Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T20:46:30.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis

Published online by Cambridge University Press:  19 April 2006

R. M. Clever  and
F. H. Busse
R. M. Clever
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles 90024
F. H. Busse
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles 90024
Get access Rights & Permissions [Opens in a new window]

Abstract

Steady finite amplitude two-dimensional solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotating about its vertical axis. Rigid boundaries with prescribed constant temperatures are assumed and the solutions are obtained numerically by the Galerkin method. The existence of steady subcritical finite amplitude solutions is demonstrated for Prandtl numbers P < 1. A stability analysis of the finite amplitude solutions is performed by superimposing arbitrary three-dimensional disturbances. A strong reduction in the domain of stable rolls occurs as the rotation rate is increased. The reduction is most pronounced at low Prandtl numbers. The numerical analysis confirms the small amplitude results of Küppers & Lortz (1969) that all two-dimensional solutions become unstable when the dimensionless rotation rate Ω exceeds a value of about 27 at P ≃ ∞. A brief discussion is given of the three-dimensional time-dependent forms of convection which are realized at rotation rates exceeding the critical value.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 4 , 29 October 1979 , pp. 609 - 627
Copyright
© 1979 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

Busse, F. H. 1967a The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Busse, F. H. 1967b On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140–150.Google Scholar
Busse, F. H. 1972 The oscillatory instability of convection rolls in a low Prandtl number fluid. J. Fluid Mech. 52, 97–112.Google Scholar
Busse, F. H. & Clever, R. M. 1979a Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319–335.Google Scholar
Busse, F. H. & Clever, R. M. 1979b Nonstationary convection in a rotating system. In Recent Developments in Theoretical and Experimental Fluid Mechanics — Compressible and Incompressible Flows (ed. U. Müller, K. G. Roesner & B. Schmidt), pp. 376–385. Springer.
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Busse, F. H. & Whitehead, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. Fluid Mech. 66, 67–79.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Clever, R. M. & Busse, F. H. 1974 Transition to time dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Clever, R. M. & Busse, F. H. 1977 Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech. 81, 107–127.Google Scholar
Clever, R. M. & Busse, F. H. 1978 Large wavelength convection rolls in low Prandtl number fluids. Z. angew. Math. Phys. 29, 711–714.Google Scholar
Hart, J. E. 1971 Transition to a wavy vortex regime in convective flow between inclined plates. J. Fluid Mech. 48, 265–271.Google Scholar
Koschmieder, E. L. 1967 On convection on a uniformly heated rotating plane. Beitr. Phys. Atmosph. 40, 215–225.Google Scholar
Krishnamurti, R. 1971 On the transition to turbulent convection. 8th Symp. on Naval Hydrodyn., rep. ARC-179, pp. 289–310. Office of Naval Research.
Küppers, G. 1970 The stability of steady finite amplitude convection in a rotating fluid layer. Phys. Latt. 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
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. H. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Somerville, R. C. J. 1971 Bénard convection in a rotating fluid. Geophys. Fluid Dyn. 2, 247–262.Google Scholar
Somerville, R. C. J. & Lipps, F. B. 1973 A numerical study in three space dimensions of Bénard convection in a rotating fluid. J. Atmos. Sci. 30, 590–596.Google Scholar
Veronis, G. 1966 Motions at subcritical values of the Rayleigh number in a rotating fluid. J. Fluid Mech. 24, 545–554.Google Scholar
Veronis, G. 1968 Large-amplitude Bénard convection in a rotating fluid. J. Fluid Mech. 31, 113–139.Google Scholar