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The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

Published online by Cambridge University Press:  04 November 2015

Michael A. Calkins ,
Kevin Hale ,
Keith Julien ,
David Nieves ,
Derek Driggs  and
Philippe Marti
Michael A. Calkins
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Kevin Hale
Affiliation:
Harvey Mudd College, Claremont, CA 91711, USA
Keith Julien
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
David Nieves
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Derek Driggs
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Philippe Marti
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
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Abstract

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading-order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers that adjusts the normal derivative of the temperature fluctuation to zero. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading-order reduced system of governing equations is therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading-order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

JFM classification

Convection: Convection Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Quasi-geostrophic flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 784 , 10 December 2015 , R2
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Copyright
© 2015 Cambridge University Press 

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