Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T15:00:24.676Z Has data issue: false hasContentIssue false
  • English
  • Français

The self-similar rise of a buoyant thermal in very viscous flow

Published online by Cambridge University Press:  10 July 2008

ROBERT J. WHITTAKER  and
JOHN R. LISTER
ROBERT J. WHITTAKER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
JOHN R. LISTER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

An exact similarity solution is obtained for the rise of a buoyant thermal in Stokes flow, in which both the rise height and the diffusive growth scale like t1/2 as time t increases. The dimensionless problem depends on a single parameter Ra = B/(νκ) – a Rayleigh number – based on the (conserved) total buoyancy B of the thermal, and the kinematic viscosity ν and thermal diffusivity κ of the fluid. Numerical solutions are found for a range of Ra. For small Ra there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For large Ra, the temperature distribution becomes elongated vertically, with a long wake containing most of the buoyancy left behind the head. Passive tracers, however, are advected into a toroidal structure in the head. A simple asymptotic model for the large-Ra behaviour is obtained using slender-body theory. The width of the thermal is found to increase like (κt)1/2, while the wake length and rise height both increase like (RalnRa)1/2(κt)1/2, consistent with the numerical results. Previous experiments suggest that there is a significant transient regime.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 606 , 10 July 2008 , pp. 295 - 324
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Boyet, M. & Carlson, R. W. 2005 142Nd evidence for early (>4.53 Ga) global differentiation of the silicate earth. Science 309, 576–581.CrossRefGoogle ScholarPubMed
Buffett, B. A. 2002 Estimates of heat flow in the deep mantle based on the power requirements for the geodynamo. Geophys. Res. Lett. 29, 7.CrossRefGoogle Scholar
Bunge, H.-P. 2005 Low plume excess temperature and high core heat flux inferred from non-adiabatic geotherms in internally heated mantle circulation models. Phys. Earth Planet. Inter. 153, 3–10.CrossRefGoogle Scholar
Campbell, I. H. & Griffiths, R. W. 1990 Implications of mantle plume structure for the evolution of flood basalts. Earth Planet. Sci. Lett. 99, 79–93.CrossRefGoogle Scholar
Davaille, A., Girard, F. & Le Bars, M. 2002 How to anchor hot spots in a convecting mantle. Earth Planet. Sci. Lett. 203, 621–634.CrossRefGoogle Scholar
Davies, G. F. 1999 Dynamic Earth: Plates, Plumes, and Mantle Convection. Cambridge University Press.CrossRefGoogle Scholar
Farnetani, C. G. 1997 Excess temperature of mantle plumes: The role of chemical stratification across D″. Geophys. Res. Lett. 24, 1583–1586.CrossRefGoogle Scholar
Garcimartin, A., Mancini, H. & Perezgarcia, C. 1992 2D dynamics of a drop falling in a miscible fluid. Europhys. Lett. 19, 171–176.CrossRefGoogle Scholar
Griffiths, R. W. 1986 a Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity. J. Fluid Mech. 166, 115–138.CrossRefGoogle Scholar
Griffiths, R. W. 1986 b Particle motions induced by spherical convective elements in stokes flow. J. Fluid Mech. 166, 139–159.CrossRefGoogle Scholar
Griffiths, R. W. 1991 Entrainment and stirring in viscous plumes. Phys. Fluids A 3, 1233.CrossRefGoogle Scholar
Griffiths, R. W. & Campbell, I. H. 1990 Stirring and structure in mantle starting plumes. Earth Planet. Sci. Lett. 99, 66–78.CrossRefGoogle Scholar
Griffiths, R. W. & Campbell, I. H. 1991 On the dynamics of long-lived plume conduits in the convecting mantle. Earth Planet. Sci. Lett. 103, 214–227.CrossRefGoogle Scholar
Hamblin, W. K. & Christiansen, E. H. 1998 Earth's Dynamic Systems, 8th edn.Prentice–Hall.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Jellinek, M. & Manga, M. 2002 The influence of a chemical boundary layer on the fixity, spacing and lifetime of mantle plumes. Nature 418, 760–763.CrossRefGoogle ScholarPubMed
Jellinek, M. & Manga, M. 2004 Links between long-lived hot spots, mantle plumes, D″, and plate tectonics. Rev. Geophys. 42, RG3002.CrossRefGoogle Scholar
Kaminski, E. & Jaupart, C. 2003 Laminar starting plumes in high-Prandtl-number fluids. J. Fluid Mech. 478, 287–298.CrossRefGoogle Scholar
Kerr, R. C. & Mériaux, C. 2004 Structure and dynamics of sheared mantle plumes. Geochem. Geophys. Geosyst. 5, Q12009.CrossRefGoogle Scholar
Koh, C. J. & Leal, L. G. 1989 The stability of drop shapes for translation at zero Reynolds number through a quiescent fluid. Phys. Fluids A 1, 1309–1313.CrossRefGoogle Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.CrossRefGoogle Scholar
Kumagai, I. 2002 On the anatomy of mantle plumes: Effect of the viscosity ratio on entrainment and stirring. Earth Planet. Sci. Lett. 198, 211–224.CrossRefGoogle Scholar
Lee, S. H. & Leal, L. G. 1982 The motion of a sphere in the presence of a deformable interface: II. A numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87, 81–106.CrossRefGoogle Scholar
Loper, D. E. & Stacey, F. D. 1983 The dynamical and thermal structure of deep mantle plumes. Phys. Earth Planet. Inter. 33, 304–317.CrossRefGoogle Scholar
Machu, G., Meile, W., Nitsche, L. & Schaflinger, U. 2001 Coalescence, torus formation and breakup of sedimenting drops: Experiments and computer simulations. J. Fluid Mech. 447, 299–336.CrossRefGoogle Scholar
Metzger, B., Nicolas, M. & Guazzelli, E. 2007 Falling clouds of particles in viscous fluids. J. Fluid Mech. 580, 283–302.CrossRefGoogle Scholar
Morton, B. R. 1960 Weak thermal vortex rings. J. Fluid Mech. 9, 107–118.CrossRefGoogle Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintined and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Moses, E., Zocchi, G. & Libchaber, A. 1993 An experimental study of laminar plumes. J. Fluid Mech. 251, 581–601.CrossRefGoogle Scholar
Nataf, H.-C. 2000 Seismic imaging of mantle plumes. Annu. Rev. Earth Planet. Sci. 28, 391–417.CrossRefGoogle Scholar
Olson, P., Schubert, G. & Anderson, C. 1993 Structure of axisymmetric mantle plumes. J. Geophys. Res. 98 (B4), 6829–6844.CrossRefGoogle Scholar
Olson, P. & Singer, H. 1985 Creeping plumes. J. Fluid Mech. 158, 511–531.CrossRefGoogle Scholar
Pozrikidis, C. 1990 The instability of a moving viscous drop. J. Fluid Mech. 210, 1–21.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.Google Scholar
Steinberger, B. & O'Connell, R. J. 1998 Advection of plumes in mantle flow: Implications for hotspot motion, mantle viscosity and plume distribution. Geophys. J. Intl 132, 412–434.CrossRefGoogle Scholar
Tolstikhin, I. & Hofmann, A. W. 2005 Early crust on top of the Earth's core. Phys. Earth Planet. Inter. 148, 109–130.CrossRefGoogle Scholar
Turner, J. S. 1969 Buoyant plumes and thermals. Annu. Rev. Fluid Mech. 1, 29–44.CrossRefGoogle Scholar
Whitehead, J. A. & Luther, D. S. 1975 Dynamics of laboratory diapir and plume models. J. Geophys. Res. 80 (5), 705–717.CrossRefGoogle Scholar
Whittaker, R. J. 2007 Theoretical solutions for convective flows in geophysically motivated regimes. PhD Thesis, University of Cambridge. Available online at http://robert.mathmos.net/research/phd/.Google Scholar
Whittaker, R. J. & Lister, J. R. 2006 a Steady axisymmetric creeping plumes above a planar boundary. Part 1. A point source. J. Fluid Mech. 567, 361–378.CrossRefGoogle Scholar
Whittaker, R. J. & Lister, J. R. 2006 b Steady axisymmetric creeping plumes above a planar boundary. Part 2. A distributed source. J. Fluid Mech. 567 379–397.CrossRefGoogle Scholar