Modeling the superstorm in the 24th solar cycle
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Abstract
Keywords
Mathematical modeling Zonal geomagnetic indices Solar wind parametersIntroduction
The St. Patrick’s Day geomagnetic storm is one of the most remarkable storms in the 24th solar cycle. The phenomenon has caused serious negative effects on the Earth. One of the reasons that make the storm interesting and important is the magnitude of the storm, and the other one is it has not been forecasted.
If one tries to reveal scientific results about geomagnetic storms, he/she should determine the relationship between solar wind parameters and zonal geomagnetic indices. With these two types of variables, the model may be established and the storm can be discussed provided that it obeys physical principles. The zonal geomagnetic indices, which are caused by solar parameter variables such as magnetic field, electric field, dynamic pressure, proton density due to the storm, have been in use since ancient times. Based on these variables, scientists can characterize the magnetosphere (Mayaud 1980; Fu et al. 2010a, b; Rathore et al. 2014). The geomagnetic storms, which have three phases including a sudden commencement, a main phase and a recovery phase, are one of the most important actions involving dynamic structures (Akasofu 1964; Burton et al. 1975). The storm reaction of the dynamic structure starts with coronal mass ejection (CME). During the CME pulse, large solar plasma clouds with an average speed of 800 km/s seriously affect magnetosphere, leaving its place to define the magnetic activity indices that determine the reflex of the geomagnetic storm. Magnetic activity indices such as AE (auroral electrojet), ap, Kp (planetary index) and Dst (disturbance storm time) are described to specify the effects of the geomagnetic storm. AE is the hourly auroral electrojet index, ap is the planetary index derived from Kp, and Kp is the quasi-logarithmic planetary index. The author utilizes hourly versions of AE and Kp indices. Dst, which exhibits the level of the magnetic storm (Hanslmeier 2007), is the hourly index related to the ring current. Kp, ap and Dst indices are generally used to define a magnetic storm (Mayaud 1980; Kamide et al. 1998; Joshi et al. 2011; Elliott et al. 2013). The St. Patrick’s Day storm started on March 17 with CME. CME usually causes sudden increases in solar wind dynamic pressure. The reason for the formation of CMEs is the regional reconnections in the solar corona (Lin and Forbes 2000). These reconnections are the result of magnetic-field-line merging (Fu et al. 2011, 2012, 2013a, b, 2015, 2017). During the eruption, the light isotopes and plasmas in the solar corona are spread throughout the solar magnetic field. The charged particles interact with the Earth’s magnetic field, causing intermittent disturbance of the ionosphere and magnetosphere (Fu et al. 2011, 2012, 2013b). Some observational (Zic et al. 2015; Manoharan et al. 2017; Subrahmanya et al. 2017) evidences suggested that the ionospheric disturbance dynamo had a significant effect on storm-time ionospheric electric fields at medium and low latitudes (Blanc and Richmond 1980). The CME leads directly to the change in solar wind parameters (Gonzalez et al. 1999).
Mathematical models give information to researchers about variables and their relationships, even if they are in different scientific areas (Ak et al. 2012; Celebi et al. 2014; Eroglu et al. 2016). In addition, they should give clues about the behavior of the variables under different circumstances and varied plasma-dense medium. Investigation of the evolution of dense plasmas over time cannot be limited to a single event. Because of their dynamic structures, establishing models will benefit scientists (Sibeck et al. 1991). Dynamic models have been used in many previous studies to describe global loading and unloading operations in storms (Burton et al. 1975; Baker et al. 1990; Dungey 1961; Gonzalez et al. 1994; Sugiura 1964; Temerin and Li 2002; Tsyganenko et al. 2003; Fu et al. 2014). Previously applied models can also be seen in this storm. For example, Wu and Leping (2016) have applied Gilmore et al. (2002) formula to St. Patrick’s Day storm for Dst and B_{z}.
The effects of the storm in all longitudinal sectors are characterized using spherical and regional electric current. Estimation of ionospheric current density can minimize the negative effects of substorm activity. The improvement of high-latitude ionospheric convection models aids in predicting substorm events (Chen et al. 2016). The effects of the magnetospheric convection electric field and the disturbing dynamo electric field at low latitudes were previously investigated (Fu et al. 2010a, b; Nava et al. 2016). The magnetic field oscillations of the Earth are seen at the same time in the Asian, African and American sectors during the southward orientation of the B_{z} component in the interplanetary magnetic field. The ionospheric irregularities at the high latitudes associated with auroral activities have been studied by Cherniak and Zakharenkova (2015).
The St. Patrick’s Day geomagnetic storm (Astafyeva et al. 2015; Cherniak et al. 2015; Baker et al. 2016; Gvishiani et al. 2016; Nayak et al. 2016) has been widely studied during the past 2 years. It is necessary to understand the complex effects of the geomagnetic storm and predict the event based on the solar wind and IMF parameters. We focus on the variables of the phenomenon and discuss mathematical models. Binary linear models have difficulty in explaining the exact relationship between variables. Nevertheless, the presentation of these models is important (Eroglu 2018). Weak correlation inspires scientists to search for linear and nonlinear models. All approaches have exact obedience cause–effect relationship, and the causality principle governs linear and nonlinear models (Tretyakov and Erden 2008; Eroglu et al. 2012). The cause–effect relation should be thought of as an inseparable duo. The solar wind plasma parameters [the magnetic field (B_{z}), the electric field (E), the solar wind dynamic pressure (P), the proton density (N), the flow velocity (v) and the temperature (T)] of the phenomenon are the “cause.” The zonal geomagnetic indices (Dst, ap, Kp and AE) of the storm are the “effect.”
This paper uses the solar wind parameters (P, v, E, T, N, B_{z}) and zonal geomagnetic indices (Dst, AE, Kp, ap). The author utilizes hourly versions of AE and Kp indices. In order to better interpret the first intense (− 250 nT ≤ Dst < − 100 nT) storm of the 24th solar cycle (March 17, 2015), solar wind parameters and zonal indices are analyzed in depth and linear and nonlinear models are established. The models support the previous work conducted by Eroglu (2018).
In “Data” section the solar parameters, zonal geomagnetic indices and a five-day distribution of variables are presented. In “Mathematical modeling” and “Conclusion” sections, the analyses are performed and discussion is given, respectively.
Data
Space Physics Environment Data Analysis Software (SPEDAS) is used in this research. Analysis software data are IDL based. It is accessible at the link below: http://themis.igpp.ucla.edu/software.shtml. The hourly OMNI-2 Solar Wind and IMF parameter data are accessible online. In addition, the AE and Dst indexes are taken from World Data Center for Geomagnetism, Kyoto, by using SPEDAS. Kp and ap are taken from NGDC by using SPEDAS with CDA Web Data Chooser (space physics public data). For March 2015 severe storm, solar wind dynamic pressure, IMF, electric field, flow speed and proton density were recorded in the OMNI hourly data. Geomagnetic storms are classified according to the intensity of the Dst index (Loewe and Prölss 1997). If the Dst index is between − 50 and − 30 nT this indicates a weak storm. If it is between − 100 and − 50 nT this indicates a moderate storm. The Dst index between − 200 and − 100 nT indicates a strong (intense) geomagnetic storm.
The parameters shown in Dst index, B_{z} magnetic field (nT), E electric field (mV/m), proton density N (1/cm^{3}), solar wind dynamic pressure P (nPa), flow speed v (km/s) and auroral electrojet AE (nT) for March 15–19, 2015, are obtained from NASA NSSDC OMNI data set.
Figure 1 is specifically described as follows. On March 17, 2015, at 22:00 (UT), when Dst is at its minimum (− 223 nT), B_{z} component increases to − 16.5 nT and the electric field E reaches 5.2 mV/m. Meanwhile, ap index reaches its maximum value 179 nT by increasing, proton density N is 8.6 1/cm^{3}, plasma flow speed v reaches one of the highest values of 558 km/s, and AE index catches 457 nT.
On March 17, 2015, at 14:00 (UT), when B_{z} component is minimum (− 18.1 nT), Dst index continues to decrease toward the minimum, the electric field E reaches its own maximum value of 10.5 mV/m, AE index reaches its own maximum value of 1570 nT, ap index reaches its maximum value 179 nT, and flow pressure P takes its own one of the maximum values of 16.7 nPa.
On March 17, 2015, at 05:00 (UT), when B_{z} component is maximum (20.1 nT), the electric field reaches its minimum value of − 9.9 mV/m, proton density N takes its own one of the maximum values of 38.5 1/cm^{3}, AE index decreases and falls to one of the minimum values of 50 nT, and ap index continues to increase. As this happens Dst index reaches its maximum value 56 nT.
Mathematical modeling
Descriptive analysis
N | Minimum | Maximum | Mean | SD | |
---|---|---|---|---|---|
B_{z} (nT) | 120 | − 18.1 | 20.1 | − .317 | 6.8809 |
T (K) | 120 | 21,425 | 912,227 | 142,900.23 | 162,135.845 |
N (1/cm^{3}) | 120 | 2.7 | 40.1 | 14.110 | 9.6623 |
v (km/s) | 120 | 298 | 683 | 485.23 | 123.119 |
P (nPa) | 120 | 1.68 | 20.76 | 5.6173 | 4.46443 |
E (mV/m) | 120 | − 9.97 | 10.57 | .5708 | 3.54970 |
Kp | 120 | 3 | 77 | 37.50 | 19.355 |
Dst (nT) | 120 | − 223 | 56 | − 43.40 | 63.249 |
ap (nT) | 120 | 2 | 179 | 39.60 | 48.606 |
AE (nT) | 120 | 17 | 1570 | 359.60 | 330.776 |
Pearson’s correlation matrix for the storm variables
B_{z} (nT) | T (K) | N (1/cm^{3}) | v (km/s) | P (nPa) | E (mV/m) | Kp | Dst (nT) | ap (nT) | AE (nT) | |
---|---|---|---|---|---|---|---|---|---|---|
B_{z} (nT) | 1 | .038 | .171 | − .316** | − .165 | − .889** | − .580** | .618** | − .712** | − .687** |
T (K) | 1 | − .133 | .511** | .378** | .084 | .407** | − .224* | .241** | .244** | |
N (1/cm^{3}) | 1 | − .682** | .627** | − .130 | − .274** | .658** | − .068 | − .277** | ||
v (km/s) | 1 | .060 | .339** | .702** | − .742** | .483** | .582** | |||
P (nPa) | 1 | .272** | .416** | .060 | .532** | .271** | ||||
E (mV/m) | 1 | .648** | − .616** | .783** | .757** | |||||
Kp | 1 | − .755** | .887** | .767** | ||||||
Dst (nT) | 1 | − .678** | − .655** | |||||||
ap (nT) | 1 | .754** | ||||||||
AE (nT) | 1 |
KMO and Bartlett’s test
Kaiser–Meyer–Olkin measure of sampling adequacy | .762 |
Bartlett’s test of sphericity | |
Approx. Chi square | 1385.342 |
df | 45 |
Sig. | .000 |
Total variance explained
Component | Initial eigenvalues | Extraction sums of squared loadings | ||||
---|---|---|---|---|---|---|
Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | |
1 | 5.352 | 53.515 | 53.515 | 5.352 | 53.515 | 53.515 |
2 | 2.049 | 20.487 | 74.002 | 2.049 | 20.487 | 74.002 |
3 | 1.432 | 14.325 | 88.327 | 1.432 | 14.325 | 88.327 |
Rotated component matrix
Component | B_{z} (nT) | T (K) | N (1/cm^{3}) | v (km/s) | P (nPa) | E (mV/m) | Kp | Dst (nT) | ap (nT) | AE (nT) |
---|---|---|---|---|---|---|---|---|---|---|
1 | − .920 | − .001 | − .141 | .400 | .315 | .938 | .766 | − .702 | .887 | .828 |
2 | .069 | .054 | .956 | − .518 | .805 | .012 | − .070 | .543 | .137 | − .115 |
According to Fig. 2, the physical reaction of zonal geomagnetic indices to the change in solar wind parameters in the storming process can be summarized as follows. The response of Dst to the magnetic field B_{z} component, the electric field (E), proton density (N) and temperature (T) is linear, and the response to the dynamic pressure (P) and flow speed (v) is nonlinear. While the response of the ap index to B_{z}, electric field, flow speed and temperature is linear, its response to dynamic pressure and proton density is nonlinear. While the response of the AE index to B_{z}, electric field, dynamic pressure and temperature is linear, its response to proton density and flow speed is nonlinear.
Linear and nonlinear model
ANOVA (analysis of variance)
Model | Sum of squares | df | Mean square | F | Sig. |
---|---|---|---|---|---|
Regression | 131237.711 | 4 | 32,809.428 | 26.445 | .000 |
Residual | 83,124.164 | 67 | 1240.659 | ||
Total | 214,361.875 | 71 |
Regression coefficients
Model | Unstandardized coefficients | Standardized coefficients | t | Sig. | |
---|---|---|---|---|---|
B | SE | Beta | |||
(Constant) | − 244.925 | 73.544 | − 3.330 | .001 | |
B_{z} (nT) | 3.497 | .670 | .471 | 5.217 | .000 |
N (1/cm3) | 10.321 | 2.396 | 1.617 | 4.308 | .000 |
P (nPa) | − 11.814 | 3.457 | − 1.201 | − 3.417 | .001 |
v (km/s) | .256 | .123 | .265 | 2.078 | .042 |
ANOVA (analysis of variance)
Model | Sum of squares | df | Mean square | F | Sig. |
---|---|---|---|---|---|
Regression | 163,844.271 | 3 | 54,614.757 | 90.643 | .000 |
Residual | 40,971.729 | 68 | 602.525 | ||
Total | 204,816.000 | 71 |
Regression coefficients
Model | Unstandardized coefficients | Standardized coefficients | t | Sig. | |
---|---|---|---|---|---|
B | SE | Beta | |||
(Constant) | 24.093 | 4.244 | 5.676 | .000 | |
E (mV/m) | 4.653 | 1.477 | .349 | 3.151 | .002 |
P (nPa) | 3.771 | .554 | .392 | 6.801 | .000 |
B_{z} (nT) | − 2.769 | .780 | − .382 | − 3.551 | .001 |
Table 8 indicates that the model is significant, while Table 9 shows that the ap index is: \({\text{ap}} = \left( {24.093} \right) + \left( {4.653} \right)E + \left( {3.771} \right)P - \left( {2.769} \right)B_{\text{z}}\), where multiple determination coefficient R is 0.894.
Regression coefficients
Unstandardized coefficients | Standardized coefficients | t | Sig. | ||
---|---|---|---|---|---|
B | SE | Beta | |||
B_{z} (nT) | 5.677 | .665 | .618 | 8.532 | .000 |
(Constant) | − 41.602 | 4.565 | − 9.114 | .000 |
Unstand. coeff. | Stand. coeff. | t | Sig. | ||
---|---|---|---|---|---|
B | SE | Beta | |||
B_{z} (nT) | 5.418 | .691 | .589 | 7.846 | .000 |
\(B_{\text{z}}^{ 2}\) (nT) | − .078 | .058 | − .101 | − 1.344 | .081 |
(Constant) | − 38.016 | 5.274 | − 7.209 | .000 |
Regression coefficients
Unstandardized coefficients | Standardized coefficients | t | Sig. | ||
---|---|---|---|---|---|
B | SE | Beta | |||
B_{z} (nT) | − 5.029 | .457 | − .712 | − 11.014 | .000 |
(Constant) | 38.007 | 3.132 | 12.134 | .000 |
Unstand. coeff. | Stand. coeff. | t | Sig. | ||
---|---|---|---|---|---|
B | Std. Err | Beta | |||
B_{z} (nT) | − 3.913 | .302 | − .554 | − 12.970 | .000 |
\(B_{\text{z}}^{ 2}\) (nT) | .336 | .025 | .567 | 13.273 | .000 |
(Constant) | 22.542 | 2.304 | 9.786 | .000 |
Regression coefficients
Unstandardized coefficients | Standardized coefficients | t | Sig. | ||
---|---|---|---|---|---|
B | SE | Beta | |||
B_{z} (nT) | − 33.012 | 3.217 | − .687 | − 10.262 | .000 |
(Constant) | 349.146 | 22.066 | 15.823 | .000 |
Unstand. coeff. | Stand. coeff. | t | Sig. | ||
---|---|---|---|---|---|
B | SE | Beta | |||
B_{z} (nT) | − 29.434 | 3.148 | − .612 | − 9.350 | .000 |
\(B_{\text{z}}^{ 2}\) (nT) | 1.077 | .264 | .267 | 4.075 | .000 |
(Constant) | 299.596 | 24.039 | 12.463 | .000 |
We know the importance of linear relationship between Dst and B_{z} step by step (Kane 2010). In addition to this approach, it is useful to investigate the relationship between ap and B_{z}, and between AE and B_{z} using both linear and nonlinear models. Table 10 and Fig. 3 display the linear and quadratic relationships of the magnetic field component B_{z} with the Dst index. \({\text{Dst}} = - \left( {41.602} \right) + \left( {5.677} \right)B_{\text{z}}\) where R is 0.618, and \({\text{Dst}} = - \left( {38.016} \right) + \left( {5.418} \right)B_{\text{z}} - \left( {0.078} \right)B_{\text{z}}^{2}\) where R is 0.625.
ANOVA (analysis of variance)
Source | Sum of squares | df | Mean squares |
---|---|---|---|
Regression | 2494.129 | 3 | 831.376 |
Residual | 384.379 | 117 | 3.285 |
Uncorrected total | 2878.509 | 120 | |
Corrected total | 1209.483 | 119 |
Parameter estimates
Parameter | Estimate | SE | 95% Confidence interval | |
---|---|---|---|---|
Lower bound | Upper bound | |||
a | − 7.209 | .762 | − 8.719 | − 5.700 |
b | 2.460 | .189 | 2.086 | 2.834 |
c | .376 | .022 | .332 | .420 |
We believe that this nonlinear mathematical model allows a unique expression of pressure and density for plasma-dense medium (underground or atmosphere).
Conclusion
The St. Patrick’s Day geomagnetic storm is the most severe storm in the 24th solar cycle. Every model that can be established about the storm should be meticulously analyzed. In particular, the mathematical models involving magnetic field, solar wind pressure and proton density give ideas of the dynamic nature of the different plasmatic structures. This study has focused on the March 2015 severe storm by using the St. Patrick’s Day severe storm data (120 h). The data have been analyzed mathematically, and the models have been established. The models strictly obeying to physical principles have been consistently introduced in this study as well. These models support the previous studies of the author. The zonal geomagnetic indices produced by solar wind parameters are displayed in the correlations based on the cause–effect relationship. Graphs and tables have presented the relationship between zonal geomagnetic indices and solar wind parameters, as well as their interactions with each other. All results are in the 95% confidence interval. Even though some models have discussed the various results of the storm with low precision (statistically), they have been included in this paper for comparison.
Notes
Authors’ contributions
The manuscript has one author. Data are collected and analyzed by the author. All interpretations and explanations belong to the author. The author read and approved the final manuscript.
Acknowledgements
I thank the NASA CDA Web for OMNI Database (http://themis.igpp.ucla.edu/software.shtml) and Kyoto World Data Center for providing AE index and Dst index. I acknowledge the usage of ap and Kp index from the National Geophysical Data Center. The Dst index and AE data were provided by the World Data Center for Geomagnetism at Kyoto University. I would like to thank Kirklareli University, Professor Ali Yigit, Dereyayla and Akyuz families for their valuable support for this study. I thank Professor Huishan S. Fu for his very helpful comments, corrections and suggestions. I thank Professor Halil Atabay for very supportive corrections.
Competing interests
The author declares no competing interests.
Availability of data and materials
The data used in this article are available at the Data Center of NASA https://omniweb.gsfc.nasa.gov/form/dx1.html
Ethics approval and consent to participate
This study does not require ethical approval.
Funding
We declare no funding.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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