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A nonlinear model for rotationally constrained convection with Ekman pumping

Published online by Cambridge University Press:  31 May 2016

Keith Julien ,
Jonathan M. Aurnou ,
Michael A. Calkins ,
Edgar Knobloch ,
Philippe Marti ,
Stephan Stellmach  and
Geoffrey M. Vasil
Keith Julien*
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Jonathan M. Aurnou
Affiliation:
Department of Earth, Planetary and Space Sciences, University of California, Los Angeles, CA 90095, USA
Michael A. Calkins
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA
Edgar Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
Philippe Marti
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Stephan Stellmach
Affiliation:
Institut für Geophysik, Westfälische Wilhelms Universität Münster, D-48149 Münster, Germany
Geoffrey M. Vasil
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
*
†Email address for correspondence: julien@colorado.edu
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Abstract

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 50 - 87
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Copyright
© 2016 Cambridge University Press 

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References

Aubert, J., Gillet, N. & Cardin, P. 2003 Quasigeostrophic models of convection in rotating spherical shells. Geochem. Geophys. Geosyst. 4 (7), 1–19; 1052.Google Scholar
Aurnou, J. M., Calkins, M. A., Cheng, J. S., Julien, K, King, E. M., Nieves, D., Soderlund, K. M. & Stellmach, S. 2015 Rotating convective turbulence in Earth and planetary cores. Phys. Earth Planet. Inter. 246, 52–71.CrossRefGoogle Scholar
Barcilon, V. 1965 Stability of a non-divergent Ekman layer. Tellus 17, 53–68.CrossRefGoogle Scholar
Bassom, A. P. & Zhang, K. 1994 Strongly nonlinear convection cells in a rapidly rotating fluid layer. Geophys. Astrophys. Fluid Dyn. 76, 223–238.Google Scholar
Calkins, M. A., Aurnou, J. M., Eldredge, J. D. & Julien, K. 2012 The influence of fluid properties on the morphology of core turbulence and the geomagnetic field. Earth Planet. Sci. Lett. 359–360, 55–60.Google Scholar
Calkins, M. A., Julien, K. & Marti, P. 2013 Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls. J. Fluid Mech. 732, 214–244.Google Scholar
Cash, J. R. & Singhal, A. 1982 High order methods for the numerical solution of two-point boundary value problems. BIT Numer. Math. 22, 183–199.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Charney, J. G. 1948 On the scale of atmospheric motions. Geofys. Publ. 17, 3–17.Google Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Cheng, J. S., Stellmach, S., Ribeiro, A., Grannan, A., King, E. M. & Aurnou, J. M. 2015 Laboratory-numerical models of rapidly rotating convection in planetary cores. Geophys. J. Intl 201, 1–17.Google Scholar
Dawes, J. H. P. 2001 Rapidly rotating thermal convection at low Prandtl number. J. Fluid Mech. 428, 61–80.CrossRefGoogle Scholar
Dudis, J. J. & Davis, S. H. 1971 Energy stability of the Ekman boundary layer. J. Fluid Mech. 47, 405–413.Google Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1, 33–52.Google Scholar
Ecke, R. E. 2015 Scaling of heat transport near onset in rapidly rotating convection. Phys. Lett. A 379, 2221–2223.Google Scholar
Ecke, R. E. & Niemela, J. J. 2014 Heat transport in the geostrophic regime of rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 113, 114301.Google Scholar
Faller, A. J. & Kaylor, R. E. 1966 A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Sci. 23, 466–480.2.0.CO;2>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.Google Scholar
Greenspan, H. P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36, 257–264.Google Scholar
Grooms, I. 2015 Asymptotic behavior of heat transport for a class of exact solutions in rotating Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 109, 145–158.Google Scholar
Grooms, I., Julien, K., Weiss, J. B. & Knobloch, E. 2010 Model of convective Taylor columns in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 224501.Google Scholar
Grooms, I. & Whitehead, J. P. 2015 Bounds on heat transport in rapidly rotating Rayleigh–Bénard convection. Nonlinearity 28, 29–42.Google 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
Heard, W. B. & Veronis, G. 1971 Asymptotic treatment of the stability of a rotating layer of fluid with rigid boundaries. Geophys. Fluid Dyn. 2, 299–316.Google Scholar
Henrici, P. 1962 Discrete Variable Methods in Ordinary Differential Equations. Wiley.Google Scholar
Julien, K. & Knobloch, E. 1998 Strongly nonlinear convection cells in a rapidly rotating fluid layer: the tilted f-plane. J. Fluid Mech. 360, 141–178.CrossRefGoogle Scholar
Julien, K. & Knobloch, E. 1999 Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 11, 1469–1483.Google Scholar
Julien, K. & Knobloch, E. 2007 Reduced models for fluid flows with strong constraints. J. Math. Phys. 48, 065405.Google Scholar
Julien, K., Knobloch, E., Milliff, R. & Werne, J. 2006 Generalized quasi-geostrophy for spatially anistropic rotationally constrained flows. J. Fluid Mech. 555, 233–274.Google Scholar
Julien, K., Knobloch, E., Rubio, A. M. & Vasil, G. M. 2012a Heat transport in low-Rossby-number Rayleigh–Bénard convection. Phys. Rev. Lett. 109, 254503.CrossRefGoogle ScholarPubMed
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11, 251–261.Google Scholar
Julien, K., Rubio, A. M., Grooms, I. & Knobloch, E. 2012b Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 106, 392–428.Google Scholar
King, E. M., Soderlund, K. M., Christensen, U. R., Wicht, J. & Aurnou, J. M. 2010 Convective heat transfer in planetary dynamo models. Geochem. Geophys. Geosyst. 11 (6), 1–19.CrossRefGoogle Scholar
King, E. M., Stellmach, S. & Aurnou, J. M. 2012 Heat transfer by rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 691, 568–582.Google Scholar
King, E. M., Stellmach, S., Noir, J., Hansen, U. & Aurnou, J. M. 2009 Boundary layer control of rotating convection systems. Nature 457, 301–304.Google Scholar
Liu, Y. & Ecke, R. E. 1997 Heat transport scaling in turbulent Rayleigh–Bénard convection: effects of rotation and Prandtl number. Phys. Rev. Lett. 79, 2257–2260.Google Scholar
Marshall, J. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 1–64.Google Scholar
Miesch, M. S. 2005 Large-scale dynamics of the convection zone and tachocline. Living Rev. Solar Phys. 2 (1).Google Scholar
Nayfeh, A. H. 2008 Perturbation Methods. Wiley.Google Scholar
Nieves, D., Rubio, A. M. & Julien, K. 2014 Statistical classification of flow morphology in rapidly rotating Rayleigh–Bénard convection. Phys. Fluids 26, 086602.Google Scholar
Niiler, P. P. & Bisshopp, F. E. 1965 On the influence of the Coriolis force on onset of thermal convection. J. Fluid Mech. 22, 753–761.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.Google Scholar
Proudman, J. 1916 On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. A 92, 408–424.Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Rubio, A. M., Julien, K., Knobloch, E. & Weiss, J. B. 2014 Upscale energy transfer in three-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett. 112, 144501.Google Scholar
Sakai, S. 1997 The horizontal scale of rotating convection in the geostrophic regime. J. Fluid Mech. 333, 85–95.Google Scholar
Schaeffer, N. & Cardin, P. 2005 Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers. Phys. Fluids 17, 104111.Google Scholar
Sprague, M., Julien, K., Knobloch, E. & Werne, J. 2006 Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141–174.Google Scholar
Stellmach, S., Lischper, M., Julien, K., Vasil, G., Cheng, J. S., Ribeiro, A., King, E. M. & Aurnou, J. M. 2014 Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics. Phys. Rev. Lett. 113, 254501.Google Scholar
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–218.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar