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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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References

Achatz, U., Klein, R. & Senf, F. 2010 Gravity waves, scale asymptotics and the pseudo-incompressible equations. J. Fluid Mech. 663, 120–147.Google Scholar
Berkoff, N. A., Kersale, E. & Tobias, S. M. 2010 Comparison of the anelastic approximation with fully compressible equations for linear magnetoconvection and magnetic buoyancy. Geophys. Astrophys. Fluid Dyn. 104, 545–563.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1995 Equations governing convection in earth’s core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97.Google Scholar
Braginsky, S. I. & Roberts, P. H. 2007 Anelastic and Boussinesq approximations. In Encyclopedia of Geomagnetism and Paleomagnetism (ed. Gubbins, D. & Herrero-Bervera, E.), pp. 11–19. Springer.Google Scholar
Brown, B. P., Vasil, G. M. & Zweibel, E. G. 2012 Energy conservation and gravity waves in sound-proof treatments of stellar interiors. Part I. Anelastic approximations. Astrophys. J. 756, 109.Google Scholar
Calkins, M. A., Julien, K. & Marti, P. 2015 The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems. Proc. R. Soc. Lond. A 471, 20140689.Google Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to Coriolis forces. Proc. R. Soc. Lond. A 217, 306–327.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Cotter, C. J. & Holm, D. D. 2014 Variational formulations of sound-proof models. Q. J. R. Meteorol. Soc. 140, 1966–1973.CrossRefGoogle Scholar
Drew, S. J., Jones, C. A. & Zhang, K. 1995 Onset of convection in a rapidly rotating compressible fluid spherical shell. Geophys. Astrophys. Fluid Dyn. 80, 241–254.CrossRefGoogle Scholar
Durran, D. R. 1989 Improving the anelastic approximation. J. Atmos. Sci. 46, 1453–1461.Google Scholar
Durran, D. R. 2008 A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech. 601, 365–379.Google Scholar
Eckart, C. & Ferris, H. G. 1956 Equations of motion of the ocean and atmosphere. Rev. Mod. Phys. 28, 48–52.Google Scholar
Gough, D. O., Moore, D. R., Spiegel, E. A. & Weiss, N. O. 1976 Convective instability in a compressible atmosphere. II. Astrophys. J. 206, 536–542.CrossRefGoogle Scholar
Gray, D. D. & Giorgini, A. 1976 The validity of the boussinesq approximation for liquids and gases. Intl J. Heat Mass Transfer 19, 545–551.Google Scholar
Jones, C. A., Kuzanyan, K. M. & Mitchell, R. H. 2009 Linear theory of compressible convection in rapidly rotating spherical shells, using the anelastic approximation. J. Fluid Mech. 634, 291.Google Scholar
Klein, R., Achatz, U., Bresch, D., Knio, O. M. & Smolarkiewicz, P. K. 2010 Regime of validity of soundproof atmospheric flow models. J. Atmos. Sci. 67, 3226–3237.CrossRefGoogle Scholar
Klein, R. & Pauluis, O. 2012 Thermodynamic consistency of a pseudoincompressible approximation for general equations of state. J. Atmos. Sci. 69, 961–968.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1980 Statistical Physics. Part 1, 3rd edn. Course of Theoretical Physics, vol. 5. Pergamon Press.Google Scholar
Lantz, S. R.1992 Dynamical behavior of magnetic fields in a stratified, convecting fluid layer. PhD thesis, Cornell University.Google Scholar
Lecoanet, D., Brown, B. P., Zweibel, E. G., Burns, K. J., Oishi, J. S. & Vasil, G. M. 2014 Conduction in low Mach number flows. I. Linear and weakly nonlinear regimes. Astrophys. J. 797, 94.CrossRefGoogle Scholar
Lipps, F. B. & Hemler, R. S. 1982 A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 39, 2192–2210.2.0.CO;2>CrossRefGoogle Scholar
Mihaljan, J. M. 1962 A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J. 136, 1126.Google Scholar
Ogura, Y. & Phillips, N. A. 1962 Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179.Google Scholar
O’Neill, W. P. & Klein, R. 2014 A moist pseudo-incompressible model. Atmos. Res. 142, 133–141.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442–447.Google Scholar
Vasil, G. M., Lecoanet, D., Brown, B. P., Wood, T. S. & Zweibel, E. G. 2013 Energy conservation and gravity waves in sound-proof treatments of Stellar interiors. II. Lagrangian constrained analysis. Astrophys. J. 773, 169.Google Scholar
Veronis, G. 1962 The magnitude of the dissipation terms in the Boussinesq approximation. Astrophys. J. 135, 655–656.Google Scholar