Quantifying the Effects of Atmospheric Stability on the Multifractal Spectrum of Turbulence

Abstract

Over the past decade, several studies suggested possible analogy between price dynamics in the foreign exchange market and atmospheric turbulent flows. Such analogies suggest that applications in business and industry can directly benefit from detailed quantification of the scale-hierarchy existing in fully developed turbulent flows. Numerous studies have already demonstrated the multifractal properties of rough-wall boundary layer turbulence. How atmospheric stability (i.e. boundary conditions) alters the multifractal spectrum (MFS) of turbulent velocity and temperature fluctuations in the atmospheric surface layer remains to be investigated. A challenge of estimating the MFS from time series via traditional regression approaches is the heteroskedastic problem because the variance of the error term are shown to depend on scale. Using a combination of Discrete Wavelet Transforms (DWT) and a Weighted Least Squares (WLS) scheme, heteroskedastic effects are minimized and robust estimation of the scaling parameters needed to compute the MFS is derived. Next, to quantify the effects of atmospheric stability on the MFS, discriminative measures that utilize the left slope (rise), Hurst exponent (maxima), and broadness are employed. Distributional property of the estimators for these discriminative measures are investigated based on Monte Carlo simulations. These summary measures are applied to the MFS of velocity and temperature time series collected in the atmospheric surface layer for a wide range of atmospheric stability conditions. The criteria for success in evaluating how atmospheric stability alters the MFS of a single flow variable time series is formulated as a statistical classification model that properly infers the stability regime when these three discriminative measures are used as input vectors.