The experiment was conducted at the National Oceanic and Atmospheric Administration (NOAA) Grid 3 area (station ID: GRI) situated on the western edge of the Snake River Plain in southeastern Idaho, USA (43.59° N, 112.94° W; 1500 m above mean sea level; Fig. ). The closest mountains are located approximately 13 km northwest from GRI. Based on the data from multiple automated meteorological observation stations in the area, southwesterly and northeasterly winds prevail during the day and night, respectively (Finn et al., 2016). Under these prevailing winds, GRI has a relatively flat and uniform upwind fetch (Finn et al., 2018; Lan et al., 2018). The vegetation primarily comprises shrubs and grasses, each with a roughness length and displacement height of a few centimeters (Finn et al., 2016).

The experiment utilized a 62 m tower and a 10 m aluminum tower (UT30, Campbell Scientific, Inc.) at the Grid 3 area to mount the sensors (Fig. ). The 62 m tower was guyed at eight levels, and the 10 m tower was guyed at one level. Retractable square booms measuring 3.6 m (12 ft) were horizontally braced to the 62 m tower to attach the sensors. The CSAT3s and IRGAs were mounted on 1 ft pipes, which were securely attached to the end of each boom. As a result, the CSAT3s and IRGAs were positioned at least 2.0 m away from the tower structure. On the 10 m tower, 1.8 m (6 ft) poles were utilized to mount the CSAT3s and IRGAs, positioning the sensors approximately 1.5 m away from the tower's frame structures.

Throughout the experiment, multiple levels of EC systems were deployed (Fig. ). These EC systems included two models of 3D sonic anemometers from the same manufacturers (model CSAT3B and CSAT3, Campbell Scientific, Inc.) and three models of infrared gas analyzers from the same manufacturers (IRGA; model LI7500RS, LI7500A, and LI7500, LICOR, Inc.). CSAT3s measured the three-dimensional wind velocity components (u, v, and w) and the sonic air temperature (Ts_m), while IRGAs measured the densities of water vapor (ρv) and CO2 (ρc). Co-located with CSAT3s and IRGAs, Type-E fine-wire thermocouples (model FW3, Campbell Scientific, Inc.) were used to measure air temperature (Tc) at six heights of 2.0, 8.2, 12.8, 15.8, 23.0, and 40.2 m. The FW3 thermocouple is composed of a chromel wire and a constantan wire with a diameter of 0.0762 mm. The FW3 determines Tc by measuring the voltage potential differences created at the junction of the two wires due to the temperature difference, whereas Ts_m is determined based on the relationship between sonic virtual temperature and the speed of sound (Liu et al., 2001). This distinction in temperature measurement principles between CSAT3s and FW3 implies that the measured Ts_m and Tc are independent of each other. Furthermore, Tc is expected to remain unaffected by vibrations of the tower and mounting arms. In this study, we also recorded the inclination of the CSAT3B (e.g., pitch and roll angle measurements) given by an integrated inclinometer in the CSAT3B. Pitch angle is defined as the angle between the gravitationally horizontal plane and the CSAT3B x axis, and roll angle is defined as the angle between the gravitationally horizontal plane and the CSAT3B y axis. The sonic roll and pitch angles were stored as 30 min averages.

Three dataloggers (model CR1000X, Campbell Scientific, Inc.) were employed to sample the high-frequency instruments (i.e., sonic anemometer, fine-wire thermocouple, and gas analyzer) at 10 Hz. Each datalogger was equipped with a GPS receiver (model GPS16X-HVS, Garmin International, Inc.) to synchronize the datalogger clocks. Additionally, a variety of meteorological measurements were conducted at GRI, including net radiation, air temperature, relative humidity, soil moisture, soil temperature, and soil heat fluxes. In this study, we examined the data collected at three different heights (40.2, 23.0, and 12.8 m) from 25 April to 31 July 2021, considering structural differences between the tall tower and short tower setups as well as potential influences from the roughness sublayer on measurement at lower levels.

The data processing mainly entailed despiking, double rotation for the wind components (Wilczak et al., 2001), sonic temperature conversion (Liu et al., 2001; Schotanus et al., 1983), and application of DEC to the raw fluxes of latent heat and CO2 (Webb et al., 1980). However, corrections for the effects of humidity and density fluctuations in this study are applied to the turbulent fluctuations of sonic temperature, ρv, and ρc, respectively (Detto and Katul, 2007; Gao et al., 2020; Sahlée et al., 2008; Schotanus et al., 1983; Webb et al., 1980). For each 30 min interval, the corrected turbulent fluctuations of sonic temperature (Ts′), ρv′, and ρc′ are determined by 1Ts′(t)=Ts_m′(t)-0.51ρv′(t)ρ‾aT‾

2ρv′(t)=(1+μσ)ρv_m′(t)+(1+μσ)ρ‾vT‾Ts′(t)3ρc′(t)=ρc_m′(t)+ρ‾c(1+μσ)Ts′(t)T‾+μρ‾cρ‾aρv′(t), where Ts_m, ρv_m, and ρc_m represent measured sonic temperature, water vapor, and CO2 densities, respectively; T‾, ρ‾a, ρ‾v, and ρ‾c are averages of air temperature, air density, water vapor, and CO2 densities, respectively; μ=md/mv (md and mv are the molecular mass of dry air and water vapor, respectively); and σ=ρ‾v/ρ‾d (ρ‾d is the density of dry air). The prime symbol denotes the turbulent fluctuations relative to the 30 min block average. As shown in the equations above, there is interdependence between Ts′ and ρv′, and thus Ts′ and ρv′ must be determined iteratively. In this study, the corrections are iterated twice. Note that in semiarid sites like ours, the adjustments to Ts′ due to fluctuations in specific humidity (ρv′(t)ρ‾a) are typically negligible (not shown here) as compared to the adjustment to ρv′ due to fluctuations in Ts′ (as demonstrated in Sect. ). The corrected time series of fluctuations facilitate the investigation of coherent structures and scalar similarity between temperature and other scalars (Detto and Katul, 2007; Sahlée et al., 2008). Here and throughout, Ts refers to the air temperature measured by sonic anemometers after humidity corrections, Ts_m the sonic temperature directly measured before corrections, and Tc the air temperature measured by fine-wire thermocouples (FW3).

Ensemble empirical mode decomposition (EEMD) (Huang et al., 1998; Huang and Wu, 2008) is applied to decompose the 30 min turbulence time series into three subsequences, corresponding to the high-, middle-, and low- frequency ranges, respectively. EEMD is a favored method in analyzing non-linear and non-stationary turbulence data (Gao et al., 2018; Hong et al., 2010; Liu et al., 2021; Huang et al., 1998; Huang and Wu, 2008). Through the sifting process in EEMD, a 30 min time series is decomposed into 13 oscillatory components Cj(t) (j=1,2,…,13) and an overall residual r13(t). Each oscillatory component generally exhibits one characteristic frequency (Hong et al., 2010; Gao et al., 2018), while the overall residual either is monotonic or contains only one extremum, from which no more oscillatory components can be further decomposed. Hence, 4x(t)=r13(t)+∑j=113Cj(t).

As detailed in Sect. , after comparing the power spectra of Ts and Tc, two frequency boundaries, 0.02 and 0.2 Hz, are identified. The oscillatory components are then categorized into three frequency ranges (I, II, and III, respectively). The oscillatory components with mean frequencies falling within the corresponding ranges are added together to generate the three subsequences. Specifically, oscillatory components with mean frequencies smaller than 0.02 Hz are summed and labeled as regime I (i.e., xI′=∑j=1013Cj,x+r13, where x=w, Ts, and Tc), between 0.02 and 0.2 Hz as regime II (i.e., xII′=∑j=69Cj,x), and larger than 0.2 Hz as regime III (i.e., xIII′=∑j=15Cj,x).

Figure 2 illustrates the comparisons between the variances and fluxes obtained using Ts and Tc at different heights. Here, we use σTc2 and w′Tc′‾ as reference values since Tc is less sensitive to the effects of humidity and wind speeds than Ts (Burns et al., 2012; Smedman et al., 2007). Generally, the variances of Ts (σTs2) are smaller than variances of Tc (σTc2), typically by 2 %–5 % (Fig. 2a–c). These lower variances result in a lower sensible heat flux (w′Ts′‾). Specifically, at 23.0 and 40.2 m, w′Ts′‾ is underestimated by approximately 2 % and 4 %, respectively, as compared to the sensible heat fluxes derived from the FW3 (i.e., w′Tc′‾) (Fig. 2d and e). However, at 12.8 m, w′Ts′‾ and w′Tc′‾ are quite comparable (Fig. 2f). These results confirm that errors in H were not entirely due to errors in the vertical velocity (Frank et al., 2013; Horst et al., 2015). Further examination of Fig. 2 reveals that the differences between σTs2 and σTc2, as well as between w′Ts′‾ and w′Tc′‾, increase with increasing measurement heights. Additionally, our tests indicate that the influence of solar heating on measurements of FW3 is negligible, primarily due to the fine-wire diameter of 0.0762 mm (Text S1 and Fig. S1 in the Supplement).

What mechanisms could have caused the observed differences? Previous studies found that CSAT3 sonic anemometers with a previous version of firmware could cause errors in Ts (Burns et al., 2012). However, this should not be the case for our study because the CSAT3B modes were used at these heights, and they have an improved design compared to the original CSAT3. Our results suggest that the differences are dependent on the measurement heights. As the measurement heights increase, the dominant length scale of coherent structures is also enlarged (Zhang et al., 2011). Therefore, we conjecture that the processes that lead to such differences are most likely scale-dependent, motivating us to examine the spectra and cospectra in the next subsection.

To gain further insight into the differences between σTs2 and σTc2 as well as the associated fluxes (i.e., w′Ts′‾ and w′Tc′‾), the spectra of u, w, Ts, and Tc and the w–Ts and w–Tc cospectra at different heights are examined. Figure a–c show the mean normalized Fourier power spectra of u, w, Ts, and Tc as a function of natural frequency (f). Note that the power spectra were computed every half hour using the fast Fourier transform and normalized by the corresponding variances before averaging. It is also interesting to note that the u, w, and Ts spectra deviate from the well-known -5/3 power law in the high-frequency range of f > 0.2 Hz, exhibiting similar features to those of previous studies (e.g., Burns et al., 2012). For f > 2 Hz, fSu, fSw, and fSTs appear to follow the f+1 slope, likely due to white noise and/or aliasing (Kaimal and Finnigan, 1994). In the 0.2 Hz < f < 2 Hz range, the distortion of fSu, fSw, and fSTs from the -5/3 power law is enhanced as the measurement heights decrease. The upturned distortion for 0.2 Hz < f < 2 Hz might be associated with spikes (Gao et al., 2020; Stull, 1988) that are not excluded from the 10 Hz time series during despiking. The Tc spectra appear to follow the -5/3 power law for 0.2 Hz < f < 1 Hz, although they are slightly attenuated for f > 1 Hz, likely because the thermal mass of the thermocouple wire limits its response time (Burns et al., 2012).

It is also noted that the magnitude of the Tc spectra is higher than that of the Ts spectra in the low-frequency range of f < 0.02 Hz, but their magnitude is comparable in the middle-frequency range of 0.02 Hz < f < 0.2 Hz. These results indicate that turbulent eddies with scales less than 0.02 Hz contribute more to σTc2 than σTs2. Hence, the underestimation of σTs2 is mostly caused by the lower magnitude of the Ts spectra in the low-frequency range, whereas σTs2 is overestimated to some extent due to the upturned distortion of the Ts spectra in the high-frequency range.

Figure d–f show the distribution of the mean normalized w–Ts and w–Tc cospectra as a function of f. Each one of the cospectra is normalized by the corresponding covariances before averaging. For f > 0.2 Hz, especially when f > 0.5 Hz, the w–Ts cospectra are generally higher than the w–Tc cospectra, consistent with the higher spectra of Ts. Compared to the power law of -3/4, the calculated w′Ts′‾ values are overestimated, while w′Tc′‾ is slightly underestimated. For 0.002 Hz < f < 0.2 Hz, the normalized w–Ts cospectra are slightly lower than the normalized w–Tc cospectra, indicating that turbulent eddies with scales in the frequency range of 0.002 Hz < f < 0.2 Hz contribute more to w′Tc′‾. Overall, the underestimation of the CSAT3B-derived fluxes is also scale-dependent, and the underestimation of w′Ts′‾ in the middle- to low-frequency range is offset by the overestimation in the high-frequency range to some extent.

According to the comparison of the Ts and Tc spectra, the whole frequency domain can be divided into three ranges: (I) f < 0.02 Hz, (II) 0.02 Hz < f < 0.2 Hz, and (III) f > 0.2 Hz. For f < 0.02 Hz, the magnitude of the Tc power spectra is slightly higher than that of the Ts spectra at all levels, but the magnitude of the Ts and Tc spectra is comparable in the middle-frequency range of 0.02 Hz < f < 0.2 Hz. For f > 0.2 Hz, the Tc spectra first follow the -5/3 power law and are then attenuated, whereas the Ts spectra are distorted upward. Based on this division, in Sect. , the 30 min time series is divided into the three frequency ranges, and the contributions of these different scales to σTs2, σTc2, w′Ts′‾, and w′Tc′‾ at different heights are then quantified.

To quantify the contributions of different scales to the corresponding temperature variances and fluxes, we apply the EEMD approach to decompose the 30 min time series of w′, Ts′, and Tc′ into various oscillatory components, which are then categorized into the three frequency ranges discussed earlier (i.e., I, II, and III). Figure a–c depict that for regime I, the ratios between the variances of Ts and Tc are generally lower than 1.0 (approximately 0.89 on average). This suggests that turbulent eddies with scales less than 0.02 Hz contribute approximately 11 % more to σTc,I2 than σTs,I2. With these turbulent eddies contributing about 41 %–57 % to the total variances (Fig. a–c and Table ), the 11 % difference between σTc,I2 and σTs,I2 would cause 4 %–6 % difference between the total variances of Ts and Tc. As for fluxes, turbulent eddies with scales less than 0.02 Hz contribute approximately 6 % more to w′Ts′‾I than w′Tc′‾I. With these turbulent eddies accounting for about 26 %–45 % of the total fluxes (Fig. d–f and Table ), the 6 % difference in w′Ts′‾I and w′Tc′‾I would cause 2 %–3 % difference in the total fluxes. Further, given that the contribution of regime I to the total temperature variances and fluxes increases with measurement height (Fig.  and Table ), the underestimation becomes more significant at higher levels.

For regime II, the ratios between the variances of Ts and Tc and the associated fluxes are close to 1.0 with minimal scatter. This indicates that turbulent eddies with scales between 0.02 and 0.2 Hz contribute consistently to the temperature variances and fluxes. These turbulent eddies contribute 32 %–38 % and 42 %–50 % to their total variances and fluxes, respectively (Fig.  and Table ). For regime III, the ratios between the variances of Ts and Tc average about 1.45 at the three heights due to the distorted spectra in this regime for both Ts and Tc. This indicates that turbulent eddies with scales larger than 0.2 Hz contribute roughly 45 % less to σTc,III2 than σTs,III2. With these turbulent eddies contributing about 12 %–26 % to the total variances (Fig. a–c and Table ), the 45 % difference in σTc,III2 and σTs,III2 would cause 5 %–7 % difference in the total variances of Ts and Tc. As for fluxes, turbulent eddies with scales larger than 0.2 Hz contribute approximately 20 % less to w′Tc′‾III than w′Ts′‾III. With these turbulent eddies accounting for 12 %–26 % of the total fluxes (Fig. 5d–f and Table ), the 20 % difference in w′Tc′‾III and w′Ts′‾III would cause 2 %–4 % difference in the total fluxes. These results suggest that the observed underestimation of the Ts variances and fluxes is primarily attributed to the large turbulent eddies with frequencies less than 0.02 Hz, which is offset to some extent by the contribution from small turbulent eddies with frequencies larger than 0.2 Hz.

The potential causes for the scale-dependent differences between the Ts and Tc spectra are measurement errors, solar heating of thermocouples, and tower and mounting arm vibrations, among others (e.g., the deficiency in the design of sonic anemometers in response to different wind speed conditions). The Tc spectra follow similar declining features in the high-frequency range independent of wind speed (Text S2 and Fig. S2 in the Supplement), suggesting that measurements of the fine-wire thermocouples were not noticeably affected by increased wind speed, and therefore operational errors could be excluded from the causes for the observed difference. Additionally, the consistent differences between the power spectra of Ts and Tc under nighttime and daytime conditions (Text S1 and Fig. S1) suggest that the impact of solar heating on thermocouples was also not the cause for the differences between the Ts and Tc variances and fluxes.

As the wind speed increases, the tower vibrations become more pronounced, especially at higher levels, as indicated by the more energetic peaks in the power spectra of sonic roll and pitch angles (Fig. ). For wind speed below approximately 10 ms-1 at 40.2 m, the ratios of the Ts and Tc variances for regime I show no obvious change with wind speed. However, for wind speed above 10 ms-1, the ratios decrease further as wind speed increases (Fig. a). The ratios of the temperature fluxes of Ts and Tc exhibit a similar pattern to the variances (Fig. d). For regime II, the ratios of both the variance and fluxes of Ts and Tc show no obvious relations with wind speed (Fig. b and e). For regime III, the ratios of both the variance and fluxes of Ts and Tc illustrate large scatter, especially when wind speed is below 10 ms-1, but still show no obvious trends as wind speed increases (Fig. c and f).

We thus hypothesize that the enhanced tower vibrations under strong winds lead to an early or delayed detection of the sonic pulse. This results in overestimations or underestimations of the speed of sound and thus errors in the 10 Hz time series of sonic temperature. In this study, for wind speeds below 10 ms-1, the ratios of the variances (and fluxes) of Ts to those of Tc are scattered around 1.0 for regime I. However, for wind speed above 10 ms-1, the ratios decrease as wind speed further increases (Fig. ). Therefore, tower and mounting arm vibrations were most likely the cause for the differences in the temperature variance and fluxes. Under this circumstance, such vibrations may also affect sonic-measured wind components along with Ts, resulting in errors in all the calculated fluxes. However, rigorous tests of this hypothesis seem necessary through testing sonic anemometers in wind tunnels or fields with different mounting strategies.

Given that Eqs. (2) and (3) are used to adjust the measured densities of water vapor and CO2 by open-path CO2–H2O gas analyzers in EC systems, any errors in temperature measurements would be propagated to water vapor and CO2 time series through these two equations. In Eqs. (2) and (3), Ts can be replaced by Tc to achieve the adjusted time series of densities of water vapor and CO2 for FW3. The adjusted time series of water vapor and CO2 by the fluctuating parts of Ts and Tc, respectively, are then decomposed into three frequency ranges to quantify the influence of Ts on variances and fluxes of water vapor and CO2.

The variances of ρv are not influenced by the use of either Ts or Tc for the density effect corrections (Table S1 in the Supplement) because the fluctuations in ρv are not very sensitive to density effects, especially at semiarid sites like ours (Gao et al., 2020). Therefore, for the water vapor fluxes, there are only minor differences (< 1 %) between the fluxes corrected by Ts and Tc for the three frequency ranges, while the overall water vapor fluxes corrected by Ts and Tc are comparable at the three heights (Fig. a–c).

As for CO2, the differences between Ts and Tc in the three frequency ranges are propagated differently to the adjusted variances of ρc (Table S2 in the Supplement). More importantly, there are relatively large differences (but still within 10 %) between the CO2 fluxes adjusted by Ts and Tc (Fig. d–f). In general, CO2 fluxes are more sensitive to errors or uncertainties in temperature measurements than water vapor fluxes (Liu et al., 2006). Therefore, precise measurements of air temperature are critical for quantifying ecosystem CO2 fluxes.