# Mathematical Analysis of the 08 May 2014 Weak Storm

Authors: Kevser Köklü
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Since the time scale of weak storms is about half the time scale of intense storms, it is troublesome and important to examine the solar wind parameters/interplanetary magnetic field (IMF) (E, v, P, T, N, and Bz) to evolve and affect to zonal geomagnetic indices (Kp, Dst, AE, and ap). In a severe storm, which usually has two main phases, solar parameters have enough time to react, but weak storms cannot find this time. They have to yield their reaction in a short time. One can find a weak storm in order to reveal and discuss the consistency of models that have proven themselves in severe and moderate storms in this study. I discuss weak storm (Dst = −46) on May 8, 2014, via solar wind parameters and zonal geomagnetic indices. The goal of the work is to realize the models applicable to the moderate and the strong storms for a weak storm. Hereby, all possible correlations between solar parameters and zonal indices are discussed in depth. I tried to obey the cause-effect relationship while creating mathematical models while not ignoring the physical principles. Therefore, the physical principles govern the study. The results are visualized with tables and graphs for the understanding of the dynamic structure of the storm.


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## Body

## 1. Introduction

A geomagnetic storm [1–23] is a characteristic event of nature that continues for 1–3 days and covers all the magnetosphere from the Earth’s surface to the magnetotail. If the variables that cause such recurring phenomena are well analyzed, the event processor can be understood easily. Good analyses can also lead to meaningful discussion in other phenomena. In a geomagnetic storm, the cause-effect relation and the causality principle [15–18, 24–26] are physically restrictive. The solar wind plasma parameters of the phenomenon: electric field (E), flow velocity v, solar wind dynamic pressure P, temperature (T), proton density (N), and magnetic field (Bz), are the “causes.” Kp, ap, Dst, and AE, the zonal geomagnetic indices of the storm, are the “effects” [27–32]. Dst (disturbance storm time) is a geomagnetic index that confirms the magnitude of the geomagnetic storms that cover the Earth. It is the average of the horizontal component of the geomagnetic field obtained from stations in different geographical regions of the world. Kp (planetary index) is obtained by a weighted average of kind exes in 13 subauroral observations. A linear index is obtained from ap and Kp. The AE index is a snapshot general index as a measure of changes in global Auroral Electrojet activity as defined by Davis and Sugiura [33].Solar wind parameters and zonal geomagnetic indices are the main elements of the dynamic structure of the storm. This plasma-dense dynamic medium has energy-charged particles that flow out at high speed from the Sun [27]. Such an intense dynamic structure causes rapid changes in the magnetic field of the Sun’s plasma-intensive environment. The coronal mass ejection (CME) cloud swallows the Earth’s magnetosphere. Meanwhile, Bz component of B magnetic field of CME cloud is negatively charged to the southward. Geomagnetic disturbances occur with the magnetic reconnection [32, 34, 35], between this component and the Earth’s magnetic field. The CME causes direct solar wind parameters to change, and then the storm process begins [36, 37].In order to discuss the dynamic structure of the storm, it is necessary to establish models by looking at the relationships between the variables and to keep the consistency of these correlations. The mathematical models can guide researchers about data and their relationships whichever in the science area [15, 20]. In addition, they may yield clues about the behavior of the variables under different terms [15, 16, 18, 26, 38, 39].In this study, the models established with the deterministic and pragmatic approach of Eroglu [15] for geomagnetic storms are discussed for a weak phenomenon for the first time. 8 May 2014 weak storm is analyzed with this mathematical perspective. Models [15, 16] have had a new identity with the 08 May 2014 geomagnetic storm [40]. The formulated models are exactly dependent on the stochastic processes [41]. Visualized models give the readers opportunities to compare correlation data. Although binary linear models may find it difficult to explain the exact relationship between variables, the presentation of these models is necessary [15, 16]. In the paper, it reveals, with obeying to the physical perspective, that the variables of solar wind parameters and zonal geomagnetic indices do not interact randomly, although they exhibit normal distribution. Weak correlation encourages scientists to seek nonlinear models.Section2 presents the Sun parameters, regional geomagnetic indexes, and the five-day distributions of variables. In Section 3 the analyses are performed. Section 4 provides a discussion.


## 2. Data

In this study, Space Physics Environmental Data Analysis Software (SPEDAS) is used. Analysis Software data are IDL-based. It is reachable at this link:https://themis.igpp.ucla.edu/software.shtml.Hourly OMNI-2 Solar Wind and IMF parameter data can be accessed online. In addition, Dst and AE indices were obtained from the World Geomagnetism Data Center Kyoto using SPEDAS. From the NGDC by using SPEDAS with CDA Web Data Chooser (space physics public data), Kp and ap are obtained. For 2014 May weak storm, solar wind dynamic pressure, IMF, electric field, flow speed, and proton density were recorded in the OMNI hourly data. Geomagnetic classification [42] is shown in Table 1.Table 1
Geomagnetic storm Dst index.
ClassNumber%Dst range (nT)Weak48244−30 to −50Moderate34632−50 to −100Strong (i.e., intense)20619−100 to −200Severe (very intense)454−200 to −350Great61<−350The characteristic of the phenomenon at the weak level (Dst = −46 nT) on 08 May 2014 have been investigated. Figure1 demonstrates the OMNI data set from 00:00 UT on 06 May 2014 to 00:00 UT on 10 May 2014. The plot interval includes the storm day (2014 May 08), two days before and two days after the storm (120 hours). The May storm started on May 08th with CME at 06:00 UT. The gradually decreasing magnetic field component (Bz) reaches the negative peak value −9.2 nT at 07:00 UT. Meanwhile, as the plasma flow speed v catches 329 km/s, its maximum value, the electric field (E) reaches 3.03 mV/m its maximum value and the geomagnetic aurora electrojet index (AE) raises to its maximum value of 767 nT. Finally, Dst hits its peak value of −46 nT at 08:00 UT.Figure 1
From top of the bottom parameters shown in Dst index, electric fieldE (mV/m), magnetic field Bz (nT), flow speed v (km/s), proton density N (1/cm3), solar wind dynamic pressure P (nPa), and aurora electrojet AE (nT) index for 2014 May 06–10 (from NASA NSSDC OMNI data set).Figure1 can be shortly detailed as follows: On 06.05.2014 at 07:00 UT when Bz component is at its minimum (−9.2 nT), Dst index dwindled to −39 nT, and the electric field E attains its maximum value of 3.03 mV/m. Meantime, plasma flow speed v minimum value becomes 329 km/s, and AE index catches its peak value 767 nT. Within a few hours, Dst index indicates its minimum value −46 nT, ap index hits its maximum value 39 nT, and proton density N reaches its peak value 21.51/cm3.On 09.05.2014 at 11:00 UT whenBz component is maximum (6 nT), the electric field E catches −2.25 mV/m, proton density N takes 6.31/cm3, AE index diminishes to 69 nT, and ap index decreases to 4 nT. As this comes to cross Dst index reaches −10 nT.


## 3. Mathematical Modeling

The expressive analysis valuation of the geomagnetic storm in May 2014 is shown in Table2. Specifically, the peak and deviation values give the reader information about the data range. The more the increase in the standard deviation of a variable, the more it become distant to average. This weakens the effect of the variable. When Table 2 is taken into account, the most stable variables are E, P, Bz, ap, N, and Dst, respectively. Physically, ap and Dst zonal geomagnetic indices are a result of P, E, Bz, and N solar wind parameters by reason of cause-effect principle. In Table 3, binary instantaneous correlations of the data are given. Correlation analysis is an auxiliary regression analysis method. As the table values approach ±1, the bond between the variables is stronger. The high relation level in Table 3 plays an important role in modeling [15, 16].Table 2
Descriptive analysis.
NMinimumMaximumMeanStd. deviationBz (nT)120−9.26.0−0.3283.5401T (K)1201023515010859373.9431491.411N (1/cm3)1201.921.56.3654.1058v (km/s)120329390359.2517.974P (nPa)1200.54.51.5130.9131E (m V/m)120−2.33.00.0911.2649Kp12014715.0813.082Dst (nT)120−460−14.3311.538ap (nT)1201397.788.264AE (nT)12017767189.89207.013Table 3
Pearson’s correlation matrix for the storm variables.
Bz (nT)T (K)N (1/cm3)v (km/s)P (nPa)E (mV/m)KpDst (nT)ap (nT)AE (nT)Bz (nT)10.225∗−0.457∗∗0.426∗∗−0.428∗∗−0.999∗∗−0.652∗∗0.381∗∗−0.541∗∗−0.745∗∗T (K)10.0800.522∗∗0.190∗−0.212∗0.128−0.521∗∗0.092−0.057N (1/cm3)1−0.258∗∗0.984∗∗0.452∗∗0.768∗∗−0.566∗∗0.723∗∗0.638∗∗v (km/s)1−0.126−0.418∗∗−0.223∗−0.026−0.245∗∗−0.312∗∗P (nPa)10.425∗∗0.771∗∗−0.612∗∗0.707∗∗0.617∗∗E (m V/m)10.654∗∗−0.381∗∗0.543∗∗0.738∗∗Kp1−0.744∗∗0.944∗∗0.824∗∗Dst (nT)1−0.700∗∗−0.674∗∗ap (nT)10.776∗∗AE (nT)1∗.and ∗∗.correlation are significant at the 0.05 level (2-tailed) and at the 0.01 level (2-tailed), respectively.KMO and Bartlett’s test table in Table4 gives an idea of the data for factor analysis. It can be mentioned that the variables are normally distributed if the value obtained by measuring the Kaiser–Meyer–Olkin sampling capability is close to 1.0. As can be seen from Table 4, the variable set of this storm is suitable for factor analysis.Table 4
KMO and Bartlett’s test.
Kaiser–Meyer–Olkin measure of sampling adequacy0.731Bartlett’s test of sphericityApprox. chi-square2144.413Df45Sig.0.000Variables cluster is displayed in Figure2(a) hierarchically. Figure 2(b) displays scattering of variables. The varimax method indicates sprinkling of the variables along two main axes. E, AE, ap, Kp N, and P lie on one side, and Dst, Bz, v, and T lie on the other side in Figure 2(b). If one pays attention, E and Bz are split symmetrically into two opposing sides in Figure 2(b).Figure 2
(a). Dendrogram of hierarchical cluster analysis. (b) Factor loadings for solution.
(a)(b)In the factor analysis with no composite variables, each variable is handled separately. Variables are examined more specifically (with the principal component) by the principal component analysis. In Table5, the maximum two eigenvalues of the covariance matrix constitute 78% of the total change when changed in the ten-variable data reduction method. This means that in Table 5, 78% of the phenomenon can be explained by modeling with the variables at hand.Table 5
Total variance explained.
ComponentInitial eigenvaluesExtraction sums of squared loadingsTotal% of varianceCumulative (%)Total% of varianceCumulative (%)15.75157.5 1557.5155.75157.51557.51522.05020.4 9878.0132.05020.49878.013Scattering plot of these variables is presented in Figure3. It is ranked from the maximum between the variables, with the showing a factor between every two points.Figure 3
Scattering for eigenvalues.For the rotation matrix, varimax examines the linear grouping of event variables by the Kaiser normalization method. The method that approaches each variable as a factor indicates the contribution and weight of these factors to linear clustering. Table6 is the table of these weights.Table 6
Rotated component matrix.
ComponentBz (nT)T (K)N (1/cm3)v (km/s)P (nPa)E (mV/m)KpDst (nT)ap (nT)AE (nT)1−0.5990.3220.846−0.1090.8690.6000.941−0.8590.8860.82320.6670.816−0.1040.7740.012−0.659−0.175−0.241−0.153−0.375Hence, these models can be written as follows with factor weights from Table6:(1)Axes1=−0.599Bz+0.322T+0.846N−0.109v+0.869P+0.600E+0.941Kp−0.859Dst+0.886ap+0.823AE,Axes2=0.667Bz+0.816T−0.104N+0.774v+0.012P−0.659E−0.175Kp−0.241Dst−0.153ap−0.375AE.Trying to catch an overview with the general linear models will aid to better understand the storm form. Binary views with some zonal geomagnetic indices (Dst, ap, and AE) and some solar wind parameters (magnetic field component (Bz), the electric field (E), temperature (T), flow velocity v, proton density (N), and flow pressure P) can be seen in Figures 4–6, respectively. Figures 4–6 help for visualizing correlation of variables.Figure 4
Binary views of Dst andBz, T, P.Figure 5
Binary views of ap andN, E.Figure 6
Binary views of AE andBz, N.The analysis of variance of Dst index is shown in Table7. It can be seen that regression coefficients are significant. This table shows how much of the residuals are explained by the variables in the linear regression model. Table 8 shows the model of the Dst index.Table 7
Anova (analysis of variance) of Dst.Dst=4.501−4.597v+0.001T+1.123Bz, where multiple determination coefficients R are 0.793.
ModelSum of squaresdfMean squareFSig.Regression9970.00033323.33365.6670.000Residual5870.66711650.609Total15840.667119Table 8
Regression coefficients of Dst.Dst=4.501−4.597v+0.001T+1.123Bz, where multiple determination coefficients R are 0.793.
ModelUnstandardized coefficientsStandardized coefficientstSig.95% confidence interval forBBStd. errorBetaLower boundUpper boundConstant4.5011.6392.7470.0071.2567.747P (nPa)−4.5970.835−0.364−5.5030.000−6.252−2.943T (K)0.0010.000−0.529−8.6290.0000.0000.000Bz (nT)1.1230.2170.3455.1710.0000.6931.553Binary views with Dst andBz, T, P can be seen in Figure 4.Table9 shows that the model is significant, while Table 10 indicates the ap index.Table 9
Anova (analysis of variance) of ap.ap=−0.070+1.207N+1.779E, where multiple determination coefficients R are 0.762.
ModelSum of squaresdfMean squareFSig.Regression4725.01722362.50981.2530.000Residual3401.90811729.076Total8126.925119Table 10
Regression coefficients of ap.ap=−0.070+1.207N+1.779E, where multiple determination coefficients R are 0.762.
ModelUnstandardized coefficientsStandardized coefficientstSig.95% confidence interval forBBStd. errorBetaLower boundUpper boundConstant−0.0700.975−0.0720.943−2.0011.861N (1/cm3)1.2070.1350.6008.9440.000−0.9401.474E (mV/m)1.7790.4380.2724.0610.0000.9112.646Binary views with ap andN, E can be seen in Figure 5.Table11 indicates that the model is significant, while Table 12 shows the AE index.Table 11
Anova (analysis of variance) of AE index.AE=58.028−33.471Bz+18.990N, where multiple determination coefficients R are 0.816.
ModelSum of squaresdfMean squareFSig.Regression3399775.03421699887.517116.9990.000Residual1699904.55811714529.099Total5099679.592119Table 12
Regression coefficients of AE index.AE=58.028−33.471Bz+18.990N, where multiple determination coefficients R are 0.816.
ModelUnstandardized coefficientsStandardized coefficientstSig.95% confidence interval forBBStd. errorBetaLower boundUpper boundConstant58.02821.7512.6680.00914.951101.106Bz (nT)−33.4713.510−0.572−9.5360.000−40.422−26.520N (1/cm3)18.9903.0260.3776.2750.00012.99724.984Binary views with AE andBz, N can be seen in Figure 6.For many years,Bz-Dst linear relationship has been an important part of the researchers’ discussions [43]. In addition to these discussions, the author tries to add depth to the meaning of the relationship between nonlinear models Dst, ap, and AE with Bz in addition to linear models. It can be seen in Tables 13 and 14 and Figure 7, the linear and quadratic relationships of the magnetic field component Bz with the Dst index.Table 13
Regression coefficients and analysis of variance.Dst=−13.926+1.242Bz, where R is 0.381.Table 14
Regression coefficients and analysis of variance.Dst=−10.803+0.696Bz−0.263Bz2 where R is 0.481.Figure 7
Linear and quadratic relation of Dst andBz.In Tables15 and 16 and Figure 8, the linear and quadratic relationships between the magnetic field component Bz and the ap index is presented.Table 15
Regression coefficients and analysis of variance.ap=7.360−1.263Bz where R is 0.541.Table 16
Regression coefficients and analysis of variance.ap=5.386−0.918Bz+0.167Bz2, where R is 0.600.Figure 8
Linear and quadratic relation of ap andBz.In Tables17 and 18 and Figure 9, the linear and nonlinear relationships between the magnetic field component Bz and the AE index is presented.Table 17
Regression coefficients and analysis of variance.AE=175.595−43.544Bz, where R is 0.745.Table 18
Regression coefficients and analysis of variance.AE=116.734−33.267Bz+4.965Bz2, where R is 0.806.Figure 9
Linear and quadratic relation of AE andBz.Ultimately, it would be appropriate to declare the proven [15, 16] model for this weak storm. This nonlinear model is a model that includes parameters P, N, and ap index. It is quite interesting that the model introduced by Eroglu for the first time demonstrates its consistency for a weak geomagnetic storm. The short reaction times of weak storms have been a difficult test for the stability of the model.The nonlinear model is in the formP=a+blnap+cN, where a, b, and c are constants. The analysis of variance values of flow pressure P are shown in Table 19. The magnitudes of coefficients are a = 0.101, b = 0.053, and c = 0.209. Table 20 shows that all parameter estimation is in confidence interval of 95%. The model explaining this storm with 97% accuracy is:Table 19
Anova (analysis of variance) of flow pressureP. P=0.101+0.053lnap+0.209N.
SourceSum of squaresdfMean squaresRegression370.7483123.853Residual2.9961170.026Uncorrected total373.744120Corrected total99.225119Table 20
Parameter estimates.P=0.101+0.053lnap+0.209N.
ParameterEstimateStd. error95% confidence intervalLower boundUpper bounda0.1010.0280.0450.157b0.0530.0220.0090.097c0.2090.0050.1980.220


## 4. Conclusion

The strength and consistency of the results obtained in moderate and strong geomagnetic events are also the goals of this weak geomagnetic storm study. Although the response time of the Dst (nT) index to the magnetic field is very short, it is noteworthy that the models are as efficient as in other kinds of storms. The 08 May 2014 geomagnetic storm is one of the weakest storms in the 24th solar cycle. I focus on the May 2014 weak storm. In the storm, solar wind parameters and zonal geomagnetic indices are discussed and the cause-effect relationship is obeyed. All possible binary or multiple relationship and model are presented to the reader even if not statistically significant. Every model that produced about the storm has been meticulously analyzed. In particular, the mathematical models involving solar wind pressure and proton density give an idea of the dynamic nature of the different plasmatic structures. In order for the model to be consistent, it is necessary to be validated by other storms. The last nonlinear model among flow pressureP, ap, and proton density (N) explains the storm with 97% accuracy. All results are in the 95% confidence interval. Graphs and tables have visualized the correlation between zonal geomagnetic indices and solar wind parameters, as well as their interactions with each other.



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