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^′2₁⟩^τ to be converged if the systematic error, Eq. (1.63), is much smaller than the random error

q2⟨u^′2₁⟩²T11/τ being defined in [56] forτ ≫T11. Both systematic and random errors decrease monotonically with τ. Thus, it is favorable to choose the maximally available averaging windowτ. To illustrate the effect of the averaging windowτ,u1(t) is assumed to be stationary in time. Later, this assumption will be checked as well.

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Figure 4.1 Overview of M161 Flight 12 on RV Meteor over night from February 15 2020 to February 16 2020 during EUREC⁴A field campaign. Barometric altitude time-recordz_PSS8(t) of the micro-MPCK flown at approximately constant altitude above MSL (A). Time-records of air temperatureT (B) and relative wind speedu1(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^′2₁⟩^τ on the temporal averaging window size τ. Therefore, ⟨u^′2₁⟩^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^′2₁⟩^τ⟩N is the ensemble average of⟨u^′2₁⟩^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^′2₁⟩^τ⟩N and the predicted variance estimate, Eq. (1.61) with τ ≲ 5T11

(T11 = 46 s), is expected. In the range of 40T11 ≤τ ≤70T11, ⟨⟨u^′2₁⟩^τ⟩N overlaps with the predicted variance estimate, Eq. (1.61) with τ = 65T11, within the standard error.

In Fig. 4.2A, τ = 50T11 is in the center of the overlap region between the empirical and theoretical curve. For τ >80T11, ⟨⟨u^′2₁⟩^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≈78T11 where the ensemble consists of S segments. In each segment, ⟨⟨u^′2₁⟩^τ⟩N is evaluated similar to the previous procedure. Consequently, ⟨⟨⟨u^′2₁⟩τ⟩N⟩_S is the ensemble average of ⟨⟨u^′2₁⟩T⟩N

over all segments S. Then, the predicted variance estimate, Eq. (1.61) with τ = 65T11, is within the standard error of ⟨⟨⟨u^′2₁⟩^τ⟩N⟩S in the range 10T11≤τ ≤78T11 (Fig. 4.2B).

In Fig. 4.2A, τ = 39T11 is at the beginning of the overlap region while τ = 50T11 is in the center of the overlap region between the empirical and theoretical curve, as mentioned before. For τ = 1800 s ≈ 39T11, ⟨⟨⟨u^′2₁⟩^τ⟩N⟩S is underestimated by ≈ 5%

compared to ⟨u^′2₁⟩^65T11. Using ^q2⟨u^′2₁⟩²T11/τ for the random error of the variance estimate [56], the random error of ⟨u^′2₁⟩^τ is 18% which is significantly larger than the systematic error of the variance estimate. Hence, ⟨u^′2₁⟩^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 = 39T11, which is the shortest reasonable averaging window possible.

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Figure 4.2 Variance estimates of longitudinal velocity time record ⟨u^′2₁⟩τ as a function of the temporal averaging window size τ re-scaled by the integral time scale T₁₁ = 46 s.

⟨u^′2₁⟩τ

N represents the ensemble average based on the averaging windowτ. 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 = 39T11 is at the beginning of the overlap region whileτ = 50T11 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^′2₁⟩τ

N is evaluated for eachτ. ⟨u^′2₁⟩^τ_NS is the average of⟨u^′2₁⟩T

N over all segmentsS. The black dashed line illustrates

⟨u^′2₁⟩65T11, i.e. the variance estimate based on an averaging window of 65T11. The theoretical expectation, Eq. 1.63, is within the standard error of the variance estimate with an averaging windowτ = 39T11denoted by⟨u^′2₁(t)⟩39T11. In the following, the shortest reasonable averaging window is chosen toτ = 39T11 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 u1(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 u1(t) asstatistically stationary if changes of⟨u^′2₁(t)⟩τ under time shifts of the averaging window τ are smaller than the random error of the variance estimate ^q2⟨u^′2₁⟩²T11/τ [56]. In the following, I will refer to ⟨u^′2₁⟩^65T11 as the reference for estimating errors because τ = 65T11 is significantly shorter than τ = 78T11 and still agrees with the prediction, Eq. (1.61), as shown in Fig. 4.2A. In doing so, the random error ⟨u^′2₁⟩^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^′2₁⟩65T11.

Figure 4.3 shows ⟨u^′2₁(t)⟩^τ as a function of time indicating that ⟨u^′2₁(t)⟩^τ fluctuates significantly. This might hint at the fact that the stationarity assumption is not fulfilled globally. As an example, ⟨u^′2₁(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 ^q2⟨u^′2₁⟩²T11/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 whichu1(t) appears to be approximately statistically stationary. In that time interval,⟨u^′2₁(t)⟩1800 sis constant within 12%, which is smaller than the random error of ≈ 18%. Furthermore, ⟨u^′2₁(t)⟩^τ fluctuates about 17% for τ = 600 s, which is lower than the random error, too. Remarkably, ⟨u^′2₁(t)⟩^{600 s} tends to be lower than ⟨u^′2₁(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.