# Contrasting Scaling Properties of Near-Sun Sub-Alfvénic and Super-Alfvénic Regions

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

## 4. Results

- 1.
- It provides an accurate determination of scaling exponents;
- 2.
- It holds down to the dissipative regime;
- 3.
- It holds for both high and mid-to-low Reynolds numbers.

- If $\zeta {\left(q\right)}^{sub}=\zeta {\left(q\right)}^{super}$, then the two intervals are characterized by the same fractal topology, i.e., they share the same geometrical structures and the same symmetries;
- If $\zeta {\left(q\right)}^{sub}=\alpha \phantom{\rule{0.166667em}{0ex}}\zeta {\left(q\right)}^{super}$, then the two intervals are characterized by a fractal topology belonging to the same class, i.e., they share the same geometrical structures but with different symmetries;
- If $\zeta {\left(q\right)}^{sub}\ne \alpha \phantom{\rule{0.166667em}{0ex}}\zeta {\left(q\right)}^{super}$, then the two intervals are characterized by a different fractal topology, i.e., they do not share neither the same geometrical structures nor the same symmetries.

## 5. Discussions and Conclusions

- 1.
- An extended self-similarity is observed for both the MHD/inertial and the sub-ion/kinetic regimes during both intervals;
- 2.
- A multifractal nature of field fluctuations is reported across inertial scales for both solar wind intervals;
- 3.
- A mono-fractal character is observed for field fluctuations at small scales during both solar wind intervals.

#### 5.1. Extended Self-Similarity

#### 5.2. Multifractality of Inertial Range

#### 5.3. Mono-Fractality at Sub-Ion/Kinetic Scales

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AU | Astronomical Unit |

ESS | Extended Self-Similarity |

FGM | Flux-Gate Magnetometer |

MHD | Magnetohydrodynamic |

PSD | Power Spectral Density |

PSP | Parker Solar Probe |

RTN | Radial–Tangential–Normal |

SCM | Search Coil Magnetometer |

SL | She–Leveque |

UT | Universal Time |

## Note

1 | The RTN is a spacecraft-centered reference system in which the radial (R) direction is identified as the spacecraft-Sun line, the tangential (T) direction is identified as the tangent to the orbit of the spacecraft, and the N direction is obtained as the curl product of R and T, completing a right-handed triad. |

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**Figure 1.**The three magnetic field components in the radial–tangential–normal (RTN) reference system for the two intervals considered in this study: (

**upper panel**) the super-alfvénic, and (

**lower panel**) the sub-alfvénic solar wind, respectively.

**Figure 2.**The trace of the 2nd-order structure function, Tr$\left({S}_{2}\left(\tau \right)\right)={\sum}_{i}{S}_{2}^{\left(i\right)}\left(\tau \right)$, for the super-alfvénic (black circles) and the sub-alfvénic (blue circles) intervals, respectively. The two scale-invariant curves, ${\tau}^{1/2}$ (dashed-dotted black) and ${\tau}^{3/2}$ (dotted black), are drawn as reference for the scalings. The vertical gray dashed line marks the proton cyclotron scale.

**Figure 3.**The ESS analysis: the trace of the 2nd-order structure function, Tr$\left({S}_{2}\left(\tau \right)\right)={\sum}_{i}{S}_{2}^{\left(i\right)}\left(\tau \right)$, vs. the trace of the 4th-order structure function, Tr$\left({S}_{4}\left(\tau \right)\right)={\sum}_{i}{S}_{4}^{\left(i\right)}\left(\tau \right)$, for the super-alfvénic (black circles) and the sub-alfvénic (blue circles) intervals, respectively.

**Figure 4.**The scaling exponents $\zeta \left(q\right)$ derived via ESS analysis for each magnetic field component (${B}_{R}$: red symbols, ${B}_{T}$: green symbols, ${B}_{N}$: blue symbols) during both the super-alfvénic (circles) and the sub-alfvénic (asterisks) intervals: inertial range (left panel) and sub-ion/kinetic range (right panel). The dashed-dotted and the dotted lines in the left panel refer to the Iroshnikov-Kraichnan $\zeta \left(q\right)=q/4$ and the Kolmogorov $\zeta \left(q\right)=q/3$ predictions. The dashed–dotted line in the right panel is used as a reference for linear scaling.

**Figure 5.**The scaling exponents of the sub-alfvénic solar wind vs. those of the super-alfvénic interval for each magnetic field component (${B}_{R}$: red symbols, ${B}_{T}$: green symbols, ${B}_{N}$: blue symbols): inertial range (left panel) and sub-ion/kinetic range (right panel). The dashed–dotted lines mark the bisector of the plane where the equality $\zeta {\left(q\right)}^{sub}=\zeta {\left(q\right)}^{super}$ holds.

**Table 1.**The best-fit values of the parameters m, p, and C based on fitting the inertial range scaling exponents $\zeta {\left(q\right)}^{iner}$ for each magnetic field component. The 95% significance range is shown as subscript and superscript.

Sub-Alfvénic | Super-Alfvénic | |||||
---|---|---|---|---|---|---|

$\mathit{m}$ | $\mathit{p}$ | $\mathit{C}$ | $\mathit{m}$ | $\mathit{p}$ | $\mathit{C}$ | |

${B}_{R}$ | $4.{06}_{3.88}^{4.23}$ | $0.{74}_{0.71}^{0.77}$ | $0.{97}_{0.91}^{1.04}$ | $3.{93}_{3.72}^{4.27}$ | $0.{72}_{0.69}^{0.74}$ | $0.{81}_{0.79}^{0.84}$ |

${B}_{T}$ | $3.{97}_{3.78}^{4.11}$ | $0.{78}_{0.73}^{0.82}$ | $0.{96}_{0.90}^{1.02}$ | $4.{11}_{3.95}^{4.21}$ | $0.{74}_{0.71}^{0.76}$ | $0.{83}_{0.69}^{0.77}$ |

${B}_{N}$ | $3.{98}_{3.94}^{4.08}$ | $0.{73}_{0.70}^{0.76}$ | $1.{05}_{1.01}^{1.12}$ | $4.{07}_{3.92}^{4.12}$ | $0.{86}_{0.82}^{0.90}$ | $0.{69}_{0.64}^{0.73}$ |

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**MDPI and ACS Style**

Alberti, T.; Benella, S.; Carbone, V.; Consolini, G.; Quattrociocchi, V.; Stumpo, M.
Contrasting Scaling Properties of Near-Sun Sub-Alfvénic and Super-Alfvénic Regions. *Universe* **2022**, *8*, 338.
https://doi.org/10.3390/universe8070338

**AMA Style**

Alberti T, Benella S, Carbone V, Consolini G, Quattrociocchi V, Stumpo M.
Contrasting Scaling Properties of Near-Sun Sub-Alfvénic and Super-Alfvénic Regions. *Universe*. 2022; 8(7):338.
https://doi.org/10.3390/universe8070338

**Chicago/Turabian Style**

Alberti, Tommaso, Simone Benella, Vincenzo Carbone, Giuseppe Consolini, Virgilio Quattrociocchi, and Mirko Stumpo.
2022. "Contrasting Scaling Properties of Near-Sun Sub-Alfvénic and Super-Alfvénic Regions" *Universe* 8, no. 7: 338.
https://doi.org/10.3390/universe8070338