Turbulence models predict low droplet-collision rates in stratocumulus
clouds, which should imply a narrow droplet size distribution and little
rain. Contrary to this expectation, rain is often observed in stratocumuli.
In this paper, we explore the hypothesis that some droplets can grow well
above the average because small-scale turbulence allows them to reside at
cloud top for a time longer than the convective-eddy time

Stratocumulus clouds are characterized by small droplets (radius,

In this paper, we explore the hypothesis of whether stratocumulus rain
formation can be enhanced if a small fraction of the droplets spend long
residence times at the cloud top, as proposed, for example, by

The enhancement of rain in parcels with different histories has been studied
with trajectory ensemble models (TEM). In those models, the evolution of the
droplet size distribution (DSD) is calculated in several parcels that are
driven by an external model, typically a large-eddy simulation (LES). The
parcels vary their position following the flow dynamics, so that the TEM can
be considered as a Lagrangian model.

In this paper, we use direct numerical simulations (DNS) to quantify the time
that droplets reside at the top of a cloudy mixed layer driven by radiative
and evaporative cooling. Our purpose is to complement past Lagrangian studies

Today's computational resources do not allow us yet to simulate at the same
time all physical processes relevant for the stratocumulus dynamics plus the
turbulent flow length scales that are relevant for the droplets' movement.
Since the focus of this paper is on turbulence, we decided to use a
relatively high resolution to simulate a simple configuration that mimics the
stratocumulus dynamics: a cloudy mixed layer. This configuration consists of
a cloud layer (moist and cool) that lays below a layer of dry and warm air
that represents the free atmosphere. Both layers are characterized by their
total water content

The flow dynamics are calculated by solving the evolution equations for momentum, total water, and enthalpy on a fixed grid. We use the name Eulerian to refer to this first part of the calculations. We also track the evolution of 1 billion droplets that follow the tendencies dictated by the Eulerian calculations. We use the name Lagrangian to refer to this second part of the calculations. We consider only one-way coupling, which means that the Eulerian calculations are independent from the Lagrangian ones. This means that the Lagrangian calculations can be considered as complex diagnostics statistics from the Eulerian simulation. In this section, we present the Eulerian and Lagrangian formulations employed in this study.

We use the Eulerian formulation described in

We define the normalized liquid water as

The first term on the right-hand side of
Eq. (

Parallel to the Eulerian calculations, we track the trajectories and mass of 1 billion individual droplets. We call these droplets Lagrangian because the equation of motion is solved independently for each droplet. The Lagrangian droplets represent only a small fraction of the number of cloud droplets in a real cloud, and are introduced to obtain extra information from the simulations.

We impose that the Lagrangian droplet dynamics follow the mean local
tendencies given by the velocity and source terms from the Eulerian
formulation. This assumption neglects all fluctuations that arise from length
scales between the simulated Kolmogorov length scale (

We initially place each Lagrangian droplet at a random position inside the
cloud (defined by

In order to study the evolution of the droplet size distribution, we assign a
mass

Droplets grow as they ascend through the clouds and shrink when they descend,
due to the change of temperature in the atmospheric boundary layer.
Equations (

The Lagrangian equation for the liquid mass (Eq.

Our simulations show too little evaporation of the Lagrangian droplets, when
compared with the Eulerian mean values (reduction up to 90 %). This strong
difference is caused by neglecting the diffusion term in
Eq. (

In order to explain the differences introduced by diffusion, we sketch in
Fig.

The left side represents a cloudy moist parcel and an unsaturated parcel before mixing. The right side represents the mixing processes in the Eulerian and Lagrangian formulations. In the Eulerian formulation (top) there is diffusion of liquid from the saturated into the unsaturated parcel. In the Lagrangian formulation (bottom) there is no diffusion, and the initially saturated parcel remains free of liquid. As a consequence, liquid water evaporates in both parcels in the Eulerian formulation, but only in the initially saturated parcel in the Lagrangian formulation. Consequently, the evaporation of in the Eulerian formulation is consistently higher.

Our reference case is based on the diurnal measurements from the reference
flight 11 in the VERDI campaign, taken on 15 May 2012 in the Beaufort Sea
area

The cloud and dry layer in the simulations mimic the cloud-top
thermodynamical properties measured by this flight: cloud layer at
temperature of

The reference case is characterized by a relatively wet free atmosphere that
limits the impact of evaporation on turbulence

The main simulation was performed in a horizontally periodic mixed layer

The numerical algorithm is based on high-order, spectral-like compact
finite differences

All the simulations were done in a cubic grid with

Eulerian fields are extrapolated into the Lagrangian droplets' positions by
using a trilinear interpolation from the eight closest grid points. The
Lagrangian equations are integrated in time with the same low-storage
fourth-order Runge–Kutta scheme

We first look at the trajectories of 20 Lagrangian cloud droplets during

Trajectories of Lagrangian cloud droplets from two different viewing
points. The droplets have been tracked during

Red trajectories correspond to cloud droplets that circulate quickly through
the cloud, and spend a short time at cloud top. Figure

Green trajectories correspond to cloud droplets that escape the downwards
movements of the convective eddies, and stay at the cloud-top region for the
whole tracing period. We use the name “long-resident droplets” for droplets
that follow these trajectories. Long-resident droplets grow on average more
than convective droplets, although they are less in number.
Figure

Orange trajectories correspond to droplets that are driven out of cloud top by intermediate eddies, so that they keep circulating through the cloud bulk. Their near future is still undecided. They can either go back to the cloud top, and eventually become a long-resident droplet, or be driven downwards by a convective eddy. We use the name “erratic droplets” for those droplets that are neither convective nor long-resident. This is indeed poor naming because erratic droplets can have quite different dynamics, but it just shows how difficult it is to characterize the movement of all droplets in a turbulent flow.

In order to quantify the different droplet dynamics we assign to each droplet
a cloud-top residence time

The residence times are initialized

At each time step, the residence time advances only for droplets that are at a distance
of less than

The residence time of cloud droplets that are distanced

The residence time of the cloud droplets whose normalized mass is below

The objective of this definition is to discriminate long-resident droplets
from convective ones. Convective droplets circulate through the mixed layer
in a time comparable to the convective-eddy time, and their maximum residence
time is a fraction of

Figure

Figure

After

In the interval

In the interval

From

Next, we look at where long-resident droplets can be found. In order to
visualize our results, we first extrapolate the resident times to the Eulerian
grid, for which we use the trilinear interpolation.
Figure

Visualization of the stratocumulus cloud top at

The different droplet dynamics discussed in previous sections should be
reflected on the DSD. In particular we expect that long-resident droplets
grow larger than average due to the combined action of radiative cooling and
collisions, thus broadening the DSD. In this section, we investigate how the
DSD broadens due to condensation induced by radiative cooling alone. Droplet
evaporation is also included in the calculations, but we expect it to have
little effect on the DSD because our Lagrangian algorithm considerably
underestimates the evaporation of droplets (see Sect.

Radiative and evaporative cooling in our simulation overcome the influx of
heat at cloud top. As a result the cloud continuously cools down and the
total liquid water linearly increases with time. This introduces a continuous
shift in the DSD equal to

Figure

We compare the DSDs in our simulations to the in situ measurements from the
VERDI campaign described in Sect.

Even when considering that there is some degree of chance in the agreement between observations and simulations (such as in the final time of the simulations), the comparison strongly suggests that condensation due to longwave radiation was an important factor for broadening the DSD in the stratocumulus observed in the VERDI campaign. Since the radiative properties of this cloud are quite common, we conclude that longwave radiation is probably also relevant for the DSD evolution in many other radiatively driven stratocumuli.

Evolution of the cloud-top DSD. The droplets in the simulations grow only by condensation due to radiative cooling. The points correspond to measurements from the VERDI campaign.

Our simulations show cloud droplets that reside at the stratocumulus top for
times considerably longer than the convective-eddy time, supporting some
previous LES studies

We observe that 15 % of the droplets partially escape the stratocumulus
large-scale convective motions and reside at the cloud-top region for periods
longer than the convective-eddy time. This percentage is in rough agreement
with cloud-top resident times in a drizzling stratocumulus LES

The large number of droplets in our simulations (

Let us provide a simple argumentation to show how long-resident droplets can
experience multiple collisions and contribute to the formation of rain
droplets at cloud top. The gravitational-settling collision rate for droplets
of radius

We observe that long-resident droplets prefer to concentrate in the thin
downdraft regions of the flow. This behavior agrees with the pattern of
in-cloud residence times (different from the cloud-top residence times in
this paper) found by

Long-resident droplets also grow on average considerably larger than
convective droplets due to condensation induced by longwave radiative
cooling, which can speed up rain formation

It has been often noted that a better knowledge of turbulence is necessary to
solve the size-gap problem in shallow-clouds rain formation

The source code and statistics from the main simulation can be found in

In the first subsection, we summarize the formulation introduced in

The original formulation is based on two scalars,

In the limit

Solving turbulence close to a rigid boundary in DNS typically requires a higher resolution than in other regions. For this reason most DNS use refined grids close to the boundaries. The problem is that using refined grids in our simulations considerably slows down the calculations because the time stepping needs to be reduced and the vectorization of the Lagrangian calculations becomes more difficult. The gain for this cost is to properly resolve the flow in a region that does not alter considerably the cloud-top dynamics in which we are interested. For this reason we decided to use an alternative strategy. The objective is to create a soft boundary that does not allow the flow to penetrate much beyond cloud base, while modifying the cloudy thermodynamical state as least as possible.

The soft boundary is created by adding a second stratification at cloud base.
This is achieved by modifying the buoyancy as

DSD dependence on viscosity (quantified by the reference Reynolds
number

We investigate the influence of the unresolved small scales by performing
simulations with different viscosities, which in DNS implies changing the
smallest resolved length scale (Kolmogorov length

We observe that changing the viscosity does not appreciably alter the cloud
convective movements. The mean turbulence dissipation rate varies by

In Fig.

If significant, the small increase of the large droplets' size with reducing
viscosity can be attributed to different causes. One possible cause is that
the radiative-forcing maximum decreases by 5 % when reducing the viscosity
from

The authors are grateful to Markus Klingebiel and Stephan Borrmann for providing the data from the VERDI campaign. Comments on the original manuscript by Juan Pedro Mellado and Axel Seifert are gratefully acknowledged. Computational time was provided by the Jülich Supercomputing Centre. The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: C. Hoose