Arbitrary-order Hilbert Spectral Analysis and Intermittency in Solar Wind Density Fluctuations

Abstract

The properties of inertial and kinetic range solar wind turbulence have been investigated with the arbitrary-order Hilbert spectral analysis method, applied to high-resolution density measurements. Due to the small sample size, and to the presence of strong non-stationary behavior and large-scale structures, the classical structure function analysis may prove to be unsuccessful in detecting the power law behavior in the inertial range, and may underestimate the scaling exponents. However, the Hilbert spectral method provides an optimal estimation of the scaling exponents, which have been found to be close to those for velocity fluctuations in fully developed hydrodynamic turbulence. At smaller scales, below the proton gyroscale, the system loses its intermittent multiscaling properties, and converges to a monofractal process. The resulting scaling exponents, obtained at small scales, are in good agreement with those of classical fractional Brownian motion, indicating a long term memory in the process, and the absence of correlations around the spectral break scale. These results provide important constraints on models of kinetic range turbulence in the solar wind.

Scaling Properties of Atmospheric Wind Speed in Mesoscale Range

“…The resulting L n (ω) demonstrates power-law behavior L n (ν) ∝ ω −β n for all n, in the range of frequencies ω ∈ [1.6 × 10 −1 , 4 × 10 1 ] days −1 (approximately from 30 min to 6 days), and the slope β 2 ≈ 1.67 ± 0.02 is compatible with the slope of the Fourier spectrum (Figure 4). This range is wider than the previous estimate ( Figure 6), due to the local nature of EMD and HSA, the strong daily modulation, as well as the possible "non-stationarity" due to ramp-cliff structures, can be constrained, isolating the properties of the cascade from the possible effects of the larger scale forcing and residual structures [15,16,39]. The last panel of Figure 9 shows the evolution of the slope β 2 as a function of time, with the associated average value and standard deviation: β 2 = 1.64 ± 0.03, compatible with the β ≈ 5/3 characteristic of the inertial sub-range turbulence [21,48].…”

“…When the decomposition is applied on dataset possessing certain feature, such as noise time series, fractional Gaussian noise, turbulent time series, fractal time series, random walks, EMD acts intrinsically as a dyadic filter bank [31,[64][65][66]. Each IMF captures a narrow spectral band in frequency space [15,16,39,67] and their superposition behaves as Figure 4). By comparison with the Fourier PSD, each IMF can be interpreted according to its characteristic time scale.…”

“…Finally, the arbitrary-order Hilbert spectral analysis (HSA) [15,16,39] was applied to identify the scaling of higher order moments, by minimizing all the effects of non-stationarity and large-scale periodic cycles embedded in the data set.…”

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

“…The instantaneous frequency directly comes out as ω i,j = 2π −1 dΦ i,j (t)/dt, and the mean period τ i,j = ω i,j −1 t , with · t representing time average [42]. When the EMD is applied on turbulent fields; fractal or multifractal processes; and random walks, it intrinsically acts as a dyadic filter bank [44][45][46][47], where each IMF captures a narrow band in frequency space, and the superposition of their Fourier spectra behaves as a power-law [14,[48][49][50].…”

Analytical representation of Gaussian processes in the A−T plane

“…Some other interesting instances include, but are not restricted to: ferro-and paramagnetic states of the Heisenberg model that exhibit H ∼ 1 and H ∼ 0.5, respectively [52], and should be easily distinguishable in the A − T plane; a photonic integrated circuit yields 0.2 H 0.8 for varied electric field of the feedback, coupled with chaotic behavior [53]; cataclysmic variable stars observed in X-rays exhibit long-term memory, H > 0.5, suggesting the accretion is driven by magnetic fields [54]; football matches can follow the rules of fBm with H ∼ 0.7 [55]; persistence of amoeboid motion [56] as well as Nitzschia sp. diatoms [57]; Solar wind proton density fluctuations are characterized by H ∼ 0.8, placing constraints on the models of kinetic turbulence [58]; values H > 0.5 were computed for epileptic patients' brain activity, quantified via magnetoencephalographic recordings, and appear to be a promising additional diagnostic tool for identifying epileptogenic zones in presurgical evaluation [59]. Recall that epilepsy has been already investigated in the A − T plane as well [17].…”