Dimensionless parameters for cloudy Rayleigh-Bénard convection: Supersaturation, Damköhler, and Nusselt numbers

In steady-state Rayleigh-Bénard convection, heat is transported by turbulent thermal convection from the bottom, hot surface to the top, cold surface, leading to a height-independent sensible heat flux. When water vapor is present and cloud formation occurs, there is also an additional latent heat flux. Heat transport in cloudy Rayleigh-Bénard convection depends on turbulent flow as well as the microphysical state of the clouds: specifically, whether substantial supersaturations exist and whether cloud liquid water is removed through sedimentation/precipitation. In this article we bridge between the Rayleigh-Bénard convection literature and the atmospheric literature. We express the governing equations for cloudy convection in dimensionless form, thereby explicitly identifying the governing parameters relevant to the cloudy case, including Schmidt, Damköhler, supersaturation, and sedimentation numbers. We further connect to the atmospheric literature by obtaining a Nusselt number (dimensionless heat flux) for a cloud-convection system, directly from the conservation equations for temperature and water vapor. This flux has the same form as that identified by Zhang et al. [L. Zhang, K. L. Chong, and K.-Q. Xia, J. Fluid Mech. 874, 1041 (2019)] for convection with water vapor, but is extended to the cloudy case. For equal thermal and water vapor diffusivities, the flux corresponds to the widely used atmospheric quantities equivalent temperature and moist static energy. Using large eddy simulation (LES) of an idealized cloudy Rayleigh-Bénard convection system with fixed boundary conditions, we find that the equivalent heat flux (Nusselt number) is only weakly dependent on the microphysical details of the system, such as liquid water mixing ratio and cloud droplet number concentration. From the results, we show the vertical profiles of sensible and latent heat fluxes depend on the liquid water content, whereas the equivalent heat flux remains a constant throughout the height of the chamber.

This article appears in the following collection:

Figure 1

Vertical profiles of liquid water mixing ratio for five of the cases. These profiles were obtained by horizontal averaging of the 3D output obtained every 5 min within a span of 2 h, after reaching a steady state. The colors refer to different CCN injection rates (for details refer to Table 2).

Figure 2

Averaged profiles of (a) temperature, (b) water vapor mixing ratio, and (c) equivalent temperature [Eq. (21)]. These profiles were obtained by horizontal averaging of the 3D output obtained every 5 min within a span of 2 h, after reaching a steady state. The colors refer to different CCN injection rates (for details refer to Table 2).

Figure 3

Time-averaged profiles of (a) sensible heat flux ($\rho C_p\overline{w^{\prime }{T}^{\prime }}$), (b) latent heat flux ($\rho L_v\overline{w^{\prime }{Q}_v^{\prime }}$), and (c) equivalent temperature flux ${\mathrm{\Phi }}_{\mu }$, as defined in Eq. (19), from 3D outputs sampled at a frequency of 5 min for 2 h. The shaded region shows the turbulent variability in the data. The line colors refer to the different CCN injection rates, as defined in Table 2.

Figure 4

Averaged profile of $\overline{nRs}$. These profiles were obtained by horizontal averaging of the 3D output obtained every 5 min within a span of 2 h, after reaching a steady state. The colors refer to different CCN injection rates (for details refer to Table 2).