Abstract
Abstract. The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a single-station observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin–Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.
Highlights
The basic principle of average of the turbulence measurements is based on ensembles averaged over space, time and state
The results show that the eddy local stability z/Lc below 2 min at temporal scale at time 03:00–04:00 (CST) during the nighttime time frame is 0.59; it is stable stratification
While the respective magnitudes of function value of the temperature and humidity are 10−2 and 10−3; at 13:00–14:00 (CST) during midday time frame, magnitude of mean ergodic function of the vertical velocity is 10−4, while the magnitudes of function value of the temperature and humidity are both 10−2. These results show that the dimensionless mean ergodic function of vertical velocity converges to 0 much more than respective function value of the temperature and humidity, and that the vertical velocity satisfies the condition of mean ergodic theorem to overmatch more than the temperature and humidity
Summary
The basic principle of average of the turbulence measurements is based on ensembles averaged over space, time and state. It is impossible to make an actual turbulence measurement with enough observational instruments in space for sufficient time to obtain all states of turbulent eddies to achieve the goal of an ensemble average. Based on the ergodic hypothesis, the time average of one spatial point, taken over a sufficiently long observational time, is used as a substitute for the ensemble average for temporally steady and spatially homogeneous surfaces (Stull, 1988; Wyngaard, 2010; Aubinet et al, 2012). The ergodic hypothesis is a basic assumption in turbulence experiments in the atmospheric boundary layer (ABL) and atmospheric surface layer (ASL). The success of MOST under the conditions of stationary and homogeneity implies
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