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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: thierry.alboussiere@ens-lyon.fr
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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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