**Appendix 3.B Supplementary Figures**

**4.2 Atmospheric Turbulence Characteristics of the Marine Boundary Layer 126**

**4.2.3 Turbulence Characteristics in the Trade-wind Boundary Layer . 130**

According to the previous Sec. 4.2.2, it is expected that the boundary layer is turbulent across the entire diurnal cycle and across its vertical extent. As mentioned in Sec. 3.1,

the mean energy dissipation rate is a central turbulence characteristic. The purpose of this section is to estimate the most relevant features of a turbulent flow in terms of the mean energy dissipation rate obtained from one-dimensional time-records of the longitudinal velocity.

The general procedure for estimating the mean energy dissipation rate is illustrated
in Fig. 3.1 where, similar to Sec. 4.1.3, *ϵ**I*2 and an averaging window of *τ* = 1800 s is
used to estimate the mean energy dissipation rate. This procedure is based on the
assumption of locally isotropic turbulence and the applicability of Taylor’s hypothesis,
which is generally fulfilled as the turbulence is 7% on average with a standard deviation
of 4%. The isotropy of the turbulent flow is assessed in Sec. 4.3. Additionally, it should
be noted that all flights with unphysical velocity signals are ignored, i.e. M161 Flight 1,
4 and 6. Furthermore, due to platform motion, the longitudinal velocity time-records
are high-pass filtered with *f**c*= 0*.*5 Hz where the effect is assessed in Sec. 4.B. In this
section, error bars are given by Eq. (3.45) for ⟨*ϵ**I*2⟩^{τ} and by the re-scaled standard
deviation of *z* during the averaging window *τ*. As the flight altitude of the helikite
is not exactly constant, the error (uncertainty) of the re-scaled barometric altitude
*z*PSS8*/*TBL is captured by the standard deviation. Here, ⟨*ϵ**I*2⟩^{τ}-values with an altitude
uncertainty larger than 50 m are discarded. We concentrate on measurements of the
mini-MPCK during M161 (RV Meteor) because the profiling of the boundary layer is
dedicated to studying its turbulent structure. On MSM89, cloud measurements were
prioritized which is why these flights are analyzed in more detail later (Sec. 4.4). A
summary of the bulk turbulence features during MSM89 is delivered in Fig. 4.35 and
Table 4.8. The averaged relative random errors*e* in Tables 4.1 and 4.8 are given by
Eqs. (3.45), (1.61), (3.42), (4.3) and (4.14) for⟨*ϵ**I*2⟩^{τ},*U*,*σ**u*^{′}_{1}, R*λ*, and*L*11, respectively.

The averaged relative random errors*e* for *η**K* and *λ**f* are given by:

*e*(*η**K*) =

*∂η**K*

*∂*⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩^{τ}

= *η**K*

4⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩^{τ} = *η**K*

4 *δ**I*2(*R*)*,* (4.11)
*e*(*λ**f*) =

*∂λ**f*

*∂*⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩^{τ}

= *λ**f*

2⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩^{τ} = *λ**f*

2 *δ**I*2(*R*)*,* (4.12)
(4.13)
invoking Gaussian error propagation with respect to the relative random error in⟨*ϵ**I*2⟩*τ*.

To assess the daily variability of the mean energy dissipation rate, an overview of
the mean energy dissipation rate estimates ⟨*ϵ**I*2⟩^{τ} is shown as a function of re-scaled
barometric altitude *z*PSS8*/*TBL in Fig. 4.10 under the assumption of a stationary flow
for that period (time scale of days to weeks). The order of magnitude of ⟨*ϵ**I*2⟩^{τ} ranges
from 1×10^{−5}W*/*kg to 1×10^{−3}W*/*kg, which possibly contradicts the assumption of
global stationarity on a weekly scale. To test this hypothesis, mean scaling exponents
*ζ*2 of *D*^{τ}_{LL}(*r*) within the fit range are classified by their deviation from the K62-
prediction where a deviation of 0.1 is accepted. Thus, *ζ*2 is in agreement with K62 if

|*ζ*2−2*/*3 + 1*/*36| *<* 0*.*1 (group 1) corresponding to the highly opaque ⟨*ϵ**I*2⟩*τ* -values
in Fig. 4.10. Otherwise, |*ζ*2 −2*/*3 + 1*/*36| ≥ 0*.*1 and *ζ*2 is not in agreement with

### 10

^{−}

^{5}

### 10

^{−}

^{4}

### 10

^{−}

^{3}

### 10

^{−}

^{2}

### h

_{I2}

### i

^{τ}

### [W kg

^{−}

^{1}

### ]

### 0*.*0 0*.*4 0*.*8 1*.*2 1*.*6

*z*

PSS8*/* TBL

3 5

7 8

9 10

### 0 10 20

### #

h*ζ*2i=0.69±0.01
h*ζ*_{2}i=0.57±0.02

**Figure 4.10** Overview of mean energy dissipation rate estimates ⟨*ϵ*_{I2}⟩*τ* obtained from one-
dimensional time-records of the longitudinal velocity during EUREC^{4}A with the mini-MPCK
on M161 (RV Meteor) as a function of the re-scaled barometric altitude*z*_{PSS8}. TBL is the
abbreviation for top of the boundary layer and the averaging window *τ* = 30 min. Error
bars are given by Eq. (3.45) for ⟨*ϵ*_{I2}⟩*τ* and by the re-scaled standard deviation of *z*_{PSS8}
during the averaging window*τ*. Highly opaque⟨*ϵ**I*2⟩*τ* -values are obtained from longitudinal
second-order structure functions *D*^{τ}_{LL}(*r*) with a mean scaling exponent *ζ*_{2}, which satisfies

|*ζ*_{2}−2*/*3 + 1*/*36|*<*0*.*1 (in agreement with K62) in the fit range. In contrast, weakly opaque

⟨*ϵ**I*2⟩*τ*-values stem from*D*^{τ}_{LL}(*r*) with |*ζ*2−2*/*3 + 1*/*36| ≥0*.*1. Furthermore,⟨*ϵ**I*2⟩*τ* is highly
opaque only if the standard deviation of the altitude is lower than 50 m. The right panel
shows the number counts of⟨*ϵ*_{I2}⟩*τ* -values agreeing with K62 in blue and the number counts
of⟨*ϵ**I*2⟩*τ*-values disagreeing with K62 in (weakly opaque) red. The ensemble average of each
*ζ*_{2}-group is denoted by ⟨*ζ*_{2}⟩.

K62 (group 2) where ⟨*ϵ**I*2⟩*τ*-value are weakly opaque in Fig. 4.10. Thus, if the inertial
range is well pronounced and if the mean scaling exponent is in accordance with K62,
*ϵ**I*2 is considered to be a valid method to estimate the mean energy dissipation rate.

Focusing on time periods where *ζ*2 is in agreement with K62, ⟨*ϵ**I*2⟩*τ* ranges only from
4×10^{−4}W*/*kg to 2×10^{−3}W*/*kg except for only a few significant outliers. Hence, the
mean energy dissipation rate is fairly constant on a daily to weekly time scale which
is investigated in more detail below. Furthermore, the right panel of Fig. 4.10 shows
the number counts of averaging windows *τ* during which the ensemble-averaged mean
scaling exponent⟨*ζ*2⟩= 0*.*69±0*.*01 is agreeing with K62 in blue. The agreement between
K62 and ⟨*ζ*2⟩ is remarkable. Likewise, the number counts of averaging windows *τ*
during which the ensemble-averaged mean scaling exponent ⟨*ζ*2⟩= 0*.*57±0*.*02 disagrees
with K62 is shown in (weakly opaque) red. Hence, the ensemble-averaged mean scaling
exponent is systematically lower than predicted by K62. In the following, only those
time periods are considered during which |*ζ*2−2*/*3 + 1*/*36| *<* 0*.*1 and the altitude
uncertainty below 50 m.

Even though the mean energy dissipation rate did not reveal a trend on a daily to weekly time scale, the atmosphere changes on hourly time scales. As an example, the air temperature changes at time scales below one hour (Sec. 4.1.1). Likewise, a signature of that time scale is found in the cloud fraction time-evolution over the diurnal cycle increasing after sunset [192, 193]. The dynamics of shallow cumuli are coupled to heat and moisture fluxes, hence to convection [68]. In general, atmospheric turbulence is driven by wind shear from the surface and buoyancy driven convection.

As convection, hence the energy injection, varies on hourly time scales, it is expected
that the mean energy dissipation rate varies over the diurnal cycle, too. Overall, the
profile of ⟨*ϵ**I*2⟩^{τ} does not reveal a significant trend with the local time of the day in
Fig. 4.11A. Nevertheless, considering Flight 9 on M161, ⟨*ϵ**I*2⟩^{τ} slightly increases from

≈ February 05 2020 19:00 AST to February 06 2022 00:00 AST but also with altitude.

Hence, it is not unambiguously that the increase is purely due to the effects of the
diurnal cycle. However, the vertical profile of *θ**v* has a negative slope for that time
frame (February 05 2020 19:00 AST to February 06 2022 00:00 AST) during M161
Flight 9, as shown in Fig. 4.33C, suggesting an unstable stratification which is critical
for convection. Therefore the increase in ⟨*ϵ**I*2⟩*τ* is most likely related to the diurnal
cycle. Similarly, the variance ⟨*u*^{′2}_{1}⟩^{τ} shows an increase with time for the same time
frame (Fig. 4.11B). Both *L*11, Eq. (1.29), and *R**λ*, Eq. (1.38), are derived from ⟨*ϵ**I*2⟩^{τ}
and⟨*u*^{′2}_{1}⟩*τ* and, therefore, reveal a similar behavior (Figs. 4.11C and D). The error bars
for⟨*u*^{′2}_{1}⟩^{τ} and R*λ* are given by Eqs. (3.42) and (4.3), respectively. The error of *L*11 due
to *δ**I*2(*R*), the random error of ⟨*ϵ**I*2⟩^{τ}, reads:

*e*(*L*11) =

*∂L*11

*∂*⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩*τ*

= *L*11

⟨*ϵ**I*2⟩^{τ}*δ**I*2(*R*)⟨*ϵ**I*2⟩*τ* =*L*11*δ**I*2(*R*)*.* (4.14)
Considering the error bars being derived from the random error *δ**I*2(*R*) , ⟨*ϵ**I*2⟩^{τ}
, ⟨*u*^{′2}_{1}⟩^{τ}, *L*11 and R*λ* only, if at all, weakly depend on altitude. A general trend
with altitude is not significant although, for*L*11 and R*λ*, it seems that two branches

10^{−5} 10^{−4} 10^{−3} 10^{−2}
_{I2}*τ* [W kg^{−1}]
0*.*0

0*.*4
0*.*8
1*.*2
1*.*6

*z*PSS8*/*TBL

3

5 7

8 9

10

00:00 06:00 12:00 18:00 24:00

timeofday

0*.*0 0*.*3 0*.*6 0*.*9 1*.*2 1*.*5
*u*^{2}_{1}*τ* [m^{2}s^{−2}]
0*.*0

0*.*4
0*.*8
1*.*2
1*.*6

*z*PSS8*/*TBL

3

5 7

8 9

10

00:00 06:00 12:00 18:00 24:00

timeofday

0 300 600 900 1200 1500
*L*_{11} [m]

0*.*0
0*.*4
0*.*8
1*.*2
1*.*6

*z*PSS8*/*TBL

3 5

7 8

9 10

00:00 06:00 12:00 18:00 24:00

timeofday

0 10000 20000 30000
*R*_{λ}

0*.*0
0*.*4
0*.*8
1*.*2
1*.*6

*z*PSS8*/*TBL

3 5

7 8

9 10

00:00 06:00 12:00 18:00 24:00

timeofday

### A B

### D C

**Figure 4.11** Influence of the diurnal cycle on bulk turbulence. The color code corresponds
to the local time of the day (UTC - 4h). TBL is the abbreviation for top of the boundary
layer and the averaging window*τ* = 1800 s. The error bars for *z*are given by the standard
deviation of*z* during the averaging window*τ*. A: The mean energy dissipation rate estimates

⟨*ϵ*_{I2}⟩*τ* are obtained from one-dimensional time-records of the longitudinal velocity during
EUREC^{4}A with the mini-MPCK on M161 (RV Meteor). The error bars are given by Eq. (3.45)
for⟨*ϵ**I*2⟩*τ*. B: The variance of velocity fluctuations⟨*u*^{′2}_{1}⟩*τ* is based on unfiltered *u*^{′}_{1}. Error bars
are only considered for the barometric altitude. C: The longitudinal integral length scale*L*_{11}
is estimated by Eq. (1.29). Error bars are obtained via Eq. (4.14). D: The Taylor micro-scale
Reynolds number R*λ* is obtained by Eq. (1.38). Error bars are obtained via Eq. (4.3).

emerge (Figs. 4.11C and D) where the left branch is constant with altitude in the
limit of accuracy. The left branch is mostly populated by M161 Flight 8 where the
boundary layer was stably stratified (Fig. 4.33B) but with vertical shear of the wind
speed (Fig. 4.33D). This is confirmed by the Richardson numbers which are mostly
positive for *z*PSS8*/*TBL *<* 1 (Fig. 4.9). The right branch consists predominantly of
M161 Flight 9 and is possibly related to larger convective structures with slightly
negative Richardson numbers for*z*PSS8*/*TBL*<*1 (Fig. 4.9)). A more profound analysis,
including vertical fluxes in the Earth frame of reference, is required for supporting
this hypothesis. Ignoring the effect of the diurnal cycle, it is possible to summarize
bulk turbulence properties in terms of PDFs in Fig. 4.13 and in terms of statistical
quantities in Table 4.1. The above considerations explain the presence of multiple
peaks in the PDFs of *L*11 and R*λ* (Fig. 4.13).

The PDFs of the systematic and random errors*δ*_{I2}^{sys}(*R*) and*δ**I*2(*R*), respectively, is
shown in Fig. 4.12. As both errors are inferred in retrospect from, i.a., the estimated
*L*11, the scatter is expected. Therefore, both *δ*^{sys}_{I2} (*R*) and*δ**I*2(*R*) should be interpreted
as rough estimates on the accuracy of ⟨*ϵ**I*2⟩^{τ} where the systematic error is only half as
large as the random error on average. The random error *δ**I*2(*R*) is the largest compared
to*δ*_{I2}^{sys}(*R*) , the systematic error*δ**I*2(*θ*) due to misalignment and due to finite turbulence
intensity *β**I*2(*I*) (compare Fig. 4.34).

0*.*0 0*.*2 0*.*4 0*.*6

−*δ*_{I2}^{sys}(*R*)
0*.*0

2*.*5
5*.*0
7*.*5
10*.*0

0*.*0 0*.*2 0*.*4 0*.*6

*δ**I*2(*R*)
0

2 4 6

### A B

**Figure 4.12** PDFs of the systematic error*δ*^{sys}_{I2} (*R*) (A) and the random error*δ**I*2(*R*) (B) of
the energy dissipation rate estimate ⟨*ϵ*_{I2}⟩*τ* during EUREC^{4}A on M161 aboard RV Meteor.

*R* =*U τ* is the spatial averaging window with *τ* = 1800 s. Both errors are estimated from
*L*11 and⟨*u*^{′2}_{1}⟩^{R} based on the same *R* and, hence, have to be regarded rather as statistical
estimates.

The turbulence features are calculated based on the mean energy dissipation
rate, which is obtained from a filtered time-record of the longitudinal velocity with a
cutoff frequency *f**c*= 0*.*5 Hz. As shown in Sec. 4.B, both the energy dissipation rate

⟨*ϵ**I*2⟩^{τ} as well as the variance⟨*u*^{′2}_{1}⟩^{τ} decrease with *f**c* (e.g. Fig. 4.32B). Hence, *L*11 is
underestimated if ⟨*u*^{′2}_{1}⟩^{τ} is evaluated from the filtered time-record of the longitudinal
velocity. In contrast,*L*11obtained from the unfiltered⟨*u*^{′2}_{1}⟩^{τ} should be rather interpreted
as an estimation of the upper bound on*L*11. Similarly, it is expected that R*λ*, Eq. (1.38),
is overestimated. Assuming that *σ**u*^{′}_{1} =^{q}⟨*u*^{′2}_{1}⟩*τ* from unfiltered *u*1(*t*) might be ∼30%

too high and the estimate of the mean energy dissipation rate ∼30% too low results
in an overestimation of *L*11 by a factor of 3 and of R*λ* by 2. As an example, a biased
estimate *L*11∼ 500 m would amount to*L*^{c}_{11} ≈160 m and R*λ* ∼20000 would amount
to R^{c}_{λ} ≈10000 where *c*denotes “correct”.

unit mean median std *e* [%]

*U* m*/*s 12.29 12.22 1.02 1.68

*σ**u*^{′}_{1} m*/*s 0.74 0.73 0.14 -

⟨*ϵ**I*2⟩^{τ} mW*/*kg 0.88 0.73 0.57 40.84

*η**K* mm 1.50 1.48 0.25 10.21

*λ**f* m 0.59 0.60 0.17 20.42

*L*11 m 462 411 333 40.84

R*λ* 16057 16336 6178 20.42

**Table 4.1** Bulk turbulence features obtained from mini-MPCK time-records of the longitu-
dinal velocity component *u*_{1}(*t*) on M161 (RV Meteor) during EUREC^{4}A .*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**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*_{11} the
integral length scale and R*λ* the Taylor micro-scale Reynolds number. For each turbulence
feature, the mean, median, standard deviation (std) and relative random error*e*in % are
given. *e* is obtained by Eqs. (3.45), (1.61), (4.3), (4.14), (4.11) and (4.12) for⟨*ϵ*_{I2}⟩^{τ}, *U*, R*λ*,
*L*_{11},*η*_{K} and *λ*_{f}, respectively. The averaging window is*τ* = 1800 s. *u*_{1}(*t*) is not corrected for
platform motion.

**4.3** **Reynolds stress Anisotropy**

The estimation of the mean energy dissipation rate according to Kolmogorov’s second
similarity hypothesis is based on the assumption of statistically (and locally) isotropic
turbulence. At least in the fit range (*r* ∈ [2 m*,*6 m]), the mean scaling exponents of
*D**LL*(*r*) are close to Kolmogorov’s prediction of *ζ*2 ≈ 0*.*69 in the inertial range for
isotropic turbulence. Although this agreement between observed and predicted scaling
exponents hints at isotropic turbulence in the inertial range, Kolmogorov’s prediction
of *ζ*2 ≈0*.*69 relies on the assumption of local isotropic turbulence conditioned on the
local energy dissipation rate in the first place [49]. Hence, an independent measure of
anisotropy is needed for confirming the inertial range isotropy, which would justify the
estimation of the mean energy dissipation rate as described above.