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Oscillatory convection and limitations of the Boussinesq approximation

Published online by Cambridge University Press:  30 August 2016

T. S. Wood Open the ORCID record for T. S. Wood [Opens in a new window]  and
P. J. Bushby
T. S. Wood*
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
P. J. Bushby
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
*
†Email address for correspondence: [email protected]
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Abstract

We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly used ‘sound-proof’ approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.

JFM classification

Compressible Flows: Compressible Flows Convection: Convection Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 502 - 515
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Copyright
© 2016 Cambridge University Press 

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