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A simple system for moist convection: the Rainy–Bénard model

Published online by Cambridge University Press:  09 January 2019

Geoffrey K. Vallis*
Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK
Douglas J. Parker
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
Steven M. Tobias
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Email address for correspondence:


Rayleigh–Bénard convection is one of the most well-studied models in fluid mechanics. Atmospheric convection, one of the most important components of the climate system, is by comparison complicated and poorly understood. A key attribute of atmospheric convection is the buoyancy source provided by the condensation of water vapour, but the presence of radiation, compressibility, liquid water and ice further complicate the system and our understanding of it. In this paper we present an idealized model of moist convection by taking the Boussinesq limit of the ideal-gas equations and adding a condensate that obeys a simplified Clausius–Clapeyron relation. The system allows moist convection to be explored at a fundamental level and reduces to the classical Rayleigh–Bénard model if the latent heat of condensation is taken to be zero. The model has an exact, Rayleigh-number-independent ‘drizzle’ solution in which the diffusion of water vapour from a saturated lower surface is balanced by condensation, with the temperature field (and so the saturation value of the moisture) determined self-consistently by the heat released in the condensation. This state is the moist analogue of the conductive solution in the classical problem. We numerically determine the linear stability properties of this solution as a function of Rayleigh number and a non-dimensional latent-heat parameter. We also present some two-dimensional, time-dependent, nonlinear solutions at various values of Rayleigh number and the non-dimensional condensational parameters. At sufficiently low Rayleigh number the system converges to the drizzle solution, and we find no evidence that two-dimensional self-sustained convection can occur when that solution is stable. The flow transitions from steady to turbulent as the Rayleigh number or the effects of condensation are increased, with plumes triggered by gravity waves emanating from other plumes. The interior dries as the level of turbulence increases, because the plumes entrain more dry air and because the saturated boundary layer at the top becomes thinner. The flow develops a broad relative humidity minimum in the domain interior, only weakly dependent on Rayleigh number when that is high.

JFM Papers
© 2019 Cambridge University Press 

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Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.Google Scholar
Ambaum, M. H. P. 2010 Thermal Physics of the Atmosphere. Wiley.Google Scholar
American Meteorological Society2018 Glossary of meteorology. Available online at Scholar
Berlengiero, M., Emanuel, K. A., Von Hardenberg, J., Provenzale, A. & Spiegel, E. A. 2012 Internally cooled convection: a fillip for Philip. Comm. Nonlin. Sci. Num. Sim. 17 (5), 19982007.Google Scholar
Bretherton, C. S. 1987 A theory for nonprecipitating moist convection between two parallel plates. Part I. Thermodynamics and ‘linear’ solutions. J. Atmos. Sci. 44 (14), 18091827.Google Scholar
Bretherton, C. S. 1988 A theory for nonprecipitating convection between two parallel plates. Part II. Nonlinear theory and cloud field organization. J. Atmos. Sci. 45 (17), 23912415.Google Scholar
Brun, A. S. & Browning, M. K. 2017 Magnetism, dynamo action and the solar–stellar connection. Living Rev. Solar Phys. 14, 4.Google Scholar
Bryan, G. H., Wyngaard, J. C. & Fritsch, J. M. 2003 Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev. 131 (10), 23942416.Google Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B.2016 Dedalus: flexible framework for spectrally solving differential equations. Astrophysics Source Code Library: Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press; reprinted by Dover Publications, 1981.Google Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 58.Google Scholar
Christensen, U. 1995 Effects of phase transitions on mantle convection. Annu. Rev. Earth Planet. Sci. 23 (1), 6587.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Emanuel, K. A. 1994 Atmospheric Convection. Oxford University Press.Google Scholar
Emanuel, K. A., Neelin, J. D. & Bretherton, C. S. 1994 On large-scale circulations in convecting atmospheres. Q. J. R. Meteor. Soc. 120, 11111143.Google Scholar
Frierson, D. M. W., Held, I. M. & Zurita-Gotor, P. 2006 A gray radiation aquaplanet moist GCM. Part 1. Static stability and eddy scales. J. Atmos. Sci. 63, 25482566.Google Scholar
Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh–Bénard convection. Physica D 10 (3), 249276.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.Google Scholar
Hernandez-Duenas, G., Majda, A. J., Smith, L. M. & Stechmann, S. N. 2013 Minimal models for precipitating turbulent convection. J. Fluid Mech. 717, 576611.Google Scholar
Khain, A. P., Beheng, K. D., Heymsfield, A., Korolev, A., Krichak, S. O., Levin, Z., Pinsky, M., Phillips, V., Prabhakaran, T., Teller, A. et al. 2015 Representation of microphysical processes in cloud-resolving models: spectral (bin) microphysics versus bulk parameterization. Rev. Geophys. 53 (2), 247322.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 13741389.Google Scholar
Lakkaraju, R., Stevens, R., Oresta, P., Verzicco, R., Lohse, D. & Prosperetti, A. 2013 Heat transport in bubbling turbulent convection. Proc. Natl Acad. Sci. USA 110 (23), 92379242.Google Scholar
Lohse, D. & Toschi, F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90 (3), 034502.Google Scholar
Ludlam, F. H. 1966 Cumulus and cumulonimbus convection. Tellus 18, 687698.Google Scholar
Ludlam, F. H. 1980 Clouds and Storms: The Behavior and Effect of Water in the Atmosphere. Penn. State University Press.Google Scholar
Mahrt, L. 1986 On the shallow motion approximations. J. Atmos. Sci. 43, 10361044.Google Scholar
Marshall, J. C. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 164.Google Scholar
Mellor, G. L. & Yamada, T. 1974 A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci. 31 (7), 17911806.Google Scholar
Mitchell, J. L. & Lora, J. M. 2016 The climate of Titan. Annu. Rev. Earth Planet. Sci. 44, 353380.Google Scholar
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the surface layer of the atmosphere. Tr. Akad. Nauk SSSR Geophiz. Inst. 24, 163187.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 123.Google Scholar
O’Gorman, P. A., Allan, R. P., Byrne, M. P. & Previdi, M. 2012 Energetic constraints on precipitation under climate change. Surv. Geophys. 33 (3–4), 585608.Google Scholar
O’Gorman, P. A. & Schneider, T. 2006 Stochastic models for the kinematics of moisture transport and condensation in homogeneous turbulent flows. J. Atmos. Sci. 63, 29923005.Google Scholar
Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205214.Google Scholar
Parodi, A., Emanuel, K. A. & Provenzale, A. 2003 Plume patterns in radiative–convective flows. New J. Phys. 5 (1), 106.Google Scholar
Parsons, B. & McKenzie, D. 1978 Mantle convection and the thermal structure of the plates. J. Geo. Res.: Sol. Earth 83 (B9), 44854496.Google Scholar
Pauluis, O. & Schumacher, J. 2010 Idealized moist Rayleigh–Bénard convection with piecewise linear equation of state. Commun. Math. Sci. 8 (1), 295319.Google Scholar
Pauluis, O. & Schumacher, J. 2011 Self-aggregation of clouds in conditionally unstable moist convection. Proc. Natl Acad. Sci. USA 108 (31), 1262312628.Google Scholar
Pierrehumbert, R. T., Brogniez, H. & Roca, R. 2007 On the relative humidity of the atmosphere. In The Global Circulation of the Atmosphere: Phenomena, Theory, Challenges (ed. Schneider, T. & Sobel, A.), pp. 143185. Princeton University Press.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinb. Dublin Phil. Mag. J. Sci. 32 (192), 529546.Google Scholar
Roche, P.-E., Castaing, B., Chabaud, B. & Hébral, B. 2001 Observation of the 1/2 power law in Rayleigh–Bénard convection. Phys. Rev. E 63 (4), 045303.Google Scholar
Romps, D. M. 2014 An analytical model for tropical relative humidity. J. Clim. 27 (19), 74327449.Google Scholar
Schmidt, L. E., Oresta, P., Toschi, F., Verzicco, R., Lohse, D. & Prosperetti, A. 2011 Modification of turbulence in Rayleigh–Bénard convection by phase change. New J. Phys. 13 (2), 025002.Google Scholar
Schubert, G. & Soderlund, K. M. 2011 Planetary magnetic fields: observations and models. Phys. Earth Plan. Interiors 187, 92108.Google Scholar
Schubert, W. H., Hausman, S. A., Garcia, M., Ooyama, K. V. & Kuo, H.-C. 2001 Potential vorticity in a moist atmosphere. J. Atmos. Sci. 58, 31483157.Google Scholar
Schumacher, J. & Pauluis, O. 2010 Buoyancy statistics in moist turbulent Rayleigh–Bénard convection. J. Fluid Mech. 648, 509519.Google Scholar
Scorer, R. S. & Ludlam, F. H. 1953 Bubble theory of penetrative convection. Q. J. R. Meteor. Soc. 79, 94103.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Part I. The basic experiment. Mon. Wea. Rev. 91 (3), 99164.Google Scholar
Smith, R. K.(Ed.) 2013 The Physics and Parameterization of Moist Atmospheric Convection. Springer.Google Scholar
Spiegel, E. A. 1971 Convection in stars. Part I. Basic Boussinesq convection. Annu. Rev. Astron. Astrophys. 9 (1), 323352.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442447. (Correction: Astrophys. J., 135, 655–656.)Google Scholar
Spyksma, K., Bartello, P. & Yau, M. K. 2006 A Boussinesq moist turbulence model. J. Turbul. 7, N32.Google Scholar
Sukhatme, J. & Young, W. R. 2011 The advection–condensation model and water-vapour probability density functions. Q. J. R. Meteor. Soc. 137, 15611572.Google Scholar
Tompkins, A. M. 2001 Organization of tropical convection in low vertical wind shears: the role of cold pools. J. Atmos. Sci. 58 (13), 16501672.Google Scholar
Tsang, Y.-K. & Vanneste, J. 2017 The effect of coherent stirring on the advection–condensation of water vapour. Proc. R. Soc. Lond. A 473, 20170196.Google Scholar
Vallis, G. K. 2017 Atmospheric and Oceanic Fluid Dynamics, 2nd edn. Cambridge University Press.Google Scholar
Weidauer, T., Pauluis, O. & Schumacher, J. 2011 Rayleigh–Bénard convection with phase changes in a Galerkin model. Phys. Rev. E 84 (4), 046303.Google Scholar
White, B., Gryspeerdt, E., Stier, P., Morrison, H., Thompson, G. & Kipling, Z. 2017 Uncertainty from the choice of microphysics scheme in convection-permitting models significantly exceeds aerosol effects. Atmos. Chem. Phys. 17, 1214512175.Google Scholar
Whitehead, J. P. & Doering, C. R. 2011 Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106 (24), 244501.Google Scholar
Zhao, M., Golaz, J.-C., Held, I. M., Ramaswamy, V., Lin, S.-J., Ming, Y., Ginoux, P., Wyman, B., Donner, L. J., Paynter, D. et al. 2016 Uncertainty in model climate sensitivity traced to representations of cumulus precipitation microphysics. J. Clim. 29 (2), 543560.Google Scholar

Vallis et al. supplementary movie 1

The evolution from the initial conditions for the four variables, b, q, T and u for Ra = 2 x 10^5 (i.e., Rayleigh number = 200,000). See text for more description.

Download Vallis et al. supplementary movie 1(Video)
Video 9.8 MB

Vallis et al. supplementary movie 2

Same but for Ra = 2 x 10^7 (i.e., Rayleigh number = 20,000,000).

Download Vallis et al. supplementary movie 2(Video)
Video 2.5 MB

Vallis et al. supplementary movie 3

Same but for Ra = 5 x 10^7 (i.e., Rayleigh number = 50,000,000).

Download Vallis et al. supplementary movie 3(Video)
Video 15.5 MB

Vallis et al. supplementary movie 4

Same but for Ra = 2.5 x 10^8 (i.e., Rayleigh number = 250,000,000).

Download Vallis et al. supplementary movie 4(Video)
Video 8.4 MB