The Max Planck Cloudkites record the velocity in the platform frame of reference.

Similar to wind velocity measurements on aircraft, the measured wind vector has to be transformed to the Earth frame of reference. This procedure is known for wind velocity measurements, e.g. on aircraft [201] or tethered balloons [98]. However, the helikite motions are driven by the turbulent flow across a wide range of scales (Sec. 2.4.2), which has been observed also by Egerer et al. deploying a helium-filled balloon [96].

Further corrections with respect to misalignment can be applied to the wind velocity
data in the Earth frame of reference, by assuming ⟨*u*^{′}_{3}⟩^{τ} = 0 [202]. Corrections for
platform motion require that the measurements of orientation, rotation rates and
acceleration are not drifting, noisy and synchronized in time with wind measurement
data. In principle, the SBG Ellipse-N is able to provide such a data set. However, due
to non-ideal configuration, mediocre GNSS reception and signal noise, it is likely that
these corrections will not succeed on EUREC^{4}A data. That is why a cutoff frequency
*f**c* = 0*.*5 Hz is chosen in order to suppress the influence of platform motion on the
turbulence analysis. Here, I want to discuss the effect of filtering on the RMS velocity
*σ**u*^{′}_{1} and the energy dissipation rate *ϵ**I*2.

The effect of filtering is illustrated by three different cases: (I) the uniformly
moving virtual probes in DNS (Fig. 4.31A and B), (II) the keel-strapped micro-MPCK
(Fig. 4.31C and D) and (III) the tether-mounted mini-MPCK on M161 Flight 10
(Fig. 4.31E and F). In case (I), DNS 3.3 is chosen due to the highest R*λ* and a turbulence

intensity of *I* = 10%, which is close to atmospheric observations. Furthermore,
uniformly moving probes in DNS 3.3 (Table 3.2) are taken because “weather vane”-like
probes measure a similar mean energy dissipation rate for *I* = 10%. However, the
range of cutoff frequency is smaller compared to cases (II) and (III) because of the
smaller separation of large and small scales as a consequence of R*λ* being only ∼300.

In all cases, the energy dissipation rate is estimated by *ϵ**I*2 with a constant fit range
across the entire range of applied cutoff frequencies (case (I): *r* ∈[20*η**K**,*200*η**K*], case
(II): *r*∈[2 m*,*4 m], case (III): *r*∈[2 m*,*4 m]).

All three cases show a strong dependency of both*σ**u*^{′}_{1} and*ϵ**I*2 on the cutoff frequency
as shown in Fig. 4.31 where *σ**u*^{′}_{1}(*f**c*) and *ϵ**I*2(*f**c*) are given relative to the case with
the lowest *f**c* ≳ 0 and referred to as “0”. Both *σ**u*^{′}_{1} and *ϵ**I*2 are ensemble averages
over all virtual probes in case (I) and over all averaging windows ⟩^{τ} = 1800 s in
case (II) and (III). Starting with case (I), both *σ**u*^{′}_{1}(*f**c*) and *ϵ**I*2(*f**c*) decay with *f**c*

only indicating a plateau for small *f**c*. Considering cases (II) and (III), *σ**u*^{′}_{1} decreases
moderately for *f**c* *<* 0*.*1 Hz and decays much faster between 0*.*1 Hz *< f**c* *<*0*.*2 Hz as
shown in Figs. 4.31C and E. In contrast, *ϵ**I*2 shows a longer plateau in Figs. 4.31D
and F, where *ϵ**I*2 only minimally changes up to*f**c*∼0*.*1 Hz. Assuming that *L*11 100 m
and *U* 10 m*/*s in the atmosphere, *f**c* ∼ 0*.*1 Hz corresponds to a cutoff length scale
*l**c* ∼*L*11. Hence, cutoff frequencies corresponding to length scales larger than *L*11 are
only slightly affected by filtering. This is supported by derivatives d*ϵ**I*2*/*d*f**c* ≈0 (red
lines in Figs. 4.31B, D, and F).

However, *f**c*= 0*.*5 Hz corresponds to a length scale *l**c*∼20 m with *L*11 100 m and
*U* 10 m*/*s in the atmospheric boundary layer. In contrast to the plateau region in
Figs. 4.31D and F, *ϵ**I*2 decays between 0*.*1 Hz *< f**c* ≲ 0*.*5 Hz and the rate of change
d*ϵ**I*2*/*d*f**c* is non-zero. This suggests that *f**c*= 0*.*5 Hz is in a regime that strongly varies
with*f**c*, which is unfavorable. At *f**c* = 0*.*5 Hz, the underestimation of *ϵ**I*2(*f**c*) is 40% in
case (II) compared to *ϵ**I*2(0). In cases (II) and (III),*σ**u*^{′}_{1}(*f**c*) is underestimated by 70 -
80% in comparison to *σ**u*^{′}_{1}(0). However, taking into account that the references*σ**u*^{′}_{1}(0)
and *ϵ**I*2(0) are also affected by platform motions for cases (II) and (III), the relative
comparison is not ideal.

To compare case (I) with cases (II) and (III), the cutoff frequency of*f**c*= 0*.*5 Hz in
(II) and (III) has to be re-scaled. In code units (c.u.) of DNS where *L*11 ≈1 c.u., *f**c* is
50 c.u. At *f**c* = 50 c.u., *ϵ**I*2 nearly vanishes. As mentioned above, the separation of
scales in DNS 3.3 is much lower than in the atmosphere which makes it impossible to
relate *f**c* of cases (II) and (III) to *η**K* of DNS. Therefore, case (I) cannot be invoked to
justify or to falsify the choice of *f**c*. Case (I) rather supports the dependence of *σ**u*^{′}_{1}

and *ϵ**I*2 on*f**c* in general.

A complementary approach for investigating the effect of filtering involves the fact
that the mean energy dissipation rate⟨*ϵ*⟩can be obtained by integrating a pre-multiplied
energy spectrum *E*(*κ*) [5]

⟨*ϵ*⟩= 2*ν* lim

*κ**c*→0

Z ∞
*κ**c*

d*κκ*^{2}*E*(*κ*)*,* (4.20)

10^{0} 10^{1} 10^{2}
*f**c*[c.u.]

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

*σ**u* 1(*f**c*)*/σ**u* 1(0)

10^{−2} 10^{−1} 10^{0} 10^{1}
*f**c*[Hz]

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

*σ**u* 1(*f**c*)*/σ**u* 1(0)

10^{−2} 10^{−1} 10^{0} 10^{1}
*f**c*[Hz]

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

*σ**u* 1(*f**c*)*/σ**u* 1(0)

10^{−2} 10^{−1} 10^{0} 10^{1}
*f**c*[Hz]

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

*I*2(*f**c*)*/**I*2(0)

−0*.*010

−0*.*005
0*.*000

*d**I*2*/df**c*

10^{−2} 10^{−1} 10^{0} 10^{1}
*f**c*[Hz]

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

*I*2(*f**c*)*/**I*2(0)

−0*.*02

−0*.*01
0*.*00

*d**I*2*/df**c*

10^{0} 10^{1} 10^{2}
*f**c*[c.u.]

0*.*0
0*.*5
1*.*0

*I*2(*f**c*)*/**I*2(0)

−0*.*04

−0*.*02
0*.*00

*d**I*2*/df**c*

### A

### F E

### D C

### B

**Figure 4.31**Effect of filtering on the RMS velocity fluctuations*σ*_{u}^{′}

1 and the energy dissipation
rate *ϵ**I*2 as a function of the cutoff frequency *f**c*. (A, B) Virtual probes sample DNS of
homogeneous isotropic turbulence at random directions but constant speed for a turbulence
intensity of ≈10% and R*λ* ≈295. Hence, the dynamics of virtual probes are statistically
uncorrelated with the flow. (C, D) The micro-MPCK is mounted directly to the keel of
the 34 m^{3} helikite. This configuration is similar to a weather vane. *D*_{LL}(*r*) is fitted for
*r* ∈ [2 m*,*6 m]. (E, F) The mini-MPCK is always attached to the main tether of a 75 m^{3}
helikite. The helikite behaves like a weather vane driving the platform motion of the
tether-mounted mini-MPCK. *D*_{LL}(*r*) is fitted for*r*∈[2 m*,*6 m].

10^{−}^{6} 10^{−}^{4} 10^{−}^{2} 10^{0}
*κη**K*

10^{−6}
10^{0}
10^{6}
10^{12}

*E*(*κ*)*/*(*η**K**u*2 *K*)

∝*κ*^{−5/3}

10^{−}^{2} 10^{−}^{1} 10^{0} 10^{1}
*κ**c**η**K*

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

1*.*00 _{}(*κ**c*)*/*(0)

*f**c*= 0*.*5 Hz
*k*(*κ**c*)*/k*(0)

### A B

**Figure 4.32** The effect of filtering using a model spectrum. A: The model spectrum,
Eq. (4.22), follows with*C**K* = 1*.*5,*p*0 = 2,*c**L*= 6*.*78, *β*= 2*.*1,*L*∼200 m,⟨*ϵ*⟩ ∼0*.*001 W*/*kg,
*U* ∼ 10 m*/*s and *ν* = 1*.*552×10^{−5}m^{2}*/*s. B: The effect of filtering on ⟨*ϵ*⟩ is modeled by
introducing a finite integration boundary in Eq. (4.20). The dashed line corresponds to
*f**c*= 0*.*5 Hz with*κ**c*= 2*πU/f**c*.

where *ν* is the kinematic viscosity, *κ* = 2*π/l*, *l* a length scale, *κ**c* = 2*πU/f**c* and *f**c* a
cutoff frequency. Similarly, the turbulent kinetic energy *k* can be obtained by [5]

*k* = lim

*κ**c*→0

Z ∞
*κ**c*

d*κE*(*κ*)*.* (4.21)

In homogeneous isotropic turbulence, *k* = 3*/*2⟨*u*^{′2}_{1}⟩= 3*/*2*σ*_{u}^{2}^{′}

1. A model spectrum of homogeneous isotropic turbulence with an exponential decay [203] is defined in [5, Eq.

6.246 ff.]:

*E*(*κ*) = *C**K*⟨*ϵ*⟩^{2/3}*κ*^{−5/3} *κL*
[(*κL*)^{2} +*c**L*]^{1/2}

!5*/*3+*p*0

exp(−*βκη**K*)*,* (4.22)
where *C**K* = 1*.*5 the Kolmogorov constant, *p*0 = 2 and *c**L* = 6*.*78 and *β* = 2*.*1 are
positive constants. Here, *L* ∼200 m is the energy injection scale, ⟨*ϵ*⟩ ∼ 0*.*001 W*/*kg,
*U* ∼ 10 m*/*s and *ν* = 1*.*552×10^{−5}m^{2}*/*s. This set of parameters corresponds to
R*λ* = ^{}^{20}_{3}^{}^{1/2}^{}_{η}^{L}

*K*

4*/*3

≈ 7100. The non-dimensional model spectrum is shown in
Fig. 4.32A. The influence of filtering on the estimation of ⟨*ϵ*⟩ can be modeled by
integrating Eq. (4.20) for various *κ**c*, i.e. neglecting the limit *κ**c*→0. Similarly to case
(I) - (III), ⟨*ϵ*⟩(*κ**c*) decays with*κ**c* as shown in Fig. 4.32B.⟨*ϵ*⟩(2*πU/f**c*) with *f**c*= 0*.*5 Hz
is underestimated by ∼20%. As *σ**u*^{′}_{1} =^{q}⟨*u*^{′2}_{1}⟩=^{q}2*k/*3 is dominated by large scales,
the effect of *σ*_{u}^{′}

1 is larger. Regarding the model spectrum in Fig. 4.32, *k*(2*πU/f**c*) is
underestimated by 95% which corresponds to underestimating *σ**u*^{′}_{1}(2*πU/f**c*) by about
78% in case of homogeneous isotropic turbulence at*f**c*= 0*.*5 Hz.

To summarize, the effect of filtering on *ϵ**I*2 is most likely between 20% and 40%

whereas the effect on *σ**u*^{′}_{1} is about 80% at large R*λ*. Note that R*λ* ∝^{}_{η}^{L}_{K}^{}^{4/3} implying
that the separation of scales grows with R*λ*. As *k* is mostly stored in large scale and

⟨*ϵ*⟩ mostly stored in small scales, *k* decays much faster with *κ**c* as shown in Fig. 4.32B.

As R*λ* is very high in the atmosphere, small R*λ* are not further discussed and the
interested reader is referred to [5].