# Statistical Analysis of Field-Aligned Alfvénic Turbulence and Intermittency in Fast Solar Wind

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## Abstract

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## 1. Introduction

## 2. Data Selection Criteria for ${k}_{\Vert}$ Magnetic Fluctuation

- Magnetic field and velocity must be aligned within ${15}^{\circ}$ at all scales in the turbulent inertial range for the whole interval, i.e., both local and global mean magnetic field must be aligned with the bulk velocity;
- The total duration cannot be less than 1 hour, with maximal percentage of missing data ≤20%;
- The whole interval must be enclosed within a fast SW stream ($V\phantom{\rule{0.166667em}{0ex}}\ge \phantom{\rule{0.166667em}{0ex}}550$ km s${}^{-1}$);
- It must have low magnetic compressibility (${C}_{B}\le \phantom{\rule{0.166667em}{0ex}}25\%$), measured in the inertial range at scale of 20 min, where ${C}_{B}$ is defined as the ratio between the variance of the magnetic field intensity fluctuations and the total variance of the fluctuations, ${C}_{B}={\sigma}_{\left|B\right|}^{2}/{\sum}_{i=x,y,z}{\sigma}_{{B}_{i}}^{2}$ [23];
- The total pressure ${P}_{m}=2n{k}_{B}T+{B}^{2}/\left(8\pi \right)$ (with n, T, and B are, respectively, the proton number density, the temperature, and magnetic field intensity) must satisfy the condition ${P}_{m}<0.05$ nPa, in order to avoid discontinuities or shocks [24].

## 3. Empirical Mode Decomposition and Arbitrary Order Hilbert Spectra

## 4. Statistical Analysis of HSA Slopes and Scaling Exponents for Alfvénic Turbulence

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Left panel: Temporal evolution of the SW bulk speed V (upper plots), zoom of the velocity on the 1-hour region of interest (central plots), and the magnitude of the magnetic field $\left|B\right|$ (bottom plots) for the sample dated $2005/01/03$. The velocity is steady on the whole interval. Central panels: same as previous plot but for sample dated $2007/09/03$. Right panel: Power spectral density (in dimensional form) for the three components of the magnetic field ${B}_{x}$, ${B}_{y}$, and ${B}_{z}$ in the wavevector space, obtained via the traditional Taylor frozen hypothesis [1]. The dashed line represents the theoretical Kolmogorov scaling ${k}^{-5/3}$.

**Figure 2.**Left panel: Intrinsic Mode Functions (IMFs) $j\in [3,13]$ and the associated residual ${r}_{\varphi}$, obtained through Empirical Mode Decomposition (EMD) for sample dated $2005/01/03$. Modes $j=1,2$ have been excluded from the plot for readability. Central panel: Log-linear plot of the average IMF period as a function of the mode j (sample 2005/01/03); error bars represent the $95\%$ confidence bounds; the dashed lines represent the relation ${\tau}_{j}=\alpha \times {\gamma}^{j}$, with $\gamma =1.97\pm 0.01$. Right panel: Comparison of Fourier power spectral density (black line) for sample $2005/01/03$ with the Fourier power spectrum of different IMFs ${\varphi}_{j}\left(\omega \right)$ ($j\in [1,13]$), as a function of frequency $\omega $ (the curves have been vertically shifted for clarity); the band-like structure of each IMF shows the dyadic nature of the decomposition.

**Figure 3.**Left panel: Hilbert marginal spectrum, ${\mathcal{L}}_{2}\left(\omega \right)$, for the components ${B}_{x}\left(t\right)$, ${B}_{y}\left(t\right)$, and ${B}_{z}\left(t\right)$ of the magnetic field (sample $2005/01/03$). In all cases the slope of ${\mathcal{L}}_{2}\left(\omega \right)$ is perfectly comparable with the slope of the PSD. The power anisotropy among the parallel and perpendicular direction is observed in all samples. Central Panel: value of the average ratio $\mathcal{R}\left({\omega}^{\star}\right)$ (with the $95\%$ of confidence bounds) between the parallel and perpendicular components of the magnetic field $\mathbf{B}\left(t\right)$, for all frequency in the inertial range ${\omega}^{\star}\in [{10}^{-2},{10}^{-1}]$ (sample $2005/01/03$). The horizontal dashed line represent the theoretical value for the isotropy ratio from the Kolmogorov’s second similarity hypothesis $\mathcal{R}\left({\omega}^{\star}\right)=3/4$ for homogeneous and isotropic turbulence [51]. Right panel: Comparison of the Hilbert spectrum, ${\mathcal{L}}_{2}\left(\omega \right)$ (circles) with the classical Fourier PSD $E\left(\omega \right)$ (solid line), and the second-order structure function ${S}_{2}\left({\ell}_{t}^{-1}\right)$ (squares, ${\ell}_{t}^{-1}$ represent the inverse time scale), relative to the component ${B}_{z}\left(t\right)$ of sample $2006/07/28$. The power-law behavior in the inertial range is clearly observable with all methods. The scaling exponent, obtained via least square fit are: ${\beta}_{2}=1.61\pm 0.10$ (dashed line) and $\zeta \left(2\right)=0.57\pm 0.02$. The curves have been vertically shifted for clarity.

**Figure 4.**Distribution of slopes ${\beta}_{2}$ for two samples (component ${B}_{z}$), and for three different number of replicas ${N}_{samp}$. The PDFs have been constructed by using the residual resampling method.

**Figure 5.**Distribution of the slopes ${\beta}_{q}$ constructed with all the results obtained from the resampling, and from all intervals, relatives to the three component of the magnetic field $\mathbf{B}\left(t\right)$. The spectral slope ${\beta}_{2}$ (analogous of the Fourier spectral slope) is characterized by a slope ${\beta}_{2}\approx 5/3$ fully compatible with the classical Kolmogorov spectrum.

**Figure 6.**Histograms of the measured scaling exponents $\xi \left(q\right)$ for the first three orders $q=1,2,3$ (form left to right columns), for the three magnetic components x, y, z (top to bottom), for sample $2005/01/03$. Vertical bars indicate the various percentiles: ${p}_{2.5}$ (dashed line), ${p}_{50}$ (dotted line), and ${p}_{97.5}$ (dash-dotted line), respectively.

**Figure 7.**Scaling exponents $\xi \left(q\right)$ obtained through the HSA, in the inertial range, for three differents samples (solid symbols).

**Table 1.**Parameters of wind data considered for the analysis: the SW bulk speed, proton number density and temperature were measured by the Solar Wind Experiment (SWE) instrument [25] at $\Delta \omega \approx 0.01$ Hz resolution; magnetic field at resolution $\Delta \omega \approx 11$ Hz was measured by the Magnetic Field Investigation (MFI) magnetometer [26]. The magnetic compressibility ${C}_{B}$ has been evaluated at scale of 20 min, within in the inertial range.

Date | Start Time | End Time | $\langle \mathit{V}\rangle $ [km s${}^{-1}$] | $\langle {\mathit{P}}_{\mathit{m}}\rangle $ [nPa] | $\langle {\mathit{C}}_{\mathit{B}}\rangle $ [%] | $\langle {\mathit{\theta}}_{\mathbf{VB}}\rangle $ [deg] |
---|---|---|---|---|---|---|

2005/01/03 | 13:36:56 | 14:36:43 | 631.7 | 0.026 | 22 | 10.07 |

2005/05/09 | 10:06:10 | 11:16:43 | 639.2 | 0.033 | 8 | 7.13 |

2006/06/18 | 14:02:36 | 15:13:07 | 579.6 | 0.014 | 23 | 13.20 |

2006/07/28 | 21:13:08 | 22:28:15 | 564.0 | 0.007 | 19 | 9.60 |

2006/12/23 | 23:33:56 | 00:33:43 | 640.9 | 0.017 | 7 | 9.75 |

2007/09/03 | 07:39:48 | 08:50:19 | 592.8 | 0.014 | 10 | 10.64 |

2007/10/27 | 00:24:16 | 01:51:39 | 576.9 | 0.028 | 16 | 9.63 |

2011/04/13 | 01:31:56 | 02:31:43 | 554.9 | 0.020 | 10 | 10.04 |

2014/06/09 | 01:06:48 | 02:55:39 | 588.8 | 0.016 | 18 | 10.41 |

2015/03/22 | 14:00:40 | 15:28:03 | 647.0 | 0.044 | 13 | 9.99 |

2016/07/13 | 01:14:48 | 02:17:39 | 575.4 | 0.021 | 21 | 9.67 |

2016/07/13 | 05:17:04 | 07:25:51 | 574.7 | 0.018 | 15 | 9.10 |

**Table 2.**Various percentile obtained from PDFs of Figure 5, for the three order Hilbert spectra ${\mathcal{L}}_{q}$: ${p}_{2.5}$, ${p}_{50}$, and ${p}_{97.5}$.

Component | Order q | Percentile p | ||
---|---|---|---|---|

${\mathit{p}}_{\mathbf{2}.\mathbf{5}}$ | ${\mathit{p}}_{\mathbf{50}}$ | ${\mathit{P}}_{\mathbf{97}.\mathbf{5}}$ | ||

1 | 1.21 | 1.31 | 1.39 | |

${B}_{x}$ | 2 | 1.55 | 1.65 | 1.75 |

3 | 1.83 | 2.01 | 2.31 | |

1 | 1.26 | 1.36 | 1.47 | |

${B}_{y}$ | 2 | 1.55 | 1.70 | 1.80 |

3 | 1.75 | 2.00 | 2.24 | |

1 | 1.26 | 1.33 | 1.44 | |

${B}_{z}$ | 2 | 1.56 | 1.68 | 1.83 |

3 | 1.81 | 2.02 | 2.24 |

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**MDPI and ACS Style**

Carbone, F.; Telloni, D.; Sorriso-Valvo, L.; Zank, G.; Zhao, L.; Adhikari, L.; Bruno, R.
Statistical Analysis of Field-Aligned Alfvénic Turbulence and Intermittency in Fast Solar Wind. *Universe* **2020**, *6*, 116.
https://doi.org/10.3390/universe6080116

**AMA Style**

Carbone F, Telloni D, Sorriso-Valvo L, Zank G, Zhao L, Adhikari L, Bruno R.
Statistical Analysis of Field-Aligned Alfvénic Turbulence and Intermittency in Fast Solar Wind. *Universe*. 2020; 6(8):116.
https://doi.org/10.3390/universe6080116

**Chicago/Turabian Style**

Carbone, Francesco, Daniele Telloni, Luca Sorriso-Valvo, Gary Zank, Lingling Zhao, Laxman Adhikari, and Roberto Bruno.
2020. "Statistical Analysis of Field-Aligned Alfvénic Turbulence and Intermittency in Fast Solar Wind" *Universe* 6, no. 8: 116.
https://doi.org/10.3390/universe6080116