Figure 1

Estimation of the intermittency coefficient for (a) a log-normal MRW multifractal measure and (b) a log-normal MRW process. In both cases the sample size is $32768$ points, ${\lambda }^2=0.025$, and $T=512$. In (c) are plotted the magnitude covariance (●) and magnitude variance (○) as a function, respectively, of the logarithm of the lag and the logarithm of the scale. The slope of both curves provides an estimation of ${\lambda }^2$. One can see that for measure $M\left(dt\right)$, the errors in the estimation are comparable. In (d) are displayed the same plots as in (c) but for the magnitude of the MRW process. Because of the additive noise and the small number of independent points at large scales, the estimation relying upon the magnitude variance turns out to be much more altered.Reuse & Permissions

Figure 2

Comparison of the two estimation methods of the intermittency coefficient for a log-normal MRW process (a) Error ratio $r={\mathcal{E}}_{II}/{\mathcal{E}}_I$ as a function of ${\lambda }^2$ for $T=256$ and $N=8192$. (b) Error ratio $r={\mathcal{E}}_{II}/{\mathcal{E}}_I$ as a function of $T$ for ${\lambda }^2=0.03$ and $N=8192$. In (a) and (b) the errors have been estimated using 2048 Monte Carlo samples. Symbols (●) represent the observed error ratios while the solid line stand for the analytical expressions derived from Eqs. (15, 16). (c) Histograms of estimated values of ${\lambda }^2$ using method I (solid line) and method II (bars) for ${\lambda }^2=0.02$, $T=512$, and $N=8192$. (d) Same plots as in (c) for ${\lambda }^2=0.2$.Reuse & Permissions

Figure 3

Power spectrum density of $v_x$ wind components 10 min rate Vignola series (solid line) and hourly Eindhoven series (dotted line) in log-log representation. (a) One clearly identifies the peaks associated with diurnal oscillation superimposed to an overall scaling regime where $P\left(f\right)\sim f^{-\beta }$ with $\beta \simeq 1.6$. This regime roughly extends from few days to few minutes. (b) Plot of the same spectra where the estimated additive local seasonal components have been removed.Reuse & Permissions

Figure 4

Time fluctuations of the $v_x\left(t\right)$ component of the Vignola velocity field (original data). (a) At large time scales, one does not see any structure; we are in the flat (white noise) regime of the power spectrum. (b) At a finer time resolution, the series appears as a superposition of turbulent gusts and a more regular component.Reuse & Permissions

Figure 5

Semilogarithmic representation of standardized velocity increment pdf at various scales. From top to bottom one goes from small to coarse time scales. All graphs have been vertically shifted for the sake of clarity. (a) Logarithm of the pdf of the deseasonalized wind velocity increments ${\delta }_{\tau }{v}_x$ for the Eindhoven wind series. Time scales $\tau$ go from 1 h to 5 days. (b) Same plots as in (a) but for the increments of the longitudinal velocity field ${\delta }_{\tau }{v}_{\parallel }$ in a high Reynolds number turbulence experiment (see text). Scales $\tau$ extend from the Kolmogorov scale to the integral scale.Reuse & Permissions

Figure 6

Square root of magnitude covariance functions estimated for all wind series (additive and multiplicative seasonal components have been removed). The three bottom graphs correspond to Netherlands hourly series while the top graph is correlation of magnitudes associated with the “high-frequency” Vignola data series. All the graphs have been arbitrary shifted vertically for clarity purpose. One can observe comparable values of the parameters ${\beta }^2$ and $T$ (see text).Reuse & Permissions

Figure 7

Mean magnitude correlation functions associated with wind series in Corsica (○) and Netherlands (●) (all seasonalities have been removed). The solid line represents the magnitude covariance of the Vignola series (10 min rate). (a) The graphs are in linear scales. (b) The square root of the covariances are represented as function of the logarithm of the time lag $\Delta t$.Reuse & Permissions