  The impact of turbulent mixing on the droplet size distribution is studied
deep inside a warm ice-free cloud. A simplified cloud mixing model was
implemented therefore which summarizes the balance equations of water vapor
mixing ratio and temperature to an effective advection-diffusion equation for
the supersaturation field $s(\textbf{x},t)$. Our three-dimensional direct
numerical simulations connect the scalar supersaturation field to the cloud
droplet dynamics, in particular to the droplet size distribution for different
box sizes. In addition, finite-time Lyapunov exponents are monitored such that
we can relate regions of higher compressive strain to those of high local
supersaturation amplitudes. We find that the mixing process in terms of the
droplet evaporation is always homogeneous in the bulk of the cloud, while being
inhomogeneous in view to the relaxation of the supersaturation field. The
probability density function of $\lambda_3$ is related to the one of $s$ by a
simple one-dimensional aggregation model. The distributions of the compressive
finite-time Lyapunov exponents $\lambda_3$, the supersaturation field, and the
droplet size are found to be Gaussian.
