**Appendix 3.B Supplementary Figures**

**4.1.2 Statistical Convergence in atmospheric turbulence**

Characterizing a stationary turbulent flow requires statistical convergence of at least
the mean and variance estimates. As mentioned in Sec. 1.2.6, I define ⟨*u*^{′2}_{1}⟩^{τ} to be
converged if the systematic error, Eq. (1.63), is much smaller than the random error

q2⟨*u*^{′2}_{1}⟩^{2}*T*11*/τ* being defined in [56] for*τ* ≫*T*11. Both systematic and random errors
decrease monotonically with *τ*. Thus, it is favorable to choose the maximally available
averaging window*τ*. To illustrate the effect of the averaging window*τ*,*u*1(*t*) is assumed
to be stationary in time. Later, this assumption will be checked as well.

Feb 15 18:48 Feb 16 03:37 Feb 16 12:25 UTC time

298 300 302

*T*[K]

Feb 15 18:48 Feb 16 03:37 Feb 16 12:25 UTC time

0 24 48 72 96 120

*z*PSS8(*t*)[m]

### A B

Feb 15 18:48 Feb 16 03:37 Feb 16 12:25 UTC time

0 3 6 9 12 15

*u*1(*t*)[m/s]

### C

0 24 48 72 96 120

*z*_{PSS8}[m]

0 2 5 7 10 12

*u*1(*t*)*z*[m/s]

### D

**Figure 4.1** Overview of M161 Flight 12 on RV Meteor over night from February 15 2020 to
February 16 2020 during EUREC^{4}A field campaign. Barometric altitude time-record*z*_{PSS8}(*t*)
of the micro-MPCK flown at approximately constant altitude above MSL (A). Time-records
of air temperature*T* (B) and relative wind speed*u*1(*t*) (C) are shown for the entire flight. D:

The mean wind speed averaged over altitude increments of 5 m saturates for altitudes higher than≈40 m above MSL.

At first, I investigate the dependence of variance estimates from the longitudinal
velocity time record ⟨*u*^{′2}_{1}⟩^{τ} on the temporal averaging window size *τ*. Therefore, ⟨*u*^{′2}_{1}⟩^{τ}
is evaluated for each *τ* for the entire time series following the procedure in [56] where
each averaging window *τ* corresponds to one realization of the flow. Then, ⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N* is
the ensemble average of⟨*u*^{′2}_{1}⟩^{τ} as shown in Fig. 4.2A. The shaded region is given by the
standard error of the mean in order to capture statistical scatter. The large deviation
between ⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N* and the predicted variance estimate, Eq. (1.61) with *τ* ≲ 5*T*11

(*T*11 = 46 s), is expected. In the range of 40*T*11 ≤*τ* ≤70*T*11, ⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N* overlaps with
the predicted variance estimate, Eq. (1.61) with *τ* = 65*T*11, within the standard error.

In Fig. 4.2A, *τ* = 50*T*11 is in the center of the overlap region between the empirical
and theoretical curve. For *τ >*80*T*11, ⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N* deviates strongly from the predicted
variance estimate, Eq. (1.61), suggesting non-stationarity. This is in accordance with
the time scale over which the air temperature changes (M161 Flight 12: ∼1 h due to
diurnal cycle, Fig. 4.1B). Hence, *τ* ∼1 h poses an upper limit on reasonable averaging
windows although it has to be emphasized that this is a time *scale*, which itself is
influenced by environmental conditions unlike in laboratory experiments or simulations.

To focus on time scales below 1 h, i.e. the time scale of non-stationarity, the
longitudinal velocity time record is divided into segments of 3600 s≈78*T*11 where the
ensemble consists of S segments. In each segment, ⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N* is evaluated similar to the
previous procedure. Consequently, ⟨⟨⟨*u*^{′2}_{1}⟩*τ*⟩*N*⟩_{S} is the ensemble average of ⟨⟨*u*^{′2}_{1}⟩*T*⟩*N*

over all segments *S*. Then, the predicted variance estimate, Eq. (1.61) with *τ* = 65*T*11,
is within the standard error of ⟨⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N*⟩*S* in the range 10*T*11≤*τ* ≤78*T*11 (Fig. 4.2B).

In Fig. 4.2A, *τ* = 39*T*11 is at the beginning of the overlap region while *τ* = 50*T*11 is
in the center of the overlap region between the empirical and theoretical curve, as
mentioned before. For *τ* = 1800 s ≈ 39*T*11, ⟨⟨⟨*u*^{′2}_{1}⟩^{τ}⟩*N*⟩*S* is underestimated by ≈ 5%

compared to ⟨*u*^{′2}_{1}⟩^{65T}11. Using ^{q}2⟨*u*^{′2}_{1}⟩^{2}*T*11*/τ* for the random error of the variance
estimate [56], the random error of ⟨*u*^{′2}_{1}⟩^{τ} is 18% which is significantly larger than the
systematic error of the variance estimate. Hence, ⟨*u*^{′2}_{1}⟩^{τ} is reasonably converged for
*τ* = 1800 s in the limit of accuracy. To resolve turbulence characteristics in time, I
choose the averaging window *τ* = 39*T*11, which is the shortest reasonable averaging
window possible.

0 25 50 75 100 125

*τ /T*_{11}
0*.*00

0*.*25
0*.*50
0*.*75
1*.*00

*u*

2 12−2 [ms]*τ**N*

*u*^{}_{1}^{2}^{τ}*N*

*u*^{2}_{1}^{65T}11(1−2*T*11*/τ*)
*u*^{}_{1}^{2}^{65T}11

### A

0 20 40 60 80

*τ /T*_{11}
0*.*00

0*.*25
0*.*50
0*.*75
1*.*00

*u*

2 12−2 [ms]*τ**N**S*

*u*^{}_{1}^{2}*τ**N**S*

*u*^{2}_{1}^{65T}11(1−2*T*11*/τ*)
*u*^{}_{1}^{2}^{65T}11

### B

**Figure 4.2** Variance estimates of longitudinal velocity time record ⟨*u*^{′2}_{1}⟩*τ* as a function
of the temporal averaging window size *τ* re-scaled by the integral time scale *T*_{11} = 46 s.

⟨*u*^{′2}_{1}⟩*τ*

*N* represents the ensemble average based on the averaging window*τ*. The shaded
blue region is given by the standard error of the mean. The systematic underestimation of
the variance estimate is shown by the solid black line and given by Eqs. (1.63) and (1.61).

A: *u*^{′}_{1}(*t*) is assumed to be stationary. *τ* = 39*T*11 is at the beginning of the overlap region
while*τ* = 50*T*11 is in the center of the overlap region between the empirical and theoretical
curve. B: Taking into account the non-stationarity, the longitudinal velocity time record
is divided into segments of *S* = 3600 s. In each segment, ^{}⟨*u*^{′2}_{1}⟩*τ*

*N* is evaluated for each*τ*.
⟨*u*^{′2}_{1}⟩^{τ}^{}_{N}^{}_{S} is the average of^{}⟨*u*^{′2}_{1}⟩*T*

*N* over all segments*S*. The black dashed line illustrates

⟨*u*^{′2}_{1}⟩65*T*11, i.e. the variance estimate based on an averaging window of 65*T*11. The theoretical
expectation, Eq. 1.63, is within the standard error of the variance estimate with an averaging
window*τ* = 39*T*11denoted by⟨*u*^{′2}_{1}(*t*)⟩39*T*11. In the following, the shortest reasonable averaging
window is chosen to*τ* = 39*T*11 in order to resolve turbulence characteristics in time.

However, atmospheric flows are generally non-stationary due to, i.a., the diurnal
cycle. Above, we assumed *u*1(*t*) to be a statistically stationary time-record of the
longitudinal velocity. According to [5], a time series of a random process is statistically

stationary if all statistics are independent of a shift in time. Here, a less strict criterion
has to be applied because the mean velocity of the flow is not accurately known due
to time-dependent platform motions as a result of ship motions. Therefore, I regard
*u*1(*t*) as*statistically stationary* if changes of⟨*u*^{′2}_{1}(*t*)⟩*τ* under time shifts of the averaging
window *τ* are smaller than the random error of the variance estimate ^{q}2⟨*u*^{′2}_{1}⟩^{2}*T*11*/τ*
[56]. In the following, I will refer to ⟨*u*^{′2}_{1}⟩^{65T}11 as the reference for estimating errors
because *τ* = 65*T*11 is significantly shorter than *τ* = 78*T*11 and still agrees with the
prediction, Eq. (1.61), as shown in Fig. 4.2A. In doing so, the random error ⟨*u*^{′2}_{1}⟩^{τ} is
18% for *τ* = 1800 s and 32% for *τ* = 600 s. The systematic errors are 4% and 15%,
respectively, in comparison to ⟨*u*^{′2}_{1}⟩65*T*11.

Figure 4.3 shows ⟨*u*^{′2}_{1}(*t*)⟩^{τ} as a function of time indicating that ⟨*u*^{′2}_{1}(*t*)⟩^{τ} fluctuates
significantly. This might hint at the fact that the stationarity assumption is not
fulfilled globally. As an example, ⟨*u*^{′2}_{1}(*t*)⟩*τ* drops by 60% on February 16 2020 at 04:30
UTC (dotted line in Fig. 4.3), which is larger than the random error of the variance
estimate ^{q}2⟨*u*^{′2}_{1}⟩^{2}*T*11*/τ* for both *τ* = 600 s and *τ* = 1800 s. However, there are local
time intervals (e.g. February 15 2020 at 23:45 UTC to February 16 2020 at 01:25
UTC, the gray shaded region in Fig. 4.3) over which*u*1(*t*) appears to be approximately
statistically stationary. In that time interval,⟨*u*^{′2}_{1}(*t*)⟩1800 sis constant within 12%, which
is smaller than the random error of ≈ 18%. Furthermore, ⟨*u*^{′2}_{1}(*t*)⟩^{τ} fluctuates about
17% for *τ* = 600 s, which is lower than the random error, too. Remarkably, ⟨*u*^{′2}_{1}(*t*)⟩^{600 s}
tends to be lower than ⟨*u*^{′2}_{1}(*t*)⟩1800 s, which is due to the underestimated variance as
illustrated in Fig. 4.2. To summarize, the choice of an averaging window *τ* = 1800 s
compromises between statistical convergence and still being temporally shorter than
characteristic time scales of non-stationarities of ∼1 h. This is further consistent with
Risius et al. [54] and Stull [40] as mentioned in Sec. 1.2.6.