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Reflections on dissipation associated with thermal convection

Published online by Cambridge University Press:  28 May 2013

Thierry Alboussière  and
Yanick Ricard
Thierry Alboussière*
Affiliation:
Laboratoire de Géologie de Lyon, UMR 5276, CNRS, ENS-Lyon, Université Lyon 1, 69622 Villeurbanne, France
Yanick Ricard
Affiliation:
Laboratoire de Géologie de Lyon, UMR 5276, CNRS, ENS-Lyon, Université Lyon 1, 69622 Villeurbanne, France
*
†Email address for correspondence: [email protected]
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Abstract

Buoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$. Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$, independent of the dissipation number.

JFM classification

Convection: Convection Mathematical Foundations: General fluid mechanics Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , R1
Copyright
©2013 Cambridge University Press 

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References

Anufriev, A. P., Jones, C. A. & Soward, A. 2005 The Boussinesq and anelastic liquid approximations for convection in the Earth core. Phys. Earth Planet. Inter. 152, 163–190.Google Scholar
Backus, G. E. 1975 Gross thermodynamics of heat engines in deep interior of Earth. Proc. Natl Acad. Sci. 72 (4), 1555–1558.CrossRefGoogle ScholarPubMed
Bercovici, D. 1996 Plate generation in a simple model of lithosphere-mantle flow with dynamic self-lubrication. Earth Planet. Sci. Lett. 144, 41–51.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1995 Equations governing convection in earth’s core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97.Google Scholar
Buffett, B. A. 2002 Estimates of heat flow in the deep mantle based on the power requirements for the geodynamo. Geophys. Res. Lett. 29 (12), 1566–1569.Google Scholar
Carnot, S. 1824 Réflexions sur la Puissance Motrice du Feu. Bachelier Libraire.Google Scholar
Christensen, U. & Tilgner, A. 2004 Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos. Nature 429, 169–171.CrossRefGoogle ScholarPubMed
Hewitt, J. W., McKenzie, D. P. & Weiss, N. O. 1975 Dissipative heating in convective flows. J. Fluid Mech. 68 (4), 721–738.Google Scholar
Jarvis, G. T. & McKenzie, D. P. 1980 Convection in a compressible fluid with infinite Prandtl number. J. Fluid Mech. 96 (3), 515–583.CrossRefGoogle Scholar
Leng, W. & Zhong, S. 2008 Viscous heating, adiabatic heating and energetic consistency in compressible mantle convection. Geophys. J. Intl 173, 693–702.CrossRefGoogle Scholar
McKenzie, D. P. & Jarvis, G. T. 1980 The conversion of heat into mechanical work by mantle convection. J. Geophys. Res. 85 (B11), 6093–6096.Google Scholar
Murnaghan, F. D. 1951 Finite Deformation of an Elastic Solid. John Wiley and Sons.Google Scholar
Ogura, Y. & Phillips, N. A. 1961 Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179.Google Scholar
Schubert, G., Turcotte, D. & Olson, P. 2001 Mantle Convection in the Earth and Planets. Cambridge University Press.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442–447.CrossRefGoogle Scholar
Tackley, P. 1996 Effects of strongly variable viscosity on three-dimensional compressible convection in planetary mantles. J. Geophys. Res. 101 (B2), 3311–3332.Google Scholar
Tan, E. & Gurnis, M. 2007 Compressible thermochemical convection and application to lower mantle structures. J. Geophys. Res. 112, B06304.Google Scholar
Verhoogen, J. 1980 Energetics of the Earth. National Academy of Science.Google Scholar