Numerical Studies of Small-Scale Turbulent Mixing and Entrainment

The turbulent entrainment and subsequent mixing of clear and cloudy air determines the overall lifetime and shape of clouds. Mixing at the cloud boundary as well as inside the cloud occurs over a broad range of spatial and temporal scales reaching from the smallest eddies at the size of the Kolmogorov length with eta_k ~ 1mm to the outer scale of L ~ 100m and beyond. The time scales vary between Kolmogorov times of tau_k ~ 10ms up to the lifetime of the order of T_L ~ 1h. These multiple scales which are involved in the dynamics of a turbulent cloud cannot be covered by direct numerical simulations. They require studies by means of large-eddy or even coarser simulations which parametrize the action of the smaller turbulent eddies and the cloud microphysics in sub-grid scale models or other parametrizations. In this work we want to study the initial phase of the entrainment of clear air into a filament of cloudy air as present at the edge of a cloud by three-dimensional direct numerical simulations without any parametrizations of the small-scale turbulence and cloud microphysics. These simulations resolve the turbulence down to the Kolmogorov scale. We combine the Eulerian description of the turbulence fields (velocity, temperature and vapor content) with the Lagrangian description of the ensemble of cloud water droplets. Condensation rate, liquid water content and local supersaturation determine the growth of the individual cloud water droplets. Typical cloud droplet concentrations of the order of 10^2 cm^-3 result in about 10 to 100 million individual droplets. They are treated as inertial point particles with Stokes drag and gravitational settling. We monitor their position, their velocity and the droplet radius. Our focus is on the initial stage of the mixing at the cloud edge and neglects droplet collisions and coalescence. The simulation domain consists of an equally spaced mesh grid with up to 1024 x 512 x 512 grid points for a box size of up to 2m x 1m x 1m. We apply periodic boundary conditions in all three directions. Therefore, all fields of our equations can be expanded into Fourier series and the equations are solved with a pseudospectral method using a parallel code. Our results can be summarized as follows. First, we find that the Damkoehler number Da, a dimensionless parameter defined as the ratio of the fluid time scale tau_{fluid} and the phase relaxation time scale tau_{phase} (the reaction time scale for the present problem), covers essential aspects of the mixing process. This was shown by a collapse of the probability density functions of the droplet radius r which have been obtained from a series of simulations starting with different initial droplet radii R_0 and number densities n_d (Kumar et al. , Theor. Comp. Fluid Dyn., 2012). Two limits of the entrainment coexist on different spatial scales: the homogeneous mixing at smaller scales of turbulence (Da << 1) and the inhomogeneous mixing at the largest scales (Da >> 1). Second, our numerical simulations confirm that the phase relaxation time tau_{phase} is the appropriate time scale that describes the evaporation process in the cloud droplet ensemble (Kumar et al., New J. Phys., 2013). Third, the microphysical response depicted in n_d- < r^3 > space shows the characteristics of both, homogeneous and inhomogeneous mixing, depending on the Damkoehler number. Fourth, we demonstrated that the variability within the n_d - < r^3 > space increases with decreasing sample volume, especially during the mixing transients which will have implications for the interpretation of measurements in clouds (Kumar et al., J. Atmos. Sci., in revision, 2014). Currently, we improve the match of our initial configuration with real cloud conditions. Our effort is based on high-resolution measurements of trade-wind cumuli in the CARIBA campaign (Siebert et al., Atmos. Phys. Chem., 2013). These data allow us to study the variation of the turbulent fluctuations of all involved fields across the cloud boundary with a resolution significantly finer than a meter and to adjust the kinetic energy dissipation rates in our simulations correspondingly. The principal configuration is that of a shearless mixing layer (Tordella and Iovieno, Phys. Rev. Lett., 2011). The attached Figure 1. shows the initial slablike configuration along the inhomogeneous direction. The upper and mid panels show profiles of characteristic turbulence quantities. The lower panel displays a two-dimensional cut through the initial vapor content field. The color coding is blue for q_v=0.0 to red for q_v=0.031.