pure and applied geophysics

, Volume 117, Issue 5, pp 958–987 | Cite as

An approximate method for determining the stability of two-scale flow in the mantle

  • John N. Skilbeck
  • P. Dan McKenzie


An approximate method of calculating the stability of two-dimensional convection rolls to cross-roll disturbances is evaluated by comparison with the results from exact analyses and good agreement is obtained. In particular, investigation of the second mode of the disturbance provides a qualitative estimate of the terms excluded in making the approximation. We conclude that this approximate method, sensibly used, gives a good indication of the stability of rolls to cross-roll disturbances.

Application of this method is made to convection rolls heated partially or wholly from within, and to rolls in the presence of a longitudinal shear flow. The approximation indicates that, at high Rayleigh Numbers, growth rates are underestimated and so the amount of shear calculated to stabilise longitudinal rolls is a lower bound. Our results suggest that two-dimensional rolls are unlikely to be a stable flow pattern beneath even the fastest moving plates.

Key words

Mantle convection Plate tectonics 


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  1. Anderson, D. L. (1962),The plastic layer of the earth's mantle, Sci. Amer.207, 52.Google Scholar
  2. Atwater, T. andMolnar, P. (1973),Relative motion of hot spots in the mantle, Nature246, 288.Google Scholar
  3. Boussinesq, J. (1903),Théorie Analytique de la Chaleur,2 172, Gauthier-Villars, Paris.Google Scholar
  4. Busse, F. H. (1967),On the stability of two-dimensional convection in a layer heated from below, J. Math. and Phys.46, 140.Google Scholar
  5. Busse, F. H. andWhitehead, J. A. (1971),Instabilities of convection rolls in a high Prandtl number fluid, J. Fluid Mech.47, 305.Google Scholar
  6. Chapple, W. M. andTullis, T. E. (1977),Evaluation of the forces that drive the plates, J. Geophys. Res.82 (14), 1967.Google Scholar
  7. Chen, M. M. andWhitehead, J. A. (1968),Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wavenumbers, J. Fluid Mech.31, 1.Google Scholar
  8. Gilbert, F. andBackus, G. E. (1966),Propagator matrices in elastic wave and vibration problems, Geophysics31, (2), 326.Google Scholar
  9. Isacks, B. andMolnar, P. (1971),Distribution of stresses in the descending lithosphere from a global survey of focal mechanism solutions of mantle earthquakes, Rev. Geophys. Space Phys.9, 103.Google Scholar
  10. Langmuir, I. (1938),Surface motions of water induced by wind, Science87, 119.Google Scholar
  11. McKenzie, D. P. andRichter, F. M. (1976),Convection currents in the earth's Mantle, Sci. Amer.235, 72.Google Scholar
  12. McKenzie, D. P., Roberts, J. M. andWeiss, N. O. (1974),Convection in the earth's mantle: towards a numerical simulation. J. Fluid Mech.62, 465.Google Scholar
  13. McKenzie, D. P. andWeiss, N. O. (1975),Speculations on the thermal and tectonic history of the earth, Geophys. J. R. Astr. Soc.42, 131.Google Scholar
  14. Minster, J. B., Jordan, T. H., Molnar, P. andHaines, E. (1974),Numerical modelling of instantaneous plate tectonics, Geophys. J. R. Astr. Soc.36, 541.Google Scholar
  15. Moore, D. R., Peckover, R. S. andWeiss, N. O. (1974),Difference methods for time-dependent two-dimensional convection, Comp. Phys. Comm.7, 198.Google Scholar
  16. Moore, D. R. andWeiss, N. O. (1973),Two-dimensional Rayleigh-Benard convection, J. Fluid Mech.58, 289.Google Scholar
  17. Richter, F. M. (1973),Convection and the large-scale circulation of the mantle, J. Geophys. Res.78, 8735.Google Scholar
  18. Richter, F. M., andParsons, B. (1975),On the interaction of two-scales of convection in the mantle, J. Geophys. Res.80, 2529.Google Scholar
  19. Richter, F. M., andMcKenzie, D. P. (1978),Simple plate models of mantle convection, J. Geophysics44, 441.Google Scholar
  20. Schneck, P. andVeronis, G. (1967),Comparison of some recent experimental and numerical results in Benard Convection, Phys. of Fluids10, 927.Google Scholar
  21. Skilbeck, J. N. (1976),The stability of mantle convection, Ph.D. Thesis, U. of Cambridge, U. K.Google Scholar
  22. Skilbeck, J. N. andWhitehead, J. A. (1978),A possible mechanism for the formation of discrete islands in linear island chains, Nature272, 499.Google Scholar
  23. Straus, J. M. (1972),Finite amplitude doubly diffusive convection, J. Fluid Mech.56, 353.Google Scholar

Copyright information

© Birkhäuser Verlag 1979

Authors and Affiliations

  • John N. Skilbeck
    • 1
  • P. Dan McKenzie
    • 2
  1. 1.Department of Geological SciencesDurhamU.K.
  2. 2.Department of Geodesy and GeophysicsCambridge

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