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Earth, Planets and Space

, 71:26 | Cite as

Modeling the superstorm in the 24th solar cycle

  • Emre ErogluEmail author
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  1. 3. Space science

Abstract

The St. Patrick’s Day phenomenon is a geomagnetic storm that deserves serious discussion because of its intensity and effectiveness. This study focuses on the St. Patrick’s Day storm on March 17, 2015, which is the first big storm of the 24th solar cycle. The data obtained from various spacecrafts observing the ionosphere reveal the reputation and the strength of the storm. The author tries to discuss the event as a whole with all its parameters. Variables of the study are the solar wind parameters and zonal geomagnetic indices. Models with solar wind pressure, proton density and magnetic field may aid in making the dynamic structure of the phenomenon more understandable. The obtained models are able to give the reader an idea of the results even if the storm prediction percentage is low. The author has endeavored to obey the cause–effect relationship without ignoring the physical principles when establishing mathematical models. Despite the fact that the relations between variables have poor correlation or have low statistical significance, in order to introduce the physical point of view they have not been ignored. This study puts forth a new mathematical perspective by discussing and visualizing what happened in the phenomenon.
s

Keywords

Mathematical modeling Zonal geomagnetic indices Solar wind parameters 

Introduction

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 Bz.

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 Bz 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 (Bz), 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, Bz) 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 characteristic storm at the intense level (Dst = − 223 nT) on March 17, 2015, has been analyzed. Figure 1 shows the OMNI data set from 00:00 UT on March 15, 2015, to 00:00 UT on March 19, 2015. The plot interval covers the storm day (March 17, 2015), 2 days before and 2 days after the storm (120 h). The St. Patrick’s Day storm started on March 17 with CME. The solar wind pressure (P) suddenly rose to one of the highest values of 17.91 nPa (min.: 1.68; max.: 20.76 nPa), the magnetic field component (Bz) reached its maximum value of 20.1 nT, and the proton density (N) increased to 38.5 1/cm3, one of its greatest values (min.: 2.7; max.: 40.1 cm−3) between 04:00 and 05:00 UT on March 17. The plasma flow speed (v) rose to 609 km/s 5 h later. It may be useful to observe variables at the maximum or minimum values before reviewing the literature. During the St. Patrick’s Day storm, the Dst index reached the minimum value of − 223 nT and the geomagnetic aurora electrojet index (AE) increased to reach its maximum value of 1570 nT. The magnetic field component of Bz decreased to − 18 nT, and ap index increased to 179 nT. The aurora appears in both hemispheres.
Fig. 1

From top to bottom the parameters shown are Dst index, \(B_{\text{z}}\) magnetic field (nT), E electric field (mV/m), proton density N (1/cm3), solar wind dynamic pressure P (nPa), flow speed v (km/s) and auroral electrojet AE (nT) index for March 15–19, 2015 (from NASA NSSDC OMNI data set). The characteristic storm at the intense level (Dst = − 223 nT) on March 17, 2015, has been analyzed. Figure 1 shows the OMNI data set from 00 UT March 15, 2015, to 00 UT March 19, 2015. The plot interval covers the storm day (March 17, 2015), 2 days before and 2 days after (120 h) the storm

The parameters shown in Dst index, Bz magnetic field (nT), E electric field (mV/m), proton density N (1/cm3), 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), Bz 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/cm3, 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 Bz 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 Bz 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/cm3, 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

The descriptive analysis values of the geomagnetic storm on March 2015 are displayed in Table 1. The reason for applying the descriptive analysis is to control the change interval of the variables and to acquire an idea about the standard deviations. The effect of the variable with a high standard deviation will be reduced. Accordingly, the most powerful variables statistically are P, E, N, Bz, ap, respectively. It is expected that these variables will shape the storm. However, because of the causality principle and the cause–effect relationship, solar parameters are the causes and zonal geomagnetic indices are the results of the storm. The instant correlation samples between each coefficient of the storm are shown in Table 2. Pearson’s correlation analysis is a parametrical statistical method which shows the direction, degree and importance of the relationship between variables. The correlation analysis is a complementary method of regression analysis. While the value between the two variables approaches ± 1, the relationship is strengthening. Physically, in this storm, the models in which take place Bz with ap, Dst, AE and T with v and N with v, P and v with Kp, Dst, AE and P with ap and E with Kp, Dst, ap, AE may be considered as preferential.
Table 1

Descriptive analysis

 

N

Minimum

Maximum

Mean

SD

Bz (nT)

120

− 18.1

20.1

− .317

6.8809

T (K)

120

21,425

912,227

142,900.23

162,135.845

N (1/cm3)

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

Table 2

Pearson’s correlation matrix for the storm variables

 

Bz (nT)

T (K)

N (1/cm3)

v (km/s)

P (nPa)

E (mV/m)

Kp

Dst (nT)

ap (nT)

AE (nT)

Bz (nT)

1

.038

.171

− .316**

− .165

− .889**

− .580**

.618**

− .712**

− .687**

T (K)

 

1

− .133

.511**

.378**

.084

.407**

− .224*

.241**

.244**

N (1/cm3)

  

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

* and ** Correlations are significant at the 0.05 level (two-tailed) and at the 0.01 level (two-tailed), respectively

KMO and Bartlett’s test tables (Table 3) reveal the suitability of the data for factor analysis and show the strength of the relationship between variables. The Kaiser–Meyer–Olkin measure of sampling adequacy is a statistic and shows the commensurate of the variance in data that can be caused by main factors. High values (close to 1.0) imply that data are appropriate for a satisfactory factor analysis method. If the test value of variables is appropriate to the method, they exhibit a normal distribution. In order to deny the hypothesis H0 (null hypothesis), the significance of the test should be less than 0.05. The attitudes of the data released as a result of a physical phenomenon can be determined by this test. If the data of the physical event indicate normal distribution, the variables show how they can be coordinated with each other and with the event. Thus, the linear or nonlinear relations can be discussed and models can be argued with obeying the mathematical approaches. As can be seen from Table 3, the variable set of this storm is suitable for factor analysis.
Table 3

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

Factor analysis is used with the principle component analysis and varimax with Kaiser normalization for the rotation (converged in 3 iterations) to divide the variables into subgroups and to distinguish those who have the highest contribution to the event. In this analysis, which does not include composite variables, each variable is handled separately. The variables are examined in a more specific (by heap) way with basic component analysis. In Table 4, when the ten variables are substituted into the data reduction method, three maximum eigenvalues of the covariance matrix describe 88% of the total change, which means that it can be explained by modeling the 88% of the phenomenon with the variables at hand.
Table 4

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

Varimax with Kaiser normalization method for the rotation matrix examines the linear grouping of variables of the event. The method approaching each variable as a factor indicates the contribution and weight of these factors in the linear clustering. Table 5 summarizes these weights.
Table 5

Rotated component matrix

Component

Bz (nT)

T (K)

N (1/cm3)

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

Hence, these models can be written as follows with factor weights from Table 5.
$$\begin{aligned} {\text{Axes}}\,1 & = - \left( {0.920} \right)B_{\text{z}} - \left( {0.001} \right)T - \left( {0.141} \right)N + \left( {0.400} \right)v + \left( {0.315} \right)P + \left( {0.938} \right)E \\ & \quad + \,\left( {0.766} \right){\text{Kp}} - \left( {0.702} \right){\text{Dst}} + \left( {0.887} \right){\text{ap}} + \left( {0.828} \right){\text{AE}} \\ \end{aligned}$$
$$\begin{aligned} {\text{Axes}}\,2 & = \left( {0.069} \right)B_{\text{z}} + \left( {0.054} \right)T + \left( {0.956} \right)N - \left( {0.518} \right)v + \left( {0.805} \right)P + \left( {0.012} \right)E \\ & \quad - \,\left( {0.070} \right){\text{Kp}} + \left( {0.543} \right){\text{Dst}} + \left( {0.137} \right){\text{ap}} - \left( {0.115} \right){\text{AE}} \\ \end{aligned}$$
Figure 2 illustrates the physical scattering of Dst, ap and AE zonal geomagnetic indices according to Bz, P, v, N solar wind parameters. In this work, the solar wind propagation time from bow shock to Earth is not taken into account when Dst, ap and AE data from ground stations are used. As the time is too short to take these into account, one can find in Fig. 2 only the magnitudes of the values of Bz, E, P, N, v, T solar wind parameters and Dst, ap, AE zonal geomagnetic indices, no matter how and when these arose with respect to each other.
Fig. 2

Appearance between Dst, ap, AE indices and Bz, E, P, N, v, T solar wind parameters. Relation between zonal geomagnetic indices (Dst, ap and AE) and solar wind parameters (magnetic field component (Bz), the electric field (E), dynamic pressure (P), proton density (N), flow velocity (v), temperature (T)) can be seen in Fig. 2. The relationships in the correlation are visualized

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 Bz 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 Bz, 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 Bz, electric field, dynamic pressure and temperature is linear, its response to proton density and flow speed is nonlinear.

Linear and nonlinear model

The regression model is:
$$y_{i} = f_{i} (x,b) + \varepsilon_{i} = \sum\limits_{j = 1}^{n} {b_{j} x_{j} } + \varepsilon_{i}$$
where yi is dependent variable, xj is n-dimensional independent variable, and \(\varepsilon_{i}\) is error. \(f_{i} (x,b)\) is called the expectation function for the regression model.
The sample covariance \(s_{jk}^{2}\) is:
$$s_{jk}^{2} \equiv \frac{{\frac{1}{n - 1}\sum\nolimits_{i = 1}^{n} {\left[ {\frac{1}{{\sigma_{i}^{2} }}\left( {x_{ij} - \bar{x}_{j} } \right)\left( {x_{ik} - \bar{x}_{k} } \right)} \right]} }}{{\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\frac{1}{{\sigma_{i}^{2} }}} \right)} }}$$
where j, k = 1, 2; \(\sigma_{i}^{2}\) is the standard deviation; n is the number of data points; and \(\bar{x}_{j}\) is
$$\bar{x}_{j} \equiv \frac{{\sum\nolimits_{i = 1}^{n} {\left( {\frac{{x_{ij} }}{{\sigma_{i}^{2} }}} \right)} }}{{\sum\nolimits_{i = 1}^{n} {\left( {\frac{1}{{\sigma_{i}^{2} }}} \right)} }}.$$
The sample variance is given by \(s_{j}^{2} \equiv s_{jj}^{2}.\) The correlation coefficient can be expressed in terms of \(r_{jk} \equiv \frac{{s_{jk}^{2} }}{{s_{j} s_{k} }}.\) Square of multiple correlation coefficient R2 is:
$$R^{2} \equiv \sum\limits_{j = 1}^{n} {\left( {b_{j} \frac{{s_{jy}^{2} }}{{s_{y}^{2} }}} \right)} = \sum\limits_{j = 1}^{n} {\left( {b_{j} \frac{{s_{j} }}{{s_{y} }}r_{jy} } \right)}.$$
R2 is the percentage of the event defined by the model. The closer the R2 is to one (1) in the established model, the greater the percentage of the model’s description of the event is (Freund 1979; Saba et al. 1997).
Before discussing the binary relations of the zonal geomagnetic indices governed by the solar wind parameters, it would be appropriate to see the linear compositions of the indices. According to independent variables (solar wind parameters), the linear compounds of the dependent variables Dst and ap (zonal geomagnetic indices) are given in Tables 6, 7, 8 and 9, respectively. The coefficients can be seen from the tables. This table (Table 6) demonstrates how much of the residuals are explained by the variables in the linear regression model. One may realize that regression coefficients are significant. Table 7 shows the model of Dst index as: \({\text{Dst}} = - \left( {244.925} \right) + \left( {3.497} \right)B_{\text{z}} + \left( {10.321} \right)N - \left( {11.814} \right)P + \left( {0.256} \right)v\), where multiple determination coefficient R is 0.782.
Table 6

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

   
Table 7

Regression coefficients

Model

Unstandardized coefficients

Standardized coefficients

t

Sig.

B

SE

Beta

(Constant)

− 244.925

73.544

 

− 3.330

.001

Bz (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

Table 8

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

   
Table 9

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

Bz (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.

Physically, the magnetospheric activity is nonlinearly proportional to the proton density (N) and plasma flow speed (v) and linearly proportional to the interplanetary magnetic field (IMF) (Temerin and Li 2006; Agopyan 2010). Changes in solar wind pressure and CME cause nonlinear behavior, fluctuations and changes in the density of particles. The serious (> 10 nT) orientation of the Bz component of the magnetic field to the southward for more than a few hours causes depression in the Dst and gets Dst directed to the negative direction. This depression on the Dst demonstrates a severe storm. Visualizing the response of such a storm to the solar wind parameters (especially Bz component) of the Dst index will give the reader a clearer idea. The linear and nonlinear models between the Dst, ap, AE indices and Bz are shown in Figs. 3, 4, 5 and Tables 10, 11, 12, respectively. The low correlation coefficients in these models should not be overlooked. Statistically, these are generally middle levels of models.
Fig. 3

Linear and quadratic relation of Dst and Bz. In Fig. 3, the linear and quadratic relationships of the magnetic field component Bz with the Dst index are shown

Table 10

Regression coefficients

 

Unstandardized coefficients

Standardized coefficients

t

Sig.

B

SE

Beta

Bz (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

Bz (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

Table 11

Regression coefficients

 

Unstandardized coefficients

Standardized coefficients

t

Sig.

B

SE

Beta

Bz (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

Bz (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

Table 12

Regression coefficients

 

Unstandardized coefficients

Standardized coefficients

t

Sig.

B

SE

Beta

Bz (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

Bz (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 Bz step by step (Kane 2010). In addition to this approach, it is useful to investigate the relationship between ap and Bz, and between AE and Bz using both linear and nonlinear models. Table 10 and Fig. 3 display the linear and quadratic relationships of the magnetic field component Bz 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.

In Table 11 and Fig. 4, the linear and nonlinear relationships between the magnetic field component Bz and the ap index are presented. \({\text{ap}} = \left( {38.007} \right) - \left( {5.029} \right)B_{\text{z}}\) where R is 0.712, and \({\text{ap}} = \left( {22.542} \right) - \left( {3.913} \right)B_{\text{z}} + \left( {0.336} \right)B_{\text{z}}^{2}\) where R is 0.896. Mathematically, the height level of the nonlinear correlation with the Bz component and ap index should not be overlooked.
Fig. 4

Linear and quadratic relation between ap and Bz. In figure, the nonlinear relationship between the magnetic field component Bz and the ap index can be seen

In Table 12 and Fig. 5, the linear and nonlinear relationships between the magnetic field component Bz and the AE index are presented. \({\text{AE}} = \left( {349.146} \right) - \left( {33.012} \right){{B}}_{z}\) where R is 0.687, and \({\text{AE}} = \left( {299.596} \right) - \left( {29.434} \right)B_{\text{z}} + \left( {1.077} \right)B_{\text{z}}^{2}\) where R is 0.733.
Fig. 5

Linear and quadratic relation of AE and Bz. In figure, the nonlinear relationship between the magnetic field component Bz and the AE index can be seen

Physically, fluctuations in the magnetic field indicate similar linear effects in flow pressure (P) and proton density (N), while ap index responds to these fluctuations nonlinearly. This nonlinear relationship and model are shown in Table 11 and Fig. 4. The nonlinear model is in the form of \(P = a + b\ln \,{\text{ap}} + cN\), where a, b, c are constants. The analysis of variance values of the model are shown in Table 13. The magnitudes of coefficients are a = − 7.209, b = 2.460 and c = 0.376. Table 14 shows that all parameter estimation is in the confidence interval of 95%. The model explaining this storm with 75.2% accuracy is
$$P = - \left( {7.209} \right) + \left( {2.460} \right)\ln \,{\text{ap}} + \left( {0.376} \right)N.$$
Table 13

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

 
Table 14

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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Authors and Affiliations

  1. 1.Department of MathematicsKirklareli UniversityKirklareliTurkey

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