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Nonlinear penetrative convection

Published online by Cambridge University Press:  29 March 2006

D. R. Moore  and
N. O. Weiss
D. R. Moore
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Convection in water above ice penetrates into the stably stratified region above the density maximum at 4 °C. Two-dimensional penetrative convection in a Boussinesq fluid confined between free boundaries has been studied in a series of numerical experiments. These included cases with a constant temperature at both boundaries as well as cases with a fixed average flux at the lower boundary. Steady convection occurs at Rayleigh numbers below the critical value predicted by linear theory. At high Rayleigh numbers, resonant coupling between convection and gravitational modes in the stable layer excites finite amplitude oscillations. The problem can be described by a simplified model which allows for distortion of the mean temperature profile and balances the convected and conducted flux. This model explains the finite amplitude instability and predicts the Nusselt number as a function of Rayleigh number. These predictions are in excellent agreement with the computed results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 61 , Issue 3 , 20 November 1973 , pp. 553 - 581
Copyright
© 1973 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bodenheimer, P. 1965 Astrophys. J. 142, 451.
Böhm, K.-H. 1963 Astrophys. J. 138, 297.
Böhm, K.-H. 1967 In Aerodynamic Phenomena in Stellar Atmospheres (ed. R. N. Thomas), p. 366. Academic.
Busse, F. H. 1967a J. Math. & Phys. 46, 140.
Busse, F. H. 1967b J. Fluid Mech. 28, 223.
Busse, F. H. 1967c J. Fluid Mech. 30, 625.
Busse, F. H. & Whitehead, J. A. 1971 J. Fluid Mech. 47, 305.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, chap. 2. Clarendon Press.
Deardorff, J. W. 1964 J. Atmos. Sci. 21, 419.
Deardorff, J. W., Willis, G. E. & Lilly, D. K. 1969 J. Fluid Mech. 35, 7.
Debler, W. R. 1966 J. Fluid Mech. 24, 165.
Elder, J. W. 1969 J. Fluid Mech. 35, 417.
Faller, A. J. & Kaylor, R. 1970 J. Geophys. Res. 75, 521.
Fromm, J. E. 1965 Phys. Fluids, 8, 1757.
Furumoto, A. & Rooth, C. 1961 Geophysical fluid dynamics. Woods Hole Oceanographic Inst. RepGoogle Scholar
Herring, J. R. 1963 J. Atmos. Sci. 20, 325.
Kohl, K. 1966 Z. Astrophys. 64, 472.
Krishnamurti, R. 1968a J. Fluid Mech. 33, 445.
Krishnamurti, R. 1968b J. Fluid Mech. 33, 457.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Latour, J. 1973 To be published.
Malkus, W. V. R. 1960 In Aerodynamical Phenomena in Stellar Atmospheres (ed. R. N. Thomas), p. 346. Bologna: Zanichelli.
Malkus, W. V. R. 1963 Proc. 3rd Tech. Conf. on Hurricanes & Tropical Met., p. 89. Mexico.
Moore, D. R., Peckover, R. S. & Weiss, N. O. 1973 Difference methods for time-dependent two-dimensional convection. Comp. Phys. Comm. (in press).Google Scholar
Moore, D. R. & Weiss, N. O. 1973 J. Fluid Mech. 58, 289.
Moore, D. W. 1967 In Aerodynamic Phenomena in Stellar Atmospheres (ed. R. N. Thomas), p. 405. Academic.
Musman, S. 1968 J. Fluid Mech. 31, 343.
Myrup, L., Gross, D., Hoo, L. S. & Goddard, W. 1970 Weather, 25, 150.
Rossby, H. T. 1969 J. Fluid Mech. 36, 309.
Spiegel, E. A. 1967 In Aerodynamic Phenomena in Stellar Atmospheres (ed. R. X. Thomas), p. 348. Academic.
Spiegel, E. A. 1968 Highlights of Astronomy (ed. L. Perek), p. 261. Dordrecht: Reidel.
Spiegel, E. A. 1971 Ann. Rev. Astron. Astrophys. 9, 323.
Spiegel, E. A. 1972 Ann. Rev. Astron. Astrophys. 10, 261.
Thomas, J. H. 1972 Solar Phys. 24, 262.
Thomas, J. H., Clark, P. A. & Clark, A. 1971 Solar Phys. 16, 51.
Townsend, A. A. 1964 Quart. J. Roy. Met. Soc. 90, 248.
Townsend, A. A. 1966 J. Fluid Mech. 24, 307.
Veronis, G. 1963 Astrophys. J. 137, 641.
Veronis, G. 1965 J. Mar. Res. 23, 1.
Veronis, G. 1966a J. Fluid Mech. 24, 545.
Veronis, G. 1966b J. Fluid Mech. 26, 49.
Veronis, G. 1968a J. Fluid Mech. 31, 113.
Veronis, G. 1968b J. Fluid Mech. 34, 315.
Welander, P. 1967 J. Fluid Mech. 29, 17.
Whitehead, J. A. & Chen, M. M. 1970 J. Fluid Mech. 40, 549.