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Dissipative heating in convective flows

Published online by Cambridge University Press:  29 March 2006

J. M. Hewitt
Affiliation:
Department of Geodesy and Geophysics, University of Cambridge
D. P. Mckenzie
Affiliation:
Department of Geodesy and Geophysics, University of Cambridge
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge

Abstract

Dissipative heating is produced by irreversible processes, such as viscous or ohmic heating, in a convecting fluid; its importance depends on the ratio d/HT of the depth of the convecting region to the temperature scale height. Integrating the entropy equation for steady flow yields an upper bound to the total rate of dissipative heating in a convecting layer. For liquids there is a regime in which the ratio of dissipative heating to the convected heat flux is approximately equal to c(d/HT), where the constant c is independent of the Rayleigh number. This result is confirmed by numerical experiments using the Boussinesq approximation, which is valid only if d/HT is small. For deep layers the dissipative heating rate may be much greater than the convected heat flux. If the earth's magnetic field is maintained by a convectively driven dynamo, ohmic losses are limited to 5% of the convected flux emerging from the core. In the earth's mantle viscous heating may be important locally beneath ridges and behind island arcs.

Type
Research Article
Copyright
© 1975 Cambridge University Press

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References

Ackroyd, J. A. D. 1974 Stress work effects in laminar flat-plate natural convection. J. Fluid Mech. 62, 677.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
BraginskiÎ, S. I. 1964 Magnetohydrodynamics of the earth's core. Geomag. Aeron. 4, 898.Google Scholar
Bullard, E. C. 1949 The magnetic field within the earth. Proc. Roy. Soc. A 197, 433.Google Scholar
Bullard, E. C. & Gellman, H. 1954 Homogeneous dynamos and terrestrial magnetism. Phil. Trans. A 247, 213.Google Scholar
Gebhart, B. 1962 Effects of viscous dissipation in natural convection. J. Fluid Mech. 14, 225.Google Scholar
Gebhart, B. & Mollendorf, J. 1969 Viscous dissipation in external natural convection flows. J. Fluid Mech. 38, 97.Google Scholar
Graham, E. 1975 Numerical simulation of compressible convection. J. Fluid Mech (in the Press).Google Scholar
Griggs, D. T. 1972 The sinking lithosphere and the focal mechanism of deep earthquakes. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 361. McGraw-Hill.
Gubbins, D. 1974 Theories of the geomagnetic and solar dynamos. Rev. Geophys. SpacePhys. 12, 137.Google Scholar
Hide, R. 1956 The hydrodynamics of the earth's core. In Physics and Chemistry of theEarth, vol. 1 (ed. L. H. Ahrens, K. Rankama & S. K. Runcorn), p. 94. Pergamon.
Higgins, G. & Kennedy, G. C. 1971 The adiabatic gradient and the melting point gradient of the core of the earth. J. Geophys. Res. 76, 1870.Google Scholar
Jacobs, J. A. 1973 Physical state of the earth's core. Nature Phys. Sci. 243, 113.Google Scholar
Karig, D. E. 1971 Origin and development of the marginal basins in the Western Pacific. J. Geophys. Res. 76, 2542.Google Scholar
Kennedy, G. C. & Higgins, G. 1972 The core paradox. J. Geophys. Res. 78, 900.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, chap. 5. Pergamon.
Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of Continuous Media, chap. 8. Pergamon.
McKenzie, D. P. 1969 Speculations on the causes and consequences of plate motions. Geophys. J. Roy. Astr. Soc. 18, 1.Google Scholar
McKenzie, D. P., Roberts, J. M. & Weiss, N. O. 1974 Convection in the earth's mantle: towards a numerical solution. J. Fluid Mech. 62, 465.Google Scholar
McKenzie, D. P. & Weiss, N. O. 1975 Speculations on the thermal and tectonic history of the earth. Geophys. J. Roy. Astr. Soc. (in the Press).Google Scholar
Malkus, W. V. R. 1964 Boussinesq equations. Geophysical Fluid Dynamics. p. 1. WoodsHole Oceanog. Inst. Rep. no. 64-46.Google Scholar
Malkus, W. V. R. 1973 Convection at the melting point: a thermal history of the earth's core. Geophys. Fluid Dyn. 4, 267.Google Scholar
Nisbet, E. & Pearce, J. A. 1973 TiO2 and a possible guide to past Oceanic spreading rates. Nature, 246, 468.Google Scholar
Oxburgh, E. R. & Turcotte, D. L. 1968 Problems of high heat flow and volcanism associated with zones of descending mantle convective flow. Nature, 218, 1041.Google Scholar
Rice, A. R. 1971 Mechanism of dissipation in mantle convection. J. Geophys. Res. 76, 1450.Google Scholar
Spiegel, E. A. 1971a Turbulence in stellar convection zones. Comm. Astrophys. Space Sci. 3, 53.Google Scholar
Spiegel, E. A. 1971b Convection in stars. I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323.Google Scholar
Spiegel, E. A. 1972 Convection in stars. II. Special effects. Ann. Rev. Astron. Astrophys. 10, 261.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442.Google Scholar
Sugimura, A. & Uyeda, S. 1973 Island arcs: Japan and Its Environs. Elsevier.
Tozer, D. C. 1965 Heat transfer and convection currents. Phil. Trans. A 258, 252.Google Scholar
Turcotte, D. L., Hstji, A. T., Torrance, K. E. & Schubert, G. 1974 Influence of viscous dissipation on Bénard convection. J. Fluid Mech. 64, 369.Google Scholar
Vacquier, V., Uyeda, S., Yasui, M., Sclater, J., Corry, C. & Watanabe, T. 1966 Studies of the thermal state of the earth. The 19th paper: heat flow measurements in the north-western Pacific. Bull. Earthquake Res. Inst. 44, 1519.Google Scholar