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Convectons in a rotating fluid layer

Published online by Cambridge University Press:  01 February 2013

Cédric Beaume,
Alain Bergeon,
Hsien-Ching Kao and
Edgar Knobloch
Cédric Beaume*
Affiliation:
INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse, Allée Camille Soula, F-31400 Toulouse, France and CNRS, IMFT, F-31400 Toulouse, France
Alain Bergeon
Affiliation:
INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse, Allée Camille Soula, F-31400 Toulouse, France and CNRS, IMFT, F-31400 Toulouse, France
Hsien-Ching Kao
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
Edgar Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: ced.beaume@gmail.com
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Abstract

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Two-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time-independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical SystemsNonlinear Dynamical Systems: Pattern formationGeophysical and Geological Flows: Rotating flows

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013, pp. 417 - 448
Copyright
©2013 Cambridge University Press

References

Assemat, P., Bergeon, A. & Knobloch, E. 2008 Spatially localized states in Marangoni convection in binary mixtures. Fluid Dyn. Res. 40, 852–876.CrossRefGoogle Scholar
Bajaj, K. M. S., Ahlers, G. & Pesch, W. 2002 Rayleigh–Bénard convection with rotation at small Prandtl numbers. Phys. Rev. E 65, 056309.CrossRefGoogle ScholarPubMed
Batiste, O., Knobloch, E., Alonso, A. & Mercader, I. 2006 Spatially localized binary-fluid convection. J. Fluid Mech. 560, 149–156.CrossRefGoogle Scholar
Beaume, C., Bergeon, A. & Knobloch, E. 2011 Homoclinic snaking of localized states in doubly diffusive convection. Phys. Fluids 23, 094102.CrossRefGoogle Scholar
Bergeon, A., Burke, J., Knobloch, E. & Mercader, I. 2008 Eckhaus instability and homoclinic snaking. Phys. Rev. E 78, 046201.CrossRefGoogle ScholarPubMed
Bergeon, A. & Knobloch, E. 2008 Spatially localized states in natural doubly diffusive convection. Phys. Fluids 20, 034102.CrossRefGoogle Scholar
Blanchflower, S. 1999 Magnetohydrodynamic convectons. Phys. Lett. A 261, 74–81.CrossRefGoogle Scholar
Burke, J. & Knobloch, E. 2007 Snakes and ladders: localized states in the Swift–Hohenberg equation. Phys. Lett. A 360, 681–688.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.CrossRefGoogle ScholarPubMed
Cox, S. M. 2004 The envelope of a one-dimensional pattern in the presence of a conserved quantity. Phys. Lett. A 333, 91–101.CrossRefGoogle Scholar
Cox, S. M. & Matthews, P. C. 2000 Instability of rotating convection. J. Fluid Mech. 403, 153–172.CrossRefGoogle Scholar
Cox, S. M. & Matthews, P. C. 2001 New instabilities in two-dimensional rotating convection and magnetoconvection. Physica D 149, 210–229.CrossRefGoogle Scholar
Dawes, J. H. P. 2007 Localized convection cells in the presence of a vertical magnetic field. J. Fluid Mech. 570, 385–406.CrossRefGoogle Scholar
Dawes, J. H. P. 2008 Localized pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7, 186–206.CrossRefGoogle Scholar
Dawes, J. H. P. & Lilley, S. 2010 Localized states in a model of pattern formation in a vertically vibrated layer. SIAM J. Appl. Dyn. Syst. 9, 238–260.CrossRefGoogle Scholar
Doedel, E. J., Champneys, A. R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X. & Zhang, C. 2008 AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations.Google Scholar
Eckhaus, W. & Iooss, G. 1989 Strong selection or rejection of spatially periodic patterns in degenerate bifurcations. Physica D 39, 124–146.CrossRefGoogle Scholar
Elmer, F. J. 1988 Nonlinear and non-local dynamics of spatially extended systems: stationary states, bifurcations and stability. Physica D 30, 321–342.CrossRefGoogle Scholar
Elmer, F. J. 1992 Parallels between pattern formation in high-power ferromagnetic resonance and fluid convection. In Pattern Formation in Complex Dissipative Systems (ed. Kai, S.). pp. 309–313. World Scientific.Google Scholar
Evonuk, M. & Glatzmaier, G. A. 2007 The effects of rotation rate on deep convection in giant planets with small solid cores. Planet. Space Sci. 55, 407–412.CrossRefGoogle Scholar
Firth, W. J., Columbo, L. & Scroggie, A. J. 2007 Proposed resolution of theory-experiment discrepancy in homoclinic snaking. Phys. Rev. Lett. 99, 104503.CrossRefGoogle ScholarPubMed
Goldstein, H. F., Knobloch, E. & Silber, M. 1990 Planform selection in rotating convection. Phys. Fluids A 2, 625–627.CrossRefGoogle Scholar
Goldstein, H. F., Knobloch, E. & Silber, M. 1992 Planform selection in rotating convection: hexagonal symmetry. Phys. Rev. A 46, 4755–4761.CrossRefGoogle ScholarPubMed
Guckenheimer, J. & Knobloch, E. 1983 Nonlinear convection in a rotating layer: amplitude expansions and normal forms. Geophys. Astrophys. Fluid Dyn. 23, 247–272.CrossRefGoogle Scholar
Hall, P. 1984 Evolution equations for Taylor vortices in the small-gap limit. Phys. Rev. A 29, 2921–2923.CrossRefGoogle Scholar
Kao, H.-C. & Knobloch, E. 2012 Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking. Phys. Rev. E 85, 026211.CrossRefGoogle ScholarPubMed
Knobloch, E. 1998 Rotating convection: recent developments. Intl J. Engng Sci. 36, 1421–1450.CrossRefGoogle Scholar
Lo Jacono, D., Bergeon, A. & Knobloch, E. 2010 Spatially localized binary fluid convection in a porous medium. Phys. Fluids 22, 073601.CrossRefGoogle Scholar
Lo Jacono, D., Bergeon, A. & Knobloch, E. 2011 Magnetohydrodynamic convectons. J. Fluid Mech. 687, 595–605.CrossRefGoogle Scholar
Lo Jacono, D., Bergeon, A. & Knobloch, E. 2012 Spatially localized magnetoconvection. Fluid Dyn. Res. 44, 031411.CrossRefGoogle Scholar
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.CrossRefGoogle Scholar
Marcus, P. S., Kundu, T. & Lee, C. 2000 Vortex dynamics and zonal flows. Phys. Plasmas 7, 1630–1640.CrossRefGoogle Scholar
Marshall, J. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 1–64.CrossRefGoogle Scholar
Matthews, P. C. & Cox, S. M. 2000 Pattern formation with a conservation law. Nonlinearity 13, 1293–1320.CrossRefGoogle Scholar
Norbury, J., Wei, J. & Winter, M. 2002 Existence and stability of singular patterns in a Ginzburg–Landau equation coupled with a mean field. Nonlinearity 15, 2077–2096.CrossRefGoogle Scholar
Norbury, J., Wei, J. & Winter, M. 2007 Stability of patterns with arbitrary period for a Ginzburg–Landau equation with a mean field. Eur. J. Appl. Maths 18, 129–151.CrossRefGoogle Scholar
Petersen, M. R., Julien, K. & Stewart, G. R. 2007 Baroclinic vorticity production in protoplanetary disks. I. Vortex formation. Astrophys. J. 658, 1236–1251.CrossRefGoogle Scholar
Proctor, M. R. E. 2001 Finite amplitude behaviour of the Matthews–Cox instability. Phys. Lett. A 292, 181–187.CrossRefGoogle Scholar
Riecke, H. 1999 Localized structures in pattern-forming systems. In Pattern Formation in Continuous and Coupled Systems (ed. Golubitsky, M., Luss, D. & Strogatz, S. H.). pp. 215–229. Springer.CrossRefGoogle Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.CrossRefGoogle ScholarPubMed
Silber, M. & Knobloch, E. 1990 Travelling wave convection in a rotating layer. Geophys. Astrophys. Fluid Dyn. 51, 195–209.Google Scholar
Silber, M. & Knobloch, E. 1993 Oscillatory convection in a rotating layer. Physica D 63, 213–232.Google Scholar
Vega, J. M. 2005 Instability of the steady states of some Ginzburg–Landau-like equations with real coefficients. Nonlinearity 18, 1425–1441.CrossRefGoogle Scholar
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–435.CrossRefGoogle Scholar
Wei, J. & Winter, M. 2004 On a cubic-quintic Ginzburg–Landau equation with global coupling. Proc. Am. Math. Soc. 133, 1787–1796.CrossRefGoogle Scholar
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