Statistical analysis of wind speed fluctuation and increments of non-stationary atmospheric boundary layer turbulence

We study the statistics of the horizontal component of atmospheric boundary layer wind speed. Motivated by its non-stationarity, we investigate which parameters remain constant or can be regarded as being piece-wise constant and explain how to estimate them. We will verify the picture of natural atmospheric boundary layer turbulence to be composed of successively occurring close to ideal turbulence with different parameters. The first focus is put on the fluctuation of wind speed around its mean behaviour. We describe a method estimating the proportionality factor between the standard deviation of the fluctuation and the mean wind speed and analyse its time dependence. The second focus is put on the wind speed increments. We investigate the increment distribution and use an algorithm based on superstatistics to quantify the time dependence of the parameters describing the distribution. Applying the introduced tools yields a comprehensive description of the wind speed in the atmospheric boundary layer.

💡 Research Summary

The paper presents a thorough statistical investigation of horizontal wind speed measured at 10 m height with an 8 Hz sampling rate at the Lammefjord site (1987). Recognizing that atmospheric boundary‑layer (ABL) turbulence is strongly non‑stationary due to diurnal cycles, surface heating, and topographic influences, the authors aim to identify which statistical parameters can be treated as constant or piece‑wise constant over appropriate time windows and to develop robust estimation procedures for them.

The first part of the analysis focuses on the fluctuations of wind speed around a moving average. For a chosen window length m = 101 (≈12.5 s), the series is decomposed into a local mean (\bar x^{(m)}_n) and a fluctuation (f^{(m)}_n = x_n - \bar x^{(m)}_n). By conditioning on the local mean V, the conditional variance (\sigma_f^{(m)}(V)^2) is computed. Empirical results (Fig. 2) show a clear linear relationship (\sigma_f^{(m)}(V) = a(\vartheta,m),V), where the proportionality factor a depends on the time of day (\vartheta) but varies only slowly. To estimate a, the authors normalize the fluctuations, (g^{(m)}_n = f^{(m)}_n / \bar x^{(m)}_n), and compute the standard deviation of g over a chosen interval Δθ (e.g., 8 h, 24 h). The resulting a(θ,m) is interpreted as the turbulence intensity (TI) commonly used in wind‑energy literature. Figure 3 displays a(θ,m) as a step‑wise function over eleven days, revealing both diurnal patterns and day‑to‑day variability.

The conditional distributions (q^{(m)}(f|V)) are then compared with Gaussian PDFs having standard deviation a(θ,m) V (Eq. 12). The histograms (Fig. 4) match the Gaussian curves remarkably well, confirming that, within each Δθ window, the fluctuations are essentially Gaussian with a variance proportional to the local mean wind speed. This validates the assumption that the normalized fluctuations have a V‑independent variance, a property that does not hold for generic stationary processes.

The second major focus is on wind‑speed increments (x_{s;n}=x_{n+s}-x_n) with lag s≫m. According to the Kolmogorov‑Obukhov 2‑intermittency hypothesis, short‑time increments should exhibit leptokurtic (fat‑tailed) statistics. The authors demonstrate that the increment distribution can be modeled as a superposition of Gaussian PDFs with different variances, reflecting the slow temporal evolution of the underlying variance parameter. To quantify this, they apply a superstatistics framework: for each time segment Δθ they estimate the local variance β⁻¹ of the increments and construct the empirical distribution f(β). The overall increment PDF is then (p_s(x_s)=\int \mathcal{N}(0,\beta^{-1}) f(\beta),d\beta). Because β fluctuates over a wide range, p_s exhibits heavy tails, consistent with the observed non‑Gaussian behavior.

In the discussion, the authors argue that ABL turbulence can be viewed as a concatenation of quasi‑ideal turbulent episodes, each characterized by its own (approximately) constant a(θ,m) and β(θ). The temporal variability of these parameters is the source of the observed non‑stationarity and of the fat‑tailed increment statistics. This conceptual picture aligns with earlier work by Boettcher et al. (2007) but is reached here through a different methodological route that emphasizes the piece‑wise constancy of the fluctuation variance and the superstatistical description of increments.

The paper concludes that the proposed statistical tools—conditional variance analysis for fluctuations and superstatistics for increments—provide a comprehensive description of wind speed in the ABL. They are directly applicable to wind‑energy forecasting, structural load assessment, and extreme‑event risk analysis. Moreover, the methodology is readily extendable to other atmospheric variables exhibiting similar non‑stationary behavior.