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Deﬁning AW this way requires that χ(γ ) be stationary only over [t − AW, t] ¯ and not over [t − T, t]. As t increases to deﬁne successive sets of T, S(t), and AW, the ABL is approximated by a succession of (locally) stationary segments (cf. Holman and Hart, 1972) which, if necessary, can overlap one another. The time extent of the segments here, though, are different for the mean and random components. This reﬂects the adaptiveness of TDMM and is consistent with the premise that the best averaging window ‘depends on the signal, and may differ for different components’ (Jones and Parks, 1990). Typically, AW = 10L will be less than T ; but if this is not the case, T should be taken as AW since the choice of the numerical value of 10 in 10L is a decision made by the analyst. Some investigators (see Henjes, 1997) prefer to use windows smaller than 10L.
Determining the spectrum of χ(γ ) begins with t χ(γ )e−ιω(γ −t )dγ, X(t, ω, AW) = √ (7) AW t −AW where ω is in the interval [2π /AW, π (SR)]. Note (7) is a variant of the short-time Fourier transform (STFT), which is the accepted method of tracking frequency information as it changes with time (Cohen, 1995). Further note that (7) is also a real-time average. The peculiar feature of (7), though, is that its ‘shortness’ is deﬁned by a t-dependent property of χ(γ ), t − T ≤ γ ≤ t, which, aside from the factor of 10, cannot be adjusted by the user. This is what ensures that we are using AW to analyze χ(γ ), and not vice versa. Speciﬁcally, that the determined properties of X(t, ω, AW) are more closely related to χ(γ ) as opposed to AW; the mathematics of Fourier analysis makes no distinction (cf. Blackwelder and Kaplan, 1976).
Even though STFT is known to introduce high-frequency information associated with the boxcar averaging interval, we show below, see Equation (11), that these effects are negligible when determining. That is, even though TDMM produces spurious information in the STFT of χ(γ ), this information does not carry over into. The reason is that, while (7) abruptly truncates χ(γ ) at both t and t − AW, producing ‘end effects’, the function nonetheless decays gradually to zero over an interval of sufﬁcient extent to make these effects negligible. Note that X is a function of AW, the reason being that even though AW = 10L is considered long
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCE
where ωs = 2π /AW is the smallest frequency resolvable from χ(γ ), and ωl is the Nyquist frequency (= π (SR)). Note there are two t-dependencies on the right-hand side of (12); one in and the other in ωs. A meaningful analysis must identify both.
A note of caution is due here. It would seem to the unsuspecting that once AW is deﬁned, the FFT algorithm, or some variation of it, could be invoked to determine frequency information in the random part of the turbulence deﬁned over [t − AW,
240 GEORGE TREVIÑO AND EDGAR L. ANDREASt]. This is not necessarily the case since there is no guarantee that AW will be deﬁned by the turbulence to be a power of 2.
We applied TDMM to longitudinal wind speed U (γ ) measured with an ATI (Applied Technologies, Inc., Boulder, CO) three-axis sonic anemometer positioned 4 m above the ground and digitized at 10 Hz. We selected data recorded on 4 August 1991 at the Sevilletta National Wildlife Refuge in New Mexico beginning at 20:00 local time. Andreas et al. (1998) describe the measurements in more detail.
According to ATI, the accuracy of the data is ± 1% of the mean wind.
We took data beginning at 20:06 and ﬁrst detrended them according to the following procedure. We assumed that over a data block of length [t − T, t] the wind had a mean deﬁned as µ(γ ) ≈ µ1 γ + µ0, where µ1 and µ0 are constants for each block but can vary from block to block. That is, we allowed for the possibility that there may be a time-dependent trend in the wind.
The value of µ1 was then approximated from the data by the random variable
(14) That is, (13) is an unbiased estimator of the instantaneous slope of µ(γ ) (cf.
Treviño, 1985). This value was then multiplied by γ and subtracted from U (γ ) over the block in question. The remainder of that subtraction is the time series u(γ ) = U (γ ) − µ1γ, which has a mean that is approximately constant over [t − T, t], as required by (1).
For the 20:06 data block, the algorithm yielded µ1 = 0.01 m s−2, indicating a mild trend in the wind. The requisite length of the block was found to be T = 25.6 s, which required that H = 51.2 s (recall that data are available as far back as 20:00). For this 25.6 s data block, we found L = 1.46 s and, thus, AW =
14.6 s. The minimum value of the set of averages deﬁned by (1) for this block was
1.61 m s−1 and its maximum was 1.74 m s−1. This we took as compatible with the condition that the mean of the detrended series (µ0 ) is approximately constant.
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCE
where we approximated du(γ )/dγ as (SR)[u(γi+1 ) − u(γi )]. Note that η(t) is by deﬁnition also a positive number. It is characteristic of the small-scale behaviour of the ABL; and, even though it too is a time scale, it is not proportional to L.
Speciﬁcally, L can be thought of as a macroscale and η as a microscale. η is the ¯ equivalent of the Taylor microscale λ according to η = λ/U when SR is high enough (1000 Hz) and the sampling volume of the sensor is small enough.
The difference between L and η is that η is sensitive to local rearrangement of the data points while L is not. In other words, if we take the given data and display them versus time in a column on a spreadsheet and then (locally) scramble this column using a SORT command, η will likely change by a noticeable amount. L, on the other hand, because it is deﬁned as a maximum, will not. The value of β, though, where this maximum occurs will change.
The value of η we found for 20:06 is 0.25 s; our results are indicated in Figure
1. The function ω (20:06, ω) is plotted versus ω in rad s−1 (solid line). The broken line has −2/3 slope, and σ (20:06) = 0.27 m s−1. The fact that the −2/3 slope line does not pass squarely through the data indicates the inertial subrange is (slightly) contaminated by large-scale anisotropy.
We next performed the analysis on a data segment 10 min into the record (20:10). The value of µ1 there was 0.00 m s−2 indicating no trend. The value of µ0 oscillated between 2.23 m s−1 and 2.30 m s−1, and T was 51.2 s (H = 102.4 s).
For these data, though, L was 1.08 s, which gave AW = 10.8 s and values of σ and η equal to 0.21 m s−1 and 0.18 s, respectively. The related ω (20:10, ω) is shown in Figure 2.
Repeating this procedure beginning now at, say, t = 20:12 (or some later time), will yield values for σ (20:12) and L(20:12). These iterations will produce a sequence of values for σ, L, and η, as well as functions ω (γ, ω). If these collectively do not change with running time, we can conclude that the ABL is stationary. In this case, ω (γ, ω) will be similar to a spectrum obtained using FFT. If the values do change, however, the ABL is nonstationary, and the time scale over which they change is a measure of the time scale of the phenomenon forcing the changes.
A uniﬁed method for assessing the degree-of-nonstationarity in a random signal is found in Treviño (1982). It is called the generalized frequency spectrum (GFS) concept. The essence of the GFS concept is to carry out a second frequency analysis of (γ, ω) to obtain the GFS, denoted (, ω). The ‘spread’ of (, ω) about = 0 is then a measure of the degree-of-nonstationarity. Since (, ω) is in
242 GEORGE TREVIÑO AND EDGAR L. ANDREASFigure 1. The function ω (20:06, ω) versus ω in rad s−1 (solid line). The values of η and L at this time are 0.25 s and 1.46 s (AW = 14.6 s), respectively. The value of T is 25.6 s. The broken line has −2/3 slope, and σ = 0.27 m s−1 is determined by averaging χ 2 (γ ) over [20:05:45.4, 20:06].
Figure 2. The function ω (20:10, ω) versus ω in rad s−1 (solid line).
The values of η and L at this time are 0.18 s and 1.08 s (AW =10.8 s), respectively. The value of T is 51.2 s. The broken line has −2/3 slope, and σ = 0.21 m s−1 is determined by averaging χ 2 (γ ) over [20:09:49.2, 20:10].
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCEFigure 3. The function ω (20:14, ω) versus ω in rad s−1 (solid line). The values of η and L are
0.22 s and 0.88 s (AW = 8.8 s), respectively. The value of T is 51.2 s. The broken line has −2/3 slope, and σ = 0.19 m s−1 is determined by averaging χ 2 (γ ) over [20:13:51.2, 20:14].
general complex (i.e., it has real and imaginary parts), it is best to analyze the behaviour of | (, ω)| about = 0.
The time scale of changes in σ, we should note, is not the same as the time scale of changes in L (or η). That is, it is possible for σ to vary slowly and for L to vary quickly (or vice versa). Also keep in mind that instants in time when σ is ‘large’ and L is ‘small’ correspond to energetic, highly random events. Conversely, instants in time when σ is ‘small’ and L is ‘large’ correspond to quiet, less random events.
As further examples of TDMM, we also performed the analysis on a segment 14 min into the record. The value of µ1 there was again 0.00 m s−2. The value of µ0 oscillated between 1.52 m s−1 and 1.63 m s−1, and T was again 51.2 s (H =
102.4 s). This time L was 0.88 s, which gave AW = 8.8 s and values of σ and η equal to 0.19 m s −1 and 0.22 s, respectively. The related ω (20:14, ω) is shown in Figure 3.
We took yet another segment of the series, this time 18 min into the record. The value of µ1 there was again 0.00 m s−2, and the value of µ0 oscillated between
1.85 m s−1 and 1.92 m s−1. The T was 51.2 s (H = 102.4 s) and L was 3.04 s;
values of σ and η were 0.23 m s−1 and 0.28 s, respectively. The function ω (20:18, ω) is shown in Figure 4.
In Table II, we summarize the statistics that our four analyses have produced.
From µ0, L, and σ especially, we see that this time series is moderately nonstationGEORGE TREVIÑO AND EDGAR L. ANDREAS Figure 4. The function ω (20:18, ω) versus ω in rad s−1 (solid line). The values of η and L are
0.28 s and 3.04 s (AW = 30.4 s), respectively. The value of T is 51.2 s. The broken line has −2/3 slope, and σ = 0.23 m s−1 is determined by averaging χ 2 (γ ) over [20:17:29.6, 20:18].
20:06 0.01 1.61–1.74 25.6 1.46 14.6 0.21 0.27 20:10 0.00 2.23–2.30 51.2 1.08 10.8 0.18 0.21 20:14 0.00 1.52–1.63 51.2 0.88 8.8 0.22 0.19 20:18 0.00 1.85–1.92 51.2 3.04 30.4 0.28 0.23 ary. In practice, we would prepare more closely spaced samples of the statistics summarized in Table II and look for patterns in the nonstationarity. For example, intense mixing events would yield periods with smaller L and larger σ, while more quiescent periods would show larger L and smaller σ.
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCE
4. Summary Remarks
Advancing our understanding of the ABL requires obtaining meaningful statistics that quantify intermittency. We have described an algorithm (TDMM) capable of extracting, in real-time, such statistics from events lasting only a few tens of seconds. TDMM makes no assumptions about the ABL other than it has a ﬁnite memory. In other words, correlations eventually go to zero. The event itself dictates the actual length of the required averaging. The quantities we compute are the variance, memory, a scale considerably shorter than the memory akin to the Taylor microscale, and the spectrum. The variations in these with time characterize intermittency. We summarize TDMM by delineating the sequence of computations that produce the spectral result and also the advantages that TDMM has over traditional spectral analysis algorithms.