Identification of Non-Turbulent Motions for Enhanced Estimation of Land–Atmosphere Transport Through the Anisotropy of Turbulence

Abstract

Quantifying land–atmosphere transport remains crucial for advancing climate prediction and weather forecasting efforts. To improve turbulent flux estimation, the anisotropy of turbulence is taken into consideration. The parameters $x_B$ and $y_B$, which quantify anisotropy degrees across motion scales, form trajectories in the barycentric map. Using the Hilbert–Huang transform, the scale-dependent properties of anisotropy in observational data from multiple sites are investigated. Analysis reveals consistent patterns in the average $y_B-x_B$ trajectories across stratification conditions: as scale increases, $x_B$ increases from 0.4 to 0.9, while $y_B$ initially climbs from 0.5 to 0.7 before declining to 0. Meanwhile, individual case trajectories sometimes deviate from this pattern, indicating contamination by non-turbulent motions that typically cause turbulent flux overestimation. Crucially, identifying the scale at which deviations occur allows effective separation of atmospheric turbulence from non-turbulent motions, which enables the reconstruction of turbulence data. Results demonstrate that corrected fluxes reduce overestimation inherent in traditional eddy covariance systems by approximately 30%, with enhancements for CO₂ and air pollutants reaching 45–83%. Furthermore, the correlation between anisotropy and stratification suggests potential for refining similarity theories into a broader scope, such as carbon cycle assessment and pollution control. Therefore, anisotropy shows promise in quantifying the land–atmosphere transport.

1. Introduction

The dominant form of land–atmosphere transport is the turbulent flux, which is the primary mechanism for the exchange of energy, heat, greenhouse gases and air pollutants [1,2,3,4,5]. The eddy covariance (EC) system is recognized as the most universal method [6,7,8] to quantify the turbulent flux as the covariance of vertical wind and a second quantity of interest. However, turbulent transport quantification faces significant challenges under non-ideal conditions, in which the heterogeneity of the land surface (e.g., roughness, height, and humidity [9,10,11,12]) and the non-stationary nature of atmospheric conditions [13,14,15,16,17] usually affect the identification and analysis of atmospheric turbulence, causing deviations in the estimation of turbulent fluxes from traditional theories [18].

Extensive studies have confirmed that non-turbulent motions occur across stable and unstable conditions [19,20,21], significantly impacting turbulent transport quantification. While both turbulent and non-turbulent motions are crucial for land–atmosphere transport [15,22,23,24], a key challenge arises: large-scale motions (including coherent turbulence structures) may be redundantly incorporated into turbulence parameterizations within different simulation models focusing on varied scales [25,26,27]. Thus, systematic identification and removal of non-turbulent motions are essential for accurate turbulent transport quantification. Efforts like the cospectral gap [28] and spectral plateau [29] stem from spectral analysis, marking boundaries between turbulent and non-turbulent motions. Advanced techniques such as the Hilbert–Huang Transform [30,31,32] (HHT) and multiresolution decomposition [33,34,35] enhance spectral analysis effectiveness. Methods beyond spectra also exist, like anisotropy analysis for turbulence time scales [36], Eulerian autocorrelation function diagnostics [37], Mexhat-wavelet techniques [38], and amplitude probability distribution algorithms [39]. Nevertheless, significant challenges persist. Non-turbulent (or submesoscale) motions exhibit complex, superimposed, and interactive forms [15,40,41], resisting precise mathematical definition; existing methods often focus mainly on statistical characteristics/effects. Many techniques are also stratification-specific, limiting practical application. Furthermore, accurate material flux estimation (including carbon dioxide and air pollutants such as PM_2.5) remains underexplored due to complex transport mechanisms [11,42,43] and demanding observational requirements [44,45,46].

Emerging approaches leveraging novel parameters show promise in addressing these limitations. Examples include aerosol composition-visibility correlations for pollutant density [45,47,48] and fractal dimensions characterizing turbulent flow properties [49,50], highlighting the value of moving beyond traditional spectral analysis by incorporating parameters capturing turbulence’s multiscale nature for robust flux estimation. Herein, anisotropy, a key turbulence characteristic derived from the Reynolds stress tensor and quantified on a 2D barycentric map (BAM) [51], offers significant potential. It influences near-surface similarity theories for atmospheric turbulence properties [52,53]. There is a wealth of studies demonstrating the validity of anisotropy. The anisotropy of flow over two-dimensional rigid dunes proves to deviate from the traditional isotropic hypothesis and demonstrates that anisotropy can reliably represent turbulent properties in numerical models [54]. The return-to-isotropy phenomenon, in which the turbulence kinetic energy (TKE) tends to be equally redistributed at small scales, is also detected and analyzed [55]. The applicability of isotropy is evaluated through simulations of fluidized gas-particle suspensions [56] and the internal gravity wave detection in a stable boundary layer [57]. Introducing anisotropy into scaling relations of atmospheric turbulence is also justified for its potential in generalizing MOST in non-ideal conditions, which is supported by evidence from comprehensive analyses from the aspect of normalized standard deviations [58] and the flux–profile relationships [59]. While anisotropy’s role in scaling is established, few studies focus on the intrinsic connection between anisotropy, fundamental turbulence, and atmospheric boundary-layer (ABL) characteristics. The application for identifying non-turbulent motions and improving flux quantification is limited to single-point observations and lacks further exploration. Thus, examining the scale-dependent properties of anisotropy across diverse conditions holds great promise for revealing new turbulence characteristics, effectively distinguishing non-turbulent motions, and ultimately enhancing turbulent flux quantification.

This study investigates how the application of anisotropy can enhance the land–atmosphere transport estimation through the analysis of dynamic and thermodynamic properties of atmospheric turbulence. The investigation involves exploration of scale-dependent anisotropy properties using observational data from three stations representing diverse underlying surfaces, and the evaluation of the effectiveness of anisotropy incorporation across different stratification conditions. Section 2 introduces the observational sites and data, together with the data processing approaches. Section 3 describes the methods for investigating the scale-dependent properties of atmospheric turbulence. Section 4 applies anisotropy to provide criteria for distinguishing non-turbulent motions and correcting turbulent fluxes. Section 5 summarizes the methods, conclusions, and discussions of the study, exploring the application prospects of anisotropy.

2. Data

2.1. Observational Site

To investigate the effectiveness of anisotropy in reflecting characteristics of atmospheric turbulence, the data are selected from multiple observational sites across different underlying surfaces and in different seasons.

The 255 m meteorological tower in Tianjin (${39}^{\circ }{05}^{\prime } \mathrm{N},{117}^{\circ }{13}^{\prime } \mathrm{E}$) is located in the PBL Meteorological Observation Station, Tianjin Meteorological Administration, which is representative of the typical urban landscape with profound anthropogenic influences. Turbulent data are acquired from sonic anemometers (CSAT-3, Campbell Scientific, Inc., Logan, UT, USA) from the meteorological tower at 80 m orienting towards East, which can provide the three-dimensional wind speed and the sonic temperature data. Fast-response observations of gases are conducted with the Open Path CO₂/H₂O Analyzer (LI-7500, LI-COR, Inc., Lincoln, NE, USA) installed at 80 m, providing density time series of CO₂ and water vapor. The sampling frequencies of the instruments above are all 10 Hz. There are also observations of mean wind speed (Cup and vane anemometer, Changchun Meteorological Instrument Ltd., Changchun, China) and mean temperature and humidity (HMP45C, Campbell Scientific, Inc., Logan, UT, USA) at a total of 15 levels. The gradient data are calculated as the averaged finite difference in the mean variables from 5 levels, from 40 m to 120 m. The observational data were collected from 1 July to 31 August 2017. There is more information about the Tianjin Station [60,61].

The Naiman station is located in Horqin Sandy Land (${42}^{\circ }{56}^{\prime } \mathrm{N},{120}^{\circ }{42}^{\prime } \mathrm{E}$) in the Naiman County of Inner Mongolia. The underlying surface is nearly flat and homogeneous, consisting mainly of sandy land with low and open shrubs, which is typical of a semiarid continental monsoon climate, and there are few anthropogenic activities. The sonic anemometers and Open Path CO₂/H₂O Analyzer are the same as Tianjin Station at 8 m, orienting towards the west, with the same sampling frequencies of 10 Hz. The mean wind speed (010C, MetOne Instruments Inc., Grants Pass, OR, USA), air temperature, and humidity (HMP45C, Campbell Scientific, Inc., Logan, UT, USA) are observed at four levels from 2 to 20 m. The gradient data are obtained from the finite difference between the two levels of 4 m and 16 m. The observational data were collected from 1 July to 31 July 2022. More information concerning the Naiman station can be found in references [62,63].

The Dezhou station is located at the Pingyuan County Meteorological Bureau (${37}^{\circ }{09}^{\prime } \mathrm{N},{116}^{\circ }{28}^{\prime } \mathrm{E}$) in Shandong Province. The underlying surface is typical flat farmland in the North China Plain. The station is equipped with a set of instruments at a height of 2.8 m, including a three-dimensional sonic anemometer (IRGASON, Campbell Scientific, Inc., Logan, UT, USA) orienting towards North, and a continuous particle measuring instrument, E-sampler (Met One, Inc., Grants Pass, OR, USA). The sampling frequency of wind speed is 10 Hz, and that of PM_2.5 density is 1 Hz. The gradient data is acquired from a near-surface observation of a GPS low-altitude electronic sonde (XHD-403, Institute of Atmospheric Physics of the Chinese Academy of Sciences). The sonde collects data every 10 m, and the averaged finite difference within 100 m from the land surface is calculated as the gradient of each meteorological parameter. Observations are repeated every 3 h from 27 December 2018 to 24 January 2019, and the data are interpolated to match the sampling frequency of other instruments. There is a comprehensive introduction on the Dezhou station and the methodology of the measurement of PM_2.5 density [45].

The comparison among the three observational stations is given in

Table 1. Comparison of the conditions around different observational stations.

Station	Variables Observed	Underlying Surface	Anthropogenic Activity	Stratification Condition
Tianjin	$u,v,w,T,{\rho }_{{CO}_2},q$	Urban land (Medium Homogeneity)	High	Balanced
Naiman	$u,v,w,T,{\rho }_{{CO}_2},q$	Sandy land (High Homogeneity)	Low	Mainly Unstable
Dezhou	$u,v,w,T,{\rho }_{PM2.5}$	Farmland (High Homogeneity)	Medium	Mainly Stable

, and their locations are shown in Figure 1.

2.2. Data Processing

The raw turbulence observational data were pre-processed using the EddyPro software (ver. Advanced 7.0.9, LI-COR Biosciences, Inc., Lincoln, NE, USA), including error flags, despiking [10], double coordinate rotation [64], and linear detrending. The turbulent fluxes are also corrected to eliminate the influence of tilt, time lag, and WPL terms within the software and the following reconstructions. The averaging time span for Naiman and Dezhou is 30 min following the standardized calculation of EddyPro. To address non-stationarity arising from complex conditions at the Tianjin station, the averaging time span is set to 1 hr. This extended duration better captures large-scale turbulence components, notably coherent structures, while enabling comparative analysis of varying averaging spans across heights and observational sites to strengthen result robustness. The calculations in the following sections use the same time spans.

From the observational data, several important scales and parameters can be defined: The Obukhov Length $L$ is

$$L=-{\displaystyle \frac{{u_{\ast }}^3}{\kappa \frac{g}{\overline{\theta }}\overline{w^{\prime }{\theta }^{\prime }}}},$$

(1)

where $\kappa$ is the von Kármán constant, $g$ is the numerical value of gravitational acceleration, and $\overline{\theta }$ is the average potential temperature. $u_{\ast }=\sqrt[4]{{\left(-\overline{u^{\prime }{w}^{\prime }}\right)}^2+{\left(-\overline{v^{\prime }{w}^{\prime }}\right)}^2}$ is called the friction velocity, which is a characteristic scale of the turbulent wind stress generated by surface friction. The definition of sensible heat flux $H$, momentum flux $\tau$, latent heat flux $H_L$, and turbulent flux of specific materials $F_c$ are defined as follows:

$$H=\rho c_p\overline{w^{\prime }{\theta }^{\prime }},$$

(2)

$$\tau =-\rho \sqrt{{(-\overline{u^{\prime }{w}^{\prime }})}^2+(-{\overline{v^{\prime }{w}^{\prime }})}^2},$$

(3)

$$H_L=\lambda E=\lambda \overline{w^{\prime }{q}^{\prime }},$$

(4)

$$F_c=\overline{w^{\prime }{c}^{\prime }},$$

(5)

where $\rho$ and $c_p$ are the air density and specific heat capacity at constant pressure; $\lambda$ is the latent heat, $q$ is the concentration of water vapor, and $c$ is the density of a certain kind of material. To ensure data quality for anisotropy evaluation, the data where the angle between the wind direction and the sensing probe axis exceeded ±120° were excluded. This threshold minimizes errors associated with flow distortion and acoustic shadowing inherent to sonic anemometry at extreme angles. The other criteria in the standard EC system, such as the mean wind speed range $0.1~5\mathrm{m}/\mathrm{s}$ and the friction velocity $u_{\ast }>0.05\mathrm{m}/\mathrm{s}$, are NOT applied to ensure the inclusion of properties that could be reflected by anisotropy.

3. Methods

3.1. Hilbert–Huang Transform (HHT)

The HHT effectively analyzes nonlinear and non-stationary atmospheric turbulence by adaptively decomposing signals while isolating non-ideal components as residuals [30,65]. Through Empirical Mode Decomposition (EMD), the original signal separates into components spanning distinct frequency ranges, enabling scale-specific characterization. Low-index IMFs (small $k$) capture high-frequency, small-scale motions, whereas high-index IMFs (large $k$) represent low-frequency, large-scale phenomena. The residual $r_n$, being monotonic or having at most one extremum, embodies the non-oscillatory trend. The multivariate EMD extension further processes combined signals (e.g., wind components u, v, w) through redefined local mean calculations [66,67]. Applying the Hilbert transform to each IMF generates an analytical signal with instantaneous amplitude and frequency. Further analysis integrates the joint probability density function $p(\omega ,A)$ to define the arbitrary-order Hilbert marginal spectrum ${\mathcal{L}}_q\left(\omega \right)$ in amplitude-frequency space. The second-order spectrum ${\mathcal{L}}_2$ corresponds to energy density at frequency $\omega$, sharing the physical significance of the Fourier transform spectrum.

The fundamental steps of the HHT, including the EMD process and the calculation of Hilbert spectra, are detailed in Appendix A.

3.2. Anisotropy and Turbulent Motion

The Reynolds stress tensor characterizes the stress distribution caused by turbulent fluctuations in the fluid flow and is defined as

$${\tau }_{ij}=\overline{{u_i}^{\prime }{u_j}^{\prime }},$$

(6)

where $u_i=u,v,w$ are the wind speeds in the directions of prevailing wind, cross wind, and vertical wind. The non-dimensional anisotropy tensor is defined as the normalized ${\tau }_{ij}$ subtracting the isotropic components, which holds the definition as

$$b_{ij}={\displaystyle \frac{\overline{{u_i}^{\prime }{u_j}^{\prime }}}{\overline{{u_m}^{\prime }{u_m}^{\prime }}}}-{\displaystyle \frac{1}{3}}{\delta }_{ij},$$

(7)

where ${\delta }_{ij}$ is the Kronecker delta, and the index $m$ appearing twice in a single term follows the Einstein summation notation. From the three eigenvalues of the normalized anisotropy tensor, ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, and the different states can be depicted graphically through the Barycentric Anisotropy Map (BAM) with respect to the parameters $x_B$ and $y_B$ [51], which is considered in this study from a linearized point of view:

$$x_B=C_{1c}{x}_{1c}+C_{2c}{x}_{2c}+C_{3c}{x}_{3c}={\lambda }_1-{\lambda }_2+{\displaystyle \frac{1}{2}}(3{\lambda }_3+1),$$

(8)

$$y_B=C_{1c}{y}_{1c}+C_{2c}{y}_{2c}+C_{3c}{y}_{3c}={\displaystyle \frac{\sqrt{3}}{2}}(3{\lambda }_3+1),$$

(9)

where $C_{1c}={\lambda }_1-{\lambda }_2$, $C_{2c}=2({\lambda }_2-{\lambda }_3)$, $C_{3c}=3{\lambda }_3+1$ are the weights determined by the Reynolds stress, and $(x_{1c},y_{1c})=\left(\mathrm{1,0}\right)$, $(x_{2c},y_{2c})=\left(\mathrm{0,0}\right)$, $(x_{3c},y_{3c})=(1/2,\sqrt{3}/2)$ are the three vertices of a unit equilateral triangle, which corresponds to the different limiting states of turbulence anisotropy.

Figure 2 shows the distribution of the anisotropy of the observational data from three different stations. It reveals that the data from different stations collectively reflect a broad range of atmospheric turbulence conditions. For the data from Tianjin, the complex terrain, weather, and anthropogenic conditions cause the anisotropy to be widely distributed in the BAM, revealing both idealized and extreme cases. Furthermore, multiple strongly unstable conditions are observed, in which the turbulence is fully developed and reaches the isotropic limit. For the latter two stations, the turbulence is less disturbed as a result of the high homogeneity and minimal change in characteristics of the underlying surface, and the anisotropy concentrates around specific areas in the BAM. Specifically, in Dezhou, there are more stable cases, the horizontal motions are driven by the dominant wind, with vertical motions constrained by the near-surface inversion layer. Therefore, the turbulence is highly anisotropic, as reflected by the scatters at the bottom of BAM in Figure 2c,f. It can be observed that $x_B$ has positive correlations with the absolute value of the stability parameter $\zeta =z/L$. There is also a strong positive correlation between $y_B$ and $\left|\zeta \right|$, especially in Figure 2d, but there are also cases where the correlation becomes weaker or even slightly opposite.

Incorporating the multivariate EMD process in Section 3.1, the anisotropy of each IMF can also be analyzed to reveal a comprehensive view of different scales. The analysis at different scales can be applied in two ways:

(1)

By calculating the anisotropy of each IMF. Note the kth IMF of the wind speed component $u_i$ as

$$u_{i,k}=C_{i,k}\left(t\right).$$

(10)

The non-dimensional anisotropy tensor of the kth IMF can be defined by

$$b_{ij,k}={\displaystyle \frac{\overline{{C_{i,k}}^{\prime }\left(t\right){C_{j,k}}^{\prime }\left(t\right)}}{\overline{{C_{m,k}}^{\prime }\left(t\right){C_{m,k}}^{\prime }\left(t\right)}}}-{\displaystyle \frac{1}{3}}{\delta }_{ij},$$

(11)

The eigenvalues and parameters $x_{B,k}$ and $y_{B,k}$ of the kth IMF are calculated accordingly, yielding the $y_B-x_B$ trajectory.

(2)

By superimposing the IMFs of the signal from high frequencies to low frequencies (namely from small to large scale), the superimposed signal from 1st to kth IMF is given by

$$U_{i,k}=\sum _{n=1}^k{C}_{i,n}\left(t\right).$$

(12)

The anisotropy tensor, eigenvalues, and the parameters ${x^{\prime }}_{B,k}$ and ${y^{\prime }}_{B,k}$ are calculated the same way as the 1st way, and are denoted as the superimposed $y_B-x_B$ trajectory. The residuals $r_n$ are not considered as they correspond with motions around ${10}^{-4}$ Hz, which typically fall beyond the energy-containing eddies of atmospheric turbulence.

To separate and retrieve the turbulence signal from the data series, the upper limit scale of turbulence can be evaluated by an anisotropy analysis in the frequency domain [39]. Technically, atmospheric turbulence and non-turbulent motions typically differ in scale and can be distinguished using several criteria [20,39]. Non-turbulent motions generally exhibit a lower frequency of motion. In such cases, a boundary frequency, $f_B$, can be proposed to mark the transition between turbulent transport and other forms of transport. In this study, the boundary frequency is determined using the properties of anisotropy, given that all turbulent transport is closely related to the dynamics of atmospheric turbulence. The time series can then be expressed as

$$X\left(t\right)=X_{turb}\left(t\right)+X_{non-turb}\left(t\right),$$

(13)

where $X_{turb}$ contains the signal with frequencies higher than $f_B$, and $X_{non-turb}$ contains the signal with frequencies lower than $f_B$. Note the number of IMF with a frequency closest to $f_B$ as $k_B$. These two kinds of signals can be calculated as

$$X_{turb}\left(t\right)=\sum _{k=1}^{k_B}{C}_k\left(t\right),$$

(14)

$$X_{non-turb}\left(t\right)=\sum _{k=k_B+1}^n{C}_k\left(t\right)+r_n\left(t\right).$$

(15)

In this way, the turbulence data is reconstructed as $X_{turb}\left(t\right)$, and all the turbulent fluxes calculated by Equations (2)–(5) should be replaced with the reconstructed turbulent variables to eliminate other forms of transport.

4. Results

4.1. Scale-Dependent Properties of Anisotropy

Figure 3 illustrates the changes in anisotropy properties with the scale of motions through the parameters $x_B$ and $y_B$ calculated from different IMFs across the averaging time span (introduced in Section 2.2), and data from all three observational stations are used for the general analysis. Each trajectory in Figure 3a,c,e,g corresponds to the average changing pattern of anisotropy with frequency in a specific range of stratification conditions. The trajectories in Figure 3a,e represent the $y_B-x_B$ trajectory calculated from the shift in single IMFs, while the ones in Figure 3c,g represent the superimposed IMFs. As shown in Figure 3a,e, under different stratification conditions, the $y_B-x_B$ trajectory follows a regular pattern:

(1)

From small scale to large scale, the $x_B$ initially remains constant around 0.4 to 0.5. When the average frequency of motions decreases below ${10}^{-2}\mathrm{H}\mathrm{z}$, the $x_B$ starts to increase rapidly, reaching up to 0.8 to 0.9.

(2)

From small scale to large scale, the $y_B$ initially increases slightly from 0.6 to 0.7 before the average frequency decreases to $0.5\mathrm{H}\mathrm{z}$, and for larger scales (lower frequencies), $y_B$ continues to decrease until it approaches 0.

(3)

With the increase in the absolute values of stability parameters, the bending of the $y_B-x_B$ trajectory is weakened. The location where the bending occurs is roughly limited in the trapezoidal gray area in Figure 3a,e, which holds the boundary at $(1/\mathrm{3,0})$, $\left(\mathrm{1,0}\right)$, $(5/6,\sqrt{3}/6)$, and $(1/2,\sqrt{3}/6)$.

These findings align with the results of previous studies [55,68], confirming that the atmospheric turbulence approaches isotropy at small scales, and the anisotropy primarily originates from larger-scale motions. The comparable anisotropy patterns in Figure 3a,e implicates minimal variations between unstable and stable stratification conditions [55]. The fluctuation of $y_B$ near the small-scale ends exist for most $y_B-x_B$ trajectories, which can be attributed to the loss of accuracy of high-frequency observations and the decomposition of signals. However, in extremely stable conditions, the end of the trajectory appears away from the isotropic limit, implying that in stable boundary layers, the development of turbulence becomes non-Kolmogorov where local isotropy is not achieved [69,70]. It can be observed from the differentiated lines within the same panels that for different ranges of the absolute value of the stability parameter $\left|\zeta \right|\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$ the difference in the bending of $y_B-x_B$ trajectory is revealed, which originates from different physical processes. In near-neutral cases marked by dotted lines, the vertical forcing is limited, and the motions of middle scales (${10}^{-2}~{10}^{-1}\mathrm{H}\mathrm{z}$) transition to the axisymmetric two-component state near the left-bottom vertex of BAM, before bending to the one-component limit (right-bottom vertex) with the dominant parameter $x_B$ following the dominant wind speed. On the contrary, under extreme conditions ($\left|\zeta \right|\gg 1$), turbulence dynamics become vertically dominated in two opposing regimes, both resulting in the weakened bending in the middle of the $y_B-x_B$ trajectory. For strongly unstable cases, the enhanced vertical forcing and efficient vertical mixing lead to the balanced distribution of the Reynolds stress tensor. In this situation, both the TKE and the covariances of wind speed are similar, with less dominant or minor values favoring the two-component state (in the direction of bending). For strongly stable cases, the turbulence generation is mechanically weakened for a considerable time, resulting in both minimal horizontal and vertical winds with frequent turbulent intermittency. In this situation, the distribution of the Reynolds stress tensor is also balanced, making the two-component state also no longer favored. The above discussion can be corroborated from the overall density distribution shown in Figure 3b,f. There are high-density areas in the upper-middle of the BAM, indicating the high importance of the $y_B$ parameter. However, considering the anisotropy properties at larger scales, the $x_B$ parameter may also be significant. Between the frequency range ${10}^{-2}\mathrm{H}\mathrm{z}$ and ${10}^{-3}\mathrm{H}\mathrm{z}$, the change in $x_B$ is faster than that in $y_B$, suggesting its potential relevance.

Figure 3c,d,g,h depict the changes in the superimposed anisotropy properties, which are similar to Figure 3a,b; however, each point on the trajectories corresponds to the superimposition of IMFs, starting from the smallest scale. The main physical difference between $y_B-x_B$ trajectory and the superimposed $y_B-x_B$ trajectory is that the former decomposes and reveals the anisotropy properties of the motion within specific scales, while the latter visualizes the anisotropy as an aggregation of a certain scale together with all the smaller scales. As shown in Figure 3c,g, in most cases, the changing patterns of the trajectory of parameters $x_B$ and $y_B$ from superimposed IMFs reveal similar information to that from a single IMF. However, the differentiation in the converging points allows for more conclusions to be drawn. The superimposition of IMFs results in the trajectory reaching the center of the BAM, rather than the bottom, which indicates a transition from isotropic to a balanced state the three limiting states contribute equally. However, the convergence points for the large-scale end of the superimposed $y_B-x_B$ trajectory are highly dependent on the stratification condition. In stable conditions, the final $y_B$ of the trajectories are close, while the $x_B$ shows an increasing trend with the increase of $\zeta$, making the cluster of trajectories resemble branches away from the small-scale isotropic end and towards the 1-component limit [57,70]. In contrast, both the final $x_B$ and $y_B$ increase together with $\left|\zeta \right|$, indicating that the more unstable the stratification is, the turbulence becomes closer to isotropic. These properties shape the overall density distribution, as Figure 3d,h. The change in anisotropy in the superimposed $y_B-x_B$ trajectory is more influenced by the parameter $y_B$ for small-scale motions, which explains the prominence of $y_B$ in previous studies [58,59]. However, it can be judged by all the results in Figure 3 that $x_B$ also plays an important role, especially for larger scales.

The analysis above focuses on the differentiation of anisotropy at large-scale ends. Conversely, from the perspective of the scales from large to small, a return-to-isotropy pattern can also be observed, which is consistent with previous scale-dependent analysis in the BAM [71]. There are also studies concerning both the phenomenon of isotropic turn with the relaxed development of turbulence [72,73,74], and the isotropic transition of the multi-scale turbulence data samples [55,68]. The results in this study reveal a similar transition to isotropic from different anisotropic states of various stratification conditions, providing a novel view that enables the decomposition and reconstruction of turbulence data. Practically, the scale-dependent anisotropy is relevant when examining the properties of motions at a specific scale. Section 3.1 has demonstrated that a high index of IMFs and the residual of EMD correspond to large-scale motions within the observational data, where non-turbulent motions often arise. From the above analysis, the properties of these motions are more accurately reflected through the parameter $x_B$ and dominate the point distribution in BAM. Therefore, analyzing the large-scale ends of $y_B-x_B$ trajectories can provide heuristic insights into identifying non-turbulent motions within the observational data.

4.2. Distinguishing Non-Turbulent Motions

Section 4.1 has demonstrated that the anisotropy of atmospheric turbulence is scale-dependent, exhibiting a regular pattern in the $y_B-x_B$ trajectory with the shift in IMFs in the BAM. The trapezoidal area introduced in the last section warrants focus due to its $x_B$-dominant and monotonous characteristics. However, in individual cases, this pattern happens to be violated at IMFs of larger scales, as reflected by outliers from the mean $y_B-x_B$ trajectory in the trapezoidal area. In previous studies, the violation of statistical characteristics of atmospheric turbulence tends to be taken as the criterion of the existence of non-turbulent motions, such as the spectral gap [20,75] and the transport properties [39]. To distinguish non-turbulent motions from observed signals, an integrated criterion can be proposed. Note that the $y_B-x_B$ trajectory in the trapezoidal area is monotonic for all the conditions (as the left-top boundary marks the bending place of the trajectory), the criterion can be summarized by identifying the violation of the regular pattern through the following steps:

(1)

Collect the points violating the monotonicity of $y_B$ within the trapezoidal area.

(2)

Collect the points violating the monotonicity of $x_B$ within the trapezoidal area.

(3)

Determine whether the collected points are statistical outliers: if they are, the points are kept for the next step; otherwise, they are discarded.

(4)

Locate the index of IMFs for these outliers and the average frequencies.

The statistical outliers are defined as values that exceed three times the standard deviation from the local median within a sliding window of 5 points. If more than one point meets the criterion, the one with the highest frequency is considered the boundary between turbulent and non-turbulent motions. If no points satisfy the criterion, the boundary frequency is set to the IMF of the largest scale, and the residual of EMD is considered as the non-turbulent motion. The identification of statistical outliers is determined through established statistical principles such as the three-sigma rule, which aims at avoiding misinterpreting data fluctuations and enhancing the reliability of the algorithm. However, the distribution of anisotropy is not necessarily Gaussian, which requires multiple tests on this criterion. The results have proven that the anisotropy-based algorithm effectively distinguishes the existence of non-turbulent motions from the anomaly of $y_B-x_B$ trajectory under different conditions, and Figure 4 presents two typical cases of stable and unstable stratifications and visualizes the outliers as the boundaries between turbulent and non-turbulent motions. As seen in Figure 4a, the $y_B-x_B$ trajectory may appear to bend in individual cases, but generally aligns with the overall pattern observed in Figure 3a. However, certain points deviate from this pattern, mostly in the form of the violation of the monotonicity of the $y_B-x_B$ trajectory, which indicates the presence of non-turbulent motions. The right column in Figure 4 displays the time series of the two cases shown in Figure 4a. It can be seen from the time series that the original observational signal is strongly non-stationary, with both meanderings in direction and mutations in amplitude, which complies well with the definitions in the form of non-turbulent motions [15,41]. After the reconstruction through anisotropy properties, the turbulence signal is successfully separated with steadier amplitudes and smaller scales.

In stable conditions reflected in Figure 4b, the non-stationary component in the original signal, together with the local motions that affect the amplitude, can be mostly attributed to the non-turbulent motion. Moreover, in stable conditions, the complexity of non-turbulent motion is higher and the average scale is smaller, as seen from the comparison between Figure 4b,c. This is primarily due to static and steady meteorological conditions limiting motions in the stable boundary layer and often inducing intermittency in turbulence. In unstable conditions, however, the scale of both turbulence and non-turbulent motion is larger, and they happen to interact and are thus difficult to distinguish. From Figure 5c, it can be observed that non-turbulent motion is identified as the large-scale component of the original signal, conforming to the typical definitions. Meanwhile, the coherent structure, a large-scale form of atmospheric turbulence (see the ramp-like structure in the time series from 0 to 15 min [21], is not misidentified and is effectively preserved. This demonstrates the good performance of anisotropy in distinguishing non-turbulent motions without compromising the original turbulent structure in observational data. As the residual of EMD is automatically identified as non-turbulent motions in the algorithm, the application for small-size data samples (~10 min) and for intensely convective/non-stationary boundary layers should be evaluated twice to ensure the large-scale structures of atmospheric turbulence are not mistakenly filtered.

Once the non-turbulent motions are identified, the turbulent time series can be reconstructed following the instructions in Section 3.2. Figure 5 presents a comparison of turbulent fluxes, which reveals that the absolute value of the reconstructed fluxes tends to be smaller than the original fluxes (calculated using the standardized EC system), with the turbulent momentum flux overestimated by an average of 39%, sensible heat flux by 32%, latent heat flux by 38%, CO₂ flux by 45%, and the PM_2.5 flux by 83%. This result suggests that non-turbulent motions, which hold larger scales and evident non-stationary characteristics, typically lead to a significant overestimation of turbulent fluxes [39,76]. In the meantime, there are minor instances where the reconstructed flux is higher than the original flux, indicating underestimation and corresponding to counter-gradient transport [13,14]. The results reveal that non-turbulent motions typically generate large-scale transport aligned with turbulent flux directions. The corrections of turbulent flux demonstrate broader applicability for momentum and sensible heat fluxes, as these incorporate measurements from all three stations. The misjudgment of material fluxes can be attributed to non-stationarity from the sources and sinks, and for PM_2.5, the predominant stable boundary layer (and possibly, the different season of winter) within the observational span of Dezhou station also makes a great difference. Similarly to prior studies, misjudgment of greenhouse gas and pollutant transport is more profound in stable conditions, where the calm and steady synoptic conditions limit the development of turbulence and result in turbulence intermittency [20,75]. Although some of the values above are based on specific conditions, the flux corrections can be representative in an arbitrary region based on historical data under characteristic synoptic conditions, and are transferable to future observations. For example, for summer and flat underlying surfaces, sufficient mixing favors the isotropy turbulence, with smaller bending for $y_B-x_B$ trajectory and fewer misjudgments of turbulent fluxes. Correspondingly, in other seasons (and complex surfaces), the turbulent fluxes tend to need larger corrections as the locality and anisotropy of turbulence (and thereby the influence of non-turbulent motions) become higher. The analysis above reveals that the anisotropy-based turbulence reconstruction provides substantial reference for critical applications like carbon cycle assessment and pollution control, where traditional methods falter.

Once non-turbulent motions are eliminated, the spectra change accordingly, with the energy of larger scales of motion decreasing, as shown in Figure 6. Following the procedures of HHT, the 2nd order of Hilbert marginal spectra is shown to reveal the scale-dependent energy distribution. The horizontal axes in the spectra are standardized into 40 bins. In higher frequency ranges, where there is no non-turbulent motion, the points of the original and reconstructed spectra overlap, resembling hexagrams. In lower frequency ranges, however, the spectra deviate. The location of this deviation marks the boundary frequency between turbulent and non-turbulent motions and is marked in the spectra as dashed lines. Comparing the 1st and 2nd columns of Figure 6, it can be judged that non-turbulent motions tend to occur around ${10}^{-2}\mathrm{H}\mathrm{z}$, corresponding to a time scale of ${10}^2$ s. For unstable stratification conditions, most boundary frequencies $f_B$ take place at frequencies lower than ${10}^{-2}\mathrm{H}\mathrm{z}$ in most cases, while for stable conditions, $f_B$ tends to be slightly higher than ${10}^{-2}\mathrm{H}\mathrm{z}$. The boundary frequency also proves to be dependent on the stability parameter $\zeta$, as shown by the interesting dispersion of the dashed lines in Figure 6a–j. In unstable conditions, the lines are spread wider with a regular pattern of sequence, except for PM_2.5, where many lines overlap. In stable conditions, the lines are closer to each other, and the overlapping phenomenon is more evident. This serves as evidence that for different stratification conditions, the non-turbulent motions might have various forms, different scales, and potential interaction with turbulence. However, based on the scale-dependent anisotropy properties, the existence of non-turbulent motions can be effectively identified and eliminated. Therefore, the anisotropy-based turbulence reconstruction provides a robust correction framework for land–atmosphere transport.

The 3rd column in Figure 6 works in concert with the dashed lines in the left two columns and provides a more detailed illustration of the relationship between $f_B$ and $\zeta$. With the increase in the absolute values of $\zeta$, the boundary frequency usually decreases. For unstable stratifications with strong surface heating or convective instability, atmospheric turbulence intensifies and exhibits an increased characteristic scale, and the observations capture a broader spectrum of motion scales, extending from fine turbulent eddies to larger organized structures. The effects of non-turbulent motions tend to interact and blur with the coherent structures of turbulence, with overlapped spectra, favoring a larger size of data samples. On the contrary, in stable stratifications, the turbulence itself is limited in strength and scale by the static and steady meteorological conditions, and its deviation from the non-turbulent motions only depends on the maximum scale of turbulence, making the decreasing trend of $f_B$ less significant. This behavior makes it easier to distinguish non-turbulent motions in the stable boundary layer, whose other characteristics, such as the spectral gap [20,75], are also more evident. For both scenarios, the non-turbulent motions lack involvement in the Kolmogorov energy cascade, and most of them can be resolved by large-scale models, which should not be recalculated via parameterization schemes and can be effectively eliminated from the turbulent reconstruction based on the anisotropy of atmospheric turbulence.

4.3. Similarity Relationships and Anisotropy

Compared with the direct calculation and correction of turbulent fluxes in the EC system, in which intensive computations are involved, introducing anisotropy into similarity relationships may hold significant promise for the estimation of land–atmosphere transport across multiple datasets under a substantially reduced cost. Previous studies have sufficiently analyzed the effectiveness of anisotropy in generalizing MOST in both normalized standard deviations and the flux-gradient profiles [58,59], and the analyses above suggest that both $x_B$ and $y_B$ are promising for advancing the performance of MOST in the prediction of atmospheric turbulence properties in different stratification conditions. To further investigate the joint influence of the two anisotropy parameters, Figure 7 compares the similarity relationships and the anisotropy parameters using turbulence data reconstructed in Section 4.2, where $f_u={\sigma }_u/u_{\ast }$ and $f_{\theta }={\sigma }_{\theta }/{\theta }_{\ast }=-{\sigma }_{\theta }\overline{w^{\prime }{\theta }^{\prime }}/u_{\ast }$ are the normalized standard deviations of wind speed and potential temperature. The results conform well to the M-O similarity theory (the relationships of $f_v$ and $f_w$ are similar to $f_u$ and are therefore omitted), even though the data were not strictly selected for near-ideal conditions. Figure 7a–d indicates that as the stability parameter $\zeta$ changes from small to large absolute values, the average $x_B$ varies from small to large values. However, for points that deviate significantly from the empirical curves, $x_B$ exhibits larger values. This phenomenon demonstrates that the anisotropy parameter $x_B$ is closely related to the dynamic properties of atmospheric turbulence, providing a physical basis for incorporating $x_B$ into similarity relationships. Similarly, Figure 7e–h shows increasing trends of $y_B$ with increasing absolute values of $\zeta$, and the outliers from the scaling curves correspond with particularly low $y_B$. The correlation relationships of $x_B$, $y_B$ and $\left|\zeta \right|$ align well with the results in Figure 3.

Figure 7’s heuristic approach effectively addresses limitations in traditional MOST under non-ideal conditions. By deliberately incorporating anisotropy parameters ($x_B$ and $y_B$)—both critically linked to turbulent flux characteristics—the modified similarity relationships demonstrate superior performance, even when tested against observational data selected solely by wind direction without excluding non-ideal conditions. This robust validation confirms that anisotropy integration simultaneously enhances similarity relationship precision and expands its applicability across diverse atmospheric regimes. Similar analysis can also be applied to the normalized standard deviations and flux–profile relationships concerning CO₂, water vapor, and PM_2.5, which holds promise for enhancing predictive accuracy for critical sustainability applications, including weather/climate prediction, greenhouse effect quantification, and urban pollution mitigation. Therefore, the advances above based on anisotropy enable reconstruction of pure turbulence data and significantly improve turbulent flux estimates, providing essential theoretical foundations for optimizing parameterization schemes of land–atmosphere transport.

5. Conclusions

In this study, anisotropy derived from the Reynolds stress tensor is quantified using two parameters, $x_B$ and $y_B$, based on turbulence data acquired from multiple observational stations with different underlying surfaces and stratification conditions. The properties of anisotropy are investigated across different scales using HHT, and the scale-dependent properties are summarized.

From small to large-scale turbulent motions, $x_B$ exhibits an overall increasing trend from 0.4 to 0.9, while $y_B$ initially increases to 0.7 before decreasing to 0. This regular pattern forms the circuitous curve of the $y_B-x_B$ trajectory in the BAM for both unstable and stable stratification conditions. However, in individual cases, the $y_B-x_B$ trajectory sometimes deviates from the average pattern. Using a comprehensive algorithm, these outliers can be identified and serve as the boundary between turbulent and non-turbulent motions, occurring at a time scale of approximately ${10}^2$ seconds. Once non-turbulent motions are identified, the turbulence data can be reconstructed, with the spectra at large scales being weaker than the original data. Moreover, the turbulent fluxes calculated from the reconstructed data are typically lower than those from the standardized EC system. The overestimation of turbulent momentum, sensible heat, and latent heat fluxes can vary around 30%, and even higher for CO₂ and PM_2.5, with occasional instances of flux underestimation. The application of flux correction highlights the potential of anisotropy in refining similarity theory, which is intrinsically linked to turbulent transport. It is demonstrated that $x_B$ and $y_B$ vary consistently with the stability parameter, making them suitable for incorporation into similarity relationships. Introducing anisotropy into the similarity theory outperforms direct correction of turbulent flux in both computational efficiency and reliability. These findings demonstrate that the modified similarity relationships not only improve predictions of turbulent transport characteristics but also extend their applicability across a wider range of atmospheric conditions, showing enhanced utilities for both fundamental turbulence research and practical applications.

In conclusion, the stable statistical pattern in the properties of anisotropy serves as a new indicator of the scale-dependent characteristics of atmospheric turbulence and is effective in the accurate estimation of land–atmosphere transport. On the one hand, it provides a systematic scheme for reconstructing pure-turbulent motions, allowing for more precise estimation of turbulent transport. On the other hand, the use of anisotropy in similarity relationships enables a broader application of atmospheric turbulence properties. Consequently, anisotropy-based turbulence reconstruction offers an operationally robust approach for refining land–atmosphere transport quantification and warrants further investigations, which may involve the evaluation of anisotropy’s effectiveness across spatially distributed datasets beyond single-point observations, and the transformation of these findings into parameterization schemes through atmospheric turbulence simulations. Such advancements would benefit diverse domains, including climate projection, greenhouse gas monitoring, and pollution management.