Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

References

• Bannon, P.R., On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci. 1996, 53, 3618–3628.
   Web of Science ®Google Scholar
• Baszenski, G. and Tasche, M., Fast polynomial multiplication and convolutions related to the discrete cosine transform. Lin. Alg. Appl. 1997, 252, 1–25.
   Web of Science ®Google Scholar
• Batchelor, G., An Introduction to Fluid Dynamics, 2000 (Cambridge University Press: Cambridge).
   Google Scholar
• Berkoff, N.A., Kersalé, E. and Tobias, S.M., Comparison of the anelastic approximation with fully compressible equations for linear magnetoconvection and magnetic buoyancy. Geophys. Astrophys. Fluid Dyn. 2010, 104, 545–563.
   Web of Science ®Google Scholar
• Braginsky, S. and Roberts, P., Equations governing convection in Earth’s core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 1995, 79, 1–97.
   Web of Science ®Google Scholar
• Brown, B.P., Vasil, G.M. and Zweibel, E.G., Energy conservation and gravity waves in sound-proof treatments of stellar interiors. Astrophys. J. 2012, 756.
   Web of Science ®Google Scholar
• Brummell, N.H., Hurlburt, N.E. and Toomre, J., Turbulent compressible convection with rotation. I. Flow structure and evolution. Astrophys. J. 1996, 473, 494–513.
   Web of Science ®Google Scholar
• Calkins, M.A., Julien, K. and Marti, P., Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls. J. Fluid Mech. 2013, 732, 214–244.
   Web of Science ®Google Scholar
• Cattaneo, F., Brummell, N.H., Toomre, J., Malagoli, A. and Hurlburt, N.E., Turbulent compressible convection. Astrophys. J. 1991, 370, 282–294.
   Web of Science ®Google Scholar
• Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, 1961 (Courier Dover Publications: New York).
   Google Scholar
• Davies, T., Staniforth, A., Wood, N. and Thuburn, J., Validity of anelastic and other equation sets as inferred from normal-mode analysis. Q. J. R. Meteorol. Soc. 2003, 129, 2761–2775.
   Web of Science ®Google Scholar
• Dormy, E., Soward, A.M., Jones, C.A., Jault, D. and Cardin, P., The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 2004, 501, 43–70.
   Web of Science ®Google Scholar
• Drew, S.J., Jones, C.A. and Zhang, K.K., Onset of convection in a rapidly rotating compressible fluid spherical shell. Geophys. Astrophys. Fluid Dyn. 1995, 80, 241–254.
   Web of Science ®Google Scholar
• Gastine, T. and Wicht, J., Effects of compressibility on driving zonal flow in gas giants. Icarus 2012, 219, 428–442.
   Web of Science ®Google Scholar
• Gill, A.E., Atmosphere-Ocean Dynamics Vol. 30, 1982 (Academic Press: San Diego).
   Google Scholar
• Gilman, P.A. and Glatzmaier, G.A., Compressible convection in a rotating spherical shell. Part 1. Anelastic equations. Astrophys. J. Suppl. Ser. 1981, 45, 335–349.
   Web of Science ®Google Scholar
• Glatzmaier, G.A., Evonuk, M. and Rogers, T.M., Differential rotation in giant planets maintained by density-stratified turbulent convection. Geophys. Astrophys. Fluid Dyn. 2009, 103, 31–51.
   Web of Science ®Google Scholar
• Glatzmaier, G.A. and Gilman, P.A., Compressible convection in a rotating spherical shell. Part 2. Linear anelastic model. Astrophys. J. Suppl. Ser. 1981a, 45, 351–380.
   Web of Science ®Google Scholar
• Glatzmaier, G.A. and Gilman, P.A., Compressible convection in a rotating spherical shell. Part 3. Analytic model for compressible vorticity waves. Astrophys. J. Suppl. Ser. 1981b, 45, 381–388.
   Web of Science ®Google Scholar
• Glatzmaier, G.A. and Gilman, P.A., Compressible convection in a rotating spherical shell. Part 4. Effects of viscosity, conductivty, boundary conditions and zone depth. Astrophys. J. Suppl. Ser. 1981c, 47, 103–116.
   Web of Science ®Google Scholar
• Gottlieb, D. and Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, 1993 (SIAM: Montpelier).
   Google Scholar
• Gough, D.O., The anelastic approximation for thermal convection. J. Atmos. Sci. 1969, 26, 448–456.
   Web of Science ®Google Scholar
• Gough, D.O., Moore, D.R., Spiegel, E.A. and Weiss, N.O., Convective instability in a compressible atmosphere II. Astrophys. J. 1976, 206, 536–542.
   Web of Science ®Google Scholar
• Graham, E., Numerical simulation of two-dimensional compressible convection. J. Fluid Mech. 1975, 70, 689–703.
   Web of Science ®Google Scholar
• Gray, D.D. and Giorgini, A., The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Trans. 1976, 19, 545–551.
   Web of Science ®Google Scholar
• Hathaway, D.H. and Somerville, R.C.J., Three-dimensional simulations of convection in layers with tilted rotation vectors. J. Fluid Mech. 1983, 126, 75–89.
   Web of Science ®Google Scholar
• Hathaway, D.H., Toomre, J. and Gilman, P., Convective instability when the temperature gradient and rotation vector are oblique to gravity. II. Real fluids with effects of diffusion. Geophys. Astrophys. Fluid Dyn. 1980, 15, 7–37.
   Web of Science ®Google Scholar
• Heard, W.B., Convective instability of a rapidly rotating compressible fluid. Astrophys. J. 1973, 186, 1065–1081.
   Web of Science ®Google Scholar
• Hurlburt, N.E., Toomre, J. and Massaguer, J.M., Two-dimensional compressible convection extending over multiple scale heights. Astrophys. J. 1984, 282, 557–573.
   Web of Science ®Google Scholar
• Jones, C.A., Boronski, P., Brun, A.S., Glatzmaier, G.A., Gastine, T., Miesch, M.S. and Wicht, J., Anelastic convection-driven dynamo benchmarks. Icarus 2011, 216, 120–135.
   Web of Science ®Google Scholar
• Jones, C.A., Kuzanyan, K.M. and Mitchell, R.H., Linear theory of compressible convection in rapidly rotating spherical shells, using the anelastic approximation. J. Fluid Mech. 2009, 634, 291–319.
   Web of Science ®Google Scholar
• Jones, C.A., Soward, A.M. and Mussa, A.I., The onset of thermal convection in a rapidly rotating sphere. J. Fluid Mech. 2000, 405, 157–179.
   Web of Science ®Google Scholar
• Julien, K. and Knobloch, E., Strongly nonlinear convection cells in a rapidly rotating fluid layer: the tilted f-plane. J. Fluid Mech. 1998, 360, 141–178.
   Web of Science ®Google Scholar
• Julien, K., Knobloch, E., Milliff, R. and Werne, J., Generalized quasi-geostrophy for spatially anistropic rotationally constrained flows. J. Fluid Mech. 2006, 555, 233–274.
   Web of Science ®Google Scholar
• Julien, K., Legg, S., McWilliams, J. and Werne, J., Rapidly rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech. 1996, 322, 243–273.
   Web of Science ®Google Scholar
• Julien, K. and Watson, M., Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods. J. Comp. Phys. 2009, 228, 1480–1503.
   Web of Science ®Google Scholar
• Klein, R., Achatz, U., Bresch, D., Knio, O.M. and Smolarkiewicz, P.K., Regime of validity of soundproof atmospheric flow models. J. Atmos. Sci. 2010, 67, 3226–3237.
   Web of Science ®Google Scholar
• Lantz, S.R. and Sudan, R.N., Magnetoconvection dynamics in a stratified layer. I. Two-dimensional simulations and visualization. Astrophys. J. 1995, 441, 903–924.
   Web of Science ®Google Scholar
• Lipps, F.B. and Hemler, R.S., A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 1982, 39, 2192–2210.
   Web of Science ®Google Scholar
• Mihaljan, J.M., A rigorous exposition of the Boussinesq approximations applicable for a thin layer of fluid. Astrophys. J. 1962, 136, 1126–1133.
   Web of Science ®Google Scholar
• Mizerski, K.A. and Tobias, S.M., The effect of stratification and compressibility on anelastic convection in a rotating plane layer. Geophys. Astrophys. Fluid Dyn. 2011, 105, 566–585.
   Web of Science ®Google Scholar
• Ogura, Y. and Phillips, N.A., Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 1962, 19, 173–179.
   Web of Science ®Google Scholar
• Olver, S. and Townsend, A., A fast and well-conditioned spectral method. SIAM Rev. 2013, 55, 462–489.
   Web of Science ®Google Scholar
• Spiegel, E.A., Convective instability in a compressible atmosphere I. Astrophys. J. 1965, 141, 1068–1090.
   Web of Science ®Google Scholar
• Spiegel, E.A. and Veronis, G., On the Boussinesq approximation for a compressible fluid. Astrophys. J. 1960, 131, 442–447.
   Web of Science ®Google Scholar
• Vasil, G.M., Lecoanet, D., Brown, B.P., Wood, T.S. and Zweibel, E.G., Energy conservation and gravity waves in sound-proof treatments of stellar interiors II. Lagrangian constrained analysis. Astrophys. J. 2013, 773, 169.
   Web of Science ®Google Scholar
• Verhoeven, J. and Stellmach, S., The compressional beta effect: a source of zonal winds in planets? Icarus 2014, 237, 143–158.
   Web of Science ®Google Scholar
• Vickers, G.T., On the formation of giant cells and supergranules. Astrophys. J. 1971, 163, 363–374.
   Web of Science ®Google Scholar