Connecting finite-time Lyapunov exponents with supersaturation and droplet dynamics in a turbulent bulk flow.

The impact of turbulent mixing on an ensemble of initially monodisperse water droplets is studied in a turbulent bulk that serves as a simplified setup for the interior of a turbulent ice-free cloud. A mixing model was implemented that summarizes the balance equations of water vapor mixing ratio and temperature to an effective advection-diffusion equation for the supersaturation field s(x,t). Our three-dimensional direct numerical simulations connect the velocity and scalar supersaturation fields in the Eulerian frame of reference to an ensemble of cloud droplets in the Lagrangian frame of reference. The droplets are modeled as point particles with and without effects due to inertia. The droplet radius is subject to growth by vapor diffusion. We report the dependence of the droplet size distribution on the box size, initial droplet radius, and the strength of the updraft, with and without gravitational settling. In addition, the three finite-time Lyapunov exponents λ_{1}≥λ_{2}≥λ_{3} are monitored which probe the local stretching properties along the particle tracks. In this way, we can relate regions of higher compressive strain to those of high local supersaturation amplitudes. For the present parameter range, the mixing process in terms of the droplet evaporation is always homogeneous, while it is inhomogeneous with respect to the relaxation of the supersaturation field. The probability density function of the third finite-time Lyapunov exponent, λ_{3}<0, is related to the one of the supersaturation s by a simple one-dimensional aggregation model. The probability density function (PDF) of λ_{3} and the droplet radius r are found to be Gaussian, while the PDF of the supersaturation field shows sub-Gaussian tails.