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Oscillatory convection and limitations of the Boussinesq approximation

Lookup NU author(s): Dr Toby Wood, Professor Paul BushbyORCiD



This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).


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.

Publication metadata

Author(s): Wood TS, Bushby PJ

Publication type: Article

Publication status: Published

Journal: Journal of Fluid Mechanics

Year: 2016

Volume: 803

Pages: 502-515

Online publication date: 30/08/2016

Acceptance date: 28/07/2016

Date deposited: 21/07/2016

ISSN (print): 0022-1120

ISSN (electronic): 1469-7645

Publisher: Cambridge University Press


DOI: 10.1017/jfm.2016.511


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