Mathematical Analysis of the 08 May 2014 Weak Storm

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. 'ey 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) onMay 8, 2014, via solar wind parameters and zonal geomagnetic indices.'e 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. 'erefore, the physical principles govern the study. 'e results are visualized with tables and graphs for the understanding of the dynamic structure of the storm.


Introduction
A geomagnetic storm  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][16][17][18][24][25][26] are physically restrictive. e 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 (B z ), are the "causes." Kp, ap, Dst, and AE, the zonal geomagnetic indices of the storm, are the "effects" [27][28][29][30][31][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. e 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. is 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. e coronal mass ejection (CME) cloud swallows the Earth's magnetosphere. Meanwhile, B z 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. e 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. e 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]. e 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.
Section 2 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.

Data
In this study, Space Physics Environmental Data Analysis Software (SPEDAS) is used. Analysis Software data are IDLbased. 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.
e characteristic of the phenomenon at the weak level (Dst � − 46 nT) on 08 May 2014 have been investigated. Figure 1 demonstrates the OMNI data set from 00:00 UT on 06 May 2014 to 00:00 UT on 10 May 2014. e plot interval includes the storm day (2014 May 08), two days before and two days after the storm (120 hours). e May storm started on May 08 th with CME at 06:00 UT. e gradually decreasing magnetic field component (B z ) 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 can be shortly detailed as follows: On 06.05.2014 at 07:00 UT when B z 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/cm 3 .
On 09.05.2014 at 11:00 UT when B z component is maximum (6 nT), the electric field E catches − 2.25 mV/m, proton density N takes 6.31/cm 3 , AE index diminishes to 69 nT, and ap index decreases to 4 nT. As this comes to cross Dst index reaches − 10 nT.

Mathematical Modeling
e expressive analysis valuation of the geomagnetic storm in May 2014 is shown in Table 2. Specifically, the peak and deviation values give the reader information about the data range. e more the increase in the standard deviation of a variable, the more it become distant to average. is weakens the effect of the variable. When Table 2 is taken into account, the most stable variables are E, P, B z , ap, N, and Dst, respectively. Physically, ap and Dst zonal geomagnetic indices are a result of P, E, B z , 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. e high relation level in Table 3 plays an important role in modeling [15,16].
KMO and Bartlett's test table in Table 4 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.
Variables cluster is displayed in Figure 2(a) hierarchically. Figure 2(b) displays scattering of variables. e varimax method indicates sprinkling of the variables along two main axes. E, AE, ap, Kp N, and P lie on one side, and Dst, B z , v, and T lie on the other side in Figure 2(b). If one pays attention, E and B z are split symmetrically into two opposing sides in Figure 2 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 Table 5, the maximum two eigenvalues of the covariance matrix constitute 78% of the total change when changed in the ten-variable data reduction method. is means that in Table 5, 78% of the phenomenon can be explained by modeling with the variables at hand.
Scattering plot of these variables is presented in Figure 3. It is ranked from the maximum between the variables, with the showing a factor between every two points.
For the rotation matrix, varimax examines the linear grouping of event variables by the Kaiser normalization method.
e method that approaches each variable as a factor indicates the contribution and weight of these factors to linear clustering. Table 6 is the table of these weights.
and * * . correlation are significant at the 0.05 level (2-tailed) and at the 0.01 level (2-tailed), respectively.   Mathematical Problems in Engineering Hence, these models can be written as follows with factor weights from Table 6: (1) 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 (B z ), 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. e analysis of variance of Dst index is shown in Table 7. It can be seen that regression coefficients are significant. is 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.
Binary views with Dst and B z , T, P can be seen in Figure 4. Table 9 shows that the model is significant, while Table 10 indicates the ap index.
Binary views with ap and N, E can be seen in Figure 5. Table 11 indicates that the model is significant, while Table 12 shows the AE index.
Binary views with AE and B z , N can be seen in Figure 6. For many years, B z -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 B z 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 B z with the Dst index.
In Tables 15 and 16 and Figure 8, the linear and quadratic relationships between the magnetic field component B z and the ap index is presented.
In Tables 17 and 18 and Figure 9, the linear and nonlinear relationships between the magnetic field component B z and the AE index is presented.
Ultimately, it would be appropriate to declare the proven [15,16] model for this weak storm. is 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. e short reaction times of weak storms have been a difficult test for the stability of the model. e nonlinear model is in the form P � a + b ln ap + cN, where a, b, and c are constants. e analysis of variance values of flow pressure (P) are shown in Table 19. e 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%. e model explaining this storm with 97% accuracy is:

Conclusion
e 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. e 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. e last nonlinear model among flow pressure (P), 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.
Data Availability e datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. e author utilizes the NASA CDA Web for OMNI Database (https://themis.igpp.ucla.edu/software.shtml) and Kyoto World Data Center for providing AE index and Dst index. She acknowledges the usage of ap and Kp index from the National Geophysical Data Center.
e Dst index and AE data are provided by the World Data Center for Geomagnetism at Kyoto University. e data used in this article are available at the Data Center of NASA https://omniweb.gsfc.nasa.gov/form/dx1.html.

Conflicts of Interest
e author declares no conflicts of interest.

Authors' Contributions
Data are collected and analyzed by the author. All interpretations and explanations belong to the author. e author read and approved the final manuscript.