The IN can be simply illustrated by the Brownian motion. A discrete time series of the Brownian motion is generated by cumulatively summing the independent Gaussian samples with zero mean and the same standard deviation σ : 2B(t=NΔt)=∑i=0Ngi, where gi is a Gaussian sample and Δt is the sampling interval. The Brownian motion B(t) is non-stationary because its standard deviation scales as σt.

Two discrete time series of the Brownian motion are shown in Fig. a. The two series are generated by the same Brownian motion; i.e. the statistical distributions of gi are the same for the two series. However, they have different non-stationary trends: sample A has a decreasing trend from t≈104, while sample B has a wave-like trend. We call these non-stationary trends the stochastic trends because they are not attributed to any external forcings but are only attributed to randomness of the time series. As a distinction, the non-stationary trends related to external forcings are called the dynamical trends. Although the stochastic trends are different, the power spectral densities S(f) of two time series are not changed (see Fig. b): both of them agree well with the theoretical prediction that S(f)∼f-2 . Unlike the stochastic trends, different dynamical trends indicate that systems would probably be dominated by different external forcings, and the corresponding power spectral densities could also be different.

The fluctuation analysis (FA) was firstly proposed to detect and quantify possible intrinsic non-stationarity in time series or other sequence data . However, the intrinsic non-stationarity and the non-stationarity caused by external forcing always coexist in a real time series. The FA cannot distinguish between the two kinds of non-stationarity. The detrended fluctuation analysis (DFA) method was then proposed to resolve this problem by eliminating large-scale trends in the data .

The DFA of a time series xk (k=1,2,⋯,N) is briefly listed as follows. In the first step, the profile of xk is calculated by 3Yi=∑k=1ixk-x‾, where x‾ is the time average of xk over the whole time period. In the second step, the profile Yi (i=1,2,⋯,N) is cut into Ns non-overlapping segments with equal timescale s=mΔt, where Δt is the sampling interval of xk and m is a positive integer (1≤m≤N). In the third step, the profile Yi in each segment is fitted by a polynomial pi,jn, where j is the segment index and n is the degree of the polynomial. Then, the fitted polynomial in each segment is removed from the profile: 4Yi,j=Yi-pi,jn. In this step, the dynamical trends modeled by the polynomials are removed, but the IN stochastic trends are left . Generally, the choice of degree n would affect the results when dynamical trends exist in the time series. However, we test the Brownian motion without dynamical trends and the carbon dioxide fluxes with dynamical trends already removed by the Reynolds average and find that the results are not substantially affected by the choice of n. Figure  shows the impact of the choice of n on DFA. Results show that the choice of n from 1 to 4 does not affect the conclusion of the DFA (Fig. a). For the Brownian motion, the fluctuation exponents are almost the same with different degrees of n. For the carbon dioxide turbulent fluxes, the variations of the fluctuation functions also do not vary substantially with n (Fig. b). Thus, we set n=1 in this study. In the fourth step, the variance of Yi,j in each segment is calculated by 5Fj2=1m∑i=1mY(j-1)m+i,j2. Then, the variance Fj2 is averaged over all segments: 6F(s)=1Ns∑j=1NsFj2, where F(s) is called the fluctuation function.

Generally, the fluctuation function behaves as a power function: 7F(s)∼sα, where the fluctuation exponent α can be used to diagnose and quantify IN in the time series of xk . In practice, large statistical errors will occur at large s. Thus, the largest fitting scale is normally set to the position where the F(s) begins to fluctuate around the power function significantly. If the fluctuation exponent α>1, the IN exists in the time series. The more the α deviates from 1, the more significant the IN is. If 1/2<α<1, the time series is stationary and long-term correlated. If α=1/2, the time series is stationary and independent (or short-term correlated). Figure c shows the DFA of the Brownian motion. The fluctuation exponent is close to the theoretical value of 1.5 , which is consistent with the fact that the Brownian motion has IN. Besides, the example also shows that the IN will not be removed in the third step of the DFA.

The data were collected on a 325 m meteorological tower in the downtown of Beijing, China (39.97∘ N, 116.37∘ E). Within 5 km of the tower, there are buildings with a height of about 10–60 m. About 200 m away to the west of the tower, there are a north–south highway bridge and a ring road. About 150 m away to the north of the tower, there is an east–west busy road. The 10 Hz turbulence data, including wind velocity and carbon dioxide concentration, were collected by an ultrasonic anemometer (Windmaster Pro, Gill, UK) and an open-path CO2/H2O analyser (LI-7500, LI-COR, USA) deployed at the 80 m level. Data collected from 28 July to 28 August 2020 are analysed in this study.

Based on the estimation of mean building height , the height of the inertia sublayer around the tower is about 45–135 m . The constant flux layer (i.e. the inertial sublayer) is observed to extend to 140 m, and the 80 m height is located in the constant flux layer . According to , the turbulent fluctuations (with scales less than 1 min) observed on the tower are nearly isotropic, and large-scale motions (with scales greater than 1 min and less than 10 min) are anisotropic. More details about the meteorological tower, the typical meteorological conditions, urban geometry effects, and potential sources of carbon dioxide around the observation site can also be found in and .

The quality control methods proposed by are used to find problematic data, including spikes, dropouts, data with discontinuities, data violating absolute limits, data with the amplitude resolution problem, and data with unphysical high-order moments. Their method used automated tests to identify instrumentation problems and physically plausible but unusual situations in tower time series. Besides, they also proposed automated tests to identify flux sampling problems, such as the non-stationary problem that will be discussed in the following sections. The time series seriously contaminated by the problematic data are removed in the analysis. The time series seriously contaminated by high-frequency white noises are also removed. After quality controlling, a total of 520 1 h time series are left. The instrument reference frame is transformed to the streamline reference frame by the double rotation .

The 1 h time series of carbon dioxide turbulent fluxes is obtained by Eq. (), where the ensemble average is replaced by the time average. In order to remove dynamical trends, the Reynolds average time is usually set to be equal to or smaller than 30 min . We analyse the intrinsic non-stationarity for all the 1 h time series of carbon dioxide turbulent fluxes, and a typical example is shown in Fig. . To analyse the impact of the Reynolds average time on IN in the 1 h time series of carbon dioxide turbulent fluxes, we here choose the Reynolds average times τ=900 s and 300 s that are commonly used in the eddy-covariance method . In order to show the effect of very small Reynolds average times in sharp contrast, we also choose a timescale of 6 s in the analysis.

The DFA is shown in Fig. b. Two scaling regimes are found in the fluctuation functions. At a large timescale s, the fluctuation exponent is found to be less than 1; at a small timescale s, the fluctuation exponent is found to be greater than 1. Results indicate that the time series of carbon dioxide turbulent fluxes have IN at small timescales but are stationary at large timescales, whatever the Reynolds average time is. As shown in Fig. a, the small-scale variations of these time series are evidently non-stationary, although the large-scale dynamical trends have been removed by subtracting the Reynolds average from the data. Besides, one can note that the fluctuation functions with τ=900 and 300 s are almost the same but are different from that with τ=6 s. The crossover scale in the case with τ=6 s (at s≈2 s) is smaller than that in cases with τ=900 and 300 s (at s≈20 s). The power spectral densities of these time series are shown in Fig. c. The spectra with τ=900 and 300 s are almost the same but are also different from that with τ=6 s. The case with τ=6 s is found to have a much shorter inertial sub-range than cases with τ=900 and 300 s. The inertial sub-range is recognized by the Kolmogorov -5/3 law . Results indicate that the IN is a small-scale phenomenon which is intimately related to the inertial sub-range turbulence. The choice of a very small Reynolds average time could partly remove the IN, but the turbulence contribution to fluxes is also partly removed. It is believed that if the sampling frequency is improved and the flux-averaging time is further reduced, the stationary assumption of the eddy-covariance method can be better guaranteed. Our findings indicate that the above consideration may not be right because the further reduction of the flux-averaging time would face the intrinsic non-stationarity.

The Ornstein–Uhlenbeck (OU) process, which is well studied and used to model many physical and chemical processes , is a simple model of small-scale IN. The OU process has similar crossover characteristic as carbon dioxide fluxes. Besides, many statistical properties (including the fluctuation exponents) of the OU process can be solved analytically . We here use this model to simulate the IN of carbon dioxide fluxes.

The discrete time series of the OU process is generated by the iterative equation: 8y(t+Δt)=y(t)-ay(t)Δt+bΔtξ, where a and b are model parameters, Δt is the sampling interval, and ξ is an independent random variable with the normal distribution. For the OU process, the fluctuation function F(s)∼s0.5 at large scales and F(s)∼s1.5 at small scales . This indicates that the OU process has IN at small scales but is stationary at large scales, as clearly illustrated by an example in Figs.  and . Figure a shows the 1 h time series of the OU process generated by Eq. (). Due to the small-scale IN, the time series seems to be intermittent. However, the large-scale variations of the same OU process, obtained by averaging the time series in Fig. a with an average time much greater than the crossover scale, seem to be like a stationary white noise (Fig. b). The DFA shows that the fluctuation exponent of the averaged time series is about 0.5, as the fluctuation exponent of the unaveraged time series at large scales (Fig. ). This indicates that the averaged time series with a large average time, reflecting the large-scale variations of the OU process, is stationary.

The DFA of 520 1 h time series of instantaneous carbon dioxide fluxes is shown in Fig. . The Reynolds average time is set to 5 min. Results show that the fluctuation functions of carbon dioxide turbulent fluxes typically have two scaling regimes, as shown in Fig. . The fluctuation exponents are generally greater than 1 at small scales and less than 1 at large scales. The OU process can fit the data as a first approximation, although the fluctuation exponent of data seems to be greater at large scales and less at small scales compared with the OU process. The details of the fitting procedure are listed as follows. In the first step, choose the parameters of the OU process a and b from the same set (0.1,0.2,⋯,1) and set Δt=0.1 s. The 1 h time series of the OU process is generated by Eq. () with the chosen parameters. In the second step, compute the fluctuation function of the generated 1 h time series. In the third step, go back to the first step and choose another new value of a or b in the set (0.1,0.2,⋯,1). If all possible combinations of a and b are used, go to the fourth step. In the fourth step, the root mean relative square error for the ith combination of a and b is computed: 9RMRSi=1Nj∑j=1NjFi(sj)-Fdata(sj)Fi(sj)2, where Nj is the number of discrete timescales sj, Fi is the fluctuation function of the OU process with the ith combination of a and b, and Fdata is the averaged fluctuation function of carbon dioxide turbulent fluxes (shown by the red line in Fig. ). In the fifth step, the parameters of a and b corresponding to the minimum of RMRSi are considered the optimal fitting parameters.

The fluctuation exponent of the OU process at large scales equals 0.5. The fact that the fluctuation exponent of data is greater than that of the OU process but less than 1 at large scales indicates that the data are stationary and long-term correlated at large scales. This could be related to the large-scale coherent structure of scalar turbulence . The fluctuation exponent of the OU process at small scales equals that of the Brownian motion. The fluctuation exponent of data seems to be less than that of the OU process at small scales, which indicates that the data deviate from the Brownian motion at small scales. This could be related to the non-Gaussian intermittency of turbulence in the inertial subrange . As we have discussed in Sect. 3.1, the IN is intimately related to the inertial sub-range turbulence, which is usually considered to be produced by the cascade mechanism . The OU process is a very simple mathematical model that does not include the cascade mechanism. It is believed that the fitting results would be improved by adding the cascade mechanism to the OU process. This paper focuses on the main characteristics of the IN, and further extensions of the OU process will be investigated in a future study.

There are at least two impacts of IN on the calculation of average carbon dioxide turbulent fluxes.

First, the IN could affect the short-term averaged turbulent flux normally used in the analysis of plant photosynthesis efficiency . To avoid IN at small scales, the average time-averaging instantaneous turbulent fluxes (i.e. the flux-averaging time) should be much greater than the crossover scale in the fluctuation function, because crossover scale separates the IN at small scales and stationarity at large scales. Note that the flux-averaging time is not necessarily the same as the Reynolds average time . The former is denoted by T in the following discussion. For the OU process , the crossover scale is 10s×≈5.4a. According to the fitting results in Fig. , the crossover scale of carbon dioxide turbulent fluxes is about 27 s. The errors of fluxes averaged with T≲s×≈27 s would be large due to the existence of small-scale IN.

Second, the IN could affect the diagnosis methods of non-stationarity. For example, used a dimensionless index RN to diagnose non-stationarity: 11RN=δxx‾, where δx is the difference between the beginning and the end of the linear regression trend of the diagnosed time series and x‾ is the time average of the same time series. If RN is greater than a predefined threshold, the time series is diagnosed as non-stationary and is not recommended to be averaged by time. We here use the RN method for the OU process. Because the OU process is stationary at large scales, it is meaningful to calculate its average with a large average time. Thus, we hope that the OU process can be diagnosed as stationary by the RN method. The proportion of diagnosed stationarity for the OU process is plotted as a function of threshold in Fig. . Results show that once the threshold is less than a critical value, the RN method has a certain probability of making a wrong diagnosis. With the decrease in the threshold, the probability of misdiagnosis will increase. The critical threshold increases as the parameter a decreases. In the limit case with a=0, the OU process with small-scale IN becomes the Brownian motion with full-scale IN (see Eq. ). We thus hope that the proportion of diagnosed stationarity for the Brownian motion is 0; however, the RN method has the probability of misdiagnosis almost at any threshold. In another limit case of the white noises without non-stationarity, the RN method performs well, and the probability of misdiagnosis is 0 for most thresholds. The results remind us that the parameters of diagnosis methods must be carefully chosen when diagnosing carbon dioxide fluxes with IN.