Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T05:36:25.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapidly rotating turbulent Rayleigh-Bénard convection

Published online by Cambridge University Press:  26 April 2006

K. Julien ,
S. Legg ,
J. Mcwilliams  and
J. Werne
K. Julien
Affiliation:
Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder. CO 80309, USA National Center for Atmospheric Research, Boulder, CO 80307, USA Present address: University of California, Los Angeles, CA 90007, USA.
S. Legg
Affiliation:
Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder. CO 80309, USA National Center for Atmospheric Research, Boulder, CO 80307, USA
J. Mcwilliams
Affiliation:
National Center for Atmospheric Research, Boulder, CO 80307, USA
J. Werne
Affiliation:
National Center for Atmospheric Research, Boulder, CO 80307, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulent Boussinesq convection under the influence of rapid rotation (i.e. with comparable characteristic rotation and convection timescales) is studied. The transition to turbulence proceeds through a relatively simple bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh (Ra) and Taylor numbers (Ta) and culminating in a state dominated by coherent plume structures at high Ra and Ta. Like non-rotating turbulent convection, the rapidly rotating state exhibits a simple power-law dependence on Ra for all statistical properties of the flow. When the fluid layer is bounded by no-slip surfaces, the convective heat transport (Nu − 1, where Nu is the Nusselt number) exhibits scaling with Ra2/7 similar to non-rotating laboratory experiments. When the boundaries are stress free, the heat transport obeys ‘classical’ scaling (Ra1/3) for a limited range in Ra, then appears to undergo a transition to a different law at Ra ≈ 4 × 107. Important dynamical differences between rotating and non-rotating convection are observed: aside from the (expected) differences in the boundary layers due to Ekman pumping effects, angular momentum conservation forces all plume structures created at flow-convergent sites of the heated and cooled boundaries to spin-up cyclonically; the resulting plume/cyclones undergo strong vortex-vortex interactions which dramatically alter the mean state of the flow and result in a finite background temperature gradient as Ra → ∞, holding Ra/Ta fixed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 322 , 10 September 1996 , pp. 243 - 273
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adrian, R. J., Ferreira, R. T. D. S. & Boberg, T. 1986 Turbulent thermal convection in wide horizontal fluid layers. Exps. Fluids 4, 121–141.Google Scholar
Belmonte, A., Tilgner, A. & Libchaber, A. 1993 Boundary layer length scales in thermal turbulence. Phys. Rev. Lett. 70, 4067–4070.Google Scholar
Boubnov, B. M. & Golitsyn, G. S. 1986 Experimental study of convective structures in rotating fluids. J. Fluid Mech. 167, 503–531.Google Scholar
Boubnov, B. M. & Golitsyn, G. S. 1990 Temperature and velocity field regimes of convective motions in a rotating plane fluid layer. J. Fluid Mech. 219, 215–239.Google Scholar
Brummell, N. H., Toomre, J. & Hurlbert, N. E. 1996 Turbulent compressible convection with rotation. Part I: Overview, model and flow structure. Submitted to Astrophys. J.Google Scholar
Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Busse, F. H. 1983 A model of mean zonal flows in the major planets. Geophys. Astrophys. Fluid Dyn. 23, 153–174.Google Scholar
Busse, F. H. & Clever, R. M. 1979 Nonstationary convection in a rotating system. In Recent Developments in Theoretical and Experimental Fluid Mechanics, Compressible and Incompressible Flows (ed. U. Muller, K. G. Roesner & B. Schmidt), pp. 376–385. Springer.
Cabot, W., Hubickyj, O., Pollack, J. B., Cassen, P. & Canuto, V. M. 1990 Direct numerical simulations of turbulent convection: I. Variable gravity and uniform rotation. Geophys. Astrophys. Fluid Dyn. 53, 1–42.Google Scholar
Cabot, W. & Pollack, J. B. 1992 Direct numerical simulations of turbulent convection: II. variable gravity and differential rotation. Geophys. Astrophys. Fluid Dyn. 64, 97–133.Google Scholar
Canuto, C. Hussaini, M. Y., Quateroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer.
Carrier, G. F. 1971 Swirling flow boundary layers. J. Fluid Mech. 49, 133–144.Google Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh-Benard convection. J. Fluid Mech. 204, 1–39.Google Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to Coriolis forces.. Proc. R. Soc. Lond. A 217, 306–327.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Christie, S. L. & Domaradzki, J. A. 1993 Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection. Pjtls. Fluids A 5, 412–421.Google Scholar
Christie, S. L. & Domaradzki, J. A. 1994 Scale dependence of the statistical character of turbulent fluctuations in thermal convection. Phys. Fluids 6, 1848–1855.Google Scholar
Clever, R. M. & Busse, F. H. 1979 Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis. J. Fluid Mech. 94, 609–627.Google Scholar
Clune, T. 1993 Pattern selection in convecting systems. PhD thesis, University of California, Berkley (unpublished).
Clune, T. & Knobloch, E. 1993 Pattern selection in rotating convection with experimental boundary conditions.. Phys. Rev. E 47, 2236–2550.Google Scholar
Cortese, T. & Balachandar, S. 1993 Vortical nature of thermal plumes in turbulent convection.. Phys. Fluids A 5, 3226–3232.Google Scholar
Deardorff, J. W. 1970a Preliminary Results From Numerical Integrations Of The Unstable Planetary Boundary Layer. J. Atmos. Sci. 27, 1209–1211.Google Scholar
Deardorff, J. W. 1970b Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211–1213.Google Scholar
DeLuca, E. E., Werne, J., Rosner, R. & Cattaneo, R. 1990 Numerical simulations of soft and hard turbulence: Preliminary results for 2-D convection. Phys. Rev. Lett. 64, 2370–2373.Google Scholar
Ekman, V. W. 1905 On the influence of the earth's rotation on ocean-currents. Arkiv. Matem. Astr. Fysik, Stockholm 2-11, 1–52.Google Scholar
Fernando, H. J. S., Boyer, D. L. & Chen, R-R. 1989 Turbulent thermal convection in rotating and stratified fluids. Dyn. Atmos. Oceans 13, 95–121.Google Scholar
Fernando, H. J. S., Chen, R.-R. & Boyer, D. L. 1991 Effects of rotation on convective turbulence. J. Fluid Mech. 228, 513–547.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.
Gilman, P. A. 1977 Nonlinear dynamics of Boussinesq convection in a deep rotating spherical shell. Geophys. Astrophys. Fluid Dyn. 8, 93–135.Google Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1994 Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers. J. Fluid Mech. 262, 293–324.Google Scholar
Hathaway, D. H., Toomre, J. & Gilman, P. A. 1980 Convective instability when the temperature gradient and rotation vector are oblique to gravity. II. Real fluids with effects of diffusion. Geophys. Astrophys. Fluid Dyn. 15, 7–37.Google Scholar
Helfrich, K. R. 1994 Thermals with background rotation and stratification. J. Fluid Mech. 259, 265–280.Google Scholar
Heslot, R. Castaing, B. & Libchaber, A. 1987 Transitions to turbulence in helium gas.. Phys. Rev. A 36, 5870–5873.Google Scholar
Hide, R. 1964 The viscous boundary layer at the free surface of a rotating baroclinic fluid. Tellus XVI, 523–529.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh number. In Proc. llth Intl Congr. ofAppl. Mech. Munich (Germany) (ed. H. Gortler), pp. 1109–1115. Springer.
Howells, P. A. C. Rotunno, R. & Smith, R. K. 1988 A comparative study of atmospheric and laboratory-analogue numerical tornado-vortex models Q. J. R. Met. Soc. 114, 801–822.Google Scholar
Inoersoll, A. P. 1990 Atmospheric dynamics of the outer planets. Science 248, 308–315.Google Scholar
Jones, H. & Marshall, J. M. 1993 Convection in a neutral ocean; a study of open-ocean deep convection. J. Phys. Oceanogr. 23, 1009–1039.Google Scholar
Julien, K., Legg, S., Mcwilliams, J. & Werne, J. 1996a Hard Turbulence In Rotating Rayleigh-Benard Convection. Phys. Rev. E 53, 5557R–5560R.Google Scholar
Julien, K., Legg, S., Mcwilliams, J. & Werne, J. 1996b A statistical analysis on the influence of rotation in Rayleigh-Benard convection. In preparation.
Kerr, R. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.Google Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In Proc. 3rd GAMM Conf. Numerical Methods in Fluid Mechanics (ed. E. H. Hirschel), p. 165. Vieweg, Braunschweig.
Klinger, B. A. & Marshall, J. 1995 Regimes and scaling laws for rotating deep convection in the ocean. Dyn. Atmos. Oceans 21, 227–256.Google Scholar
Küppers, G. 1970 The steady finite amplitude convection in a rotating fluid layer.. Phys. Lett. A 32, 7–8.Google Scholar
Küppers, G. & Lortz, D. 1969 Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609–620.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.
Li, N. & Ecke, R. E. 1993 Kiippers-Lortz transition at high dimensionless rotation rates in rotating Rayleigh-Benard convection. Phys. Rev. E 47, R2991–R2994.Google Scholar
Malkus, W. V. R. 1963 Outline of a theory of turbulent convection. In Theory and Fundamental Research in Heat Transfer (ed. J. A. Clark). Pergamon.
Maxworthy, T. & Narimousa, S. 1994 Unsteady deep convection in a homogeneous rotating fluid. J. Phys. Oceanogr. 24, 865–887.Google Scholar
McWilliams, J. C. 1971 The boundary layer dynamics of symmetric vortices. PhD thesis. Harvard University (unpublished).
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Melander, M. V., Zabusky, N. J. & Mcwilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Nagakawa, Y. & Frenzen, P. 1955 A theoretical and experimental study of cellular convection in rotating fluids. Tellus 7, 1–21.Google Scholar
Ohlsen, D. R., Hart, J. E. & Kittelman, S. 1995 Laboratory experiments on rotating turbulent convection. American Meteorological Society Tenth Conf. on Atmospheric and Oceanic Waves and Stability, Preprints, pp. 255–256.
Ooyama, K. 1966 On the stability of the baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci. 23, 43–53.Google Scholar
Prandtl, L. 1932 Meteorologische Anwendungen der Stromungslehre. Beitr. Physik Atmos. 19, 188–202.Google Scholar
Priestley, C. H. B. 1959 Turbulent Transfer in the Lower Atmosphere. The University of Chicago Press.
Raasch, S. & Etling, D. 1991 Numerical simulation of rotating turbulent thermal convection. Beitr. Phys. Atmosph. 64, 185–199.Google Scholar
Riley, J. J., Metcalfe, R. W. & Orszag, S. A. 1986 Direct numerical simulations of chemically reacting turbulent mixing layers. Phys. Fluids 29, 406–422.Google Scholar
Rossby, H. T. 1969 A study of Benard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Sano, M., Wu, X.-Z. & Libchaber, A. 1989 Turbulence in helium-gas free-convection.. Phys. Rev. A 40, 6421–6430.Google Scholar
Schott, R, Visbeck, M. & Fischer, J. 1993 Observations of vertical currents and convection in the central Greenland Sea during the winter of 1988-1989. J. Geophys. Res. 98, 14401–14421.Google Scholar
She, Z.-S. 1989 On the scaling laws of thermal turbulent convection.. Phys. Fluids A 1, 911–913.Google Scholar
Shraiman, B. & Siggia, E. 1990 Heat transport in high-Rayleigh-number convection.. Phys. Rev. A 42, 3650–3653.Google Scholar
Somerville, R. C. F. & Lipps, F. B. 1973 A numerical study in three space dimensions of Benard convection in a rotating fluid. J. Atmos. Sci. 30, 590–596.Google Scholar
Spalart, P. R., Moser, R. D. & Rogers, M. M. 1991 Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions. J. Comput. Phys. 96, 297–324.Google Scholar
Spiegel, E. A. 1971 Convection in stars I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352.Google Scholar
Spiegel, E. A. 1972 Convection in stars II. Special effects. Ann. Rev. Astron. Astrophys. 10, 261–304.Google Scholar
Stommel, H. 1972 Deep winter-time convection in the western Mediterranean sea. In Studies in Physical Oceanography: a tribute to Georg Wust on his 80th birthday (ed. A. L. Gordon), pp. 207–218. Gordon and Breach.
Swift, J. 1984 Convection in a rotating fluid layer. Contemp. Maths 28, 435–448.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
Thompson, W. 1867 On Vortex Atoms. Phil. Mag. 34, 20. [Papers iv.1]Google Scholar
Tilgner, A., Belmonte, A. & Libchaber, A. 1993 Temperature and velocity profiles of turbulent convection in water. Phys. Rev. E 47, R2253–R2256.Google Scholar
Tong, P. & Shen, Y. 1992 Relative velocity fluctuations in turbulent Rayleigh-Benard convection. Phys. Rev. Lett. 69, 2066–2069.Google Scholar
Veronis, G. 1968 Large amplitude Benard convection in a rotating fluid. J. Fluid Mech. 31, 113–139.Google Scholar
Werne, J. 1993 The structure of hard turbulent convection in two dimensions: numerical evidence.. Phys. Rev. E 48, 1020–1035.Google Scholar
Werne, J. 1995 Incompressibility and no-slip boundaries in the Chebyshev-tau approximation: correction to Kleiser and Schumann's influence-matrix solution. J. Comput. Phys. 120, 260–265.Google Scholar
Werne, J., Deluca, E. E., Rosner, R. & Cattaneo, F. 1991 Development of hard turbulent convection in 2-D numerical simulations. Phys. Rev. Lett. 67, 3579–3581.Google Scholar
Wilson, T. & Rotunno, R. 1986 Numerical simulation of a laminar end-wall vortex and boundary layer. Phys. Fluids 29, 3993–4005.Google Scholar
Wu, X.-Z., Castaing, B., Heslot, F. & Libchaber, A. 1988 Scaling properties of soft thermal turbulence in Rayleigh-Benard convection. In Universalities in Condensed Matter (ed. R. Jullien, L. Peliti, R. Rammal & N. Boccara), pp. 208–212. Springer.
Wu, X.-Z., Kadanoff, L. P. Libchaber, A. & Sano, M. 1990 Frequency power spectrum of temperature-fluctuations in free-convection. Phys. Rev. Lett. 64, 2140–2143.Google Scholar
Wu, X.-Z. & Libchaber, A. 1992 Scaling relations in thermal turbulence - the aspect-ratio dependence. Phys. Rev. A 45, 842–845.Google Scholar
Yakhot, V. 1992 4/5 Kolmogorov law for statistically stationary turbulence: application to high-Rayleigh-number Benard convection. Phys. Rev. Lett. 69, 769–771.Google Scholar
Zhong, F., Ecke, R. & Steinberg, V. 1993 Rotating-Benard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.Google Scholar
Zocchi, G., Moses, E. & Libchaber, A. 1990 Coherent structures in turbulent convection, an experimental-study.. Physica A 166, 387–407.Google Scholar