Skip to main content
×
Home
    • Aa
    • Aa

Transition to geostrophic convection: the role of the boundary conditions

  • Rudie P. J. Kunnen (a1), Rodolfo Ostilla-Mónico (a2), Erwin P. van der Poel (a2), Roberto Verzicco (a2) (a3) and Detlef Lohse (a2)...
Abstract

Rotating Rayleigh–Bénard convection, the flow in a rotating fluid layer heated from below and cooled from above, is used to analyse the transition to the geostrophic regime of thermal convection. In the geostrophic regime, which is of direct relevance to most geo- and astrophysical flows, the system is strongly rotating while maintaining a sufficiently large thermal driving to generate turbulence. We directly simulate the Navier–Stokes equations for two values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\times 10^{10}$, at constant Prandtl number $Pr=1$, and vary the Ekman number in the range $Ek=1.3\times 10^{-7}$ to $Ek=2\times 10^{-6}$, which satisfies both requirements of supercriticality and strong rotation. We focus on the differences between the application of no-slip versus stress-free boundary conditions on the horizontal plates. The transition is found at roughly the same parameter values for both boundary conditions, i.e. at $Ek\approx 9\times 10^{-7}$ for $Ra=1\times 10^{10}$ and at $Ek\approx 3\times 10^{-7}$ for $Ra=5\times 10^{10}$. However, the transition is gradual and it does not exactly coincide in $Ek$ for different flow indicators. In particular, we report the characteristics of the transitions in the heat-transfer scaling laws, the boundary-layer thicknesses, the bulk/boundary-layer distribution of dissipations and the mean temperature gradient in the bulk. The flow phenomenology in the geostrophic regime evolves differently for no-slip and stress-free plates. For stress-free conditions, the formation of a large-scale barotropic vortex with associated inverse energy cascade is apparent. For no-slip plates, a turbulent state without large-scale coherent structures is found; the absence of large-scale structure formation is reflected in the energy transfer in the sense that the inverse cascade, present for stress-free boundary conditions, vanishes.

Copyright
Corresponding author
Email address for correspondence: r.p.j.kunnen@tue.nl
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

G. Ahlers , S. Grossmann  & D. Lohse 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.

A. P. Bassom  & K. Zhang 1994 Strongly nonlinear convection cells in a rapidly rotating fluid layer. Geophys. Astrophys. Fluid Dyn. 76, 223238.

J. S. Cheng , S. Stellmach , A. Ribeiro , A. Grannan , E. M. King  & J. M. Aurnou 2015 Laboratory-numerical models of rapidly rotating convection in planetary cores. Geophys. J. Int. 201, 117.

T. Clune  & E. Knobloch 1993 Pattern selection in rotating convection with experimental boundary conditions. Phys. Rev. E 47, 25362550.

R. E. Ecke 2015 Scaling of heat transport near onset in rapidly rotating convection. Phys. Lett. A 379, 22212223.

R. E. Ecke  & J. J. Niemela 2014 Heat transport in the geostrophic regime of rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 113, 114301.

B. Favier , L. J. Silvers  & M. R. E. Proctor 2014 Inverse cascade and symmetry breaking in rapidly rotating Boussinesq convection. Phys. Fluids 26, 096605.

I. Grooms , K. Julien , J. B. Weiss  & E. Knobloch 2010 Model of convective Taylor columns in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 224501.

S. Grossmann  & D. Lohse 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 33163319.

S. Grossmann  & D. Lohse 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16, 44624472.

W. B. Heard  & G. Veronis 1971 Asymptotic treatment of the stability of a rotating layer of fluid with rigid boundaries. Geophys. Fluid Dyn. 2, 299316.

M. Heimpel , J. Aurnou  & J. Wicht 2005 Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model. Nature 438, 193196.

K. Julien , E. Knobloch , A. M. Rubio  & G. M. Vasil 2012a Heat transport in low-Rossby-number Rayleigh–Bénard convection. Phys. Rev. Lett. 109, 254503.

K. Julien , A. M. Rubio , I. Grooms  & E. Knobloch 2012b Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 106, 392428.

E. M. King  & J. M. Aurnou 2012 Thermal evidence for Taylor columns in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. E 85, 016313.

E. M. King , S. Stellmach , J. Noir , U. Hansen  & J. M. Aurnou 2009 Boundary layer control of rotating convection systems. Nature 457, 301304.

R. P. J. Kunnen , H. J. H. Clercx  & B. J. Geurts 2006 Heat flux intensification by vortical flow localization in rotating convection. Phys. Rev. E 74, 056306.

R. P. J. Kunnen , H. J. H. Clercx  & B. J. Geurts 2008a Breakdown of large-scale circulation in turbulent rotating convection. Europhys. Lett. 84, 24001.

R. P. J. Kunnen , H. J. H. Clercx  & B. J. Geurts 2008b Enhanced vertical inhomogeneity in turbulent rotating convection. Phys. Rev. Lett. 101, 174501.

R. P. J. Kunnen , H. J. H. Clercx  & B. J. Geurts 2010a Vortex statistics in turbulent rotating convection. Phys. Rev. E 82, 036306.

R. P. J. Kunnen , Y. Corre  & H. J. H. Clercx 2013 Vortex plume distribution in confined turbulent rotating convection. Eur. Phys. Lett. 104, 54002.

Y. Liu  & R. E. Ecke 1997 Heat transport scaling in turbulent Rayleigh–Bénard convection: effects of rotation and Prandtl number. Phys. Rev. Lett. 79, 22572260.

Y. Liu  & R. E. Ecke 2009 Heat transport measurements in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. E 80, 036314.

Y. Liu  & R. E. Ecke 2011 Local temperature measurements in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. E 84, 016311.

J. Marshall  & F. Schott 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 164.

M. S. Miesch 2000 The coupling of solar convection and rotation. Solar Phys. 192, 5989.

P. D. Mininni , A. Alexakis  & A. Pouquet 2009 Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers. Phys. Fluids 21, 015108.

Y. Nakagawa  & P. Frenzen 1955 A theoretical and experimental study of cellular convection in rotating fluids. Tellus 7, 121.

J. W. Portegies , R. P. J. Kunnen , G. J. F. van Heijst  & J. Molenaar 2008 A model for vortical plumes in rotating convection. Phys. Fluids 20, 066602.

P. Roberts  & G. Glatzmaier 2000 Geodynamo theory and simulations. Rev. Mod. Phys. 72, 10811123.

A. M. Rubio , K. Julien , E. Knobloch  & J. B. Weiss 2014 Upscale energy transfer in three-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett. 112, 144501.

S. Schmitz  & A. Tilgner 2009 Heat transport in rotating convection without Ekman layers. Phys. Rev. E 80, 015305(R).

S. Schmitz  & A. Tilgner 2010 Transitions in turbulent rotating Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 104, 481489.

B. I. Shraiman  & E. D. Siggia 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 36503653.

S. Stellmach , M. Lischper , K. Julien , G. Vasil , J. S. Cheng , A. Ribeiro , E. M. King  & J. M. Aurnou 2014 Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics. Phys. Rev. Lett. 113, 254501.

R. J. A. M. Stevens , H. J. H. Clercx  & D. Lohse 2010 Optimal Prandtl number for heat transfer in rotating Rayleigh–Bénard convection. New J. Phys. 12, 075005.

R. J. A. M. Stevens , H. J. H. Clercx  & D. Lohse 2012 Breakdown of the large-scale circulation in 𝛤 = 1/2 rotating Rayleigh–Bénard flow. Phys. Rev. E 86, 056311.

R. J. A. M. Stevens , H. J. H. Clercx  & D. Lohse 2013a Heat transport and flow structure in rotating Rayleigh–Bénard convection. Eur. J. Mech. (B/Fluids) 40, 4149.

R. J. A. M. Stevens , J.-Q. Zhong , H. J. H. Clercx , G. Ahlers  & D. Lohse 2009 Transitions between turbulent states in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 024503.

R. Verzicco  & P. Orlandi 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402413.

S. Weiss , R. J. A. M. Stevens , J.-Q. Zhong , H. J. H. Clercx , D. Lohse  & G. Ahlers 2010 Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 105, 224501.

J.-Q. Zhong , R. J. A. M. Stevens , H. J. H. Clercx , R. Verzicco , D. Lohse  & G. Ahlers 2009 Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 102, 044502.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 1
Total number of PDF views: 78 *
Loading metrics...

Abstract views

Total abstract views: 136 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd March 2017. This data will be updated every 24 hours.