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A bound on the vertical transport of heat in the ‘ultimate’ state of slippery convection at large Prandtl numbers

Published online by Cambridge University Press:  18 July 2013

Xiaoming Wang  and
Jared P. Whitehead
Xiaoming Wang
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
Jared P. Whitehead*
Affiliation:
Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
†Email address for correspondence: [email protected]
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Abstract

An upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’ $R{a}^{1/ 2} $ scaling and, in the limit of infinite $Pr$, agrees with the bound of $R{a}^{5/ 12} $ recently derived in that limit for stress-free boundaries.

JFM classification

Geophysical and Geological Flows: Mantle convection Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 103 - 122
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
©2013 Cambridge University Press.

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