, TABLE I: (Cont.) See caption below

G. Ahlers, S. Grossmann, and D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection, Reviews of Modern Physics, vol.81, pp.503-537, 2009.

P. Urban, P. Hanzelka, T. Kralik, V. Musilova, A. Srnka et al., Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh-Benard convection at very high Rayleigh numbers, Physical Review Letters, vol.109, p.154301, 2012.

S. Horn, O. Shishkina, and C. Wagner, On non-Oberbeck Boussinesq effects in threedimensional Rayleigh Bénard convection in glycerol, Journal of Fluid Mechanics, pp.175-202

, List of all 51 simulations discussed in this paper. Ra is the reference Rayleigh number (see equation (2)). Simulations are run with a stiffness parameter S, and with n x Fourier and n z Chebyshev basis functions in the horizontal and vertical directions. Timesteps were taken using the Courant-Friedrichs-Levy (CFL) condition, with a safety factor given in the table. dt is the typical time step, and each simulation was run for at least O(1) thermal time (about dt ?1 timesteps). The aspect ratio of all simulations is 2, TABLE I: (Cont.), 2013.

G. Ahlers, E. Calzavarini, F. F. Araujo, D. Funfschilling, S. Grossmann et al., Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point, Physical Review E, vol.77, pp.1-16, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00372031

, 15: Same as figure 8 but for the higher Rayleigh number Ra = 8 × 10 8 case

J. Nycander, M. Hieronymus, and F. Roquet, The nonlinear equation of state of sea water and the global water mass distribution, Geophysical Research Letters, pp.1-8, 2015.

G. Vettoretti and W. R. Peltier, Thermohaline instability and the formation of glacial North Atlantic super polynyas at the onset of Dansgaard-Oeschger warming events, Geophysical Research Letters, vol.43, pp.5336-5344, 2016.

F. Roquet, G. Madec, L. Brodeau, and J. Nycander, Defining a Simplified Yet Realistic Equation of State for Seawater, Journal of Physical Oceanography, vol.45, pp.2564-2579, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01234084

G. Veronis, Penetrative convection, The Astrophysical Journal, vol.137, pp.641-663, 1963.

M. L. Bars, D. Lecoanet, S. Perrard, A. Ribeiro, L. Rodet et al., Experimental study of internal wave generation by convection in water, Fluid Dynamics Research, vol.47, p.45502, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01274939

D. Lecoanet, M. L. Bars, K. J. Burns, G. M. Vasil, B. P. Brown et al., Numerical simulations of internal wave generation by convection in water, Physical Review E-Statistical, Nonlinear, and Soft Matter Physics, vol.91, p.63016, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01274942

A. A. Townsend, Natural convection in water over an ice surface, Q. J. R. Meteorol. Soc, vol.90, 1964.

S. Backhaus, K. Turitsyn, and R. E. Ecke, Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry, Physical Review Letters, vol.106, 2011.

J. J. Hidalgo, J. Fe, L. Cueto-felgueroso, and R. Juanes, Scaling of convective mixing in porous media, Physical Review Letters, vol.109, pp.1-5, 2012.

D. R. Hewitt, J. A. Neufeld, and J. R. Lister, Ultimate regime of high Rayleigh number convection in a porous medium, Physical Review Letters, vol.108, pp.551-586, 2013.

M. Hemmati, C. T. Moynihan, and C. A. Angell, Interpretation of the molten BeF2 viscosity anomaly in terms of a high temperature density maximum, and other waterlike features, Journal of Chemical Physics, vol.115, pp.6663-6671, 2001.

J. P. Mellado, The evaporatively driven cloud-top mixing layer, Journal of Fluid Mechanics, vol.660, pp.5-36, 2010.

N. H. Brummell, T. L. Clune, and J. Toomre, Penetration and Overshooting in Turbulent Compressible Convection, The Astrophysical Journal, vol.570, pp.825-854, 2002.

T. M. Rogers and G. A. Glatzmaier, Penetrative Convection within the Anelastic Approximation, The Astrophysical Journal, vol.620, pp.432-441, 2005.

L. Alvan, S. Mathis, and T. Decressin, Coupling between internal waves and shear-induced turbulence in stellar radiation zones: the critical layers, Astronomy & Astrophysics, vol.553, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01094469

C. Pinçon, K. Belkacem, and M. J. Goupil, Astrophysics Generation of internal gravity waves by penetrative convection, Astronomy & Astrophysics, vol.588, pp.1-21, 2016.

K. Hirose, S. Labrosse, and J. Hernlund, Composition and State of the Core, Annual Review of Earth and Planetary Sciences, vol.41, pp.657-691, 2013.
DOI : 10.1146/annurev-earth-050212-124007

B. A. Buffett, Geomagnetic fluctuations reveal stable stratification at the top of the Earth's core, Nature, vol.507, pp.484-487, 2014.

P. Goldreich and P. Kumar, Wave generation by turbulent convection, The Astrophysical Journal, vol.363, p.694, 1990.
DOI : 10.1086/169376
URL : https://authors.library.caltech.edu/37506/1/1990ApJ___363__694G.pdf

S. Wunsch, Stochastic simulations of buoyancy-reversal experiments, Physics of Fluids, vol.15, pp.1442-1456, 2003.
DOI : 10.1063/1.1572160

K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, B. P. Brown et al., Dedalus: A Flexible Pseudo-Spectral Framework for Solving Partial Differential Equations, 2017.

E. Large and C. D. Andereck, Penetrative Rayleigh-Bénard convection in water near its maximum density point, Physics of Fluids, vol.26, p.94101, 2014.
DOI : 10.1063/1.4895063

D. R. Moore and N. O. Weiss, Nonlinear penetrative convection, Journal of Fluid Mechanics, vol.61, p.553, 1973.
DOI : 10.1017/s0022112073000868

S. Grossmann and D. Lohse, Scaling in thermal convection: A unifying theory, Journal of Fluid Mechanics, vol.407, 2000.

H. V. Dosser and B. R. Sutherland, Anelastic internal wave packet evolution and stability, Journal of the Atmospheric Sciences, vol.68, pp.2844-2859, 2011.
DOI : 10.1175/jas-d-11-097.1

E. Siggia, High Rayleigh Number Convection, Annual Review of Fluid Mechanics, vol.26, pp.137-168, 1994.
DOI : 10.1146/annurev.fluid.26.1.137

R. M. Kerr, Rayleigh number scaling in numerical convection, Journal of Fluid Mechanics, vol.310, pp.139-179, 1996.

M. S. Emran and J. Schumacher, Fine-scale statistics of temperature and its derivatives in convective turbulence, Journal of Fluid Mechanics, vol.611, pp.13-34, 2008.

D. Lecoanet and E. Quataert, Internal gravity wave excitation by turbulent convection, Monthly Notices of the Royal Astronomical Society, vol.430, pp.2363-2376, 2013.
DOI : 10.1093/mnras/stt055
URL : https://academic.oup.com/mnras/article-pdf/430/3/2363/4932436/stt055.pdf

J. K. Ansong and B. R. Sutherland, Internal gravity waves generated by convective plumes, Journal of Fluid Mechanics, vol.648, p.405, 2010.

N. H. Brummell, The effect of the Prandtl number on penetrative convection, Geophysical & Astrophysical Fluid Dynamics, vol.68, pp.115-132, 1993.

P. Garaud, B. Gallet, and T. Bischoff, The stability of stratified spatially periodic shear flows at low Péclet number, Physics of Fluids, vol.27, p.84104, 2015.
DOI : 10.1063/1.4928164
URL : https://authors.library.caltech.edu/60499/1/1.4928164.pdf

N. E. Hurlburt, J. Toomre, J. M. Massaguer, and J. Zahn, Penetration below a convective zone, The Astrophysical Journal, vol.421, pp.245-260, 1994.
DOI : 10.1086/173642
URL : https://doi.org/10.1086/173642

J. H. Shiode, E. Quataert, M. Cantiello, and L. Bildsten, The observational signatures of convectively excited gravity modes in main-sequence stars, Monthly Notices of the Royal Astronomical Society, vol.430, pp.1736-1745, 2013.

J. Vidal and N. Schaeffer, Quasi-geostrophic modes in the Earth's fluid core with an outer stably stratified layer, Geophysical Journal International, vol.202, pp.2182-2193, 2015.
DOI : 10.1093/gji/ggv282
URL : https://academic.oup.com/gji/article-pdf/202/3/2182/1697074/ggv282.pdf

D. Gubbins, The rayleigh number for convection in the earths core, dynamics and Magnetic Fields of the Earth's and Planetary Interiors, vol.128, pp.3-12, 2001.

J. Marshall and F. Schott, Open-ocean convection: Observations, theory, and models, Reviews of Geophysics, vol.37, pp.1-64, 1999.

A. Parodi and K. Emanuel, A theory for buoyancy and velocity scales in deep moist convection, Journal of the Atmospheric Sciences, vol.66, pp.3449-3463, 2009.

K. M. Grise, D. W. Thompson, and T. Birner, A global survey of static stability in the stratosphere and upper troposphere, Journal of Climate, vol.23, pp.2275-2292, 2010.

B. King, M. Stone, H. P. Zhang, T. Gerkema, M. Marder et al., Buoyancy frequency profiles and internal semidiurnal tide turning depths in the oceans, Journal of Geophysical Research: Oceans, vol.117, p.4008, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00715293

M. P. Baldwin, L. J. Gray, T. J. Dunkerton, K. Hamilton, P. H. Haynes et al., The quasi-biennial oscillation, Reviews of Geophysics, vol.39, pp.179-229, 2001.

U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, vol.25, pp.151-167, 1997.