Ergodicity test of the eddy correlation method

Abstract

Abstract. The turbulent flux observation in the near-surface layer is a scientific issue which researchers in the fields of atmospheric science, ecology, geography science, etc. are commonly interested in. For eddy correlation measurement in the atmospheric surface layer, the ergodicity of turbulence is a basic assumption of the Monin–Obukhov (M–O) similarity theory, which is confined to steady turbulent flow and homogenous surface; this conflicts with turbulent flow under the conditions of complex terrain and unsteady, long observational period, which the study of modern turbulent flux tends to focus on. In this paper, two sets of data from the Nagqu Station of Plateau Climate and Environment (NaPlaCE) and the cooperative atmosphere–surface exchange study 1999 (CASE99) were used to analyze and verify the ergodicity of turbulence measured by the eddy covariance system. Through verification by observational data, the vortex of atmospheric turbulence, which is smaller than the scale of the atmospheric boundary layer (i.e., its spatial scale is less than 1000 m and temporal scale is shorter than 10 min) can effectively meet the conditions of the average ergodic theorem, and belong to a wide sense stationary random processes. Meanwhile, the vortex, of which the spatial scale is larger than the scale of the boundary layer, cannot meet the conditions of the average ergodic theorem, and thus it involves non-ergodic stationary random processes. Therefore, if the finite time average is used to substitute for the ensemble average to calculate the average random variable of the atmospheric turbulence, then the stationary random process of the vortex, of which spatial scale was less than 1000 m and thus below the scale of the boundary layer, was possibly captured. However, the non-ergodic random process of the vortex, of which the spatial scale was larger than that of the boundary layer, could not be completely captured. Consequently, when the finite time average was used to substitute for the ensemble average, a large rate of error would occur with use of the eddy correction method due to losing the low frequency component information of the larger vortex. When the multi-station observation was compared with the single-station observation, the wide sense of stationary random process originating from the multi-station observation expanded from a vortex which was about 1000 m smaller than a boundary layer scale to the turbulent vortex, which was larger than the boundary layer scale of 2000 m. Therefore, the calculation of the turbulence average or variance and turbulent flux could effectively meet the ergodic assumption, and the results would be approximate to the actual values. Regardless of vertical velocity and temperature, if the ergodic stationary random processes could be met, then the variance of the vortexes in the different temporal scales could follow M–O similarity theory; in the case of the non-ergodic random process, its vortex variance deviated from the M–O similarity relations. The exploration of ergodicity in the atmospheric turbulence measurements is doubtlessly helpful to understanding the issues in atmospheric turbulent flux observation, and provides a theoretical basis for overcoming related difficulties.