Multifractal analysis of velocity and temperature fluctuations in the atmospheric surface layer

The characterization of the structure of non-stationary, noisy fluctuations in a time series, e.g., the time series of the velocity components or temperature in turbulent flows, is a problem of central importance in fluid dynamics, nonequilibrium statistical mechanics, atmospheric physics and climate science. Over the past few decades, a variety of statistical techniques, like detrended fluctuation analysis (DFA), have been used to reveal intricate, multiscaling properties of such time series. We present an analysis of velocity and temperature time series, which have been obtained by measurements over the canopy of Hyytiälä Forest in Finland.
In our study we use DFA, its generalization, namely, multifractal detrended fluctuation analysis (MFDFA), and the recently developed multiscale multifractal analysis (MMA), which is an extension of MFDFA. These methods allow us to characterize the rich hierarchy or multi- fractality of the dynamics of the time series of the velocity components and the temperature. In particular, we can clearly distinguish these time series from white noise and the signals that display simple, monofractal, scaling with a single exponent (also called the Hurst exponent). It is useful to recall that monofractal scaling is predicted for fluid turbulence at the level of the Kolmogorov’s phenomenological approach of 1941 (K41); experiments and direct numerical simulations suggest that three-dimensional (3D) fluid turbulence must be characterised by a hierarchy of exponents for it is truly multifractal.

We present an analysis of multifractality of velocity and temperature fields that have been measured, at different heights, over the canopy of Hyytiälä Forest in Finland. In particular, we carry out a detailed study of velocity and temperature time series by using MFDFA and MMA. Results from both these methods are consistent, as they must be; but, of course, the MMA results contain more information because they account for the dependence of the multifractality on the time intervals.