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

  • Geoffrey K. Vallis (a1), Douglas J. Parker (a2) and Steven M. Tobias (a3)


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.


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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.

 Video (9.8 MB)
9.8 MB

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

 Video (2.5 MB)
2.5 MB

Vallis et al. supplementary movie 3
Same but for Ra = 5 x 10^7 (i.e., Rayleigh number = 50,000,000).

 Video (15.5 MB)
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).

 Video (8.4 MB)
8.4 MB

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

  • Geoffrey K. Vallis (a1), Douglas J. Parker (a2) and Steven M. Tobias (a3)


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