Two different collection kernels which include turbulence effects on the collision rate of liquid droplets are used as a basis to develop a parameterization of the warm-rain processes autoconversion, accretion, and self-collection. The new parameterization is tested and validated with the help of a 1-D bin microphysics model. Large-eddy simulations of the rain formation in shallow cumulus clouds confirm previous results that turbulence effects can significantly enhance the development of rainwater in clouds and the occurrence and amount of surface precipitation. The detailed behavior differs significantly for the two turbulence models, revealing a considerable uncertainty in our understanding of such effects. In addition, the large-eddy simulations show a pronounced sensitivity to grid resolution, which suggests that besides the effect of sub-grid small-scale isotropic turbulence which is parameterized as part of the collection kernel also the larger turbulent eddies play an important role for the formation of rain in shallow clouds.

The formation of rain in warm liquid clouds is a result of the condensational
growth on cloud condensation nuclei and the subsequent growth of these
droplets by binary collisions

The semi-empirical collision–coalescence kernel of Ayala and Wang is to a
large extent based on the results of direct numerical simulation (DNS) which
are necessary to quantify the turbulence effects on the collision statistics
in terms of, e.g., the radial distribution function to describe the
preferential concentration effect. As the DNS results are obtained at fairly
low Reynolds number, much lower than observed within clouds, the formulation
of the collection kernel includes an extrapolation to large Reynolds numbers.
An alternative collection kernel recently proposed by

In the following we revisit the results of

The structure of this paper very much follows in the steps of the

For pure gravitational collisions the collection kernel can be written as

Any physical model of

A turbulent flow is not yet fully characterized by

Enhancement factor of the collision–coalescence kernel for a
dissipation rate of

Various models have been suggested to parameterize

The Ayala–Wang model shows a significant increase of the collection kernel
for high Reynolds numbers for droplets smaller than 80

The evolution of the drop size distribution

Following

Enhancement factor of the autoconversion rate
for the Ayala–Wang kernel (upper row) and the Onishi kernel (lower row) at

Coefficients as a result of the nonlinear regression for

The most notable difference between the two kernels is that for the
Ayala–Wang kernel the autoconversion rate increases with

The different autoconversion enhancement factors for the two kernels and the
quality of the fits are shown by Fig.

Time

A first test of the autoconversion parameterization is obtained by
simulations of exactly the same kind as used as training data, i.e., SCE
simulations with an initial condition following a gamma distribution. As a
metric for evaluation we use the timescale

As in

The accretion rate and self-collection of rain are parameterized as

Accumulated surface precipitation of the 1-D
kinematic model as a function of the assumed in-cloud turbulent
dissipation rate

Extensive tests with the 1-D kinematic model have shown that the
parameterization compares reasonably well with the bin microphysics solution
for both collection kernels. The most important metric to evaluate the
warm-rain scheme in the 1-D kinematic model is the precipitation amount at the
surface. One could argue that the timing is almost as relevant as the
precipitation amount, but as shown by

To investigate the effect of in-cloud turbulence on rain formation in trade
wind cumulus clouds, we perform large-eddy simulations of the Rain In Cumulus
over the Ocean (RICO) case as described by

Figure

Time series of the cloud liquid water path, rainwater path, the surface rain rate, and the inversion height for four simulations using the three different collection kernels. The simulation marked “au only” applies the turbulent enhancement only to autoconversion, ignoring the effect on accretion. We have applied a running average to all time series with an averaging window of 120 min for the surface rain rate and 30 min for RWP, CWP, and inversion height.

Sensitivity of LES results to variations in the cloud droplet number density. Shown are the rainwater path, surface rain rate, inversion height, and accretion / autoconversion ratio for the three different collection kernels of the control simulations using the purely gravitational kernel (bullets, grey shading), the Ayala–Wang kernel (squares, blue shading), and the Onishi kernel (diamonds, red shading). The shaded area indicates the standard error at a 95 % confidence level.

The strong turbulence effect of both kernels suggested by
Fig.

Statistics for the large-eddy simulations assuming different
collection kernel.

We have performed a larger set of large-eddy simulations for different cloud
droplet number densities. In addition, simulations have been repeated with
different random seeds to sample the stochastic uncertainty of the system and
to reduce the standard error in the statistical evaluation.
Table

Transition timescales

For low cloud droplet numbers we do not find a significant difference for the
rainwater path and the surface rain rate between the three different kernel
during the 24 to 30 h sampling period, because all three simulations develop
a rain rate that is close to the quasi-equilibrium rainwater flux.
Nevertheless, the transient behavior is different between the three kernels
for all droplet number densities as, e.g., seen from the timescales

As Fig.

The turbulence effects on the collision rate, as postulated by the two
different turbulence models, lead to a strong increase of the autoconversion
rate and a moderate increase of accretion. This is true for both kernels,
although the Onishi model has a weaker enhancement of autoconversion and a
stronger increase in accretion, especially at high Reynolds numbers. It is
therefore interesting to check whether a significant shift in the importance
of those two warm-rain processes can be observed in the large-eddy
simulations. Figure

Previous studies, e.g., by

As previous table but for the
simulations to investigate the resolution dependency at

We have derived a warm-rain bulk two-moment scheme which incorporates the effects of small-scale isotropic turbulence on the collision rate following the two alternative models of Ayala–Wang and Onishi. The two collision kernels differ mostly in their Reynolds number dependency. While the Ayala–Wang model postulates an increase of autoconversion with Reynolds number, the Onishi model predicts a decrease of autoconversion but an increase in accretion for high Reynolds number. The two newly derived variants of the Seifert–Beheng warm-rain scheme have been tested and validated in 1-D simulations and compare favorably with the bin microphysics model that acts as a reference.

The new bulk scheme has been applied in large-eddy simulations of
precipitating shallow convection to investigate the impact of the different
collision kernels. Both turbulence kernels lead to a significant enhancement
of the rain formation in shallow convective clouds, but the turbulence effect
is much weaker for the Onishi kernel. Especially for intermediate cloud
droplet numbers – in our simulations 50 cm

As Fig.

The large-eddy simulations show a strong sensitivity to horizontal grid
spacing with a more rapid rain formation at higher resolution. This suggests
that the larger turbulent eddies like in-cloud circulations, which are
resolved by high-resolution LES, can play an important role for the growth of
rain drops. It is hypothesized that rain drops with large Stokes numbers,

Our results show that the differences between the Ayala–Wang model and the Onishi models are significant, and it needs to be clarified either by observations or by additional DNS studies which collision kernel is more realistic at high Reynolds numbers.

The UCLA-LES model is distributed under GNU General Public
License and can easily be downloaded from

We thank the computing center of ECWMF, where all simulations were performed using resources provided through DWD. We thank Ann Kristin Naumann for helpful comments on the manuscript. We thank the editor Graham Feingold and three anonymous reviewers for their comments that helped to improve the manuscript. Edited by: G. Feingold Reviewed by: three anonymous referees