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, ϵI2 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 fc= 0.5 Hz where the effect is assessed in Sec. 4.B. In this section, error bars are given by Eq. (3.45) for ⟨ϵI2⟩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 zPSS8/TBL is captured by the standard deviation. Here, ⟨ϵI2⟩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 errorse in Tables 4.1 and 4.8 are given by Eqs. (3.45), (1.61), (3.42), (4.3) and (4.14) for⟨ϵI2⟩τ,U,σu′1, Rλ, andL11, respectively.
The averaged relative random errorse for ηK and λf are given by:
e(ηK) =
∂ηK
∂⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ
= ηK
4⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ = ηK
4 δI2(R), (4.11) e(λf) =
∂λf
∂⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ
= λf
2⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ = λf
2 δI2(R), (4.12) (4.13) invoking Gaussian error propagation with respect to the relative random error in⟨ϵI2⟩τ.
To assess the daily variability of the mean energy dissipation rate, an overview of the mean energy dissipation rate estimates ⟨ϵI2⟩τ is shown as a function of re-scaled barometric altitude zPSS8/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 ⟨ϵI2⟩τ ranges from 1×10−5W/kg to 1×10−3W/kg, which possibly contradicts the assumption of global stationarity on a weekly scale. To test this hypothesis, mean scaling exponents ζ2 of DτLL(r) within the fit range are classified by their deviation from the K62- prediction where a deviation of 0.1 is accepted. Thus, ζ2 is in agreement with K62 if
|ζ2−2/3 + 1/36| < 0.1 (group 1) corresponding to the highly opaque ⟨ϵI2⟩τ -values in Fig. 4.10. Otherwise, |ζ2 −2/3 + 1/36| ≥ 0.1 and ζ2 is not in agreement with
10
−510
−410
−310
−2h
I2i
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
#
hζ2i=0.69±0.01 hζ2i=0.57±0.02
Figure 4.10 Overview of mean energy dissipation rate estimates ⟨ϵI2⟩τ obtained from one- dimensional time-records of the longitudinal velocity during EUREC4A with the mini-MPCK on M161 (RV Meteor) as a function of the re-scaled barometric altitudezPSS8. 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 ⟨ϵI2⟩τ and by the re-scaled standard deviation of zPSS8 during the averaging windowτ. Highly opaque⟨ϵI2⟩τ -values are obtained from longitudinal second-order structure functions DτLL(r) with a mean scaling exponent ζ2, which satisfies
|ζ2−2/3 + 1/36|<0.1 (in agreement with K62) in the fit range. In contrast, weakly opaque
⟨ϵI2⟩τ-values stem fromDτLL(r) with |ζ2−2/3 + 1/36| ≥0.1. Furthermore,⟨ϵI2⟩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⟨ϵI2⟩τ -values agreeing with K62 in blue and the number counts of⟨ϵI2⟩τ-values disagreeing with K62 in (weakly opaque) red. The ensemble average of each ζ2-group is denoted by ⟨ζ2⟩.
K62 (group 2) where ⟨ϵI2⟩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, ϵI2 is considered to be a valid method to estimate the mean energy dissipation rate.
Focusing on time periods where ζ2 is in agreement with K62, ⟨ϵI2⟩τ ranges only from 4×10−4W/kg to 2×10−3W/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⟨ζ2⟩= 0.69±0.01 is agreeing with K62 in blue. The agreement between K62 and ⟨ζ2⟩ is remarkable. Likewise, the number counts of averaging windows τ during which the ensemble-averaged mean scaling exponent ⟨ζ2⟩= 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 |ζ2−2/3 + 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 ⟨ϵI2⟩τ does not reveal a significant trend with the local time of the day in Fig. 4.11A. Nevertheless, considering Flight 9 on M161, ⟨ϵI2⟩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 θv 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 ⟨ϵI2⟩τ is most likely related to the diurnal cycle. Similarly, the variance ⟨u′21⟩τ shows an increase with time for the same time frame (Fig. 4.11B). Both L11, Eq. (1.29), and Rλ, Eq. (1.38), are derived from ⟨ϵI2⟩τ and⟨u′21⟩τ and, therefore, reveal a similar behavior (Figs. 4.11C and D). The error bars for⟨u′21⟩τ and Rλ are given by Eqs. (3.42) and (4.3), respectively. The error of L11 due to δI2(R), the random error of ⟨ϵI2⟩τ, reads:
e(L11) =
∂L11
∂⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ
= L11
⟨ϵI2⟩τδI2(R)⟨ϵI2⟩τ =L11δI2(R). (4.14) Considering the error bars being derived from the random error δI2(R) , ⟨ϵI2⟩τ , ⟨u′21⟩τ, L11 and Rλ only, if at all, weakly depend on altitude. A general trend with altitude is not significant although, forL11 and Rλ, it seems that two branches
10−5 10−4 10−3 10−2 I2τ [W kg−1] 0.0
0.4 0.8 1.2 1.6
zPSS8/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 u21τ [m2s−2] 0.0
0.4 0.8 1.2 1.6
zPSS8/TBL
3
5 7
8 9
10
00:00 06:00 12:00 18:00 24:00
timeofday
0 300 600 900 1200 1500 L11 [m]
0.0 0.4 0.8 1.2 1.6
zPSS8/TBL
3 5
7 8
9 10
00:00 06:00 12:00 18:00 24:00
timeofday
0 10000 20000 30000 Rλ
0.0 0.4 0.8 1.2 1.6
zPSS8/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 windowτ = 1800 s. The error bars for zare given by the standard deviation ofz during the averaging windowτ. A: The mean energy dissipation rate estimates
⟨ϵI2⟩τ are obtained from one-dimensional time-records of the longitudinal velocity during EUREC4A with the mini-MPCK on M161 (RV Meteor). The error bars are given by Eq. (3.45) for⟨ϵI2⟩τ. B: The variance of velocity fluctuations⟨u′21⟩τ is based on unfiltered u′1. Error bars are only considered for the barometric altitude. C: The longitudinal integral length scaleL11 is estimated by Eq. (1.29). Error bars are obtained via Eq. (4.14). D: The Taylor micro-scale Reynolds number Rλ 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 zPSS8/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 forzPSS8/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 L11 and Rλ (Fig. 4.13).
The PDFs of the systematic and random errorsδI2sys(R) andδI2(R), respectively, is shown in Fig. 4.12. As both errors are inferred in retrospect from, i.a., the estimated L11, the scatter is expected. Therefore, both δsysI2 (R) andδI2(R) should be interpreted as rough estimates on the accuracy of ⟨ϵI2⟩τ where the systematic error is only half as large as the random error on average. The random error δI2(R) is the largest compared toδI2sys(R) , the systematic errorδI2(θ) due to misalignment and due to finite turbulence intensity βI2(I) (compare Fig. 4.34).
0.0 0.2 0.4 0.6
−δI2sys(R) 0.0
2.5 5.0 7.5 10.0
0.0 0.2 0.4 0.6
δI2(R) 0
2 4 6
A B
Figure 4.12 PDFs of the systematic errorδsysI2 (R) (A) and the random errorδI2(R) (B) of the energy dissipation rate estimate ⟨ϵI2⟩τ during EUREC4A on M161 aboard RV Meteor.
R =U τ is the spatial averaging window with τ = 1800 s. Both errors are estimated from L11 and⟨u′21⟩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 fc= 0.5 Hz. As shown in Sec. 4.B, both the energy dissipation rate
⟨ϵI2⟩τ as well as the variance⟨u′21⟩τ decrease with fc (e.g. Fig. 4.32B). Hence, L11 is underestimated if ⟨u′21⟩τ is evaluated from the filtered time-record of the longitudinal velocity. In contrast,L11obtained from the unfiltered⟨u′21⟩τ should be rather interpreted as an estimation of the upper bound onL11. Similarly, it is expected that Rλ, Eq. (1.38), is overestimated. Assuming that σu′1 =q⟨u′21⟩τ from unfiltered u1(t) might be ∼30%
too high and the estimate of the mean energy dissipation rate ∼30% too low results in an overestimation of L11 by a factor of 3 and of Rλ by 2. As an example, a biased estimate L11∼ 500 m would amount toLc11 ≈160 m and Rλ ∼20000 would amount to Rcλ ≈10000 where cdenotes “correct”.
unit mean median std e [%]
U m/s 12.29 12.22 1.02 1.68
σu′1 m/s 0.74 0.73 0.14 -
⟨ϵI2⟩τ mW/kg 0.88 0.73 0.57 40.84
ηK mm 1.50 1.48 0.25 10.21
λf m 0.59 0.60 0.17 20.42
L11 m 462 411 333 40.84
Rλ 16057 16336 6178 20.42
Table 4.1 Bulk turbulence features obtained from mini-MPCK time-records of the longitu- dinal velocity component u1(t) on M161 (RV Meteor) during EUREC4A .U =⟨u1(t)⟩τ is the mean velocity obtained from u1(t) for the averaging window τ, σu′
1 the RMS fluctuation ofu1(t),⟨ϵI2⟩τ the estimate of the mean energy dissipation rate according to Eq. (3.15) with n = 2, ηK the Kolmogorov length scale, λf the longitudinal Taylor micro-scale, L11 the integral length scale and Rλ the Taylor micro-scale Reynolds number. For each turbulence feature, the mean, median, standard deviation (std) and relative random errorein % are given. e is obtained by Eqs. (3.45), (1.61), (4.3), (4.14), (4.11) and (4.12) for⟨ϵI2⟩τ, U, Rλ, L11,ηK and λf, respectively. The averaging window isτ = 1800 s. u1(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 DLL(r) are close to Kolmogorov’s prediction of ζ2 ≈ 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 ζ2 ≈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.
