**Appendix 3.B Supplementary Figures**

**4.4 In-situ Measurements of Trade-wind Cumuli**

**4.4.2 Statistics of Cloud Droplets in Trade-wind Cumuli**

In the atmosphere, typical cloud droplets with diameters up to 25 µm, as observed at
EUREC^{4}A , are generally two orders of magnitude smaller than the Kolmogorov length
scale of the turbulent flow in which they are embedded. Besides the gravitational force,
the dynamics of cloud droplets is coupled predominantly to the dissipative scales of
the flow being captured by the cloud particle Stokes number St [e.g. 4]. The relative

unit mean median std

*U* m*/*s 7.83 7.02 2.61

*σ**u*^{′}_{1} m*/*s 0.66 0.65 0.25

⟨*ϵ**I*2⟩^{τ} mW*/*kg 31.45 20.85 35.87

*η**K* mm 0.94 0.70 0.54

*λ**f* m 0.20 0.11 0.17

*L*11 m 62 12 109

R*λ* 4302 2714 4109

**Table 4.3** Cloud turbulence features obtained from time-records of the longitudinal velocity
component *u*_{1}(*t*) on both MSM89 (RV Maria S. Merian) and M161 (RV Meteor) during
EUREC^{4}A by both the MPCK+ and mini-MPCK.*U* =⟨*u*1(*t*)⟩30*L*11*τ* is the mean velocity
obtained from *u*1(*t*) for the averaging window*τ*, *σ*_{u}^{′}

1 the RMS fluctuation of *u*1(*t*), ⟨*ϵ**I*2⟩^{τ}
the estimate of the mean energy dissipation rate according to Eq. (3.15) with *n*= 2,*η*_{K} the
Kolmogorov length scale,*λ**f* the longitudinal Taylor micro-scale, *L*11the integral length scale
and R*λ* the Taylor micro-scale Reynolds number. For each turbulence feature, the mean,
median and standard deviation (std) are given. The averaging window is*τ* = 60 s. *u*_{1}(*t*) is
not corrected for platform motion. The MPCK+ and mini-MPCK are considered to be inside
clouds if the cloud droplet number density*n >*10*/*cm^{3}.

unit mean median std

*U* m*/*s 9.06 9.04 2.80

*σ**u*^{′}_{1} m*/*s 0.55 0.55 0.24

⟨*ϵ**I*2⟩^{τ} mW*/*kg 16.51 1.51 25.49

*η**K* mm 1.34 1.31 0.78

*λ**f* m 0.33 0.26 0.27

*L*11 m 120 37 160

R*λ* 5726 3514 4761

**Table 4.4** Bulk turbulence features obtained from time-records of the longitudinal velocity
component *u*_{1}(*t*) both on MSM 89 (RV Maria S. Merian) and M161 (RV Meteor) during
EUREC^{4}A in cloud-free air (*n*≤10*/*cm^{3}). *U* =⟨*u*1(*t*)⟩*τ* is the mean velocity obtained from
*u*_{1}(*t*) for the averaging window*τ*, *σ*_{u}^{′}

1 the RMS fluctuation of*u*_{1}(*t*), ⟨*ϵ*_{I2}⟩*τ* the estimate of
the mean energy dissipation rate according to Eq. (3.15) with *n*= 2, *η*_{K} the Kolmogorov
length scale,*λ**f* the longitudinal Taylor micro-scale,*L*11the integral length scale and R*λ* the
Taylor micro-scale Reynolds number. For each turbulence feature, the mean, median and
standard deviation (std) are given. The averaging window is*τ* = 60 s. *u*_{1}(*t*) is not corrected
for platform motion.

impact of gravity to turbulence is captured by the non-dimensional settling velocity Sv [e.g. 23, and references therein]. Both parameters identify potential cloud droplet growth due to collision-coalescence and gravitational collection, which are important to the initiation of rain in warm clouds [4, 23]. In the following, the parameter space of cloud droplets is characterized in terms of St and Sv, which both depend on the cloud droplet diameter and the energy dissipation rate.

**Figure 4.20**Cloud droplet statistics in trade-wind cumuli based on measurements of MPCK+

and mini-MPCK during EUREC^{4}A . A: The PDF of the Stokes number St is shown for all
flights based on the PbP-records acquired by FCDP and CDP-2. B: Non-dimensional settling
velocity parameter Sv as a function of cloud droplet Stokes number St for various altitude
ranges. The altitude is re-scaled by TBL (*z*_{PSS8}*/*TBL). Arrows visualize the dependence
on the mean energy dissipation rate estimate ⟨*ϵ**I*2⟩*τ* and the droplet diameter*d**p*. C: PDF
of droplet size ratio of two subsequent droplets where *d*^{>}_{p} is the larger droplet and*d*^{<}_{p} the
smaller droplet. D: Collision efficiency *E* according to [64]. *τ* = 60 s is the averaging window.

The Stokes number, Eq. (1.70), is obtained from PbP data under the assumption
of a constant kinematic viscosity of air *ν* = 1*.*5571×10^{−5}m^{2}*/*s and a constant density
of liquid water *ρ*H2O = 997 kg*/*m^{3}. The energy dissipation rate is estimated by ⟨*ϵ**I*2⟩^{τ}
with *τ* = 60 s. To calculate St, PbP-time stamps are matched with the time of ⟨*ϵ**I*2⟩*τ*,
which is implicitly assumed to be constant over the averaging window. The resulting
PDF of St obtained from PbP cloud droplet diameters during EUREC^{4}A is shown in
Fig. 4.20A. The vast majority of the encountered cloud droplets have a Stokes number
St*<*0*.*01 (86% on average, maximally 99.99%) where droplet inertia can be neglected.

Inertial effects such as the sling effect just become significant for St≳0*.*1 [28, 61]. The

fraction of cloud droplets with St*>*0*.*1 is 0.1% for all flights and 0.56% for MSM89
Flight 12 where the MPCK+ sampled precipitating clouds. This flight is investigated
in more detail in Sec. 4.4.3.

Despite that, the vast majority of cloud droplets behave most likely as tracer
particles (St*<*0*.*01). The terminal velocity of a droplet is ∝*d*^{2}_{p} in still air [4] so that
gravitational settling could dominate cloud droplet motion. Furthermore, previous
considerations have shown an altitude dependence of the droplet size PDF and the
presence of particles larger than 20 µm (Sec. 4.4). Therefore, it is necessary to condition
the Sv-St parameter space on altitude, i.e. vertical distance to cloud base, as shown in
Fig. 4.20B. Considering the regime Sv*>*1 and St *<*0*.*1, gravitational settling is an
important process compared to droplet inertia in shallow cumulus clouds particularly
in the low altitude range 0*.*8 *< z*PSS8*/*TBL *<* 1*.*2. This regime is associated with a
moderate⟨*ϵ**I*2⟩^{τ} (on average 2×10^{−3}W*/*kg),*L*11≈177 m and an average cloud droplet
diameter of 17 µm. In the regime Sv *<*1 and St*<*0*.*1, neither gravitational settling
nor droplet inertia plays a significant role. This regime is the “tracer” regime and is
related to ⟨*ϵ**I*2⟩^{τ} (on average 4×10^{−2}W*/*kg) and small particle diameters (average
*d**p* ≈ 7 µm). This regime is occupied by all altitude ranges and associated with an
average integral scale of *L*11 ≈ 45 m. A significant fraction of cloud droplets in the
medium altitude regime 1*.*2*< z*PSS8*/*TBL *<*1*.*6 is dominated by droplet inertia, i.e.

Sv *<*1 and St *>*0*.*1. Cloud droplets are attributed to this regime with an average
diameter of 15 µm and in turbulent regions with⟨*ϵ**I*2⟩^{τ} being 8×10^{−2}W*/*kg on average.

Furthermore, the Taylor-scale Reynolds number is R*λ* ∼1660 hinting at strong local
fluctuations of the energy dissipation rate due to intermittency. This is supported by
an average measured integral length scale *L*11 of only 5 m.

The regime where rain is initiated presumably the most efficiently is for Sv*>*1 and
St *>* 0*.*1 with ⟨*ϵ**I*2⟩^{τ} being 6×10^{−2}W*/*kg (R*λ* ≈ 2330, *L*11 ≈ 10 m and *d**p* ≈ 16 µm
on average). Due to finite droplet inertia (0*.*1 *<* St *<* 1), cloud droplets mostly
occupy regions of low vorticity in the turbulent flow while, simultaneously, they tend
to settle [4, 20]. As they mostly pass through strain-dominated regions of the flow
[4, 20], it is expected that the relative velocities between cloud droplets differ. It
can be concluded that gravitational collection is combined with collision-coalescence.

If Sv ≫ 1, the gravitational collection dominates over collision-coalescence. Most
of the cloud droplets in that Sv-St range (Sv *>* 1 and St *>* 0*.*1) are sampled at
1*.*2*< z*PSS8*/*TBL*<*1*.*6, which might be due to the flight strategy. Further analysis is
required to validate the explanation above where holography and PIV data promise
valuable insight. Furthermore, the gap between Sv ≳1 andO(St)∼0*.*01 is remarkable.

It can be suggested that cumuli with a sufficient depth to reach the medium altitude regime exhibit higher energy dissipation rates. To validate or falsify this hypothesis, measurements of cloud top altitudes of the sampled clouds have to be taken into account. Turbulent characteristics should be subsequently conditioned on the cloud top altitude and should reveal bimodal behavior.

The efficiency of collision-coalescence depends on the sizes of colliding droplets [64, 200]. Despite that the CDPs cannot resolve collision events, they deliver cloud droplet

sizes of two successive cloud droplets. Ignoring the turbulent flow and thermodynamics
of the background, two neighboring droplets can be considered as best candidates for
a collision. To estimate the efficiency of these potential collisions, the size ratio of
two successive droplets is calculated. *d*^{>}_{p} is the larger droplet and *d*^{<}_{p} the smaller so
the values lie within 0 and 1. The PDF of *d*^{<}_{p}*/d*^{>}_{p} is shown in Fig. 4.20C. The first
point to note is that ratios of 0.1 and smaller are very rare. This can be explained
by the fast condensational growth of cloud droplets up to a few microns and the fact
that cloud droplets larger than 20 µm are rare (e.g. Fig. 4.17B). Despite variability
among the flights, droplet size ratios between 0.2 and 0.8 are nearly equally likely. The
largest range is observed for a size ratio of 0.6. In case of MSM89 Flight 12 , a local
minimum at size ratio about 0.5 emerges. The size ratio increases again at 0.6 and
0.7. As an example, a size ratio of 0.6 to 0.7 corresponds to neighboring droplets of
20 µm and 30 µm in diameter. In the case of MSM89 Flight 12 where the droplet size
ratio is higher for 0.6 and 0.7 compared to 0.5 and 0.9. As illustrated in Fig. 4.20D,
the collision efficiency *E*(*d*^{>}_{p}*, d*^{<}_{p}) is optimal in the range 0.5 to 0.75 [64]. The observed
size ratios during the rain event in MSM89 Flight 12 (cf. Sec. 4.4.3), therefore, are in
the optimal range.

Most cloud processes, among which is collision-coalescence, are related to the
spatial distribution of cloud droplets [23]. The spatial distribution of cloud droplets in
trade-wind cumuli is investigated as a function of the inter-particle distance ∆*r*. The
inter-particle distance is obtained from the PbP-inter-arrival time and low-pass filtered
relative wind speed (cutoff frequency at 0*.*05 Hz due to platform motion). The counting
statistics is conditioned on cloudy air, i.e. *n >* 10*/*cm^{3}. The PDFs of inter-particle
distances irrespective of their size are shown in Fig. 4.21A where ∆*r* is re-scaled by
the Kolmogorov length scale for global comparison. Up to ∆*r*∼1*η**K*, the error bars
are large due to counting statistics. however, MSM89 Flight 3 and 12 as well as M161
Flight 5 exhibit different behavior than the other flights. In the case of Flight 5,
the enhanced probability of ∆*r <*1*η**K*, is due to high particle counts at low altitude
in the lowest size bin (and does not appear in Figs. 4.21B-D anymore). The CDP2
does not provide chemical information about the recorded particles. I suppose that it
measured sea spray or large aerosols which is why this flight is not further considered.

However, during MSM89 Flight 12, ∆*r < η**K* is more likely than ∆*r > η**K* which hints
at spatial clustering. In this context, it would be interesting to evaluate statistics on
inter-particle distances from 3D positions in order to avoid the projection issues of the
one-dimensional quantity ∆*r*. Between 1*η**K*−10*η**K*, the PDF is nearly constant. On
these scales, the distribution of particles in space appears to be random.

However, the probability of finding two particles separated by 10*η**K* to 100*η**K* (i.e.

0*.*6 cm to 6 cm) is 3 to 4 orders of magnitude smaller than droplets separated by ∆*r*
between 1 – 10*η**K*. In other words, shorter inter-particle distances of ∆*r*∼1*η**K*−10*η**K*

are more likely than ∆*r* ∼ 10*η**K* −100*η**K*. This suggests that scales comparable to
10*η**K* −100*η**K* are less populated, i.e. devoid of cloud droplets. These regions, that
are not occupied by cloud droplets, are also known as “cloud voids” and have been
experimentally measured on Mt. Zugspitze. Cloud voids on Mt. Zugspitze have

an average size of 70 *η**K* [39], which is in agreement with the ∆*r*-measurements in
trade-wind cumuli. However, this drop can also be associated with merging different
cloud regions in the PDF. The number density in the cloud core is higher than at the
cloud edge, where entrainment and mixing processes occur. Thus, it is possible that the
two plateaus are related to the cloud core and cloud edge. This would imply that cloud
droplets, neglecting their size, are nearly randomly distributed. Conditioning the PDF
of ∆*r* to cloud core with *n >*100*/*cm^{3}, the PDF also deviates from Poisson behavior
for ∆*r >*60*η**K* (Fig. 4.22). Therefore, the spatial distribution of cloud droplets is only
close to being random for scales ∆*r <*60*η**K*.

Furthermore, the inter-particle distances depend on the droplet size range. The
inter-particle distances increase with increasing *d**p*-threshold (Fig. 4.21B-D) where the
minimum distance is several *η**K* in the case of*d**p* *>*15 µm. Considering cloud droplets
larger than 15 µm, PDFs of MSM89 Flights 11, 12 and 18 exhibit power-law-like
behavior. In [25], this is attributed to dynamic processes such as entrainment.

To summarize, the Sv-St parameter space spans four orders of magnitude for both
parameters. Hence, diverse dynamics of cloud droplets embedded in a turbulent flow
are expected. For a rain event, the droplet size ratio has a local maximum at ≈0*.*6
where the collision efficiency is optimal. The droplet spatial distribution suggests cloud
voids but more detailed analysis, ideally paired with an analysis of 3D droplet positions,
is needed.