[Paper Review] Statistical analysis of wind speed fluctuation and increments of non-stationary atmospheric boundary layer turbulence

T. Laubrich, Fatemeh Faraji Ghasemi+2 more|ArXiv.org|Nov 20, 2008

Wind and Air Flow Studies13 references3 citations

TL;DR

This paper proposes a superstatistical framework to analyze non-stationary atmospheric boundary layer (ABL) wind speed fluctuations and increments, revealing that ABL turbulence behaves as a sequence of quasi-stationary, locally ideal turbulence states with time-varying parameters. The key contribution is demonstrating that wind speed increment distributions are leptokurtic at large scales, with parameters dynamically evolving over time, validated through empirical data from Lammefjord, Denmark, showing a critical time scale separating fast and slow dynamics.

ABSTRACT

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.

Motivation & Objective

• To understand the statistical behavior of wind speed in the non-stationary atmospheric boundary layer (ABL), where turbulence is influenced by solar heating and surface roughness.
• To determine which statistical parameters remain approximately constant or vary slowly, enabling a piecewise-stationary description of ABL turbulence.
• To test the validity of Castaing’s intermittency hypothesis—where wind speed increments are normally distributed with log-normally distributed variance—under real atmospheric conditions.
• To quantify the time dependence of fluctuation and increment statistics using superstatistics, capturing the dynamics of turbulence parameters across multiple time scales.
• To assess whether highly non-stationary wind fields produce fat-tailed increment distributions even at large increment lengths.

Proposed method

• Empirically estimates the proportionality factor between the standard deviation of wind speed fluctuations and the mean wind speed using normalized fluctuation data.
• Applies a sliding window approach to compute local mean and standard deviation of wind speed over intervals of 12.5 seconds (m=101 samples at 8 Hz).
• Uses superstatistics to model the time-dependent variance of wind speed increments by assuming that the increment process is locally Gaussian with a slowly varying variance.
• Estimates the distribution of the logarithm of the variance (Λs(ϑ)) and its shape parameter (λ²s(ϑ)) across sub-samples to detect non-stationarity in the variance dynamics.
• Compares empirical increment distributions to theoretical predictions under Castaing’s hypothesis, particularly focusing on leptokurtic behavior at large increment lengths.
• Analyzes data from a single anemometer at 10 m height in Lammefjord, Denmark, over multiple days to assess time-scale separation in turbulence parameters.

Experimental results

Research questions

• RQ1 Can the standard deviation of wind speed fluctuations be approximated as proportional to the mean wind speed, and is the proportionality factor approximately constant over time?
• RQ2 Does the increment distribution of ABL wind speed exhibit leptokurtic behavior, as predicted by Castaing’s intermittency hypothesis, even under non-stationary conditions?
• RQ3 What is the critical time scale below which increment distributions appear Gaussian and above which they become leptokurtic, indicating a separation of time scales?
• RQ4 Can superstatistics effectively capture the time-varying nature of turbulence parameters in real atmospheric wind speed data?
• RQ5 Does high non-stationarity in wind speed lead to persistent fat tails in the increment distribution at large increment lengths?

Key findings

• The standard deviation of wind speed fluctuations is proportional to the mean wind speed, with the proportionality factor estimated as the standard deviation of the normalized fluctuation, and this factor remains approximately constant over 24-hour periods.
• Wind speed fluctuations around a 12.5-second window mean are well described by a symmetric normal distribution, supporting the idea that ABL turbulence is composed of locally stationary states with varying mean.
• The superstatistical approach reveals that the logarithm of the increment variance (Λs(ϑ)) is not strictly stationary but follows a sequence of normal distributions with time-varying mean and variance, indicating a third time scale in the dynamics.
• A critical time scale of approximately 2 hours is identified, below which increment distributions are approximately Gaussian and above which they become leptokurtic, consistent with Castaing’s hypothesis.
• Highly non-stationary wind speed data, such as on day 191, show that fat-tailed increment distributions can persist at large increment lengths, challenging the assumption of stationarity in extreme wind event modeling.
• The analysis confirms that ABL turbulence can be modeled as a sequence of successively occurring, close-to-ideal turbulence states with different parameters, validating the superstatistical framework for real atmospheric data.