Detecting Near-Surface Coherent Structure Characteristics Using Wavelet Transform with Different Meteorological Elements

Although the concept of coherent structures in atmospheric turbulence was proposed a long time ago (Taylor ^[1]; Kline et al. ^[2]), it still has no clearly agreed-upon definition. Hussain defined coherent structures as connected turbulent fluid masses with instantaneously phase-correlated vorticity over their spatial extent ^[3]. Blackwelder and Kaplan described them as parcels of vortical fluids occupying a confined spatial region such that a distinct phase relationship is maintained between the flow variables associated with their constituent components during their evolution in space and time ^[4]. In a time series of atmospheric turbulence, coherent structures are usually represented as being ramp-like (either a sudden decline after a slow rise or a sudden rise after a slow fall) (Katul et al. ^[5]; Chen et al. ^[6]). These phenomena have been identified in the time series of meteorological elements such as three-dimensional wind speed (Chen and Hu ^[7]), temperature (Barthlott et al. ^[8]), water vapor (Chen et al. ^[9]), and carbon dioxide and ozone (Rummel et al. ^[10]). Ongoing research has emphasized the importance of coherent structures to the transport of momentum, water vapor, heat, and matter in the atmospheric boundary layer (Drobinski et al. ^[11]), and further studies can better reveal the nature of turbulence and develop better boundary layer models (Wang et al. ^[12]).

Coherent structures in atmospheric turbulence have been studied through theoretical analysis (Raupach et al. ^[13]; Raupach et al. ^[14]; Aksamit and Pomeroy ^[15]; Williams and Hacker ^[16]), experimental observations (Drobinski et al. ^[17]; Krusche and De Oliveira ^[18]; Sun et al. ^[19]; Fang et al. ^[20]), and numerical simulations (Foster et al. ^[21]; Drobinski et al. ^[22]; Stawiarski et al. ^[23]). In recent years, most data obtained from experimental observations have been analyzed using wavelet transforms (Farge ^[24]; Eder et al. ^[25]; Conan et al. ^[26]; Ai et al. ^[27]), along with statistical parameters, such as time scale (Mohr and Schindler ^[28]; Fesquet et al. ^[29]; Lotfy et al. ^[30]) (e. g., the number of coherent structures appearing within 30 min (Starkenburg et al. ^[31]), to better define their characteristics. These results usually depend on the wavelet function selected (Collineau and Brunet ^[32]). Unfortunately, the meteorological elements and conditions used by previous analyses were not consistent, and the time period of the analyzed turbulence data was usually short (Barthlott et al. ^[8]), resulting in distinctly different results (Starkenburg et al. ^[31]; Feigenwinter and Vogt et al. ^[33]).

Given such inconsistencies, in this study, we seek to determine whether the statistical characteristics of coherent structures detected from atmospheric turbulence were related to the selection of different meteorological elements. We used three functions, i. e., Mexican-hat, Morlet, and Wave, and six meteorological elements, i. e., three-dimensional wind speed U, V, W; temperature T; and CO₂ and H₂O concentration, to detect coherent structures and analyze atmospheric turbulence data for up to a year in order to determine relevant similarities and differences.

All data were recorded at the Zengcheng Scientific Test Station for Research of Climate, Environment and Global Change in South China, Sun Yat-sen University (ZSCEG, SYSU) (Fig. 1). A CSAT3 three-dimensional ultrasonic anemometer measured three-dimensional wind speed and ultrasonic virtual temperature, while a Licor7500 gas analyzer measured CO₂ and H₂O concentrations; these were mounted at a height of 50 m on a 70 m tall meteorological tower equipped with an eddy covariance system to measure high-frequency atmospheric turbulence data (10 Hz).

Figure 1.  Experimental data source: (a) location of observation site (red star) and (b) recording instrument at a height of 50 m on a meteorological tower (red arrow).

The observation site is located at the top of a hill about 30 meters above sea level. The underlying surface is mainly composed of shrubs and grass. There are some low buildings about one kilometer to the south and Zengjiang River approximately 500 meters to the east. Due to the complex surface boundary conditions at this observation site, and the inherent characteristics of the eddy covariance system, the raw data were pre-processed via spike removal, coordinate rotation, and linear trend removal, following a previously described procedure (Wang et al. ^[34]). In addition, since precipitation could interfere with the ultrasonic anemometer and water vapor measurements, data for days with precipitation were excluded. Finally, we applied the turbulence data quality classification standard (Foken et al. ^[35]), after which all the 2017 data was filtered as previously (Barthlott et al. ^[8]; Gao and Li ^[36]) to produce 563 sets of high-quality 30-min atmospheric turbulence data covering different time periods, meteorological conditions, and stabilities. In order to improve calculation efficiency, we performed frequency reduction processing following Chen and Hu, using the final results for subsequent analysis ^[7].

The wavelet transform is calculated as:

where W(a, b) is the wavelet coefficient, a is the stretch coefficient, b is the displacement coefficient, f(t)is the signal to be analyzed, g is the wavelet basis function, and t is the physical quantity to be analyzed.

We used the following three wavelet basis functions:

The wavelet variance for a given scale a is W(a):

The scale corresponding to the maximum wavelet variance is the time scale of the atmospheric turbulence coherent structure (Collineau and Brunet ^[32]). At this time, the time-series signal of atmospheric turbulence can be reconstructed using the wavelet reconstruction formula (Chen and Hu ^[7]):

where c_φ^* = π, and a_* is the time scale of the turbulence coherent structure.

To illustrate, we used Equation (5) to calculate the wavelet variance at different time scales in order to detect the coherent structure of U from 16:00 to 16:30 on January 8, 2017 (Fig. 2). The time scales of the coherent structures detected by the three wavelet basis functions were significantly different, ranging from 25- 146 s.

Figure 2.  Time scale of the coherent structure of U as detected by three wavelet basis functions from 16:00 to 16:30 (UTC + 8), January 8, 2017.

In order to determine the most accurate time scale, we used Equation (6) to reconstruct the preprocessed signals, and then compared this with the preprocessed signal (Fig. 3). The time scale of the coherent structure represented the scale of the turbulence vortex representing the main energy in the turbulence, while the reconstructed signal reflected the motion of the original signal on a specific time scale (Chen and Hu ^[7]). Therefore, the fitting degree of the two signals reflects the quality of the reconstruction.

Figure 3.  Reconstruction of the original U signal using three wavelet basis functions from 16:00 to 16:30 (UTC + 8), January 8, 2017.

We then performed a correlation analysis of the reconstructed and pre-processed signals for the three wavelet basis functions (Fig. 4). For each set of 30- min data, the Wave wavelet correlation coefficient was the highest, indicating that this reconstructed signal had the best fit with the pre-processed signal. Therefore, we used the Wave wavelet as the basis function for detecting the turbulence coherent structures in subsequent analyses.

Figure 4.  Correlation coefficients between reconstructed and pre-processed signals from three wavelet basis functions for 563 Correlation coefficients between reconstructed and pre-processed signals from three wavelet basis functions for 563

When the scale of the maximum wavelet variance was known, the maximum point of the wavelet coefficient on this time scale corresponded to the abrupt jump point in the coherent structure. This allowed the number of coherent structures occurring within a given 30 min to be obtained (Fig. 5).

Figure 5.  Example of Wave wavelet use for detecting the number of CO₂ coherent structures (17 in this case) from 16:00 to 16: 30 (UTC + 8) on January 8, 2017.

The time scales of the six meteorological elements, which were concentrated mostly between 20 and 25 s, ranged from 5 to 40 s (Fig. 6). The overall consistency of the six elements was good, indicating that the time scales of atmospheric turbulence coherent structures were not dependent on the choice of meteorological elements. The distributions of the three scalars were very similar, but the three wind speed vectors varied from U > V > W. This was consistent with previous studies that showed the time scale for W was smaller than that of U and V (Chen and Hu ^[7]) and that the time scale of W was smaller than that of U and T (Eder et al. ^[25]).

Figure 6.  Statistical distribution of the coherent structure time scales for three-dimensional wind speed (U, V, and W), temperature (T), CO₂ concentration (C), and H₂O concentration (Q).

The number of coherent structures appearing in the six meteorological elements within 30 min mostly ranged from 10-20 with good consistency (Fig. 7), suggesting that similar to time scale, the number of coherent structures appearing within 30 min is independent of meteorological elements. element. The three-dimensional wind speeds were more likely to have higher values than those of the three scalars, especially W. As we did not set a threshold for wavelet coefficients when detecting the number of coherent structures, it was inevitable that relatively large values would be detected for each meteorological element (Barthlott et al. ^[8]; Collineau and Brunet ^[32]).

Figure 7.  Statistical distribution of the number of coherent structures appearing within 30 min for three-dimensional wind speed (U, V, and W), temperature (T), CO₂ concentration (C), and H₂O concentration (Q).

In order to better understand the characteristics of the coherent structures of different meteorological elements, we defined the time cover that the coherent structures occupied for 30 min:

The time cover values ranged mostly between 18 and 20%, and had very good consistencies, indicating that the time cover of atmospheric turbulence coherent structures did not depend on meteorological element (Fig. 8). As the time scale and number of structures for the three-dimensional wind speed (Figs. 6 and 7) showed an opposite trend (time scale decreased from U > V > W, but the number of coherent structures increased), Equation (7) shows that the effect of time cover is weakened, resulting in greater time cover consistency for the meteorological elements than the first two statistical characteristics.

Figure 8.  Statistical distribution of the time cover of coherent structures appearing within 30 min for three-dimensional wind speed (U, V, and W), temperature (T), CO₂ concentration (C), and H₂O concentration (Q).

Table 1 presents a comparison of the results of this study with those of other studies. It shows that different studies used different elements. With respect to time scale, our results are similar to those obtained by Thomas et al. and Zhang et al. ^{[37, 38]}. With respect to the number of coherent structures detected within 30 min, Barthlott et al., Fesquet et al. and Starkenburg et al. obtained smaller values, while Thomas et al. and Zhang et al. obtained larger values, and our results are between theirs ^{[8, 29, 31, 37, 38]}. With respect to time cover, our results are similar to those obtained by Starkenburg et al. ^[31].

Of the three wavelet functions tested on the 563 data sets, the Wave wavelet's correlation coefficients (0.5-0.6) outperformed those of the Mexican-hat (0.4-0.5) and Morlet (0.3-0.4) wavelets. Using the Wave wavelet, the atmospheric turbulence coherent structures detected for the six meteorological elements showed good consistency with regard to time scale (mostly between 15-25 s), number of coherent structures within 30 min (mostly 10-20) and time cover within 30 min (mostly 18-20%).

The statistical characteristics of the coherent structures of the three-dimensional wind speeds and the three scalars were broadly similar but differed in certain details. For example, the three scalars were more consistent than the wind speeds and the time scale decreased from U > V > W, while the number followed the opposite trend. This difference may be related to the complexity of wind speed compared with scalars.

Our results show that the detection of coherent structures in atmospheric turbulence is not dependent on meteorological elements. However, the statistical characteristics of the coherent structures may be related to other parameters. According to previous studies, the coherent structure characteristics of different locations are usually different which may be related to roughness length of the surface (Eder et al. ^[25]; Fesquet et al. ^[29]). In addition, there are many other parameters that can have an effect on the coherent structure such as friction speed (Krusche and De Oliveira ^[18]), instrument measurement height (Gao and Li ^[36]), vertical wind profile (Thomas and Foken ^[39]), average wind intensity (Krusche and De Oliveira ^[18]), and inflection point (Dias Junior et al. ^[40]). These relationships still need further study.

We did not consider the multi-scale nature of coherent structures because the maximum value of the wavelet transform coefficient variance can only be used to identify the average feature scale of the signal, and various physical structures can only be assumed in a one scale model. Subsequent research should assess this issue to better ascertain the multi-scale characteristics of coherent structures.