Stationarity and low-frequency turbulent flux variability

The multiresolution analysis, introduced by Mallat (1989), allows decomposing a signal in terms of the temporal scales of its individual events. The use of this method to study turbulent fluxes allows the proper quantification of the flux for each different time scale. However, many studies that use this approach have shown that, for scales longer than a threshold (cospectral gap), the flux variability becomes appreciably large. Such large variability of the low frequency fluxes may be associated to poor data sampling at large temporal intervals or may be, at least partially, demonstrating a genuine physical process: mesoscale fluxes that are controlled by forcings on scales larger than the boundary layer. A third hypothesis for the large variability of the low-frequency fluxes is explored in the present study. In this case, the variability is caused by nonstationarity of the turbulent fields on large temporal scales. This is motivated by the fact that an average and, subsequently, the Reynolds procedure that leads to the association between covariances and fluxes, only can be defined if the original series are stationary. A series is considered to be stationary, in the weak or restrict sense, when its two first statistical moments are temporal functions. To test the nonstationarity hypothesis, we use the double mass balance (Nappo et al., 2010), Detrended Fluctuation Analysis (Peng et al., 1994) and the statistical t-test, following Andreas et al. (2008). In the first method, the cumulative sum of a given variable, which can be turbulent kinetic energy or wind speed, is determined. Stationarity is assumed to occur if the cumulative sum increases at the same rate as the time scale increases. Agiven interval is considered stationary when the difference between the cumulative sum and the value obtained from a linear extrapolation from previous periods is lower than the standard deviation of the difference between consecutive cumulative sums. The second method, Detrended Fluctuation Analysis (DFA), is largely used for studying fractal properties of systems. In this method, the existence of power-laws that identify long-range correlations may be used to find stationary intervals of a given series. The t-test is based on the fact that if two distributions have the same average and variance, the difference between the averages of two given samples of the original series follows a Studentxs t distribution. It allows the acceptance of the stationarity hypothesis, for a given uncertainty level. Furthermore, using a wavelet transform to filter the series, the large-scale fluctuations can be removed from the series. This removal is carried out until stationarity occurs, leading to the determination of the temporal scale for stationarity. In our study, we apply the three methods and the multiresolution decomposition to turbulent time series obtained from different field studies, under different stability conditions. The results for stationarity time scale are compared among themselves and to the cospectral gap time scale, allowing the identification of the extent to which the variability of the low-frequency fluxes is influenced by nonstationarity.