Appendix 3.B Supplementary Figures
4.2 Atmospheric Turbulence Characteristics of the Marine Boundary Layer 126
4.3.2 Anisotropy in high-R λ flows
− 0.2 0.0 0.2 0.4 ξ
0.0 0.2 0.4
2C - disk oblate
prolate 2C - ellipse
Figure 4.14 Graphical representation of special turbulence states in terms of the Lumley triangle in the η−ξ plane. Figure based on .
−0.2 0.0 0.2 0.4 ξ
0.0 0.1 0.2 0.3 0.4
M161 MSM89 MSM89 line DNS
−0.2 0.0 0.2 0.4 ξ
0.0 0.1 0.2 0.3 0.4
M161 MSM89 MSM89 line DNS
Figure 4.15 Lumley triangle in terms of the invariants ξ, Eq. (4.17), and η, Eq. (4.18), obtained from EUREC4A field measurements for each mini-MPCK flight and from DNS of homogeneous isotropic turbulence. The velocity wind vector is measuredin situ as a function of time by the mini-MPCK with the help of the sonic during EUREC4A field campaign.
Invariants obtained from measurements on M161 aboard RV Meteor are shown in blue, on MSM89 aboard RV Maria S. Merian in red and from single snapshots of DNS 3.1 – 3.5 of homogeneous isotropic turbulence, Table 3.2, by gray stars. Circles denote flights with a tether-mounted mini-MPCK with a separation of at least 50 m to the helikite (“M161” and
“MSM89 line”). Upright triangles represent flights where the mini-MPCK is mounted to the main spare on MSM89 (“MSM89”). Gray circles and gray lines represent special states of turbulence according to Table 4.2. A: Time-records of the wind vector are not filtered. B:
Time-records of the wind vector are filtered atfc= 0.5 Hz. The inset shows a zoom close to the origin.
obtained from flights in the tether-mounted configuration appear to be more isotropic than flights in the spare-mounted configuration. Some of the invariants obtained from filtered time-records of the three-dimensional wind vector during M161 are even close to DNS as demonstrated in the inset of Fig. 4.15B. Hence, in the platform frame of reference, the turbulence appears to be globally isotropic on scales l≲20 m. This is in agreement with the ensemble-averaged scaling exponent, e.g. ⟨ζ2⟩ = 0.74±0.06 for M161 Flight 10, overlapping with the K62-prediction. Already, it can be concluded that the estimation of the mean energy dissipation rate according to Kolmogorov’s phenomenology is justified for scales l≲20 m.
To evaluate how well the isotropy assumption of Kolmogorov’s phenomenology is fulfilled for shorter averaging windows, the Lumley triangle has been also visualized for finite averaging windows of τ = 30 min. In contrast to bulk anisotropy in Fig. 4.15, invariants obtained from mini-MPCK wind vector time-records for averaging windows of 30 min are compared to each virtual probe in DNS 3.3 as shown in Fig. 4.16A. On the one hand, reducing the averaging window enhances the statistical scatter as argued, e.g., in Sec. 4.1.2. On the other hand, sampling a three-dimensional turbulent flow along a one-dimensional trajectory might affect the invariants, too. Presumably, the combination of finite averaging windows and one-dimensional sampling causes the scatter of the invariants in the regime 0 ≤ η ≤ 0.15 and −0.15 ≤ ξ ≤ 0.2 for both DNS 3.3 and mini-MPCK. Hence, there is no significant difference between invariants obtained from virtual probes sampling DNS of homogeneous isotropic turbulence along
Figure 4.16 Lumley triangle for individual averaging windows τ = 30 min obtained from tether-mounted mini-MPCK time-records of the wind vector. A: Invariants obtained from measurements with the mini-MPCK for averaging windows τ are compared to each virtual probe of DNS 3.3 (Table 3.2). B: Invariants from airborne measurements with the mini-MPCK are compared to ground-based measurements in mountainous terrain (personal communication with Steffen Risius) where each point is colored by Rλ. C: Invariants obtained from the mini-MPCK wind vector measurements are shown as a function of barometric altitude zPSS8
above MSL re-scaled by the top of the boundary layer (TBL). D: In order to characterize the effect of static stability on the invariants, they are colored as a function of the Richardson number Ri.
a one-dimensional trajectory and mini-MPCK measurements.
Furthermore, invariants from airborne measurements in the marine boundary layer with the mini-MPCK are compared to ground-based measurements in mountainous terrain on Mt. Zugspitze [54, 197] as shown in Fig. 4.16B. The invariants obtained from ground-based measurements (upright triangles) tend to be more anisotropic than the invariants obtained from filtered airborne measurements in the marine boundary layer. On the one hand, this might be an effect of Rλ which is higher by at least a factor of 2 for measurements in the marine boundary layer compared to ground-based measurements in mountainous terrain. On the other hand, UFS measurements are conducted within the surface layer in the presence of shear . Another possible cause for this deviation is the filtering in the case of mini-MPCK measurements.
In contrast to UFS measurements in the surface layer, the mini-MPCK sampled the boundary layer in the well-mixed region and in the cloud layer as shown in Fig. 4.16C.
The invariants obtained by the mini-MPCK tend to isotropy in the well-mixed region of the marine boundary layer, i.e. where the virtual potential temperature is constant with height  as fulfilled for 0.1 ≤ zPSS8/TBL ≤ 0.8, while the flow appears to be comparable to ground-based measurements for 0.8 ≤ zPSS8/TBL. Taking into account the inaccuracy of ∼ 100 m in determining the depth of the boundary layer (TBL), the regime 0.8≲zPSS8/TBL is close to an inversion which imposes a large-scale anisotropy on the flow. The influence of decreasing line tension on the invariants remains undetermined because of the absence of a tensiometer on RV Meteor.
At last, anisotropy is investigated with respect to static stability being captured by the Richardson number Ri, Eq. (4.10). Here, the invariants are determined for the same averaging windows with τ = 30 min as Ri. As the flow is considered to be statically stable for Ri > 1, it is expected that the flow tends to be more anisotropic than for Ri<1 where the flow is expected to be fully turbulent, which can be confirmed as shown in Fig. 4.16D. To summarize, the isotropy assumption of Kolmogorov’s phenomenology can be considered as being fulfilled for scales l ∼ 20 m and in the platform frame of reference as there is no significant difference to velocity measurements of virtual probes in DNS of homogeneous isotropic turbulence. Thus, the estimation of the mean energy dissipation rate based on averaging windows with τ = 30 min is justified.