Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T05:34:28.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear response of a rotating fluid to differential heating from below

Published online by Cambridge University Press:  21 April 2006

Charles Quon
Charles Quon
Affiliation:
Atlantic Oceanographic Laboratory, Department of Fisheries and Oceans, Bedford Institute of Oceanography, Dartmouth, N.S., Canada B2Y 4A2 and National Center for Atmospheric Research, Boulder, CO 80303, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies numerically the instabilities and quasi-steady states of thermal convection in a Boussinesq fluid within a unit square which is bounded rigidly on all sides, differentially heated on the bottom, insulated above, and rotating about a vertical axis. The flows for five ascending values of the thermal Rossby number β are studied for constant Prandtl number σ and infinitesimal Ekman number E. For small values of β, either transient or stationary spatial oscillations occur. The results agree with Daniels’ (1976) and Daniels & Stewartson's (1977, 1978) linear and weakly nonlinear theories in principle if not in detail. For large values of β. the flows are highly nonlinear. They undergo two distinct stages of instability and eventually settle down to steady states. It is shown that a viscous non-diffusive boundary layer can exist at steady state for large β. For the maximum value of β under investigation, only quasi-steady states have been reached in the interior. Inertial gravity waves are observed emanating from the unstable corner of the fluid. These numerical solutions establish that the initial instability is an exchange instability rather than a catastrophic one. The question of bifurcation has not been resolved.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 233 - 263
Copyright
© 1987 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

Beardsley, R. C. & Festa, J. F. 1972 A numerical model of convection driven by a surface stress and non-uniform horizontal heating. J. Phys. Oceanogr. 2, 444–455.Google Scholar
Chandrasekhar, S. 1962 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Daniels, P. G. 1976 Thermal convection in a differentially heated rotating fluid annulus. Geophys. Fluid Dyn. 7, 297–330.Google Scholar
Daniels, P. G. & Stewartson, K. 1977 On the spatial oscillations of a horizontally heated rotating fluid. Math. Proc. Camb. Phil. Soc. 81, 325–349.Google Scholar
Daniels, P. G. & Stewartson, K. 1978 On the spatial oscillations of a horizontally heated rotating fluid. II. Q. J. Mech. Appl. Maths 31, 113–135.Google Scholar
Gill, A. E. 1966 The boundary layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536.Google Scholar
Hignett, P., Ibbetson, A. & Killworth, P. D. 1981 On rotating thermal convection driven by non-uniform heating from below. J. Fluid Mech. 109, 161–187.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh number. Proc. 11th Intl congr. of applied mechanics, Munich 1964 (ed. H. Gortler), pp. 1109–1115. Springer-Verlag.
Hunter, C. 1967 The axisymmetric flow in a rotating annulus due to a horizontally applied temperature gradient. J. Fluid Mech. 27, 753–778.Google Scholar
Kalnay De Rivas, E. 1973 Numerical models of the circulation of the atmosphere of Venus. J. Atmos. Sci. 30, 763–779.Google Scholar
Killworth, P. D. & Manins, P. C. 1980 A model of confined thermal convection driven by non-uniform heating from below. J. Fluid Mech. 98, 587–607.Google Scholar
Needler, G. T. 1967 A model for thermocline circulation in an ocean of finite depth. J. Mar. Res. 25, 329–342.Google Scholar
Quon, C. 1972 High Rayleigh number convection in an enclosure - a numerical study. Phys. Fluids 15, 12–19.Google Scholar
Quon, C. 1976 A mixed spectral and finite difference model to study baroclinic annulus waves. J. Comp. Phys. 29, 442–479.Google Scholar
Quon, C. 1977 Axisymmetric states of an internally heated rotating annulus. Tellus 29, 83–96.Google Scholar
Quon, C. 1980 Quasi-steady symmetric regimes of a rotating annulus differentially heated on the horizontal boundaries. J. Atmos. Sci. 37, 2407–2423.Google Scholar
Quon, C. 1981 In search of symmetric baroclinic instability in an enclosed rotating fluid. Geophys. Astrophys. Fluid Dyn. 17, 171–197.Google Scholar
Quon, C. 1983a Effect of grid distribution on the computation of high Rayleigh number convection in a differentially heated cavity. In Numerical Properties and Methodologies in Heat Transfer (ed. T. M. Shih), pp. 261–281. Hemisphere and Springer.
Quon, C. 1983b Convection induced by insulated boundaries in a square. Phys. Fluids 26, 632–637.Google Scholar
Quon, C. 1986 Temporal development of spatial oscillations in a confined rotating fluid - a numerical study. Geophys. Astrophys. Fluid Dyn. 36, 207–228.Google Scholar
Quon, C. 1987 Onset of spatial oscillations in a deep rotating fluid differentially heated from below. Phys. Fluids 30, 672–678.Google Scholar
Roberts, G. O. 1971 Computational meshes for boundary layer problems. In Proc. Second Intl Conf. on Numerical Methods in Fluid Dynamics (ed. M. Holt), pp. 171–177. Springer.
Robinson, A. R. 1959 The symmetric state of a rotating fluid differentially heated in the horizontal. J. Fluid Mech. 6, 599–620.Google Scholar
Rossby, H. T. L. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 9–16.Google Scholar
Stern, M. E. 1975 Horizontal convection and thermoclines. In Ocean Circulation Physics, Ch. 12. Academic.
Stewartson, K. 1957 On almost rigid rotation. J. Fluid Mech. 3, 17–26.Google Scholar
Stommel, H. 1962 On the smallness of sinking regions in the ocean. Proc. Nat. Acad. Sci. 48, 766–772.Google Scholar