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The Boussinesq approximation in rapidly rotating flows

Published online by Cambridge University Press:  15 November 2013

Jose M. Lopez ,
Francisco Marques  and
Marc Avila
Jose M. Lopez
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
Francisco Marques*
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
Marc Avila
Affiliation:
Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
*
†Email address for correspondence: marques@fa.upc.edu
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Abstract

In commonly used formulations of the Boussinesq approximation centrifugal buoyancy effects related to differential rotation, as well as strong vortices in the flow, are neglected. However, these may play an important role in rapidly rotating flows, such as in astrophysical and geophysical applications, and also in turbulent convection. Here we provide a straightforward approach resulting in a Boussinesq-type approximation that consistently accounts for centrifugal effects. Its application to the accretion-disc problem is discussed. We numerically compare the new approach to the typical one in fluid flows confined between two differentially heated and rotating cylinders. The results justify the need of using the proposed approximation in rapidly rotating flows.

JFM classification

Convection: Buoyancy-driven instability

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 56 - 77
Copyright
©2013 Cambridge University Press 

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References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.Google Scholar
Ali, M. E. & Weidman, P. D. 1990 On the stability of circular Couette-flow with radial heating. J. Fluid Mech. 220, 53–84.CrossRefGoogle Scholar
Arfken, G. B. & Weber, H. J. 2005 Mathematical Methods for Physicists, 6th edn. Academic.Google Scholar
Avila, M. 2012 Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys. Rev. Lett. 108, 124501.Google Scholar
Avila, K. & Hof, B. 2013 High-precision Taylor–Couette experiment to study subcritical transitions and the role of boundary conditions and size effects. Rev. Sci. Instrum. 84, 065106.Google Scholar
Balbus, S. A. 2003 Enhanced angular momentum transport in accretion disks. Annu. Rev. Astron. Astrophys. 41 (1), 555–597.Google Scholar
Balbus, S. A. 2011 Fluid dynamics: a turbulent matter. Nature 470 (7335), 475–476.Google Scholar
Bannon, P. R. 1996 On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci. 53 (23), 3618–3628.Google Scholar
Barcilon, V. & Pedlosky, J. 1967 On the steady motions produced by a stable stratification in a rapidly rotating fluid. J. Fluid Mech. 29, 673–690.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
Boussinesq, J. 1903 Théorie Analytique de la Chaleur, vol. II, Gauthier-Villars.Google Scholar
Brummell, N., Hart, J. E. & Lopez, J. M. 2000 On the flow induced by centrifugal buoyancy in a differentially-heated rotating cylinder. Theor. Comput. Fluid Dyn. 14, 39–54.Google Scholar
Canuto, C., Quarteroni, A., Hussaini, M. Y. & Zang, T. A. 2007 Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Choi, I. G. & Korpela, S. A. 1980 Stability of the conduction regime of natural convection in a tall vertical annulus. J. Fluid Mech. 99, 725–738.Google Scholar
Elperin, T., Kleeorin, N. & Rogachevskii, I. 1998 Dynamics of particles advected by fast rotating turbulent fluid flow: fluctuations and large-scale structures. Phys. Rev. Lett. 81, 2898–2901.Google Scholar
Hart, J. E. 2000 On the influence of centrifugal buoyancy on rotating convection. J. Fluid Mech. 403, 133–151.Google Scholar
Hide, R. & Fowlis, W. W. 1965 Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J. Atmos. Sci. 22, 541–558.Google Scholar
Homsy, G. M. & Hudson, J. L. 1969 Centrifugally driven thermal convection in a rotating cylinder. J. Fluid Mech. 35, 33–52.Google Scholar
Klahr, H. & Bodenheimer, P. 2003 Turbulence in accretion disks: vorticity generation and angular momentum transport via the global baroclinic instability. Astrophys. J. 582 (2), 869–892.Google Scholar
Klahr, H. H., Henning, Th. & Kley, W. 1999 On the azimuthal structure of thermal convection in circumstellar disks. Astrophys. J. 514 (1), 325–343.Google Scholar
Lee, Y., Korpela, S. A. & Horn, R. N. 1982 Structure of multicellular natural convection in a tall vertical annulus. In Proceedings of the 7th International Heat Transfer Conference, Munich, vol. 2, pp. 221–226.Google Scholar
Lesur, G. & Papaloizou, J. C. B. 2010 The subcritical baroclinic instability in local accretion disc models. Astron. Astrophys. 513, A60.Google Scholar
Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.Google Scholar
Maretzke, S., Hof, B. & Avila, M. 2013 Transient growth in linearly stable Taylor–Couette flows. J. Fluid Mech. (submitted).Google Scholar
Marques, F., Mercader, I., Batiste, O. & Lopez, J. M. 2007 Centrifugal effects in rotating convection: axisymmetric states and 3d instabilities. J. Fluid Mech. 580, 303–318.Google Scholar
McFadden, G. B., Coriell, S. R., Boisvert, R. F. & Glicksman, M. E. 1984 Asymmetric instabilities in buoyancy-driven flow in a tall vertical annulus. Phys. Fluids 27, 1359–1361.Google Scholar
Meseguer, A., Avila, M., Mellibovsky, F. & Marques, F. 2007 Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries. Eur. Phys. J. Special Topics 146, 249–259.Google Scholar
Meseguer, A. & Marques, F. 2000 On the competition between centrifugal and shear instability in spiral Couette flow. J. Fluid Mech. 402, 33–56.Google Scholar
Paoletti, M. S. & Lathrop, D. P. 2011 Measurement of angular momentum transport in turbulent flow between independently rotating cylinders. Phys. Rev. Lett. 106, 024501.Google Scholar
Petersen, M., Julien, K. & Stewart, G. 2007 Baroclinic vorticity production in protoplanetary disks. Astrophys. J. 658, 1236–1251.Google Scholar
Randriamampianina, A., Früh, W.-G., Read, P. L. & Maubert, P. 2006 Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus. J. Fluid Mech. 561, 359–389.Google Scholar
Regev, O. & Umurhan, O. M. 2008 On the viability of the shearing box approximation for numerical studies of mhd turbulence in accretion disks. Astron. Astrophys. 481 (1), 21–32.Google Scholar
Tassoul, J. L. 2000 Stellar Rotation. Cambridge University Press.Google Scholar
de Vahl Davis, G. & Thomas, R. W. 1969 Natural convection between concentric vertical cylinders. Phys. Fluids Suppl. II. 198–207.Google Scholar