Richter, H. & Kramer, L. 1992 The Eckhaus instability for traveling waves. Physica D 55, 269–286.Google Scholar
Kane, C. L. & Lubensky, T. C. 2014 Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45.CrossRefGoogle Scholar
Khanikaev, A. B. & Shvets, G. 2017 Two-dimensional topological photonics. Nat. Photon. 11, 763–773.CrossRefGoogle Scholar
Knobloch, E. 1994 Bifurcations in rotating systems. In Lectures on Solar and Planetary Dynamos, pp. 331–372. Cambridge University Press.CrossRefGoogle Scholar
Kunnen, R. P. J., Stevens, R. J. A. M., Overkamp, J., Sun, C., van Heijst, G. F. & Clercx, H. J. H. 2011 The role of Stewartson and Ekman layers in turbulent rotating Rayleigh–Bénard convection. J. Fluid Mech. 688, 422–442.CrossRefGoogle Scholar
Kuo, E. Y. & Cross, M. C. 1993 Traveling-wave wall states in rotating Rayleigh–Bénard convection. Phys. Rev. E 47, R2245–R2248.Google ScholarPubMed
Liao, X., Zhang, K. & Chang, Y. 2005 Convection in rotating annular channels heated from below. Part 1. Linear stability and weakly nonlinear mean flows. Geo. Astro. Fluid Dyn. 99, 445–465.CrossRefGoogle Scholar
Liao, X., Zhang, K. & Chang, Y. 2006 On boundary-layer convection in a rotating fluid layer. J. Fluid Mech. 549, 375–384.CrossRefGoogle Scholar
Liu, Y. & Ecke, R. E. 1999 Nonlinear traveling waves in rotating Rayleigh–Bénard convection: stability boundaries and phase diffusion. Phys. Rev. E 59, 4091–4105.Google Scholar
Nash, L. M., Kleckner, D., Read, A., Vitelli, V., Turner, A. M. & Irvine, W. T. M. 2015 Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500.CrossRefGoogle ScholarPubMed
Niino, H. & Misawa, N. 1984 An experimental and theoretical study of barotropic instability. J. Atmos. Sci. 41, 1992–2011.2.0.CO;2>CrossRefGoogle Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.CrossRefGoogle Scholar
Sánchez-Álvarez, J. J., Serre, E., Crespo del Arco, E. & Busse, F. H. 2005 Square patterns in rotating Rayleigh–Bénard convection. Phys. Rev. E 72, 036307.Google ScholarPubMed
Scheel, J. D., Paul, M. R., Cross, M. C. & Fischer, P. F. 2003 Traveling waves in rotating Rayleigh–Bénard convection: analysis of modes and mean flow. Phys. Rev. E 68, 066216.Google ScholarPubMed
Soni, V., Bililign, E. S., Magkiriadou, S., Sacanna, S., Bartolo, D., Shelley, M. J. & Irvine, W. T. M. 2019 The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15, 1188–1194.CrossRefGoogle Scholar
Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. 2019 Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122, 128001.CrossRefGoogle ScholarPubMed
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.CrossRefGoogle Scholar
Tauber, C., Delplace, P. & Venaille, A. 2019 A bulk-interface correspondence for equatorial waves. J. Fluid Mech. 868, R2.CrossRefGoogle Scholar
Tauber, C., Delplace, P. & Venaille, A. 2020 Anomalous bulk-edge correspondence in continuous media. Phys. Rev. Res. 2, 013147.Google Scholar
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. 1982 Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408.CrossRefGoogle Scholar
Tsai, J.-C., Ye, F., Rodriguez, J., Gollub, J. P. & Lubensky, T. C. 2005 A chiral granular gas. Phys. Rev. Lett. 94, 214301.CrossRefGoogle ScholarPubMed
Vasil, G. M., Brummell, N. H. & Julien, K. 2008 A new method for fast transforms in parity-mixed PDEs. Part II. Application to confined rotating convection. J. Comput. Phys. 227, 8017–8034.CrossRefGoogle Scholar
de Wit, X. M., Aguirre Guzman, A. J., Madonia, M., Cheng, J., Clercx, H. J. H. & Kunnen, R. P. J. 2020 Turbulent rotating convection confined in a slender cylinder: the sidewall circulation. Phys. Rev. F 5, 023502.Google Scholar
Zhang, K. & Liao, X. 2009 The onset of convection in rotating circular cylinders with experimental boundary conditions. J. Fluid Mech. 622, 63–73.CrossRefGoogle Scholar
Zhang, X., van Gils, D. P. M., Horn, S., Wedi, M., Zwirner, L., Ahlers, G., Ecke, R. E., Weiss, S., Bodenschatz, E. & Shishkina, O. 2020 Boundary zonal flow in rotating turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 124, 084505.CrossRefGoogle ScholarPubMed
Zhong, F., Ecke, R. E. & Steinberg, V. 1991 Asymmetric modes and the transition to vortex structures in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 67, 2473–2476.CrossRefGoogle ScholarPubMed
Zhong, F., Ecke, R. E. & Steinberg, V. 1993 Rotating Rayleigh–Bénard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.CrossRefGoogle Scholar