# Concurrent Effects between Geomagnetic Storms and Magnetospheric Substorms

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

#### 3.1. Time Delay Embedding

#### 3.2. Dynamical System Metrics for Univariate Time Series

#### 3.3. Dynamical System Metrics for Bivariate Time Series

## 4. Results

#### 4.1. Instantaneous Dimensions

#### 4.2. Instantaneous Stability

#### 4.3. Differences between Quiet and Storm Time Conditions

- 1.
- Group I: $\mathrm{SYM-H}>10$ nT;
- 2.
- Group II: $-20\phantom{\rule{3.33333pt}{0ex}}\mathrm{nT}\le \mathrm{SYM-H}\le 10$ nT;
- 3.
- Group III: $\mathrm{SYM-H}\le -50$ nT.

#### 4.4. Case Study: The Bastille Day Geomagnetic Storm

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CME | Coronal Mass Ejection |

GNSS | Global Navigation Satellite Systems |

HF | High Frequency |

SEP | Solar Energetic proton |

SMI | Solar wind-magnetosphere–ionosphere |

SSC | Sudden Storm Commencement |

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**Figure 1.**Time series of the high latitude geomagnetic index AL (upper panel) and low latitude geomagnetic index SYM-H (lower panel) from 1 January to 31 December 2000.

**Figure 2.**Joint probability density (frequencies of occurrences) of values of the SYM-H and AL geomagnetic indices.

**Figure 3.**Normalized auto-mutual information functions $\mathrm{MI}\left(\tau \right)/\mathrm{MI}\left(0\right)$ for AL (red) and SYM-H (blue), respectively. The stars represent the chosen embedding delays $\mathsf{\Delta}$ as those values at which $\mathrm{MI}\left(\tau \right)/\mathrm{MI}\left(0\right)={e}^{-1}$, the latter reported by the horizontal dotted line. The two dotted vertical lines from the star symbols visually highlight the choices of $\mathsf{\Delta}$.

**Figure 5.**Instantaneous dimension D as a function of the values of both geomagnetic indices. Panels (

**a**,

**b**) refer to the univariate time series, while panel (

**c**) corresponds to the bivariate one.

**Figure 6.**Instantaneous stability $\theta $ as a function of the values of both geomagnetic indices. Panels (

**a**,

**b**) refer to the individual time series, while panel (

**c**) corresponds to the joint observations. The color map is saturated above 0.5 for visualization purposes.

**Figure 7.**Histograms of the instantaneous dimensions D as a function of the different geomagnetic activity levels: Group I ($\mathrm{SYM-H}>10$ nT), Group II ($-20\phantom{\rule{3.33333pt}{0ex}}\mathrm{nT}\le \mathrm{SYM-H}\le 10$ nT), Group III ($\mathrm{SYM-H}\le -50$ nT). Blue, orange, and yellow bars refer to ${D}_{AL}$, ${D}_{SYM-H}$, and ${D}_{AL,SYM-H}$, respectively. The average values and their uncertainties are reported in each panel for each D. The values of D have been grouped into equidistant bins of size $0.2$.

**Figure 8.**Histograms of the instantaneous stability $\theta $ as a function of the different geomagnetic activity levels: Group I ($\mathrm{SYM-H}>10$ nT), Group II ($-20\phantom{\rule{3.33333pt}{0ex}}\mathrm{nT}\le \mathrm{SYM-H}\le 10$ nT), Group III ($\mathrm{SYM-H}\le -50$ nT). Blue, orange, and yellow bars refer to ${\theta}_{AL}$, ${\theta}_{SYM-H}$, and ${\theta}_{AL,SYM-H}$, respectively. The average values and their uncertainties are reported in each panel for each $\theta $. The values of $\theta $ have been grouped into equidistant bins of size $0.05$.

**Figure 9.**Behavior of the co-dimension ${D}_{AL,SYM-H}$ (upper panel), the co-persistence ${\theta}_{AL,SYM-H}$ (middle panel), and the co-recurrence ratio $\alpha $ during the Bastille Day geomagnetic storm across the SYM-H index. The time behavior of the AL index is reported in gray for comparison.

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**MDPI and ACS Style**

Alberti, T.; Faranda, D.; Consolini, G.; De Michelis, P.; Donner, R.V.; Carbone, V.
Concurrent Effects between Geomagnetic Storms and Magnetospheric Substorms. *Universe* **2022**, *8*, 226.
https://doi.org/10.3390/universe8040226

**AMA Style**

Alberti T, Faranda D, Consolini G, De Michelis P, Donner RV, Carbone V.
Concurrent Effects between Geomagnetic Storms and Magnetospheric Substorms. *Universe*. 2022; 8(4):226.
https://doi.org/10.3390/universe8040226

**Chicago/Turabian Style**

Alberti, Tommaso, Davide Faranda, Giuseppe Consolini, Paola De Michelis, Reik V. Donner, and Vincenzo Carbone.
2022. "Concurrent Effects between Geomagnetic Storms and Magnetospheric Substorms" *Universe* 8, no. 4: 226.
https://doi.org/10.3390/universe8040226