A Multifractal Random-Walk Description of Atmospheric Turbulence: Small-Scale Multiscaling, Long-Tail Distribution, and Intermittency

Abstract

The prevalent multifractal characteristics of turbulent velocity fluctuations in the atmosphere are important for estimating various wind effects in wind engineering. Here, the multifractal characteristics of turbulent velocity fluctuations, including the small-scale multiscaling, the long-tail distributions and the intermittency, are thoroughly investigated by using a highfrequency dataset of three-dimensional velocities (100 Hz) collected at three levels during one month. To reduce uncertainties in the estimate of multiscaling exponents, a new method, the sequential extended self-similarity, is proposed. Based on this method, we obtain the multiscaling exponents of qth-order moments of velocity increments as a function of q, that is the so-called multifractal spectrum. The multifractal random walk (MRW) model is then shown to describe the various multifractal spectra of turbulent velocity fluctuations. With the help of this model, we find a connection between the small-scale multiscaling and the long-tail distributions, which is generally observed in our dataset, again validating the MRW model. A non-linear multifractal spectrum is commonly considered to be related to the intermittency of turbulent velocity fluctuations at small scales and its curvature is usually used as a quantification of intermittency intensity. However, we suggest that models capturing the nonlinear multifractal spectrum may fail to represent the long-tail distribution, which is a more direct quantification of intermittency. Finally, qualitative variations of validated indicators with specific boundary-layer parameters are investigated. Results show that the intermittency of turbulent velocity fluctuations is more relevant to the friction velocity, compared with the average wind speed, the average temperature, and the surface-layer stability.

Scaling Properties of Atmospheric Wind Speed in Mesoscale Range

“…Intermittency effects were clearly present in each sample, however the values of the intermittency parameter µ varied from case to case. The average value, with the associated standard deviation, was found to be of the order of µ ≈ 0.0165 ± 0.0031, in good agreement with estimates obtained from the classical multifractal spectrum [90], and from magnitude covariance analysis [7,8]. The scaling exponents ξ(n) obtained thorough HSA are similar to the exponents obtained in other experiments through ESS, up to the fourth order n = 4.…”

“…The horizontal dashed line represents the value µ ≈ 0.02 obtained for isotropic turbulence in the inertial sub-range, the dotted line represents the average intermittency value µ ≈ 0.0165 ± 0.0031 for the MLO wind speed data. The value of µ is in good agreement with other estimates [90] obtained with different methods.…”

“…The intermittency parameter obtained for the MLO wind speed, is slightly lower the isotropic case, and the characteristic average value and standard deviation is µ ≈ 0.0165 ± 0.0031. The value of µ is in good agreement with other estimates obtained from the classical multifractal spectrum [90] and from magnitude covariance analysis [7,8]. Intermittency effects could suggest the existence of a universal cascade mechanism associated with the energy transfer between synoptic motions and turbulent microscales in the atmospheric boundary layer.…”

Finescale Clusterization Intermittency of Turbulence in the Atmospheric Boundary Layer

“…For the atmospheric turbulence, because of widely used single-point measurements, the statistics of velocity increments with time lags is mostly studied. The probability density function of velocity increments is found to change from the non-Gaussian long-tailed distribution at smaller lags to the Gaussian-like distribution at larger lags, which means that the turbulent velocity fluctuations are more likely intermittent at smaller scales (Castaing et al 1990;Noullez et al 2000;Boettcher et al 2003;Liu et al 2010;Liu and Hu 2013;Liu et al 2019). Thus, the quantities quantifying deviation from the Gaussian distribution, such as the flatness (Frisch 1995;Tabeling et al 1996;Bos et al 2007), the average variation rate of cumulants (Malécot et al 2000), and the Kullback-Leibler divergence (Granero-Belinchón et al 2018), can be considered as measures of turbulence intermittency.…”

Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer

“…Large eddy simulations have shown that isotropy recovery is a "genuine feature" of the inertial sub-range [21,22]. The statistical properties of homogeneous and isotropic turbulence are usually characterized by means of energy spectral density and by the anomalous scaling of the structure functions of the field increments, e.g., S q ∝ ζ(q) [23][24][25][26], correlations [27][28][29][30]; or in terms of multifractality from the analysis of the so called multifractal spectrum f (α) [31][32][33][34][35][36]. However, it is often difficult to clearly disentangle the turbulent fluctuations from the possible superposed large-scale motions, so that the genuine features of the cascade may be covered in the data.…”