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The nucleation, growth and settling of crystals from a turbulently convecting fluid

Published online by Cambridge University Press:  26 April 2006

Richard A. Jarvis  and
Andrew W. Woods
Richard A. Jarvis
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06511, USA Present address: Laboratoire de Dynamique des Systèmes Géologiques, Institut de Physique du Globe, 4 place Jussieu, 75252 Paris Cedex 05, France
Andrew W. Woods
Affiliation:
Institute Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

We present a new model to describe the thermal and compositional evolution of a binary alloy which is cooled from above. Explicit account is taken of the nucleation of crystals in the cold upper thermal boundary layer, the growth of crystals in the turbulently convecting interior, and their subsequent gravitational settling to the floor of the chamber. The crystallization of one solid phase only is considered. When the residence time of a typical crystal within the convecting bulk is short compared with the overall cooling time of the fluid, the crystal size distribution loses memory of earlier conditions in the fluid and the number density simply decays exponentially with the cube of the crystal size. A quasi-steady state exists in which the rate of crystal production is balanced by the rate of sedimentation at the floor, allowing the volume fraction of suspended crystals to remain small until convection ceases to be vigorous.

We focus on the situation in which the latent heat released by solidification would far exceed the heat flux extracted through convection if the melt undercooling were maintained equal to the initial temperature difference applied at the cold upper boundary. In this case, either the growth or nucleation of crystals must be limited in order that the fluid continues to cool. Both the growth-limited and nucleation-limited regimes may develop during the cooling of an individual fluid body, depending upon the thermal boundary condition at the upper boundary of the convecting portion of the fluid.

We calculate how the mean crystal size within the sedimented crystal pile evolves as the fluid cools. During the growth-limited regime, the mean crystal size in the crystal pile typically decreases with height, owing to the decrease in the extracted heat flux and the greater efficiency of crystal settling as the fluid layer becomes shallower. In contrast, during the nucleation-limited regime, the fluid undercooling may increase significantly as the fluid cools, and inverse grading (large crystals over small) is possible. We discuss the possible application of our theory to the cooling of large igneous intrusions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 273 , 25 August 1994 , pp. 83 - 107
Copyright
© 1994 Cambridge University Press

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References

Bartlett, R. W. 1969 Magma convection, temperature distribution and differentiation. Am. J. Sci. 269, 169–182.Google Scholar
Bowen, N. L. 1915 Crystal differentiation in silicate liquids. Am. J. Sci. 39, 175–191.Google Scholar
Brandeis, G. & Jaupart, C. 1986 On the interaction between convection and crystallization in cooling magma chambers. Earth Planet. Sci. Lett. 77, 345–361.Google Scholar
Brandeis, G. & Jaupart, C. 1987 The kinetics of nucleation and crystal growth and scaling laws for magmatic crystallization. Contrib. Min. Petrol. 96, 24–34.Google Scholar
Brandeis, G., Jaupart, C. & Allègre, C. J. 1984 Nucleation, crystal growth and the thermal regime of cooling magmas. J. Geophys. Res. 89, 10161–10177.Google Scholar
Bruce, P. M. 1989 Thermal convection within the Earth's crust. PhD thesis, University of Cambridge.
Campbell, I. H. 1978 Some problems with the cumulus theory. Lithos 11, 311–323.Google Scholar
Davaille, A. & Jaupart, C. 1993 Transient high-Rayleigh-number thermal convection with large viscosity variations. J. Fluid Mech. 253, 141–166.Google Scholar
Dowty, E. 1980 Crystal growth and nucleation theory and the numerical simulation of igneous crystallization. In Physics of Magmatic Processes (ed. R. Hargraves), pp. 419–485. Princeton.
Fenn, P. M. 1977 The nucleation and growth of alkali fieldspars from hydrous melts. Can. Mineral. 15, 135–161.Google Scholar
Hansen, U., Yuen, D. A. & Kroening, S. E. 1992 Mass and heat transport in strongly time-dependent thermal convection at infinite Prandtl number. Geophys. Astrophys. Fluid Dyn. 63, 67–89.Google Scholar
Hort, M. & Spohn, T. 1991 Crystallization calculations for a binary melt cooling at constant rates of heat removal: implications for the crystallization of magma bodies. Earth Planet. Sci. Lett. 107, 463–474.Google Scholar
Howard, L. N. 1964 Convection at high Rayleigh number. In Proc. 11th Intl Congr. of Applied Mechanics, Munich 1964 (ed. H. Görtler), pp. 1109–1115. Springer.
Huppert, H. E. 1990 The fluid dynamics of solidification. J. Fluid Mech. 212, 209–240.Google Scholar
Jaupart, C. & Parsons, B. 1985 Convective instabilities in a variable viscosity fluid cooled from above. Phys. Earth Planet. Inter. 39, 14–32.Google Scholar
Katsaros, K. B., Liu, W. T., Businger, J. A. & Tillman, J. E. 1977 Heat transport and thermal structure in the interfacial boundary layer in an open tank of water in turbulent free convection. J. Fluid Mech. 83, 331–348.Google Scholar
Kerr, R. C., Woods, A. W., Worster, M. G. & Huppert, H. E. 1990 Solidification of an alloy cooled from above. Part 2. Non-equilibrium interfacial kinetics. J. Fluid Mech. 217, 331–348.Google Scholar
Kirkpatrick, R. J. 1976 Towards a kinetic model for the crystallization of magma bodies. J. Geophys. Res. 81, 2565–2571.Google Scholar
Kirkpatrick, R. J. 1977 Nucleation and growth of plagioclase, Makaopuhi and Alae lava lakes, Kilauea Volcano, Hawaii. Geol. Soc. Am. Bull. 88, 78–84.Google Scholar
Kraichnan, R. H. 1962 Mixing-length analysis of turbulent thermal convection at arbitrary Prandtl numbers. Phys. Fluids 5, 1374–1389.Google Scholar
Marsh, B. D. 1988 Crystal size distribution (CSD) in rocks and the kinetics and dynamics of crystallization. Contrib. Min. Petrol. 99, 277–291.Google Scholar
Martin, D. 1990 Crystal settling and in situ crystallization in aqueous solutions and magma chambers. Earth Planet. Sci. Lett. 96, 336–348.Google Scholar
Martin, D. & Nokes, R. 1988 Crystal settling in a vigorously convecting magma chamber. Nature 332, 534–536.Google Scholar
Martin, D. & Nokes, R. 1989 A fluid-dynamical study of crystal settling in convecting magmas. J. Petrol. 30, 1471–1500.Google Scholar
Randolph, A. D. & Larson, M. A. 1971 Theory of Particulate Processes. Academic.
Richter, F. M. 1978 Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J. Fluid Mech. 89, 553–560.Google Scholar
Rudman, M. 1992 Two-phase natural convection: implications for crystal settling in magma chambers. Phys. Earth Planet. Inter. 72, 153–172.Google Scholar
Shaw, H. R. 1972 Viscosities of magmatic silicate liquids: an empirical method of prediction. Am. J. Sci. 272, 870–893.Google Scholar
Solomatov, V. S. & Stevenson, D. J. 1993a Suspension in convective layers and style of differentiation of a terrestrial magma ocean. J. Geophys. Res. 98(E3), 5375–5390.Google Scholar
Solomatov, V. S. & Stevenson, D. J. 1993b Kinetics of crystal growth in a terrestrial magma ocean. J. Geophys. Res. 98(E3), 5407–5418.Google Scholar
Sparks, R. S. J., Huppert, H. E., Koyaguchi, T. & Hallworth, M. 1993 Origin of modal and rhythmic layering by sedimentation in a convecting magma chamber. Nature 361, 246–249.Google Scholar
Spohn, T., Hort, M. & Fischer, H. 1988 Numerical simulation of the crystallization of multicomponent melts in thin dikes or sill. 1. The liquidus phase. J. Geophys. Res. 93(B5), 4880–4894.Google Scholar
Tait, S. R. & Jaupart, C. 1992 Convection and macrosegregation in magma chambers. In Interactive Dynamics of Convection and Solidification (ed. S. H. Davis et al.), pp. 241–260. Kluwer.
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.
Turner, J. S., Huppert, H. E. & Sparks, R. S. J. 1986 Komatiites II. Experimental and theoretical investigations of post-emplacement cooling and crystallization. J. Petrol. 27, 397–437.Google Scholar
Woods, A. W. & Huppert, H. E. 1989 The growth of compositionally stratified solid above a horizontal boundary. J. Fluid Mech. 199, 29–53.Google Scholar