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Ergodicity of turbulence measurements upon complex terrain in Loess Plateau

Science China Press

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IMAGE: The comparison of the ergodicities of the autocorrelation function of radial wind turbulence under different stratification and scale conditions. view more 

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Loess Plateau possesses a particular loess physiognomy with numerous ravines and slopes, and tableland is a typical landform in it. Together, the ununiform in both topographic undulation and land coverage compose the ununiform, complex underlying surface on Loess Plateau. This provides a special platform for research of turbulence above the complex underlying surface.

As the front-edge problem encountered in the atmospheric boundary layer thesis, the turbulence research for complex underlying surface has drawn extensive attention recently. Local similarity has already proven that under certain condition, theories of turbulence based on the uniform underlying surface can also be applied to which for the ununiform underlying surface. However, as there are still many inapplicable aspects for the complex underlying surface due to its complexity. For eddy-correlation technique, the basic principles of turbulence measurements are ensemble averages of certain space, time, and status, but it is impossible to set up massive equipment within a limited space and obtain all-state for turbulence eddy for a required period time while meeting the ensemble average. Therefore, this experiment bases on such assumption that the fluid field is steady and space are horizontally uniform by replacing the ensemble average with the average of a long-time measurement at one site. So it is necessary to regard the examination of ergodicity for the eddy-correlation technique as the precondition for the experimental research of turbulence in complex terrain of Loess Plateau tableland in its early stage.

Originated in statistic mechanics, the ergodic hypothesis, a principle of examination in microcosmic problem from macro-perspective, has been widely applied recently. Ergodicity theorem has given necessary and sufficient conditions for the stationary stochastic process to satisfy ergodicity. A few researches have revealled the ergodicity of turbulence from Navier-Stokes formula and qualitative researches. We have studied the ergodicity of turbulence measurements above uniform underlying surface. However, it is not researched how atmospheric stratification stability influences on the satisfaction of the ergodic hypothesis. Moreover, low-frequency processes are non-negligible in the estimation of turbulence flux. Under the existing observation conditions, what range of low-frequency processes can satisfy the ergodichy pothesis? These problems are imperative in the early stage of experimental research in turbulence and will be the basis of further researches.

Based on the analysis above, the possible main issues that influence accuracies in measurements of turbulences are the ergodicity of incomplete turbulences such as the unavoidable flows around bodies, especially for those on the Loess Plateau. So, the effects of Monin-Obukhov (M-O) stability on the ergodicity of different scale turbulence in surface layer are studied by applying turbulence data in the tableland of Loess Plateau in this paper. The main goal is to establish a proper scale for turbulence measurements and to improve the estimation accuracy on turbulence flux.

The comprehensive comparison of ergodicity for different-scale horizontal wind velocity, vertical wind velocity, and temperature turbulence under various stratification stability conditions fails to conclude a law of variation for turbulence measurement ergodicity with respect to the change of stratification stability. Turbulence measurement results show that with more steady turbulence signal, it can satisfy ergodicity more easily. Essentially, the steadiness of the turbulence signal is consistent with the uniformity of turbulence distribution. This means that during the single-point turbulence measurement, the more uniform the distribution of turbulence during a longtime interval, the higher the accuracy of the turbulence measurement can be. The steady turbulent field is more likely to appear on the Loess Plateau than on the flat underlying surface. The smallest-scale eddies inside the inertial sub-regionare unsteady since the intension of the energy region and the atmospheric stratification conditions affects the frequency and the waveband corresponding to the Kolmogorov energy cascade (fE(f)?f-2/3) part. For the large-scale eddies with a three-dimensional coherent structure, difference in energy spectrum will affect the position and energy size with corresponding frequency in the fE(f)?f-2/3 spectral coverage, and cause the ununiform distribution of eddies in the fE(f)?f-2/3 energy spectrum. However, as long as the measurement time of 1 h is long enough, though the distribution of the small-scale eddies is ununiform, statistically, they can satisfy ergodicity within certain stratification range, and it can be called as quasi-ergodicity.

The reseach suggests that ergodicity in turbulence measurement under the condition of loess plateau tableland complex terrain features following characteristics: (1) Whether the turbulence can satisfy ergodicity depends on the steadiness of various-scale turbulence distribution, and it does not show simple variation with the change of M-O stability. Generally, from the weakly stable stratification to the weakly unstable stratification, turbulence can satisfy ergodicity relatively easily. Under the condition of extremely stable stratification, turbulence does not readily satisfy ergodicity due to the change in large-scale, poor periodic temperature and wind velocity caused by turbulence intermittence. (2) The influence of terrain tends to result in a long-period steady turbulence coherent structure. Therefore, comparing with the flat underlying surface, relatively large-scale turbulence under the condition of complex terrain tends to satisfy ergodicity more easily. (3) For single-point turbulence measurement of 1h, though the small-scale turbulence can satisfy ergodicity, due to the difference in intensity in different locations of large-scale turbulence eddies, the small-scale eddies, with the scale of dozens of seconds, in the inertial sub-region is unsteady.

In sum, measurement of ergodicity can not only help understand if turbulence measurement can satisfy theory requirements, but can also help master the distribution of eddies with various scales in-depth and establish a sweeping scheme for problems such as lidar turbulence measurement. More importantly, ergodicity from single-point turbulence measurement features relatively significant differences with multipoint turbulence measurement. The structure of large-scale eddies, the distribution characteristic of small-scale eddies, and the difference in turbulence intensity and relevant flux for 1h and 30min oftime intervals and for various ergodicity conditions will be discussed with the 6 eddy-correlation system measurement in next essay.


This research was funded by the National Natural Science Foundation of China (Nos. 41675014 & 91437103); Gansu Science and Technology Program (18JR2RA005)

See the article: Chen J, Chen X, Jing X, Jia W, Yu Y, Zhao S. 2021. Ergodicity of turbulence measurements upon complex terrain in Loess Plateau. Science China Earth Sciences, 64(1): 37-51,

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