Modeling and comparison of two geomagnetic storms

Journal Pre-proofs Modeling and Comparison of Two Geomagnetic Storm Samed Inyurt PII: DOI: Reference:

S0273-1177(19)30795-1 https://doi.org/10.1016/j.asr.2019.11.004 JASR 14528

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Advances in Space Research

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8 September 2019 1 November 2019 4 November 2019

Please cite this article as: Inyurt, S., Modeling and Comparison of Two Geomagnetic Storm, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.11.004

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Modeling and Comparison of Two Geomagnetic Storm Samed Inyurt Department of Geomatics Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey [email protected]

Abstract. This paper is on the investigation and comparison of two consecutive geomagnetic storms in the 24th solar cycle. The first storm is a moderate (Dst=-50 nT) storm occurred on 22.04.2017, while the second is an intense (Dst=-125 nT) storm happened on 28.05.2017. The comparison was carried out based on solar wind (Bz, E, P, N, v, T) and zonal geomagnetic indices (Dst, ap, AE). Mathematical analysis is introduced to the reader with a data set of 9 variables obtained from NASA's observations. The descriptive statistics, demonstrating the power of binary relations with a covariance matrix, hierarchical clustering with a dendrogram, and factor analysis pointing to three separate eigenvectors are the headstone of the discussion. Considering the results, linear models of zonal geomagnetic indices caused by solar parameters are discussed governing the causality principle. Finally, the study ends with a nonlinear model which includes solar wind pressure (P), proton density (N) and ap index. In all analysis, the causality principle is obeyed rigidly and the mathematical models are discussed with the causeeffect relationship. Keywords: Mathematical modeling; zonal geomagnetic indices; solar wind parameters

1. Introduction The rapid changes in the magnetic field caused by a solar wind blasted from the sun at very high speeds cause a geomagnetic storm. The solar wind, a plasma-dense dynamic medium, has energetic particles like electron and proton. The main conditions of the geomagnetic storm are a coronal mass ejection (CME) cloud that enters the Earth's atmosphere and swallows it, and the Bz component of the magnetic field is directed to the southward negatively. Since the solar corona has a very thick layer, a sudden ejection (burst) produces a mass pulse, escaping the gravitational field of the sun at high speeds. CMEs caused by magnetic reconnection in the solar corona lead to rapid changes in solar wind pressure (Lin et al. 2000). After the sudden commencement and the main phase, with the aid of the magnetic field lines, magnetic reconnection occurs (Lin and Forbes, 2000; Fu et al., 2014; Lin and Ni, 2018). CME causes changes in solar wind parameters, resulting in disruption of ionospheric electric fields at medium and low latitudes (Blanc and Richmond, 1980; Gonzalez et al., 1987, 1989, 1999; Zic et al., 2015; Manoharan et al., 2017; Subrahmanya et al., 2017). However, it is not known exactly how CMEs interact with the ejecta velocity and the southward magnetic field to influence the electric field and change the geomagnetic storm (Yingt et al., 2015; Cherniak and Zakharenkova, 2015). The magnetic reconnection between the negative Bz component of the southbound magnetic field and the terrestrial magnetic field causes geomagnetic disturbances. The change in the strength of the terrestrial magnetic field of the earth is measured by the disturbance storm time index (Dungey, 1961; Sugiura, 1964). The relationship between the Dst index and the Bz component of the magnetic field is vital for the geomagnetic storms. Studies have shown that in almost all storms, the Dst index reaches its minimum value after Bz. Physically, this delay is interpreted as the response time of the ring current to the solar wind (Burton, 1975). While the response time of the intense storm is about 7-9 hours, that of the moderate storm goes down to about 5-6 hours and that of the weak storm goes down to about 2-4 hours. The vital relationship between Dst-Bz is not sufficient to understand what is happening in geomagnetic storms. Kp (planetary index) is taken by a weighted average of K indexes in 13 sub-auroral observations. Kp index is a quasi-logarithmical planetary index obtained from the ap index. The AE index is a snapshot general index as a measure of changes

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in global Auroral Electrojet activity as defined by Davis and Sugiura (Davis and Sugiura, 1966). The author, therefore, tries to discuss all possible models between the solar wind parameters and zonal geomagnetic indices obeying the causality principle (Akasofu, 1964; Mayaud 1980; Eroglu et al., 2012; Eroglu et al., 2012; Rathore et al., 2014; Borovsky 2012; 2018; Borovsky and Yakymenko, 2017). Large solar plasma clouds, which reach high speeds, exert pressure on the magnetosphere, creating a storm effect. The results of this impact are scaled by geomagnetic indices in the region (Mayaud 1980; Kamide et al., 1998; Tsurutani et al., 2006; Joshi et al., 2011; Elliott et al., 2013). In this study, hourly versions of indices are used. This paper rigidly obeys to physical principles and includes math-based discussions. This obedience via cause-effect relationship and causality principle governs solar wind parameters and correlations between zonal geomagnetic indices. While Bz, E, P, v, T are the "cause", Dst, ap and AE are the "effect". In this study, the author considers the data of the April and May geomagnetic storms and he discusses mathematical models relating to storms. Our results support the last works of (Eroglu 2018, 2019) and previous paper (Inyurt and Sekertekin, 2019). The paper is organized as follows. In Section 2 solar parameters, zonal geomagnetic indices and five-day distributions of data are introduced. In Section 3 the analyses are performed. In Section 4 discussion is argued. 2. Data Space Physics Environment Data Analysis Software (SPEDAS), which is IDL-based, is utilized in this study and it can be accessed via the following link http://themis.igpp.ucla.edu/software.shtml. The hourly OMNI-2 Solar Wind and IMF parameter data are accessible online. Besides, the AE, and Dst indices are taken from World Data Center for Geomagnetism Kyoto by using SPEDAS. ap is taken from NGDC using SPEDAS with CDA Web Data Chooser (space physics public data). The general classification of magnetic storms is given in Table 1. Class Weak Moderate Strong (i.e., intense) Severe (very-intense) Great

Table 1. Geomagnetic storm Dst index Number % 482 44 346 32 206 19 45 4 6 1

Dst Range (nT) -30 - -50 -50 - -100 -100 - -200 -200 - -350 < -350

For 2017 April (moderate) and May (intense) storms, solar wind dynamic pressure, magnetic field, electric field, flow speed, proton density, temperature, Dst, ap and AE indices were recorded in the OMNI hourly data. Geomagnetic storms are classified according to the intensity of the Dst index (Loewe & Prölss 1997). This paper focuses on 22 April 2017 moderate (Dst=50 nT) and 28 May 2017 intense (Dst=-125 nT) geomagnetic storms. Figure 1 demonstrates the OMNI data set from 00:00 UT on 20 April 2017 to 00:00 UT on 24 April 2017 (left side) and 00:00 UT on 26 May 2017 to 00:00 UT on 30 May 2017 (right side). The plot interval covers the storm day (2017 April 22 and May 28), two days before and two days after the storm (120 hours). The April storm launched on April 21st with CME between 15:00 and 16:00 UT. (P) suddenly peaked 7.40 nPa (max.:7.9 nPa), the Bz component oriented southward by hitting -5.6 nT (min.:6.8 nT), the proton density (N) rose up to 12.8 1/cm3 (max.:13.9 1/cm3) and afterward Dst index hit to -46 nT (min.:-50 nT).

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The May storm started on May 27th with first CME between 15:00 UT. The solar wind pressure (P) suddenly peaked 6.78 nPa (max.:15.37 nPa), the Bz component directed southward by hitting -3.5 nT (min.:-19.5 nT), the proton density rose up to 28.8 1/cm3 (max.:62.1 1/cm3) and afterward second CME burst at 20:00 UT. The solar wind pressure (P) suddenly peaked its maximum value of 15.37 nPa. Four hours later Bz component directed southward by decreasing -19.3 nT at 00:00 UT. Seven hours later Dst index responded by hitting its minimum value of 125 nT at 07:00 UT 28.05.2017.

Figure 1. From top of the bottom parameters shown in Dst index (nT), Bz magnetic field (nT), E electric field (mV/m), solar wind dynamic pressure P (nPa), flow speed v (km/s), proton density N (1/cm3), and aurora electrojet AE (nT) index for 2017 April 20-24 (left side) and May 26-30 (right side) from NASA NSSDC OMNI data set

The variables of Figure 1 may be briefly described as follows. For April storm, on 22.04.2017 at 14:00 UT when Bz component decreases -4.8 nT, Dst index decreases to -42 nT, the electric field E reaches to value of 3.24 mV/m, ap index reaches a value of 48 nT , proton density N hits a value of 5.8 1/cm3, plasma flow speed v becomes 674 km/s, AE index catches its minimum value 1043 nT. After three hours, Dst index reaches its negative peak value -50 nT.

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On 22.04.2017 at 16:00 UT when Dst index shows its minimum value -50 nT, Bz component shows -1.9 nT, the electric field E indicates 1.36 mV/m, AE index reaches 894 nT, ap index shows 39 nT and flow pressure P takes 3.43 nPa. On 20.04.2017 at 01:00 UT when Bz component is maximum (3.2 nT), the electric field reaches its one of the minimum values of -1.61 mV/m, proton density N reaches its maximum value of 13.9 1/cm3, AE index shows 638 nT and ap index continues to decrease. As this happens Dst index reaches -10 nT. On 23.04.2017 at 18:00 UT when plasma flow speed v reaches peak value (761 km/s), Bz component decreases -0.8 nT, the electric field E reaches 0.61 mV/m, AE index reaches 435 nT, ap index indicates 27 nT, flow pressure P takes 2.95 nPa and Dst index is -35 nT. The components of Figure 1 can be briefly described as follows. On 28.05.2017 at 00:00 UT when Bz component is at its minimum (-19.5 nT), Dst index decreases to -58 nT, the electric field E reaches to its maximum value of 7.7 mV/m. Meanwhile, ap index reaches a value of 80 nT by increasing, proton density N decreases value of 9.7 1/cm3, plasma flow speed v becomes 395 km/s, AE index catches 847 nT. After seven hours Dst index reaches its negative peak value -125 nT. On 28.05.2017 at 07:00 UT when Dst index shows its minimum value -125 nT, Bz component shows -11.7 nT, the electric field E reaches 4.32 mV/m, AE index reaches 1083 nT, ap index reaches 80 nT and flow pressure P takes 2.57 nPa. On 29.05.2017 at 00:00 UT when Bz component is maximum (12.0 nT), the electric field reaches a value of -4.08 mV/m, proton density N takes 10.2 1/cm3, AE index decreases 40 nT and ap index continues to decrease. As this happens Dst index reaches -23 nT. On 30.05.2017 at 20:00 UT when plasma flow speed v reaches peak value (539 km/s), Bz component decreases 0.2 nT, the electric field E reaches -0.11 mV/m, AE index reaches 72 nT, ap index reaches 2 nT, flow pressure P takes 0.62 nPa and Dst index is -15 nT. 3. Mathematical Modeling Descriptive analysis for the variables of April and May storms is shown in Table 1 and the matrix showing the correlation coefficients is discussed in Table 2a, b. The descriptive analysis discusses minimum-maximum values, standard deviations and variances of data. The Pearson correlation matrix shows the instantaneous correlation of variables and the strength of binary relationships. The ± signs in front of the coefficients indicate the direction of the relationship. When the values in Table 2a, b approach ± 1, the bilateral relations are strengthened. Physically, in April storm, T-P-ap-AE, N-P, v-Dst, P-ap groups have a strong correlation and Bz-ap-AE, N-P, E-ap-AE groups have also a strong correlation in May storm. Mathematical models involving these pairs allow a better interpretation and understanding of the event.

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Table 2. Descriptive analysis 22.04.2017 Moderate Storm

28.05.2017 Intense Storm

Min.

Max.

Mean

Median

Std. Dev.

Min.

Max.

Mean

Median

Std. Dev.

Bz(nT)

-6.8

3.2

-1.248

-1.1

2.0033

-19.5

12.0

-.797

.3

6.5340

T(K)

5084 5

411302

224479.39

232952

90095.458

13667

779693

92053.17

52960.5

117212.023

N(1/cm3)

1.5

13.9

4.068

3.1

2.5466

.6

62.1

8.913

6.1

10.6553

v(km/s)

477

761

638.63

658

79.388

296

539

370.46

355.5

66.003

P(nPa)

.8

7.4

3.202

3.1

1.6067

.3

15.4

2.198

1.19

2.6936

E(mV/m)

-1.8

3.7

.839

.7

1.2958

-4.1

7.7

.287

-.11

2.4402

Dst(nT)

-50

10

-27.72

-30.5

13.407

-125

43

-20.25

-15

34.140

ap(nT)

2

67

29.45

27

17.652

1

132

15.35

3

27.641

AE(nT)

28

1043

472.94

463.5

261.412

24

1270

230.66

83

304.060

Table 3a. Pearson’s correlation matrix for the storm variables (22.04.2017 Moderate Storm) Bz(nT)

Bz(nT)

T(K)

N(1/cm3)

v(km/s)

P(nPa)

E(mV/m)

Dst(nT)

ap(nT)

AE(nT)

1

-.254**

-.077

-.261**

-.218*

-.991**

.290**

-.358**

-.543**

1

.398**

.570**

.719**

.258**

-.580**

.777**

.646**

1

-.306**

.868**

.042

.221*

.530**

.322**

1

.132

.317**

-.766**

.399**

.494**

1

.207*

-.124

.780**

.558**

1

-.337**

.359**

.565**

1

-.449**

-.581**

1

.695**

T(K) N(1/cm3) v(km/s) P(nPa) E(mV/m) Dst(nT) ap(nT) AE(nT) *.and **.

1 Correlation are significant at the 0.05 level (2-tailed) and at the 0.01 level (2-tailed), respectively

Table 3b. Pearson’s correlation matrix for the storm variables (28.05.2017 Intense Storm) Bz(nT) T(K) N(1/cm3) v(km/s)

Bz(nT)

T(K)

N(1/cm3)

v(km/s)

P(nPa)

E(mV/m)

Dst(nT)

ap(nT)

AE(nT)

1

.133

-.116

.017

-.145

-.997**

.375**

-.818**

-.833**

1

-.163

.575**

-.105

-.160

-.072

-.102

-.110

1

-.117

.990**

.128

.311**

.191*

.094

1

-.010

-.056

-.146

.033

.018

1

.155

.243**

.234*

.146

1

-.383**

.826**

.843**

1

-.618**

-.717**

1

. .893**

P(nPa) E(mV/m) Dst(nT) ap(nT) AE(nT) *.and **.

1 Correlation are significant at the 0.05 level (2-tailed) and at the 0.01 level (2-tailed), respectively

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KMO and Bartlett's Test examines the distribution of data and its suitability for factor analysis. Test value close to 1.0 indicates that the data are appropriate for the factor analysis method. KMO values between 0.8 and 1 indicate the sampling is well-posedness. KMO values less than 0.6 indicate the sampling is not adequate and that remedial action should be taken. Some researchers put this value at 0.5, so one can utilize judgment for values between 0.5 and 0.6. April and May storms can be modeled with the help of normal distribution of data (Table 3). Table 4. KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. Approx. Chi-Square df Bartlett's Test of Sphericity Sig.

22.04 Moderate 28.05 Intense .705 .632 1413.150 1844.955 36 36 .000 .000

Hierarchical cluster discussion of the variables of the storms is given in Figure 2. In this way, there are two main blocks. The first heap is P, E, Bz, N, Kp, ap, Dst, v, AE, while the second heap consists of temperature (T).

22.04.2017 Moderate Storm

28.05.2017 Intense Storm

Figure 2. Dendrogram of hierarchical cluster analysis.

Kaiser Normalization and Principal Component Analysis is an appropriate analysis for dividing data into subgroups. The variables divided into sub-groups indicate maximum eigenvalues with utmost contribution approach. The change in the storms of April and May can be modeled at 89% and 86% with three maximum eigenvalues, respectively (Table 5a, 5b).

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Table 5a. Total variance explained (22.04.2017 Moderate Storm) Component 1 2 3

Total 4.495 2.135 1.460

Initial Eigenvalues % of Variance Cumulative % 49.943 49.943 23.726 73.670 16.219 89.889

Total 3.154 2.679 2.257

Rotation Sums of Squared Loadings % of Variance Cumulative % 35.043 35.043 29.772 64.815 25.074 89.889

Table 5b. Total variance explained (28.05.2017 Intense Storm) Component 1 2 3

Total 4.032 2.222 1.508

Initial Eigenvalues % of Variance Cumulative % 44.805 44.805 24.686 69.491 16.759 86.250

Total 3.996 2.150 1.617

Rotation Sums of Squared Loadings % of Variance Cumulative % 44.395 44.395 23.890 68.285 17.965 86.250

Varimax Rotation Matrix method indicates the linear clustering of variables. The coefficients in Table 6a, 6b are the values of the weighted contributions of the factors. It is possible to see two main axes of the total variance in Table 6a, 6b. This table is the weight rotated matrix of the variables given in 89% and 86% total variance for April and May storms, respectively. Table 6a. Rotated component matrix (22.04.2017 Moderate Storm) Component

Bz(nT)

T(K)

N(1/cm3)

v(km/s)

P(nPa)

E(mV/m)

Dst(nT)

ap(nT)

AE(nT)

1

-.109

.686

.917

-.006

.969

.087

-.045

.780

.523

2

-.124

.626

-.338

.920

.083

.178

-.908

.427

.515

The linear models that occur with the weights of the data presented in Table 6a are as follows: Axes 1 = ― (0.109)Bz + (0.686)T + (0.917)N ― (0.006)v + (0.969)P + (0.087)E ― (0.045)Dst + (0.780)ap + (0.523)AE Axes 2 = ― (0.124)Bz + (0.626)T ― (0.338)N + (0.920)v + (0.083)P + (0.178)E ― (0.908)Dst + (0.427)ap + (0.515)AE Table 6b. Rotated component matrix (28.05.2017 Intense Storm) Component

Bz(nT)

T(K)

N(1/cm3)

v(km/s)

P(nPa)

E(mV/m)

Dst(nT)

ap(nT)

AE(nT)

1

-.917

-.099

.054

.041

.104

.922

-.682

.939

.966

2

-.104

-.083

.984

-.009

.981

.110

.414

.135

.024

The linear models that occur with the weights of the data presented in Table 6b are as follows:

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Axes 1 = ― (0.917)Bz ― (0.099)T + (0.054)N + (0.041)v + (0.104)P + (0.922)E ― (0.682)Dst + (0.939)ap + (0.966)AE Axes 2 = ― (0.104)Bz ― (0.083)T + (0.984)N ― (0.009)v + (0.981)P + (0.110)E + (0.414)Dst + (0.135)ap + (0.024)AE Discussing linear models as a result of the cause-effect relationship between solar wind parameters and zonal geomagnetic indices allows a closer look at April and May storms (Table 7-12). The linear model of the Dst index in Table 7-8, the other linear model of the ap index in Table 9-10, and the last linear model of the AE index are presented to the reader in Table 1112. It can be seen that all models are significant (Table 7-12). Table 8 shows the model of Dst index as: Dst = (32.897) ― (0.080)v ― (0.0007)T + (2.829)P ― (1.261)E, (April-left side) Dst = ― (110.549) ― (3.827)E + (16.928)N ― (62.616)P + (0.211)v, (May-right side) where multiple determination coefficient R is 0.816 and 0.698, respectively. Table 7. Anova (Analysis of variance) 22.04.2017 Moderate Storm Mean df F Square

Model

Sum of Squares

Regression

14228.187

4

3557.047

Residual

7162.180

115

62.280

Total

21390.367

119

57.114

28.05.2017 Intense Storm Mean df F Square

Sig.

Sum of Squares

.000

67634.676

4

16908.669

71063.824

115

617.946

138698.500

119

Sig.

27.363

.000

Table 8. Regression coefficients

Model (Constant) v(km/s) T(K) P(nPa) E(mV/m)

22.04.2017 Moderate Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error 32.897 7.311 -.080 .013 -.475 -5 -7.764x10 .000 -.522 2.829 .752 .339 -1.261 .599 -.122

t

Sig.

Model

4.499 -6.087 -4.850 3.762 -2.105

.000 .000 .000 .000 .037

(Constant) E(mV/m) N(1/cm3) P(nPa) v(km/s)

28.05.2017 Intense Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error -110.549 21.056 -3.827 1.003 -.274 16.928 2.410 5.283 -62.616 9.509 -4.940 .211 .055 .408

Table 9 indicates that the model is significant, while Table 10 shows that the ap index is:

8

t

Sig.

-5.250 -3.816 7.025 -6.585 3.860

.000 .000 .000 .000 .000

9

ap = ― (6.352) + (4.950)P + (0.0008)T + (2.169)E, (April-left side)

ap = (11.093) + (11.888)P ― (2.740)N + (8.857)E, (May-right side) where multiple determination coefficient R is 0.854 and 0.846, respectively. Table 9. Anova (Analysis of variance) 22.04.2017 Moderate Storm Mean df F Square

Model

Sum of Squares

Regression

27032.733

3

9010.911

Residual

10046.967

116

86.612

Total

37079.700

119

104.038

28.05.2017 Intense Storm Mean df F Square

Sig.

Sum of Squares

.000

65977.380

4

16494.345

24943.920

115

216.904

90921.300

119

Sig.

76.045

.000

Table 10. Regression coefficients

Model (Constant) P(nPa) T(K) E(mV/m)

22.04.2017 Moderate Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error -6.352 2.302 4.950 .765 .451 8.079x10-5 .000 .412 2.169 .682 .159

t

Sig.

Model

-2.759 6.473 5.850 3.181

.007 .000 .000 .002

(Constant) P(nPa) N(1/cm3) E(mV/m)

28.05.2017 Intense Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error 11.093 1.785 11.888 3.638 1.158 -2.740 .916 -1.056 8.857 .577 .782

Table 11 shows that the model is significant, while Table 12 shows that the AE index is: AE = (42.690) + (28.460)P + (0.001)T + (85.233)E, (April-left side) AE = ― (64.666) + (406.376)E + (112.453)Bz + (0.724)v, (May-right side) where multiple determination coefficient R is 0.776 and 0.860, respectively.

Table 11. Anova (Analysis of variance) 22.04.2017 Moderate Storm

28.05.2017 Intense Storm

9

t

Sig.

6.213 3.268 -2.991 15.346

.000 .000 .001 .003

10

Model

Sum of Squares

df

Mean Square

F

Sig.

Sum of Squares

df

Mean Square

F

Sig.

Regression

4891268.679

3

1630422.893

58.360

.000

8134810.492

3

2711603.497

109.711

.000

Residual

3240717.913

116

27937.223

2867050.500

116

24715.953

Total

8131986.592

119

11001860.99

119

Table 12. Regression coefficients 22.04.2017 Moderate Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error

Model

28.05.2017 Intense Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error

t

Sig.

Model

1.032

.304

(Constant)

-64.666

94.550

t

Sig.

-.684

.495

(Constant)

42.690

41.350

P(nPa)

28.460

.000

.411

4.807

.000

E(mV/m)

406.376

91.014

3.261

4.465

.000

T(K)

.001

12.245

.422

6.961

.000

Bz(nT)

112.453

33.943

2.417

3.313

.001

E(mV/m)

85.233

13.733

.175

2.072

.004

v(km/s)

.724

.253

.157

2.858

.005

The linear models are a product of a good correlation between v and Dst. The guide for these models is Table 3. Multiple determination coefficient R of the April and May storms are 0.766 and 0.146, respectively. Table 13-14 shows that analysis values of Dst-v values. Table 13. Anova (Analysis of variance)

Model

22.04.2017 Moderate Storm Sum of Mean df F Squares Square

Regression

12551.513

1

12551.513

Residual

8838.853

118

74.906

Total

21390.367

119

Sig.

167.565

.000

10

28.05.2017 Intense Storm Sum of Mean df F Squares Square 2948.204

1

2948.204

135750.296

118

1150.426

138698.500

119

2.563

Sig. .112

11

Table 14. Regression coefficients

Model (Constant) v(km/s)

22.04.2017 Moderate Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error -.129 .010 -.766 54.900 6.431

t

Sig.

Model

-12.945 8.537

.000 .000

(Constant) v(km/s)

28.05.2017 Intense Storm Unstandardized Standardized Coefficients Coefficients Std. B Beta Error -.075 .047 -.146 7.687 17.724

April Storm

t

Sig.

-1.601 .434

.112 .665

May Storm

Figure 3. Linear relationship of Dst-v.

While magnetospheric activity is nonlinearly proportional with proton density (P) and plasma flow speed (v), it is linearly proportional to the magnetic field (Temerin and Li, 2006). Fluctuations in CME cause changes in proton density. Since the April and May phenomenon are moderate and intense, it is meaning to give attention to the relationship between Dst and Bz and other solar parameters (Gilmour et al., 2002; Ayush et al., 2017). The linear and nonlinear correlation between Dst, ap, AE indices and Bz are shown in Figure 4a, b.

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R=0.313

R=0.290

R=0.358

R=0.490

R=0.543

R=0.573

Figure 4a. Linear and quadratic relation of Dst-ap-AE and Bz (April storm).

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R=0.375

R=0.564

R=0.818

R=0.900

R=0.833

R=0.885

Figure 4b. Linear and quadratic relation of Dst-ap-AE and Bz (May storm)

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It would be appropriate to end the discussion with the last nonlinear models. These nonlinear models including dynamic pressure, proton density and ap index proposes by Eroglu (2018, 2019). Physically, the dynamic pressure (P) and proton density (N) are linearly affected by fluctuations of the magnetic field, while the ap index responds logarithmically. The models with P, N and ap can be seen in Tables 15 and 16. The nonlinear model is P  a  b ln ap  cN , where a, b, c are constants. The analysis of variance values of the model is shown in Table 15. The analysis of variance values of dynamic pressure (P) is shown in Table 16. For April storm, the magnitudes of coefficients are a=-0.902, b=0.784, c=0.411 and for May storm, they are a=0.214, b=0.154, c=0.244. Table 16 displays that all parameter estimation is in confidence interval of 95%. These models explaining these storms with 89.4% and 98.5% accuracy are: P = ― (0.902) + (0.784)lnap + (0.411)N (April storm-left side). P = ― (0.241) + (0.154)lnap + (0.244)N (May storm-right side). Table 15. Anova (Analysis of variance) 22.04.2017 Moderate Storm Sum of Mean Source df Squares Squares Regression 1505.101 3 501.700 Residual 32.426 117 .277 Uncorrected 1537.528 120 Total Corrected 307.191 119 Total

28.05.2017 Intense Storm Sum of Source df Squares Regression 1429.891 3 Residual 13.327 117 Uncorrected 1443.218 120 Total Corrected 863.385 119 Total

Table 16. Parameter estimates 22.04.2017 Moderate Storm 95% Confidence Interval Lower Upper Std. Bound Bound Parameter Estimate Error

Parameter

Mean Squares 476.630 .114

28.05.2017 Intense Storm 95% Confidence Interval Lower Upper Std. Bound Bound Estimate Error

a

-.902

.176

-1.250

-.555

a

-.241

.053

-.346

-.136

b

.784

.063

.660

.909

b

.154

.025

.104

.204

c

.411

.022

.367

.454

c

.244

.003

.238

.250

When one talks about these two nonlinear models and the values of the variables, (s)he may need to assert something about their internal harmony. The coefficients of Table 16 also indicate the internal consistency (and therefore reliability), the validity of the structure and the harmony between the variables. Cronbach's Alpha coefficient can guide these discussions. This constant, which indicates the internal consistency and reliability of the data in an experiment, <0.40 then

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it is low reliable, when it is between 0.41<α<0.60 then it is called moderate reliable, from 0.61 to 0.80 then it is good reliable and when 0.81<α<1.00 then it is called highly reliable. For the nonlinear models, the Alpha value of the April storm is 0.879 (high) in the first nonlinear model and the Alpha value of the May storm is 0.640 (good) in the second nonlinear model. 4. Conclusion Substantial and mysterious natural phenomena such as geomagnetic storms, which cannot be observed in the laboratory, have been eagerly discussed through their theoretical backgrounds. Many researchers have made events more understandable with their investigations. The author analyses 21.04.2017storm and 28.05.2017 storm based on solar wind parameters and zonal geomagnetic indices. In this study, boundaries and deviations are determined by descriptive analysis, bilateral relations with the correlation matrix are discussed and linear and nonlinear modeling of data showing normal distribution by KMO-Bartlett Test and component analysis is provided. While the hierarchical cluster analysis shed light on the clustering of the data, the rotation matrix gives the reader an idea for a better understanding of the event with linear models on two main axes. The zonal indices of Dst, ap and AE are presented as linear compositions by shaping the solar parameters. Due to the role of the magnetic field Bz component in the storm, linear and nonlinear graphs of the Bz component with zonal indices are visualized. Finally, the study is completed with a nonlinear model between solar wind pressure (P), proton density (N) and ap index. Modeling of April and May storms were %89 and %98 successful respectively. The model, which coincides with the studies in the literature, expresses itself with high accuracy in two different events. It is noteworthy that different events taking place in the same solar cycle, occurring at different times and with different scales, can find their place in the same model. Moreover, all results are within the 95% confidence interval. The author's discussions with consistent models and strict obey to physical facts contribute to a better understanding of the two phenomena. Acknowledgements I thank the NASA, Kyoto (University) World Data Center, National Geophysical Data Center.

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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