Stationarity is a critical assumption in the eddy-covariance method that is widely used to calculate turbulent fluxes. Many methods have been proposed to diagnose non-stationarity attributed to external non-turbulent flows. In this paper, we focus on intrinsic non-stationarity (IN) attributed to turbulence randomness. The detrended fluctuation analysis is used to quantify IN of

The vertical transport of carbon dioxide plays an important role in estimating the exchange of carbon dioxide between the atmosphere and other systems, including the land

Illustration of intrinsic non-stationarity by the Brownian motion.

The non-stationarity attributed to various non-turbulent flows or external forcings has gained much attention in the literature

In this paper, we focus on the IN of carbon dioxide turbulent fluxes in the urban boundary layer. We firstly illustrate the IN by a simple stochastic model in Sect.

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

Two discrete time series of the Brownian motion are shown in Fig.

The DFA of

The fluctuation analysis (FA) was firstly proposed to detect and quantify possible intrinsic non-stationarity in time series or other sequence data

The DFA of a time series

Generally, the fluctuation function behaves as a power function:

The intrinsic non-stationarity in the 1 h time series of carbon dioxide turbulent fluxes.

The data were collected on a 325 m meteorological tower in the downtown of Beijing, China (39.97

Based on the estimation of mean building height

The quality control methods proposed by

The 1 h time series of carbon dioxide turbulent fluxes is obtained by Eq. (

The DFA is shown in Fig.

Small-scale non-stationarity and large-scale stationarity in the same OU process.

The Ornstein–Uhlenbeck (OU) process, which is well studied and used to model many physical and chemical processes

The discrete time series of the OU process is generated by the iterative equation:

The DFA of the time series shown in Fig.

The DFA of 520 1 h time series of instantaneous carbon dioxide fluxes is shown in Fig.

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 detrended fluctuation analysis of all 1 h time series of carbon dioxide fluxes. The Reynolds average time is set to 5 min to calculate fluxes. The sample-averaged fluctuation function is shown by the red line and uncertainties estimated by the standard deviation are shown by the red shading. The fitted fluctuation function of the OU process is shown by the blue line. The fitted parameters are

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

Second, the IN could affect the diagnosis methods of non-stationarity. For example,

The impact of IN on the non-stationarity diagnosis method proposed by

We analyse the time series carbon dioxide fluxes observed by the eddy-covariance system in the downtown of Beijing and find a new kind of non-stationarity less discussed in the literature. As illustrated by the Brownian motion, the new kind of non-stationarity has nothing to do with non-stationarity attributed to non-turbulent flows or external forcings; therefore, it is called the intrinsic non-stationarity (IN). The detrended fluctuation analysis (DFA) is a useful method to measure IN in real time series where IN always coexists with non-stationarity by external forcings. The DFA shows that the instantaneous turbulent fluxes of carbon dioxide have IN at small timescales. Combined with the spectral analysis, the IN is found to be related to inertial sub-range turbulence. The small-scale IN can be simulated by the Ornstein–Uhlenbeck (OU) process as a first approximation. The potential impacts of IN on the calculation of turbulent fluxes are also discussed. According to the OU process, the crossover scale, which is the characteristic scale under which the IN cannot be ignored, is estimated to be about 27 s. Thus, the IN could contribute systematical errors to short-term averaged fluxes when the average time is not much greater than the crossover time. Besides, we also find that there may be a probability of misdiagnosis when applying some non-stationarity diagnosis method to the time series with IN. Thus, IN should be seriously considered when designing new diagnosis methods.

This work only focuses on the main characteristics of IN of carbon dioxide fluxes in the urban boundary layer. It is interesting to discuss the difference characteristics of IN between the urban and rural boundary layer. The relationships between the IN characteristics (e.g. the crossover scale and fluctuation exponents) and urban boundary layer parameters (e.g. stability, roughness, boundary-layer height) should be systematically studied. The extensions of the OU process should be tried to obtain a better fitting with data. Except for the carbon dioxide turbulent flux, is there IN in other turbulent fluxes with different terrains? The above problems remain to be resolved in the future study.

The Matlab code of the DFA is provided by Martin Magris (downloadable at

FH and LL conceived the idea. LL finished all analysis and wrote the manuscript. YS contributed to revising the manuscript and editing the plots. All the authors contributed to the interpretation of the results.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the National Natural Science Foundation of China (grant nos. 42175101 and 41975018) and the China Postdoctoral Science Foundation (grant no. 2020M670420).

This paper was edited by Harindra Joseph Fernando and reviewed by two anonymous referees.