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 EUREC4A data. That is why a cutoff frequency fc = 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 σu′1 and the energy dissipation rate ϵI2.
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 Rλ 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 Rλ being only ∼300.
In all cases, the energy dissipation rate is estimated by ϵI2 with a constant fit range across the entire range of applied cutoff frequencies (case (I): r ∈[20ηK,200ηK], case (II): r∈[2 m,4 m], case (III): r∈[2 m,4 m]).
All three cases show a strong dependency of bothσu′1 andϵI2 on the cutoff frequency as shown in Fig. 4.31 where σu′1(fc) and ϵI2(fc) are given relative to the case with the lowest fc ≳ 0 and referred to as “0”. Both σu′1 and ϵI2 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 σu′1(fc) and ϵI2(fc) decay with fc
only indicating a plateau for small fc. Considering cases (II) and (III), σu′1 decreases moderately for fc < 0.1 Hz and decays much faster between 0.1 Hz < fc <0.2 Hz as shown in Figs. 4.31C and E. In contrast, ϵI2 shows a longer plateau in Figs. 4.31D and F, where ϵI2 only minimally changes up tofc∼0.1 Hz. Assuming that L11 100 m and U 10 m/s in the atmosphere, fc ∼ 0.1 Hz corresponds to a cutoff length scale lc ∼L11. Hence, cutoff frequencies corresponding to length scales larger than L11 are only slightly affected by filtering. This is supported by derivatives dϵI2/dfc ≈0 (red lines in Figs. 4.31B, D, and F).
However, fc= 0.5 Hz corresponds to a length scale lc∼20 m with L11 100 m and U 10 m/s in the atmospheric boundary layer. In contrast to the plateau region in Figs. 4.31D and F, ϵI2 decays between 0.1 Hz < fc ≲ 0.5 Hz and the rate of change dϵI2/dfc is non-zero. This suggests that fc= 0.5 Hz is in a regime that strongly varies withfc, which is unfavorable. At fc = 0.5 Hz, the underestimation of ϵI2(fc) is 40% in case (II) compared to ϵI2(0). In cases (II) and (III),σu′1(fc) is underestimated by 70 - 80% in comparison to σu′1(0). However, taking into account that the referencesσu′1(0) and ϵI2(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 offc= 0.5 Hz in (II) and (III) has to be re-scaled. In code units (c.u.) of DNS where L11 ≈1 c.u., fc is 50 c.u. At fc = 50 c.u., ϵI2 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 fc of cases (II) and (III) to ηK of DNS. Therefore, case (I) cannot be invoked to justify or to falsify the choice of fc. Case (I) rather supports the dependence of σu′1
and ϵI2 onfc 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⟩= 2ν lim
κc→0
Z ∞ κc
dκκ2E(κ), (4.20)
100 101 102 fc[c.u.]
0.00 0.25 0.50 0.75 1.00
σu 1(fc)/σu 1(0)
10−2 10−1 100 101 fc[Hz]
0.00 0.25 0.50 0.75 1.00
σu 1(fc)/σu 1(0)
10−2 10−1 100 101 fc[Hz]
0.00 0.25 0.50 0.75 1.00
σu 1(fc)/σu 1(0)
10−2 10−1 100 101 fc[Hz]
0.00 0.25 0.50 0.75 1.00
I2(fc)/I2(0)
−0.010
−0.005 0.000
dI2/dfc
10−2 10−1 100 101 fc[Hz]
0.00 0.25 0.50 0.75 1.00
I2(fc)/I2(0)
−0.02
−0.01 0.00
dI2/dfc
100 101 102 fc[c.u.]
0.0 0.5 1.0
I2(fc)/I2(0)
−0.04
−0.02 0.00
dI2/dfc
A
F E
D C
B
Figure 4.31Effect of filtering on the RMS velocity fluctuationsσu′
1 and the energy dissipation rate ϵI2 as a function of the cutoff frequency fc. (A, B) Virtual probes sample DNS of homogeneous isotropic turbulence at random directions but constant speed for a turbulence intensity of ≈10% and Rλ ≈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 m3 helikite. This configuration is similar to a weather vane. DLL(r) is fitted for r ∈ [2 m,6 m]. (E, F) The mini-MPCK is always attached to the main tether of a 75 m3 helikite. The helikite behaves like a weather vane driving the platform motion of the tether-mounted mini-MPCK. DLL(r) is fitted forr∈[2 m,6 m].
10−6 10−4 10−2 100 κηK
10−6 100 106 1012
E(κ)/(ηKu2 K)
∝κ−5/3
10−2 10−1 100 101 κcηK
0.00 0.25 0.50 0.75
1.00 (κc)/(0)
fc= 0.5 Hz k(κc)/k(0)
A B
Figure 4.32 The effect of filtering using a model spectrum. A: The model spectrum, Eq. (4.22), follows withCK = 1.5,p0 = 2,cL= 6.78, β= 2.1,L∼200 m,⟨ϵ⟩ ∼0.001 W/kg, U ∼ 10 m/s and ν = 1.552×10−5m2/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 fc= 0.5 Hz withκc= 2πU/fc.
where ν is the kinematic viscosity, κ = 2π/l, l a length scale, κc = 2πU/fc and fc a cutoff frequency. Similarly, the turbulent kinetic energy k can be obtained by [5]
k = lim
κc→0
Z ∞ κc
dκE(κ). (4.21)
In homogeneous isotropic turbulence, k = 3/2⟨u′21⟩= 3/2σu2′
1. A model spectrum of homogeneous isotropic turbulence with an exponential decay [203] is defined in [5, Eq.
6.246 ff.]:
E(κ) = CK⟨ϵ⟩2/3κ−5/3 κL [(κL)2 +cL]1/2
!5/3+p0
exp(−βκηK), (4.22) where CK = 1.5 the Kolmogorov constant, p0 = 2 and cL = 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 = 1.552×10−5m2/s. This set of parameters corresponds to Rλ = 2031/2ηL
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 κc, i.e. neglecting the limit κc→0. Similarly to case (I) - (III), ⟨ϵ⟩(κc) decays withκc as shown in Fig. 4.32B.⟨ϵ⟩(2πU/fc) with fc= 0.5 Hz is underestimated by ∼20%. As σu′1 =q⟨u′21⟩=q2k/3 is dominated by large scales, the effect of σu′
1 is larger. Regarding the model spectrum in Fig. 4.32, k(2πU/fc) is underestimated by 95% which corresponds to underestimating σu′1(2πU/fc) by about 78% in case of homogeneous isotropic turbulence atfc= 0.5 Hz.
To summarize, the effect of filtering on ϵI2 is most likely between 20% and 40%
whereas the effect on σu′1 is about 80% at large Rλ. Note that Rλ ∝ηLK4/3 implying that the separation of scales grows with Rλ. As k is mostly stored in large scale and
⟨ϵ⟩ mostly stored in small scales, k decays much faster with κc as shown in Fig. 4.32B.
As Rλ is very high in the atmosphere, small Rλ are not further discussed and the interested reader is referred to [5].