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On penetrative convection at low Péclet number

Published online by Cambridge University Press:  26 April 2006

John R. Lister
John R. Lister
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

A new theoretical model is developed for the growth of a convecting fluid layer at the base of a stable, thermally stratified layer when heated from below. The imposed convective heat flux is taken to be comparable to the heat flux conducted down the background gradient so that diffusion ahead of the interface between the convecting and stable layers makes a significant contribution to the interfacial heat flux and to the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specified, and it is argued that this is determined by the ability of convective eddies to mix warmed fluid below the interface downwards. The interfacial velocity, which may be positive or negative, is then determined by the joint requirements of continuity of heat flux and temperature. A similarity solution is derived for the case of an initially linear temperature gradient and uniform heating. Solutions are also given for a heat flux that undergoes a step change and for a heat flux determined from a four-thirds power law with a fixed base temperature. Numerical calculations show that the predictions of the model are in good agreement with previously reported experimental measurements. Similar calculations are applicable to a wide range of geophysical problems in which the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 292 , 10 June 1995 , pp. 229 - 248
Copyright
© 1995 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Betts, A. K. 1973 Non-precipitating cumulus convection and its parameterization. Q. J. R. Met. Soc. 99, 178–196.Google Scholar
Carmack, E. C. & Farmer, D. M. 1982 Cooling processes in deep temperate lakes: A review with examples from two lakes in British Columbia. J. Mar. Res. 40, 85–111.Google Scholar
Carson, D. J. 1973 The development of a dry inversion-capped convectively unstable boundary layer. Q. J. R. Met. Soc. 99, 450–467.Google Scholar
Crapper, P. F. & Linden, P. F. 1974 The structure of turbulent density interfaces. J. Fluid Mech. 65, 45–63.Google Scholar
Deardorff, J. W., Willis, G. E. & Lilly, D. K. 1969 Laboratory investigation of non-steady penetrative convection. J. Fluid Mech. 35, 7–31.Google Scholar
Deardorff, J. W., Willis, G. E. & Stockton, B. H. 1980 Laboratory studies of the entrainment zone of a convectively mixed layer. J. Fluid Mech. 100, 41–64.Google Scholar
Denton, R. A. & Wood, I. R. 1981 Penetrative convection at low Péclet number. J. Fluid Mech. 113, 1–21 (referred to herein as DW).Google Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Ann. Rev. Fluid Mech. 23, 455–493.Google Scholar
Fernando, H. J. S. & Little, L. J. 1990 Molecular-diffusive effects in penetrative convection. Phys. Fluids A 2, 1592–1596.Google Scholar
Fernando, H. J. S. & Long, R. R. 1985 On the nature of the entrainment interface of a two-layer fluid subjected to zero-mean-shear turbulence. J. Fluid Mech. 151, 21–53.Google Scholar
Gubbins, D., Thomson, C. J. & Whaler, K. A. 1982 Stable regions in the Earth's liquid core. Geophys. J. R. Astron. Soc. 68, 241–251.Google Scholar
Hannoun, I. A. & List, E. J. 1988 Turbulent mixing at a shear-free density interface. J. Fluid Mech. 189, 211–234.Google Scholar
Hopfinger, E. J. & Linden, P. F. 1982 Formation of thermoclines in zero-mean-shear turbulence subjected to a stabilizing buoyancy flux. J. Fluid Mech. 114, 157–173.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh number. In Proc. 11th Intl. Congr. Appl. Mech., Munich, pp. 1109–1115. Springer.
Jaupart, C. & Brandeis, G. 1986 The stagnant bottom layer of convecting magma chambers. Earth Planet. Sci. Lett. 80, 183–199.Google Scholar
Kerr, R. C. 1991 Erosion of a stable density gradient by sedimentation-driven convection. Nature 353, 423–425.Google Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating. J. Fluid Mech. 21, 565–576.Google Scholar
Linden, P. F. 1973 The interaction of vortex rings with a sharp density interface: a model of turbulent entrainment. J. Fluid Mech. 60, 467–480.Google Scholar
Mory, M. 1991 A model of turbulent mixing across a density interface including the effect of rotation. J. Fluid Mech. 223, 193–207.Google Scholar
Noh, Y. & Fernando, H. J. S. 1993 The role of molecular diffusion in the deepening of the mixed layer. Dyn. Atmos. Oceans 17, 187–215.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Rouse, H. & Dodu, J. 1955 Turbulent diffusion across a density discontinuity. Houille Blanch 10, 522–532.Google Scholar
Schwarzschild, M. 1958 Structure and Evolution of Stars. Princeton University Press.
Tennekes, H. 1973 A model for the dynamics of the inversion above a convective boundary layer. J. Atmos. Sci. 30, 558–567.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Zilitinkevich, S. S. 1991 Turbulent Penetrative Convection. Avebury.