Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-25T07:36:02.543Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of geometry on the gravitational instability of a buoyant region of viscous fluid

Published online by Cambridge University Press:  26 April 2006

John R. Lister  and
Ross C. Kerr
John R. Lister
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Research School of Earth Sciences, The Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
Ross C. Kerr
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Research School of Earth Sciences, The Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The low-Reynolds-number stability of a region of buoyant fluid surrounded by denser fluid is analysed in two situations. In the first study, the buoyant fluid lies in a thin layer sandwiched between two denser and much deeper layers. The growth rate and wavelength of the most unstable sinusoidal perturbation are calculated and the effects of the viscosity ratios and density differences between the fluids are investigated. It is found that if the buoyant fluid is much less viscous than the overlying fluid then, in quite general circumstances, both the most unstable wavelength and the corresponding growth rate are inversely proportional to the cube root of the viscosity of the buoyant fluid. A physical explanation of this result is given by scaling analysis of the total dissipation. In the second study, the buoyant fluid takes the form of a cylinder rising through a uniform environment. The eigenmodes of small perturbation about this state of motion are found for each axial wavenumber in terms of Fourier series of separable solutions to the Stokes equations. In contrast to the first study, it is found that the most unstable wavelength and growth rate are asymptotically independent of the viscosity of the buoyant fluid when this viscosity is small.

The difference between the results of the two studies is of importance, particularly for geophysical applications in which viscosity ratios are very large. Previous models of linear regions of volcanism at mid-ocean ridges and at island arcs have assumed that results obtained in simple two-layered systems can be generalized to other geometries. The conclusions of these models are discussed in the light of the stability results for a cylindrical (and hence linea.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 202 , May 1989 , pp. 577 - 594
Copyright
© 1989 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

Biot, M. A. & Odé, H. 1965 Theory of gravitational instability with variable overburden and compaction. Geophys. 30, 213–227.Google Scholar
Bonatti, E.: 1985 Punctiform initiation of seafloor spreading in the Red Sea during transition from a continental to an oceanic rift. Nature 316, 33–37.Google Scholar
Braunstein, J. & O'Brien, G. D. 1968 Diapirism and Diapirs. Am. Assoc. Petrol. Geol., Mem. 8. 444 pp.
Craig, C. & Jackson, M. P. A. 1987 The spacing of salt diapirs in the Great Kavir: modelling the Rayleigh–Taylor instability with two buoyant layers. Terra Cognita 7, 199.Google Scholar
Crane, K.: 1985 The spacing of rift axis highs: dependence upon diapiric processes in the underlying asthenosphere? Earth Planet Sci. Lett. 72, 405–414.Google Scholar
Currie, I. G.: 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.Google Scholar
Howard, L. N.: 1966 Convection at high Rayleigh number. In Proc. 11th Intl Congr. Appl. Mech., Munich, pp. 1109–1115. Springer.
Kerr, R. C. & Lister, J. R., 1988 Island arc and mid-ocean ridge volcanism, modelled by diapirism from linear source regions. Earth Planet. Sci. Lett. 88, 143–152.Google Scholar
Lick, W.: 1965 The instability of a fluid layer with time dependent heating. J. Fluid Mech. 21, 565–576.Google Scholar
Marsh, B. D.: 1979 Island arc development: Some observations, experiments and speculations. J. Geol. 87, 687–713.Google Scholar
Marsh, B. D. & Carmichael, I. S. E. 1974 Benioff zone magmatism. J. Geophys. Res. 79, 1196–1206.Google Scholar
Mohr, P. A. & Wood, C. A., 1976 Volcano spacing and lithospheric attenuation in the eastern rift of Africa. Earth Planet. Sci. Lett. 33, 126–144.Google Scholar
Nettleton, L. L.: 1934 Fluid mechanics of salt domes. Bull. Am. Assoc. Petrol. Geol. 18, 175–204.Google Scholar
Ramberg, H.: 1968a Fluid dynamics of layered systems in a field of gravity, a theoretical basis for certain global structures and isostatic adjustment. Phys. Earth Planet. Inter. 1, 63–87.Google Scholar
Ramberg, H.: 1968b Instability of layered systems in a field of gravity, 1, 2. Phys. Earth Planet. Inter. 1, 427–474.Google Scholar
Ramberg, H.: 1972 Mantle diapirism and its tectonic and magmatic consequences. Phys. Earth Planet. Inter. 5, 45–60.Google Scholar
Schouten, H., Klitgord, K. D. & Whitehead, J. A., 1985 Segmentation of mid-ocean ridges. Nature 317, 225–229.Google Scholar
Selig, F.: 1965 A theoretical prediction of salt dome patterns. Geophys. 30, 633–643.Google Scholar
Sigurdsson, H. & Sparks, R. S. J. 1978 Lateral magma flow within rifted Icelandic crust. Nature 274, 126–130.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Vidal, A. & Acrivos, A., 1968 Effects of non-linear temperature profiles on the onset of convection driven by surface tension gradients. Indust. Engng Chem. Fund. 7, 53–58.Google Scholar
Whitehead, J. A., Dick, H. B. J. & Schouten, H. 1984 A mechanism for magmatic accretion under spreading centers. Nature 312, 146–148.Google Scholar
Whitehead, J. A. & Luther, D. S., 1975 Dynamics of laboratory diapir and plume models. J. Geophys. Res. 80, 705–717.Google Scholar
Woidt, W. D.: 1978 Finite-element calculations applied to salt-dome analysis. J. Geophys. Res. 80, 705–717.Google Scholar