A theoretical model for double diffusive phenomena in cloudy convection
Abstract. Using classical rheological principles, a model is proposed to depict the molecular diffusion in a moist-saturated dissipative atmosphere: due to the saturation condition existing between water vapor and liquid water in the medium, the equations are those of a double diffusive phenomenon with Dufour effect. The double diffusivity is important because of the huge diffusivity difference between the liquid phase and the gaseous phase. Reduced equations are constructed and are then applied to describe the linear free convection of a thin cloudy layer bounded by two free surfaces. The problem is solved with respect to two destabilizing parameters, a Rayleigh number Ra and a moist Rayleigh number Rh . Two instabilities may occur: (i) oscillatory modes, which exist for sufficiently large values of the Rayleigh number: these modes generalize the static instability of the medium; (ii) stationary modes, which mainly occur when the moist Rayleigh number is negative. These modes are due to the molecular diffusion, and exist even when the medium is statically stable: the corresponding motions describe, in the moist-saturated air, configurations such as "fleecy clouds". Growth rates are determined at the instability threshold for the two modes of instability occurring in the process. The case of vanishing moisture concentration is considered: the oscillatory unstable case appears as a singular perturbation (due to the moisture) of the stationary unstable state of the Rayleigh-Bénard convection in pure fluid, and, more generally, as the dynamical perturbation of the static instability. The convective behaviour of a cloud in the air at rest is then examined: the instability of the cloud is mainly due to moisture, while the instability of the surrounding air is mainly due to heating.
Key words. Atmospheric composition and structure (cloud physics and chemistry) – Meteorology and atmospheric dynamics (convective processes, mesoscale meteorology)