Time-Integral Correlations of Multiple Variables With the Relativistic-Electron Flux at Geosynchronous Orbit: The Strong Roles of Substorm-Injected Electrons and the Ion Plasma Sheet

Abstract

Time-integral correlations are examined between the geosynchronous relativistic electron flux index F_e1.2 and 31 variables of the solar wind and magnetosphere. An “evolutionary algorithm” is used to maximize correlations. Time integrations (into the past) of the variables are found to be superior to time-lagged variables for maximizing correlations with the radiation belt. Physical arguments are given as to why. Dominant correlations are found for the substorm-injected electron flux at geosynchronous orbit and for the pressure of the ion plasma sheet. Different sets of variables are constructed and correlated with F_e1.2: some sets maximize the correlations, and some sets are based on purely solar wind variables. Examining known physical mechanisms that act on the radiation belt, sets of correlations are constructed (1) using magnetospheric variables that control those physical mechanisms and (2) using the solar wind variables that control those magnetospheric variables. F_e1.2-increasing intervals are correlated separately from F_e1.2-decreasing intervals, and the introduction of autoregression into the time-integral correlations is explored. A great impediment to discerning physical cause and effect from the correlations is the fact that all solar wind variables are intercorrelated and carry much of the same information about the time sequence of the solar wind that drives the time sequence of the magnetosphere.

Key Points

• Time-integral correlations with radiation-belt electron fluxes are stronger than time-lagged correlations
• Substorm-injected electrons and the ion-plasma sheet pressure appear to play strong roles in the evolution of the electron radiation belt
• Interpretations are made of the optimal integration times obtained from an evolutionary algorithm to maximize correlation coefficients

1 Introduction

There is substantial interest in the time evolution of the electron radiation belt and in the processes and variables that control the evolution (e.g., Bourdarie et al., 1996; Denton & Borovsky, 2017; Friedel, Reeves, & Obara, 2002; Hudson et al., 2008; Li et al., 2001; Myagkova et al., 2017; Shprits, Elkington, et al., 2008 and Shprits, Subboti, et al., 2008; Vassiliadis et al., 2005; Wing et al., 2016).

The relativistic electron flux in the magnetosphere is known to be most intense after long intervals of high solar wind speed (Blake et al., 1997; Borovsky & Denton, 2006; Paulikas & Blake, 1979): indeed, the relativistic electron flux time series mathematically correlates with the solar wind velocity time series with a time lag (Borovsky & Denton, 2014; Desorgher et al., 1998; Fung & Tan, 1998; Li et al., 2001; McPherron et al., 2009; Reeves et al., 2011; Vassiliadis et al., 2002). Similarly, the relativistic electron flux is high after long intervals of elevated geomagnetic activity (Borovsky & Denton, 2010a; Lam et al., 2009; Paulikas & Blake, 1976; Reeves, 1998) and the relativistic electron flux correlates with geomagnetic indices such as Kp (Borovsky & Denton, 2014; Buhler & Desorgher, 2002; Lam, 2004). It has been pointed out that high relativistic electron flux is also associated with intervals of elevated ULF wave intensity in the magnetosphere (Degeling et al., 2011; Elkington et al., 2003; Friedel et al., 2002; Mathie & Mann, 2000; Nakamura et al., 2002; Rostoker et al., 1998) and the relativistic electron flux correlates with ULF indices (Borovsky & Denton, 2014; Kozyreva et al., 2007; McPherron et al., 2009; Potapov et al., 2012; Potapov et al., 2014; Romanova & Pilipenko, 2009). Recently, strong correlations were found between the solar wind number density and relativistic electron fluxes (again with a time lag) (Balikhin et al., 2011; Borovsky & Denton, 2014; Boynton et al., 2013; Lyatsky & Khazanov, 2008) and between the substorm-injected energetic-electron flux in the magnetosphere and relativistic electron fluxes (Hwang et al., 2004; Meredith, Cain, et al., 2003; Potapov et al., 2016; Simms et al., 2016; Turner & Li, 2008).

Mathematical correlation between two variables means that variations in the one variable are associated with variations in the other variable: variations in the one variable could be causing the variations in the other variable, or a third variable could be causing variations in both of the two correlated variables. An example of this is the high correlation between the solar wind proton temperature T_p and the AE index, which almost certainly is not physical because the information about the upstream solar wind proton temperature is lost as the solar wind crosses the bow shock (cf. Formisano et al., 1973; Tidman & Krall, 1971); this T_p-AE correlation is undoubtedly caused by the fact that the solar wind velocity v_sw physically controls the driving of the magnetosphere (as measured by AE) (e.g., Borovsky & Birn, 2014; Newell et al., 2007) and that proton temperature T_p is correlated with the solar wind velocity v_sw (e.g., Elliott et al., 2012; Lopez & Freeman, 1986) (although the physical reason for the T_p-v_sw correlation is not known (Borovsky & Gary, 2014; Marsch & Richter, 1984). In examining the solar wind driving of the Earth's magnetosphere-ionosphere system, one can find a web of correlations involving multiple solar and solar wind variables and multiple Earth-based variables. Almost all of the variables of the web are correlated with almost every other variable of the web, with different time lags on different two-variable correlations. Variable intercorrelations lead to two well-known effects: (1) confounding wherein one variable can act as a proxy for a second variable (to which it is correlated) in correlations with a third variable (Frank, 2000; Robins, 1989) and suppression wherein one variable acts in a correlation to suppress (cancel) irrelevant variance on a second variable (Conger, 1974; Tzelgov & Henik, 1991). Additionally for correlations with the radiation belt, one variable may be acting simultaneously through more than one physical mechanism to evolve the radiation belt. An example of this that will be seen in section 5 is the solar wind velocity v_sw acting on the radiation-belt flux F_e1.2, where the velocity can act (1) by driving magnetospheric convection via its control on the dayside reconnection rate, (2) by driving ULF fluctuations via Kelvin-Helmholtz instabilities, (3) by pushing in the dayside magnetopause to increase losses of radiation-belt electrons, and (4) by regulating the temperature of the magnetosheath, which regulates the temperature of the plasma sheet, which may impact the energy distribution of the substorm-injected electrons that are the seed population for the radiation belt. All of these effects make identifying cause and effect in this web of correlations very difficult. Further, using correlations to test suspected cause and effect between variables is not foolproof. The solar wind/magnetosphere web of correlations has its root cause in the intercorrelations of solar wind variables. Motivated by the present study, two sources of the systematic intercorrelations of diverse solar wind variables are explored in Borovsky (2017a): (1) plasma-type switching caused by solar rotation switching the types of solar-surface features that face the Earth and (2) the dynamic interaction between plasma types during the advection from the Sun to the Earth.

This report explores the web of correlations between the solar wind, the magnetosphere, and the relativistic electron flux of the outer electron radiation belt. Multivariable correlations with the electron radiation belt have been examined before (e.g., Borovsky & Denton, 2014; Potapov et al., 2016; Simms et al., 2014, 2015, 2016; Wei et al., 2011), but here a more-extensive and newer list of variable choices will be utilized, along with time integration of the variables. At times physical insight will be used to guide the exploration of correlations; at other times variables will be added for the purpose of maximizing correlations. The study will utilize 31 variables in correlations with the radiation-belt electron flux, including some new indices that were created to describe the state of the outer electron radiation belt, the occurrence rate of magnetospheric substorms, and the state of the substorm-injected electron population in the magnetosphere (Borovsky & Yakymenko, 2017a). Additional new indices are created here to gauge the density, temperature, and pressure of the ion plasma sheet in the magnetosphere and to gauge the intensity of the electron strahl in the solar wind. The main tool used in this study is an “evolutionary algorithm” that randomly adjusts coefficients, lag times, and integration times in complex formulas in order to arrive at optimized correlations. For correlations with the intensity of the outer electron radiation belt, Borovsky and Denton (2014) found time-integrated variables to be superior to time-lagged variables; that will be shown here also. The very strong role of substorm-injected energetic electrons on the radiation-belt evolution will be seen, as will the strong role of the ion-plasma sheet pressure.

This report is organized as follows. In section 2 the data sets are overviewed and the correlation methodologies are explained. In section 3 single-variable time-integral correlations with F_e1.2 are examined. Section 4 examines various sets of multivariable time-integral correlations with F_e1.2. Section 5 develops physics-based sets of multivariable time-integral correlations with F_e1.2 involving both magnetospheric and solar wind variables. Section 6 contains a discussion about how to interpret the optimized integration times and a discussion about the limits on the size of the correlation coefficients found in this study (r_corr ≈ 0.88). The study is summarized in section 7.

2 Data Sets and Methods

Table 1 contains a glossary of the solar wind, geomagnetic, magnetospheric, and radiation-belt quantities used in the present study, along with the source of the values. All variables are used with 1 h time resolution for the years 1995–2006.

Table 1. Glossary of the Variables Used in This Study and the Origin of the Data

Variable	Definition	Source and reference
Fe1.2	Radiation-belt 1.2 MeV electron flux index	SOPA, Ref. 1
nerb	Radiation-belt number density	SOPA, Ref. 2
Terb	Radiation-belt temperature (hardness)	SOPA, Ref. 2
Fe130	Substorm-injected 130 keV electron flux index	SOPA, Ref. 1
Fe60	Substorm-injected 60 keV electron flux index	SOPA, Sect. 2
Srate	Rate of substorm occurrence	SuperMag, Ref. 1
nips	Number density of ion plasma sheet	MPA, Sect. 2
Tips	Temperature of ion plasma sheet	MPA, Sect. 2
Pips	Pressure of the ion plasma sheet nipsTips	MPA, Sect. 2
Kp	Planetary activity index	OMNI2, Ref. 3
AE	Auroral-electrojet index	OMNI2, Ref. 3
ASY-H	Asymmetric H index	OMNI2, Ref. 3
Sym-H	Symmetric H index	OMNI2, Ref. 3
Dst	Disturbance storm time index	OMNI2, Ref. 3
Dst*	Pressure corrected Dst index	OMNI2, Ref. 4
Sgrd	ULF index, ground based	Ref. 5
Sgeo	ULF index, geosynchronous based	Ref. 5
vsw	Solar wind speed	OMNI2, Ref. 3
nsw	Solar wind number density	OMNI2, Ref. 3
Bmag	Solar wind magnetic field strength	OMNI2, Ref. 3
Tp	Solar wind proton temperature	OMNI2, Ref. 3
sin^2(θclock/2)	Solar wind magnetic clock angle function	OMNI2, Ref. 6
Sp	Proton specific entropy of solar wind	OMNI2, Ref. 7
vA	Alfven speed in the solar wind plasma	OMNI2, Ref. 3
MA	Alfven Mach number of solar wind	OMNI2, Ref. 3
Pram	Dynamic pressure of solar wind	OMNI2, Ref. 3
Rquick	Dayside reconnection function	OMNI2, Ref. 8
Rqo	Logarithm of Rquick without clock-angle term	OMNI2, Ref. 9
I272	Intensity of 272 eV strahl in solar wind	ACE, Sect. 2
dmp	Magnetopause standoff distance	No values
Aips	Temperature anisotropy T⊥i/T||i of ion plasma sheet	No values

• Note. 1 Borovsky and Yakymenko (2017a), 2 Denton et al. (2010), 3 King and Papitashvili (2005), 4 Borovsky and Denton (2010b), 5 Kozyreva et al. (2007), 6 Kan and Lee (1979), 7 Burlaga, Mish, and Whang (1990), 8 Borovsky and Birn (2014), 9 Borovsky and Yakymenko (2017b).

The study utilizes three new indices created to monitor the state of the electron radiation belt (F_e1.2), the state of the substorm-injected energetic-electron population in the magnetosphere (F_e130), and the rate of occurrence of magnetospheric substorms (S_rate). Details of the creation of these three indices and an examination of their properties appear in Borovsky and Yakymenko (2017a). F_e1.2 is the maximum of the 1.2 MeV electron flux measured each hour by multiple spacecraft in geosynchronous orbit using the Synchronous Orbit Particle Analyzer (SOPA) energetic-particle detectors (Cayton & Belian, 2007). Similarly, F_e130 is the maximum of the 130 keV electron flux measured by multiple SOPA instruments in geosynchronous orbit. (Using the same algorithm that created the F_e130 index from the full SOPA data set, an index F_e60 is created here that represents the maximum of the 60 keV electron flux at geosynchronous orbit.) S_rate is the logarithm of the temporal rate of occurrence of substorms as determined from the 1 min resolution SuperMag auroral-electrojet index SML (Gjerloev, 2012; Newell & Gjerloev, 2011) using a substorm-onset-finding algorithm described in Yakymenko and Borovsky (2016) and Borovsky and Yakymenko (2017b).

The variables n_erb and T_erb (cf. Table 1) are the multisatellite averages of the number density and temperature of the electron radiation belt at geosynchronous orbit as measured by the SOPA energetic-particle detectors with Maxwellian fits to the SOPA count rates (cf. Borovsky & Yakymenko, 2017b). The Maxwellian fit algorithm (Cayton & Belian, 2007) removes bremstrahlung backgrounds and penetrating-radiation backgrounds from the count rates before making the fits. This density-temperature description of the electron radiation belt at geosynchronous orbit is described in Cayton et al. (1989) and Denton et al. (2010). Typical values of the density n_erb are 10⁻⁴–10⁻³ cm⁻³, and typical values of the temperature (spectral hardness) T_erb are 80–250 keV. In some of the SOPA data sets these fits to the count rates are used to calculate differential fluxes; the differential fluxes that go into the F_e1.2 and F_e130 indices are not from such fits; they are calculated directly from the SOPA count rates in those energy channels using “flux conversion factors” (Cayton & Belian, 2007), which are multiplicative coefficients determined from physical and computational calibration of the SOPA detectors that convert channel count rates into differential fluxes.

Three multisatellite indices are created to monitor the state of the ion plasma sheet in the dipolar magnetosphere: n_ips representing the number density of ions in the ion plasma sheet, T_ips representing the temperature of the plasma sheet ions, and P_ips representing the particle pressure of the ion plasma sheet. Measurements from the magnetospheric plasma analyzer (MPA) instruments (Bame et al., 1993; Thomsen et al., 1999) onboard multiple spacecraft in geosynchronous orbit are used. One-hour averages of all of the number density and temperature measurements from all of the spacecraft Icourtesy of Mick Denton, private communication, 2014) are utilized. Before producing multisatellite averages the data are cleaned to remove magnetosheath intervals and to remove measurements from aged instruments. Correction factors for the temperature measurements from one of the spacecraft are applied since the long-term trends of that spacecraft deviated from the trends of the other spacecraft. Then local-time fits to the number-density measurements, and the temperature measurements are used to correct the spacecraft measurements for the local time at which each measurement was taken. Then the number-density data and the temperature data from all of the available spacecraft were averaged into one number density as a function of time (n_ips) and one temperature as a function of time (T_ips). The pressure as a function of time P_ips is taken as the product of those averages P_ips = n_ipsT_ips. Note that the ion-plasma sheet temperature measurements from the MPA instruments are incomplete since the MPA instruments only measure ions below 40 keV: the true temperatures are higher than the measured values (cf. Borovsky et al., 1998). Because of this systematic temperature error, the ion pressure obtained with the MPA temperature measurements is a partial pressure which is an underestimate of the total pressure of the ion plasma sheet.

The geomagnetic indices Kp, AE, SYM-H, ASY-H, Dst, and Dst* are used (cf. Table 1). The source of all of these indices is the OMNI2 data set (King & Papitashvili, 2005). The pressure-corrected Dst index Dst* is calculated using the algorithm in equation 1 of Borovsky and Denton (2010b), where the measured dynamic (ram) pressure P_ram of the solar wind is used to remove the contribution of the Chapman-Ferraro current from the Dst index.

The two ULF indices S_grd and S_geo (Kozyreva et al., 2007; Romanova et al., 2007) are utilized (cf. Table 1). S_grd represents the amplitude of ULF waves in the magnetosphere as measured from ground-based magnetometers, and S_geo represents the amplitude of ULF waves in the magnetosphere as measured by magnetometers (Dunham et al., 1996) onboard the GOES spacecraft in geosynchronous orbit. The properties of S_grd and S_geo and their correlations with solar wind, geomagnetic, and radiation-belt variables can be found in Romanova and Pilipenko (2009) and Borovsky and Denton (2014).

Standard solar wind quantities (v_sw, n_sw, B_mag, and T_p) are used (cf. Table 1), as given by the OMNI2 data set created from solar wind measurements at the upstream L1 point time shifted to the Earth using the solar wind advection scheme of Weimer and King (2008). The clock angle θ_clock of the solar wind magnetic field as seen by the Earth's magnetic dipole is θ_clock = arccos(B_z/(B_y² + B_z²)^1/2) in GSM coordinates. The specific entropy of protons is S_p = T_p/n_sw ^2/3. The Alfven speed in the upstream solar wind is v_A = B_mag/(4πm_pn_sw)^1/2, with m_p being the proton mass and B_mag = (B_x² + B_y² + B_z²)^1/2. The Alfven Mach number of the solar wind is MA = v_sw/v_A. The dynamic pressure is P_ram = 0.5m_pn_swv_sw².

The quick reconnection function R_quick (cf. Table 1), derived to represent the reconnection rate between the magnetosheath plasma and the dayside magnetosphere (Borovsky, 2008, 2013a; Borovsky & Birn, 2014), is

(1)

and R_qo = R_quick/sin²(θ_clock/2) (Borovsky & Yakymenko, 2017b).

Finally in Table 1 an index I₂₇₂ has been created to represent the hourly average of the logarithm of the intensity of the outward electron strahl (electron heat flux) in the solar wind. ACE Solar Wind Electron, Proton, and Alpha Monitor (McComas et al., 1998) measurements of the electron distribution function f₂₇₂ at 272 eV are utilized. The hourly average of log₁₀(f₂₇₂) in the two angular bins in the direction parallel to the measured magnetic field direction B and the average of log₁₀(f₂₇₂) in the two angular bins antiparallel to B are compared, and the larger value is taken as the outward-from-the-Sun strahl. Note that taking the outward strahl to be the strahl pointing parallel or antiparallel to B that has a positive radial direction does not work because there are regions in the solar wind where the magnetic field lines are folded back toward the Sun (Balogh et al., 1999; Borovsky, 2016; Kahler et al., 1996) and the strahl is locally pointing toward the Sun; in those cases the direction of the magnetic field gives the wrong toward-versus-away magnetic sector identification. The direction of the Strahl dotted into the vector B is a better identification of the toward-versus-away sector (cf. Crooker et al., 2004; Kahler & Lin, 1994).

Correlations (denoted by the symbol “↔”) of the form

(2)

will be calculated, where the variables v₁(t), v₂(t), v₃(t), …, are integrated into the past for various integration times t₁, t₂, t₃, …, and the formula is constructed with coefficients c₁, c₂, c₃, … Specifically, in the equation of expression 2 F_e1.2 on the left-hand side is measured at every time t and at every time t the variables on the right-hand side are integrated in the past from time t − t_x (where t_x is t₁, t₂, t₃, …, as appropriate) to time t − 1 h. Since time integrals of the variables (except the radiation-belt variables F_e1.2, n_erb, and T_erb) are desired, the years 1995–2006 are used because of the nearly continuous availability of solar wind measurements in those years. Where there are temporal data gaps in a variable, the gaps are filled by “persistence,” replacing the missing data with the last measured value available. (Using a linear interpolation to fill the gaps would be slightly superior, but the gaps are, in general, small and the variables are appearing in time intervals.) Examining expression 2, the left-hand side is represented by a time series (with 1 h resolution) F_e1.2(t) and the right-hand side is represented by a time series (with 1 h resolution) RHS(t). Because of the filling of data gaps in the variables, there are no data gaps in RHS(t). But the radiation-belt variables are not filled, so there are data gaps in the left-hand side F_e1.2(t). When calculating correlations between the left-hand side and right-hand side, times of data gaps of F_e1.2(t) are removed from the calculations.

Notably absent in this study are (1) solar wind variables related to the amplitude of fluctuations in the solar wind magnetic field, velocity, number density, and ram pressure and (2) solar wind variables related to the nature of the fluctuations in the solar wind such as the Alfvenicity, the Alfven ratio, and the intermittencies. In several studies the amplitudes of solar wind fluctuations have been found to correlate with geomagnetic activity (Borovsky, 2006; Borovsky & Funsten, 2003; D'Amicis et al., 2007, 2009, 2011; Osmane et al., 2015) and magnetospheric ULF waves are believed to be driven by temporal variations in the solar wind ram pressure (Berube et al., 2014; Eriksson et al., 2006; Kepko et al., 2002; Liu et al., 2010; Viall et al., 2009). Also absent in this study is the alpha-to-proton density ratio α/p of the solar wind, which the authors have found to be, in general, unreliable when various α/p data sets are compared with each other (e.g., Figure 7e of Borovsky, 2016)

Linear correlations are performed between the left-hand side and right-hand side of formulas (cf. expression 2), and the quality of the correlation is represented by the Pearson linear correlation coefficient r_corr (equation (11.17) of Bevington and Robinson (1992)). All variables going into the right-hand sides of the formulas are “standardized”; that is, the mean value of the time series is subtracted off and then the resultant variables are divided by the standard deviation of the resulting time series. The time series of a standardized variable has a mean value of 0 and a standard deviation of 1.

The numerical calculation of correlation coefficients is incorporated into an “evolutionary algorithm” (genetic algorithm) that randomly changes coefficients, integration times, and time lags in the formulas. If a random change produces a new formula with a larger correlation coefficient r_corr with the left-hand side, then the change is accepted: if the random change produces a lower correlation coefficient, then the change is rejected and the formula is reverted back to the prechange form. This evolutionary algorithm was used in a previous study to uncover mathematical driver functions for the magnetosphere such as v_sw + 56B_south (cf. Borovsky, 2014a) that have higher correlations with geomagnetic indices than do the driver functions used in solar wind /magnetosphere coupling studies (e.g., Borovsky & Birn, 2014; Newell et al., 2007). (Borovsky, 2014a suggested that the driver function v_sw + 56B_south had a high correlation function with geomagnetic activity because that function was efficient at describing critical variance in the solar wind and hence had a mathematical advantage to describe variance in geomagnetic activity.) These “uncovered” driver functions highlight the math-versus-physics dilemma of correlation studies of the solar wind control of the magnetosphere (cf. Borovsky, 2013b; Borovsky, 2014a).

3 Time-Integral Correlations With F_e1.2

In Table 2 correlation coefficients r_corr of the multispacecraft 1.2 MeV electron flux index F_e1.2 with the time integrals into the past of 31 different variables are collected and compared with the correlation coefficients of F_e1.2 with the same 31 variables with time lags. The integral and lagged correlations for the ith variable are of the form

Table 2. Ranked Table for Single-Variable Time-Integral Correlations With F_e1.2 and a Comparison With the Correlation Obtained for Time-Lagged Variables

Variable	rcorr for integral	Integration time	Rank	rcorr for lagged variable
Fe130	0.736	62 h	1a	0.617
Fe60	0.699	85 h	1b	0.531
vsw	0.647	98 h	2	0.602
log(vsw/nsw)	0.627	51 h	3	0.568
log(Sp)	0.620	91 h	4	0.502
log(nsw)	−0.579	38 h	5	−0.518
Kp	0.562	138 h	6	0.413
Tp	0.554	121 h	7	0.377
AE	0.544	140 h	8	0.327
Sgrd	0.535	123 h	9	0.344
Srate	0.509	118 h	10	0.415
log(Rqo)	0.501	96 h	11	0.281
Rquick	0.493	152 h	12	0.267
log(Bmag/nsw)	0.466	81 h	13	0.389
Tp/vsw	0.435	119 h	14	0.297
log(ASY-H)	0.387	138 h	15	0.259
SYM-H	−0.373	110 h	16	−0.299
Dst	−0.373	106 h	17	−0.314
Dst*	−0.358	131 h	18	−0.317
I272	−0.358	13 h	19	−0.168
log(Pips)	−0.356	6 h	20	−0.347
Tips	−0.322	4 h	21	−0.336
vA	0.309	133 h	22	0.232
Bmag	−0.306	10 h	23	−0.291
log(Pram)	−0.286	16 h	24	−0.265
log(nips)	−0.264	9 h	25	−0.247
sin^2(θclock/2)	0.252	86 h	26	0.103
Sgeo	0.251	138 h	27	0.207
Bz	−0.23	116 h	28	−0.093
Bsouth	0.143	192 h	29	0.086
log(MA)	0.101	4 h	30	0.101

(3a)

(3b)

where the magnitude of the coefficient c_i is irrelevant to the correlation but the sign of c_i is meaningful: if c_i is positive then r_corr > 0 (positive correlation), and if c_i is negative then r_corr < 0 (negative correlation or “anticorrelation”). The evolutionary algorithm is used to optimize the value t_i of the integration time or lag time to obtain the largest magnitude of r_corr. The rows of Table 2 are organized according to the magnitude of the integral correlation coefficient r_corr in the second column, and the numerical rank of the strength of the correlation is listed in the fourth column.

Two things can be noted for the entries of Table 2. (1) If the integrated-variable correlation coefficient (column 2) is positive or negative, then the lagged-variable correlation coefficient (column 4) is likewise positive or negative. (2) For every variable in the table except T_ips at rank 21 and log(M_A) at rank 30, the magnitude of the integral-variable correlation coefficient is larger than the magnitude of the lagged-variable correlation coefficient; often the difference is substantial. (In the cases of T_ips and log(M_A) the optimal lag times are negative; that is, the variations of F_e1.2 lead the variations of T_ips and log(M_A).) The time-integral correlations being greater than the time-lag correlations indicates that correlations of F_e1.2 with time-integrated solar wind and magnetospheric variables are stronger than correlations with time-lagged variables. The higher correlations with time-integrated quantities make sense mathematically since integration introduces smoothing, which reduces noise on the variables and noise is uncorrelated variance. For the radiation belt, the higher correlations with time-integrated quantities make sense physically for several reasons. First, because the radiation-belt electrons are trapped in the magnetospheric magnetic dipole and are heated (energized) during a long-duration storm the relativistic electron flux ℱ steadily increases with time as the heating persists. If for example whistler mode chorus waves are doing the heating, then an expression something like “dℱ/dt = +chorus” may describe this: integrating both sides of this expression yields “ℱ = +∫ chorus dt,” an integral expression of the form of expression 3a. A second physical process that acts on the radiation belt is pitch angle scattering into the loss cone by plasma waves (such as whistler mode hiss or electromagnetic ion cyclotron (EMIC)), where the radiation-belt fluxes decrease with time when the waves are present yielding an expression looking like the “dℱ/dt = −waves” and so (integrating both sides) “ℱ = −∫ waves dt.” Third, for ULF radial diffusion the longer that ULF waves are “on,” the stronger the effect (either positive or negative) they produce on the radiation belt: another time-integral effect. Additionally, the occurrence of substorms is undoubtedly physically important for the evolution of the radiation belt (producing seed electrons and producing plasma waves), and the underlying process of substorm occurrence has a storage-and-release aspect (e.g., Hones, 1979; Klimas et al., 1996; McPherron et al., 1973; Rostoker, 1972; Schindler & Birn, 1978) that requires time integration of inputs before a substorm instability can occur.

Of the 31 variables examined, the substorm-injected-electron flux index F_e130 has by far the strongest correlation with F_e1.2. F_e1.2 is the maximum multisatellite 1.2 MeV flux at geosynchronous orbit; F_e130 is the maximum multisatellite 130 keV flux at geosynchronous orbit. For the 1995–2006 data set the hourly values of F_e1.2 are plotted as a function of the hourly values of the 62 h integral of F_e130 in Figure 1 (left). A least squares linear-regression fit to the points is denoted by the red dashed line, and a major axis linear-regression fit (Pearson, 1901; Smith, 2009) (also known as a total least squares fit (Golub & Van Loan, 1980)) to the points is denoted by the orange dashed line. F_e130 gauges the intensity of the substorm-injected electron population in the magnetosphere; F_e130 increases at the onset of a substorm (Borovsky & Yakymenko, 2017a). For radiation-belt physics F_e130 represents (1) the strength of the seed electron population for the electron radiation belt (Friedel et al., 2002; McDiarmid & Burrows, 1965) and (2) the strength of the substorm-injected electron population that drives whistler mode chorus waves, which can energize radiation-belt electrons (He et al., 2015; Meredith et al., 2001). As seen in the third column of Table 2, the optimal integration time for F_e130 to correlate with F_e1.2 is 62 h. One might worry that the correlation between F_e130 and F_e1.2 is so high because 130 keV electrons are really part of the outer electron radiation belt as are 1.2 MeV electrons: as a test against this possibility the flux of 60 keV electrons F_e60 is also examined (second row of Table 2). The correlation of F_e60 with F_e1.2 is not quite as high (70% instead of 74%), but it is still substantially higher than any other variable in Table 2. (With N = 93947 data points, the standard deviation of the Pearson correlation coefficient due to statistical noise is N^−1/2 = 0.006 (Bendat & Piersol, 2010), so the difference between r_corr = 0.74 for F_e130 and r_corr = 0.70 for F_e60 is statistically significant.) Hence, it is concluded that the strength of the electron radiation belt is highly correlated with the time integral of the strength of the substorm-injected electron population in the magnetosphere.

Figure 1

Hourly values of F_e1.2 is plotted (blue points) (left) as a function of the strongest positive correlator F_e130 and (right) as a function of the strongest negative correlator log(n_sw). In each panel the red dashed line is a least squares linear-regression fit and the orange curve is a major axis fit.

The next five variables in Table 2 (ranks 2–5) essentially deal with v_sw and n_sw and combinations thereof. (S_p = T_p/n_sw^2/3 is very similar to v_sw/n_sw in rank 3 since T_p is very strongly correlated with v_sw (Borovsky & Steinberg, 2006a; Elliott et al., 2005, 2012; Lopez, 1987)). These rank 2–5 variables signify strong positive correlation between v_sw and F_e1.2 and strong negative correlation of n_sw with F_e1.2. The optimal integration time for v_sw (rank 2) is 98 h, and the optimal integration time for n_sw (rank 6) is 38 h. The solar wind proton temperature T_p comes in at rank 7. Since log(n_sw) presents the strongest anticorrelation (negative correlation) with F_e1.2, F_e1.2 is plotted as a function of the 38 h integral of log(n_sw) in the right-hand frame of Figure 1 for 1995–2006. A linear-regression fit is denoted by the red dashed line, and a major axis fit is denoted by the orange dashed line.

The geomagnetic indices Kp and AE come in at ranks 6 and 8 in Table 2. Kp is a measure of the strength of magnetospheric convection (Thomsen, 2004), and the auroral-electrojet index AE is a measure of the strength of high-latitude current systems and is a good indicator of the occurrence, strength, and duration of substorms (Gjerloev et al., 2004).

The ground-based ULF index S_grd comes in at rank 9 in Table 2 with r_corr = 0.535, but the geosynchronous-based ULF index S_geo comes in at rank 27 with r_corr = 0.252. Note that physically ULF waves in the magnetosphere can either increase or decrease the intensity of the electron radiation belt (cf. section 5). The single-variable correlation of S_geo with radiation-belt fluxes being much weaker than the correlation of S_grd with radiation-belt fluxes has been seen before (Borovsky & Denton, 2014). Note that when their contribution to multivariable correlations are examined, the addition of S_geo can yield a larger correlation with F_e1.2 than does the addition of S_grd.

The rate of substorm occurrence S_rate comes in at rank 10 in Table 2 with r_corr = 0.501 and an optimal integration time of 118 h. (See Rodger et al., 2016 for a study of radiation-belt intensification occurring after intervals of repeated substorm occurrence.)

In Table 2 the reconnection driver functions log(R_qo) and R_quick correlate at ranks 11 and 12. Physically, these solar wind functions should describe geomagnetic activity; hence, they should act similarly to the variables Kp and AE, but imperfectly. Hence, their correlation rank is lower than the ranks of Kp and AE. R_qo and R_quick also carry other information about the solar wind.

The two ratios B_mag/n_sw and T_p/v_sw with ranks 13 and 14 in Table 2 are interesting solar wind quantities. The ratio B_mag/n_sw is invariant to compressions and rarefactions of the evolving solar wind (cf. Borovsky, 2016), while almost all other solar wind variables are strongly affected by compression or rarefaction. The ratio T_p/v_sw is often used in classification algorithms for the different types of solar wind plasma originating from different types of regions on the solar surface (e.g., Borovsky & Steinberg, 2006a; Camporeale et al., 2017; Elliott et al., 2005; Neugebauer et al., 2003; Neugebauer et al., 2016; Reisenfeld et al., 2003; Xu & Borovsky, 2015).

The four variables with ranks 15–18 in Table 2 measure diamagnetism related to the intensity of the ring current (ion plasma sheet) in the inner magnetosphere (Dessler & Parker, 1959; Sckopke, 1966). ASY-H (the asymmetric H index) is a measure of the ring current population on open drift paths (Dubyagin et al., 2014; Liemohn et al., 2001). Dst* is the pressure-corrected Dst index.

The index I₂₇₂ at rank 19 in Table 2 is the strength of the electron strahl in the solar wind measured at 272 eV. The strahl typically extends in energy to 1 keV and beyond (Anderson et al., 2012; Pagel et al., 2005). The strahl correlations could be acting as a proxy for other solar wind variables (e.g., Feldman et al., 1976). If the superthermal electrons in the magnetotail come from the solar wind, then the intensity of the strahl could be a measure of that superthermal electron population in the Earth's magnetotail, which becomes the seed-electron population for the radiation belt when substorms deliver those magnetotail superthermal electrons into the dipolar magnetosphere.

Next in rank in Table 2 are three variables that describe the state of the ion plasma sheet in the magnetosphere: the logarithm of the ion pressure log(P_ips) at rank 20, the temperature of the plasma sheet ions T_ips at rank 21, and the logarithm of the number density of the plasma sheet ions log(n_ips) at rank 25. The rank of these variables is fairly low in terms of single-variable correlation with F_e1.2, but in section 4.3 these variables will be found to be very important for multivariable correlations with F_e1.2.

In Table 2 the Alfven speed in the solar wind v_A = B_mag/(4πm_pn_sw)^1/2 and the solar wind magnetic field strength B_mag are ranked 22 and 23 in correlating with F_e1.2. The Alfven speed of the solar wind is a good indicator of ejecta plasma (e.g., coronal mass ejections) in the solar wind (Xu & Borovsky, 2015).

The logarithm of the ram pressure of the solar wind log(P_ram) comes in at rank 24. Correlation with P_ram = n_swv_sw² is a competition of positive correlation with v_sw and negative correlation with n_sw with a resulting weak correlation for P_ram. Note in Table 2 that the correlation of log(v_sw/n_sw) comes in at a very high rank (rank 3).

The clock angle function sin²(θ_clock/2) of the solar wind magnetic field ranks 26, with ∫ sin²(θ_clock/2) dt correlating with F_e1.2 at r_corr = +0.252. The magnitude of this correlation is relatively weak, but it has a significant interpretation. Unlike other solar wind variables, the clock angle θ_clock is uncorrelated with all of the other solar wind variables explored here; hence, it does not act as a proxy for another variable. There are N = 93,947 1 h data points in the correlation; correlation at the 95% confidence level occurs for a correlation coefficient with a magnitude larger than 2/N^1/2 = 0.006 (Bendat & Piersol, 2010; Beyer, 1966), so a correlation of 0.252 is a definite correlation. The definite correlation of sin²(θ_clock/2) with F_e1.2 demonstrates that there is a Russell-McPherron effect in the driving of the electron radiation belt by the solar wind (see also Borovsky & Denton, 2014; McPherron et al., 2009); this positive correlation of F_e1.2 with sin²(θ_clock/2) is expected if the energization or the source of the radiation belt depends on geomagnetic activity (e.g., Meredith et al., 2002; Obara et al., 2000) and/or if the decay of the radiation belt depended on a lack of geomagnetic activity (e.g., Borovsky & Denton, 2009a, 2011a; Borovsky & Steinberg, 2006b; Meredith et al., 2006).

B_z and B_south (GSM coordinates) correlate surprisingly poorly with F_e1.2 at ranks 28 and 29 in Table 2. Finally, the Alfven Mach number log(M_A) has the weakest correlation of the variables examined in Table 2 at rank 30.

In Table 3, the correlation coefficients r_corr of the integrals of the 31 variables of Table 2 with F_e1.2 are compared with their correlations with the electron-radiation-belt number density n_erb and the electron-radiation-belt temperature (spectral hardness) T_erb. In general, in Table 3 the correlations with log(n_erb) and T_erb are similar to but weaker than the correlations with F_e1.2. Cases where the correlation with log(n_erb) are higher than for F_e1.2 are for the Dst-like variables ASY-H, SYM-H, Dst, and Dst* and for v_A, S_geo, B_south, and log(M_A). Cases where the correlation with T_erb is higher than the correlation with F_e1.2 are for I₂₇₂, P_ips, T_ips, log(n_ips), B_mag, B_south, and log(M_A). Note in Table 3 that the optimal integration times tend to be longer for correlations with T_erb than with correlations for log(n_erb): it is well known that n_erb can change on shorter timescales than the timescales that T_erb typically changes on (cf. Figure 3 of Borovsky & Denton, 2010a).

Table 3. Comparison of the Correlations of Time-Integrated Variables With F_e1.2, log(n_erb), and T_erb

Integrated variable	rcorr with Fe1.2	Integration time with Fe1.2	rcorr with log(nhi)	Integration time with log(nhi)	rcorr with Thi	Integration time with Thi
Fe130	0.736	62 h	0.675	12 h	0.552	99 h
Fe60	0.699	85 h	0.594	35 h	0.506	117 h
vsw	0.647	98 h	0.537	39 h	0.522	127 h
log(vsw/nsw)	0.627	51 h	0.608	11 h	0.532	80 h
log(Sp)	0.620	91 h	0.538	35 h	0.515	116 h
log(nsw)	−0.579	38 h	−0.580	8 h	−0.507	65 h
Kp	0.562	138 h	0.509	78 h	0.394	164 h
Tp	0.554	121 h	0.466	57 h	0.438	137 h
AE	0.544	140 h	0.477	75 h	0.395	162 h
Sgrd	0.535	123 h	0.488	65 h	0.366	147 h
Srate	0.509	118 h	0.459	63 h	0.349	144 h
log(Rqo)	0.501	96 h	0.495	76 h	0.312	159 h
Rquick	0.493	152 h	0.445	87 h	0.332	173 h
Log(Bmag/nsw)	0.466	81 h	0.492	26 h	0.347	103 h
Tp/vsw	0.435	119 h	0.397	69 h	0.357	150 h
log(ASY-H)	0.387	138 h	0.416	74 h	0.267	166 h
SYM-H	−0.373	110 h	−0.404	40 h	0.106	2 h
Dst	−0.373	106 h	−0.394	38 h	0.089	2 h
Dst*	−0.358	131 h	−0.363	60 h	0.168	3 h
I272	−0.358	1 h	−0.224	24 h	−0.404	24 h
Pips	−0.356	6 h	0.270	89 h	−0.479	12 h
Tips	−0.322	4 h	0.284	61 h	−0.517	8 h
vA	0.309	133 h	0.337	71 h	0.206	161 h
Bmag	−0.306	10 h	−0.244	1 h	−0.378	19 h
log(Pram)	−0.286	16 h	−0.337	3 h	−0.330	23 h
log(nips)	−0.264	9 h	−0.202	1 h	−0.316	17 h
sin^2(θclock/2)	0.252	86 h	0.202	50 h	0.168	127 h
Sgeo	0.251	138 h	−0.322	81 h	−0.237	14 h
Bz	−0.228	116 h	−0.191	68 h	−0.155	150 h
Bsouth	0.143	192 h	0.221	97 h	−0.230	22 h
log(MA)	0.101	4 h	−0.201	76 h	0.164	12 h

To gain some physical and mathematical insights into the correlations of Table 2, the time series F_e1.2(t) is separated into intervals wherein F_e1.2 is increasing in time and intervals wherein F_e1.2 is decreasing in time and then the correlations of Table 2 are recalculated separately for those two types of intervals. To determine the intervals, a 24 h running average of F_e1.2(t) is produced: denoted ⟨F_e1.2⟩(t). Times t where the 24 h average of F_e1.2 has increased by more than 0.1 in the last 24 h are marked as “increasing” (i.e., where ⟨F_e1.2⟩(t) − ⟨F_e1.2⟩(t − 24 h) > 0.1). Similarly, times t where the 24 hr average of F_e1.2 has decreased by more than 0.1 in the last 24 h are marked as “decreasing” (i.e., where ⟨F_e1.2⟩(t) − ⟨F_e1.2⟩(t − 24 h) < −0.1). In Table 4 the correlations for all times, for F_e1.2-increasing times, and for F_e1.2-decreasing times are listed. In parentheses the optimal integration time is listed for each correlation. For the “all times” column there are 93,947 h in each correlation, for the “increasing times” column there are 25,714 h in each correlation, and for the “decreasing times” column there are 27,749 h in each correlation. Note that the time integrations of the variables into the past can extend outside of the F_e1.2-increasing and the F_e1.2-decreasing intervals. In most cases in Table 5 the optimal integration time for F_e1.2-declining intervals is longer than the optimal integration time for F_e1.2-increasing intervals, with the all-interval integration times in the middle. Some of the variables in Table 5 exhibit stronger correlations for the F_e1.2-increasing intervals: these are the substorm-injected electron fluxes F_e130 and F_e60; the solar wind velocity v_sw; the geomagnetic indices Kp, AE, and ASY-H; the substorm occurrence rate S_rate; the electron strahl intensity I₂₇₂; the ion-plasma sheet temperature T_ips; and the clock-angle function sin²(θ_clock/2). The largest of the correlations is for F_e130. In the two panels of Figure 2 F_e1.2 is plotted as functions of the time integrals of F_e130 for (Figure 2, left) increasing-F_e1.2 times and (Figure 2, right) decreasing-F_e1.2 times. Linear-regression (red) and major axis (orange) fits are displayed. Comparing the two panels of Figure 2, a statistical difference between the F_e1.2-increasing data points and the F_e1.2-decreasing data points is seen. Some variables in Table 5 show stronger correlations for F_e1.2-decreasing intervals: these are the ion-plasma sheet parameters log(P_ips) and log(n_ips), the solar wind magnetic field strength B_mag, the solar wind dynamic pressure log(P_ram), the geosynchronous ULF index S_geo, and the GSM function B_south. (Note that the sign of the correlation with B_south changes in Table 4 for F_e1.2-decreasing intervals.) It may be the case that variables that correlate better during the F_e1.2-increasing intervals control or describe mechanisms that energize the electron radiation belt, and likewise, it may be the case that variables that correlate better during the F_e1.2-decreasing intervals control or describe mechanisms that produce losses of the electron radiation belt.

Table 4. Comparison of the Correlations of Time-Integrated Variables With F_e1.2 for All Times, for F_e1.2-Increasing Times, and for F_e1.2-Decreasing Times

Variable	Rank	rcorr for all intervals	rcorr for intervals of increasing Fe1.2	rcorr for intervals of decreasing Fe1.2	Dominance
Fe130	1a	0.736 (62 h)	0.798 (49 h)	0.645 (107 h)	Fe1.2 Increasing
Fe60	1b	0.699 (85 h)	0.761 (56 h)	0.594 (129 h)	Fe1.2 Increasing
vsw	2	0.647 (98 h)	0.641 (62 h)	0.567 (139 h)	Fe1.2 Increasing
log(vsw/nsw)	3	0.627 (51 h)	0.593 (34 h)	0.552 (98 h)
log(Sp)	4	0.620 (91 h)	0.584 (77 h)	0.552 (128 h)
log(nsw)	5	−0.579 (38 h)	−0.521 (27 h)	−0.517 (75 h)
Kp	6	0.562 (138 h)	0.584 (93 h)	0.432 (184 h)	Fe1.2 Increasing
Tp	7	0.554 (121 h)	0.540 (105 h)	0.477 (156 h)
AE	8	0.544 (140 h)	0.587 (94 h)	0.412 (189 h)	Fe1.2 Increasing
Sgrd	9	0.535 (123 h)	0.510 (96 h)	0.437 (149 h)
Srate	10	0.509 (118 h)	0.558 (78 h)	0.396 (153 h)	Fe1.2 Increasing
log(Rqo)	11	0.501 (96 h)	0.529 (92 h)	0.370 (164 h)	Fe1.2 Increasing
Rquick	12	0.493 (152 h)	0.507 (106 h)	0.378 (188 h)	Fe1.2 Increasing
log(Bmag/nsw)	13	0.466 (81 h)	0.435 (60 h)	0.365 (115 h)
Tp/vsw	14	0.435 (119 h)	0.414 (114 h)	0.367 (153 h)
log(ASY-H)	15	0.387 (138 h)	0.385 (94 h)	0.262 (188 h)	Fe1.2 Increasing
SYM-H	16	−0.373 (110 h)	−0.313 (96 h)	−0.317 (143 h)
Dst	17	−0.373 (106 h)	−0.316 (94 h)	−0.311 (136 h)
Dst*	18	−0.358 (131 h)	−0.319 (103 h)	−0.293 (162 h)
I272	19	−0.358 (13 h)	−0.397 (9 h)	−0.327 (22 h)	Fe1.2 Increasing
log(Pips)	20	−0.356 (6 h)	−0.435 (17 h)	−0.488 (7 h)	Fe1.2 Decreasing
Tips	21	−0.322 (4 h)	−0.476 (4 h)	−0.345 (33 h)	Fe1.2 Increasing
vA	22	0.309 (133 h)	0.272 (94 h)	0.229 (163 h)
Bmag	23	−0.306 (10 h)	−0.243 (10 h)	−0.374 (24 h)	Fe1.2 Decreasing
log(Pram)	24	−0.286 (16 h)	−0.158 (12 h)	−0.338 (28 h)	Fe1.2 Decreasing
log(nips)	25	−0.264 (9 h)	−0.316 (31 h)	−0.441 (7 h)	Fe1.2 Decreasing
sin^2(θclock/2)	26	0.252 (86 h)	0.277 (87 h)	0.204 (136 h)	Fe1.2 Increasing
Sgeo	27	0.251 (138 h)	0.198 (106 h)	−0.262 (15 h)	Fe1.2 Decreasing
Bz	28	−0.228 (116 h)	−0.233 (97 h)	−0.181 (142 h)
Bsouth	29	0.143 (192 h)	0.122 (115 h)	−0.263 (31 h)	Fe1.2 Decreasing
log(MA)	30	0.101 (4 h)	0.110 (15 h)	0.192 (9 h)

Table 5. The Information for the Correlation of Expression 6 With r_corr = 0.787

Term	Coefficient	Integration time	Added correlation
vsw	0.294	116 h	0.009
log10nsw	−0.165	14 h	0.006
Sgrd	−0.226	172 h	0.006
Kp	−0.496	31 h	0.017
Fe130	0.912	30 h	0.092

Figure 2

Hourly values of F_e1.2 is plotted (blue points) as a function of the optimized time-integral of the substorm-injected-electron flux index F_e130 for (left) F_e1.2-increasing times and for (right) F_e1.2-decreasing times. In each panel the red dashed line is a least squares linear regression fit and the orange curve is a major axis fit.

4 Multivariable Correlations With F_e1.2

In this section formulas that are the sum of the time integrals of variables are correlated with the radiation-belt-flux index F_e1.2.

4.1 The Solar Wind Velocity and the Solar Wind Density

Recent studies have examined the relative roles of the solar wind velocity v_sw and the solar wind number density n_sw in the control of the electron radiation belt (e.g., Balikhin et al., 2011; Borovsky & Denton, 2014; Boynton et al., 2013; Kellerman & Shprits, 2012; Wing et al., 2016) with an eye toward cause and effect for the two variables interacting with the radiation belt. It is pointed out here that there is a lot of redundant information in the two time series of these two solar wind variables. The single-variable correlations between F_e1.2 and the time integral of v_sw and between F_e1.2 and the time integral of log(n_sw) appear in Table 2 (at ranks 2 and 5, respectively). If the time integrals of the two variables are simultaneously correlated with F_e1.2, with the coefficients and integration times optimized for maximum correlation, the expression

(4)

is obtained, with r_corr = 0.682. Removing the ∫ log(n_sw) dt term from expression 4 results in a decrease in the correlation r_corr by an amount 0.033, and removing the ∫ v_sw dt term from expression 4 results in a decrease in r_corr by an amount 0.103. Hence, adding the information in n_sw only increases the correlation by 0.033 over the correlation driven by the information in v_sw alone. Hence, n_sw is not adding much useful information for F_e1.2.

In Figure 3 three time series are plotted for 75 days in the year 2005. The red curve (right axis) is the solar wind velocity v_sw, the green curve (left axis) is log₁₀(n_sw), and the blue curve (left axis) is −log₁₀(n_sw) time shifted 17 h to the left (to earlier in time). Note that a lot of the major changes in the red v_sw curve are matched by changes in the blue −log10(n_sw) curve; hence, a lot of the temporal information carried in the v_sw time series is also carried in the log(n_sw) time series. Diagnosing cause and effect for the radiation belt in all-important geomagnetic storms is of great interest. In Figure 4 three curves are plotted, all of which are superposed averages for 70 high-speed-stream-driven geomagnetic storms, with the zero epoch being the onset of storm levels of magnetospheric convection as measured by the MBI index. (The selection of the storms is discussed in Borovsky & Denton, 2016.) The green curve (left axis) is the superposed average of F_e1.2 for the 70 storms, the red curve (right axis) is the superposed average of v_sw for the 70 storms, and the blue curve (left axis) is the superposed average of -log10(n_sw) time shifted 17 h to the left (to earlier times). There is a well-known temporal pattern to the relativistic electron fluxes in high-speed-stream-driven storms (i.e., Figure 5c of Borovsky & Denton, 2011b) (e.g., the green curve in Figure 4). Note that there are similar patterns in both v_sw and −log(n_sw) (e.g., the red and blue curves): redundant information in the superposed averages of the two time series. Redundant information in n_sw and v_sw makes it difficult to determine the cause and effect on the radiation belt (see also Wing et al., 2016; the similarities of the temporal patterns in v_sw and n_sw make it no surprise that both would correlate well with radiation-belt fluxes.

Figure 3

For 75 days in 2005 the solar wind velocity v_sw is plotted (blue, right axis) along with the base-10 logarithm of the solar wind number density log10(n_sw) (green, left axis). To match the velocity plot, −log10(n_sw) is plotted (red, left axis) time-shifted 17 h to the left.

Figure 4

Triggered (time t = 0) on the onset times of 70 high-speed-stream-driven storms, the superposed average of the multisatellite average of the 1.2 MeV electron flux at geosynchronous orbit F_e1.2 is plotted (green curve, left axis), the superposed average of the solar wind velocity v_sw is plotted (blue curve, right axis), and the negative base-10 logarithm of the solar wind number density −log₁₀(n_sw) is plotted time shifted 17 h to the left (blue curve, left axis).

4.2 Combinations of the Well-Known Radiation-Belt Correlators

In the literature (see section 1) there are four well-known correlations with radiation-belt flux: v_sw, Kp, S_grd, and n_sw. Those four variables are among the highest correlators in Tables 2 and 3 (ranks 2, 6, 9, and 5, respectively). Combinations of these variables in correlating with F_e1.2 are explored in Table 6. Correlating all four of these variables simultaneously with F_e1.2, adjusting the coefficients and the integration times to optimize the correlations, yields

Table 6. Correlation Coefficients r_corr Between F_e1.2 and Combinations of the Well-Known Drivers of the Radiation Belt

	No. of inputs	vsw	log10nsw	Sgrd	Kp	Fe130	rcorr
1	1	x					0.647
2	1		x				0.559
3	1			x			0.535
8	1				x		0.562
4	2	x	x				0.681
5	2	x		x			0.655
6	2	x			x		0.652
7	2		x	x			0.673
9	2		x		x		0.669
10	2			x	x		0.569
11	3		x	x	x		0.676
12	3	x		x	x		0.678
13	3	x	x		x		0.684
14	3	x	x	x			0.696
15	4	x	x	x	x		0.703
16	1					x	0.736
17	5	x	x	x	x	x	0.783

• Note. An “x” in each row indicates that the variable in that column is included in the integral correlations with F_e1.2.

(5)

with r_corr = 0.703 (cf. line 15 of Table 6). Comparing line 15 of Table 6 with line 11, one can see that removing v_sw from the four-variable correlation reduces the total r_corr by 0.008. Hence, the unique contribution of v_sw to the four-variable correlation of expression 5 is 0.008, or 0.8%. Likewise, comparing lines 12, 13, and 14 with line 15 yields contributions of 0.026 for log(n_sw), 0.019 for S_grd, and 0.008 for Kp. The variable log(n_sw) is contributing 0.026 to r_corr above the three-variable value, which is not very significant. The contributions of the other three variables to the four-variable correlations are even less significant. This is because all of the four variables are strongly correlated with each other so that any one of them carries only a small amount of unique information needed to describe the variance of F_e1.2. That is, there is a lot of redundant information in the four variables on the right-hand side of expression 5. In Table 7 the intercorrelations between the four variables are examined, with the integration time for each variable taken as the optimal integration time for single-variable correlation with F_e1.2 (i.e., from Table 2). The median value of the magnitudes of the values in Table 7 (excluding the unity values) is 0.718. The variable log(n_sw) has the weakest correlations, with the others in Table 7, but these are still strong. Note the very high correlation of the time integral of S_grd with the time integral of Kp with r_corr = 0.887.

Table 7. The Correlation Coefficients r_corr Between Time-Integrated Pairs of the Well-Known Drivers of the Radiation Belt

	∫^98 vsw dt	∫^38 log(nsw) dt	∫^123 Sgrd dt	∫^138 Kp dt	∫^62 Fe130 dt
∫^98 vsw dt	1	−0.672	0.719	0.806	0.766
∫^38 log(nsw) dt	−0.672	1	−0.373	−0.458	−0.567
∫^123 Sgrd dt	0.719	−0.373	1	0.887	0.718
∫^138 Kp dt	0.806	−0.458	0.887	1	0.777
∫^62 Fe130 dt	0.766	−0.567	0.718	0.777	1

Adding the substorm-injected-electron flux index F_e130 to the four variables v_sw, Kp, S_grd, and n_sw yields the five-variable correlation

(6)

with r_corr = 0.787 (cf. row 17 of Table 6). This is a significantly higher correlation than the four-variable case in expression 5. In Table 5 the unique contribution that each of the five variables brings to r_corr = 0.787 is listed in the fourth column. (This number is the drop in r_corr from the value 0.787 that results when that single variable is removed from the five variables.) Note that the contribution of F_e130 is 0.092 (i.e., 9.2%), which dwarfs the other contributions. The unique contribution of Kp is less than 2% in Table 5, and the unique contributions of v_sw, log(n_sw), and S_grd are each less than 1%.

If the five-variable correlation of expression 6 is constructed with time-lagged variables rather than time-integrated variables, the optimized result is

(7)

with r_corr = 0.718. This value 0.718 is a substantially weaker correlation than r_corr = 0.787 for the integral formula of expression 6. The unique contributions of F_e130, log(n_sw), v_sw, S_grd, and Kp to expression 7 are 0.059, 0.030, 0.016, 0.002, and 0.001 respectively.

4.3 Creating an Optimized Correlation Formula With F_e1.2

In Table 2 the strongest single-variable correlation with F_e1.2 is

(8)

with r_corr = 0.736. To produce an optimized two-variable correlation with F_e1.2, the other variables in Table 2 are added one at a time to the right-hand side of expression 8 and the added correlation to r_corr is calculated. The results are collected in the first column of Table 8. In Table 8 it is seen that the largest contribution corresponds to the addition of the variable log(P_ips) with an additional contribution of 0.084 to r_corr. Note in Table 2 that log(P_ips) does not have one of the strongest direct correlations with F_e1.2 (it has rank 20); however, log(P_ips) must carry some information that is unique beyond the information in F_e130 while other variables in Table 2 do not carry that unique information to the same degree. Adding the integral of log(P_ips) to the integral of F_e130, the resulting optimized formula is

Table 8. The Added Correlation With F_e1.2 Obtained by Adding an Integral of a Variable to c₁ ∫ ^t1 F_e130 dt (First Column) and to c₁ ∫ ^t1 F_e130 dt + c₂ ∫ ^t2 log(P_ips) dt (Second Column)

Added variable	Addition to rcorr for the addition of the variable to ∫ Fe130 dt	Addition to rcorr for the addition of the variable to ∫ Fe130 dt + ∫ log(Pips) dt
Fe60	0.001	0.017
vsw	0.011	0.001
log(vsw/nsw)	0.025	0.001
log(Sp)	0.024	0.005
log(nsw)	0.028	0.005
Kp	0.024	0.002
Tp	0.017	0.005
AE	0.017	0.001
Sgrd	0.010	0.004
Srate	0.002	0.005
log(Rqo)	0.011	0.002
Rquick	0.021	0.000
log(Bmag/nsw)	0.000	0.001
Tp/vsw	0.015	0.006
log(ASY-H)	0.024	0.000
SYM-H	0.023	0.003
Dst	0.023	0.002
Dst*	0.034	0.002
I272	0.023	0.000
log(Pips)	0.084
Tips	0.057	0.010
vA	0.012	0.002
Bmag	0.054	0.005
log(Pram)	0.028	0.004
log(nips)	0.055	0.006
sin^2(θclock/2)	0.002	0.007
Sgeo	0.029	0.000
Bz	0.000	0.003
Bsouth	0.022	0.001
log(MA)	0.013	0.003

(9)

with r_corr = 0.820. In Figure 5 the hourly values of F_e1.2 are plotted (green) as a function of the hourly values of the right-hand side of expression 9 for the years 1995–2006. A linear-regression fit is drawn as the red dashed line, and a major axis fit is drawn as the orange dashed line.

Figure 5

Hourly values of the multispacecraft-averaged 1.2 MeV electron flux F_e1.2 are plotted as functions of the two time-integral expression on the right-hand side of expression 9. The red dashed line is a least squares linear regression fit, and the orange curve is a major axis fit.

To produce a three-variable formula, the variables of Table 2 are added one at a time to the right-hand side of expression 9, the coefficients and integration times are optimized, and the results are collected in the final column of Table 8. The most significant increase of r_corr (by 0.017) comes from the addition of F_e60. (Note in the first column of Table 8 that the increase of r_corr for the addition of the integral of F_e60 to the integral of F_e130 alone was only 0.001.) Adding the integral of F_e60 to the right-hand side of expression 9 results in the optimized formula

(10)

with r_corr = 0.838. (If an equivalent formula is constructed with time-lagged variables rather than time-integrated variables, the optimized correlation coefficient is r_corr = 0.751.) Note that the three-variable formula of expression 10 has a superior correlation coefficient with F_e1.2 than does the five-variable formula (expression (6) where r_corr = 0.787) constructed from the strong correlators recognized in the literature.

Continuing this process for one more round of adding variables from Table 2 would arrive at the addition of the time integral of SYM-H to the right-hand side of expression 10 with an increase of 0.008 to r_corr for r_corr = 0.846. But this increase of 0.008 is barely statistically significant since 2/N^1/2 = 0.006 (Bendat & Piersol, 2010; Beyer, 1966) for the N = 93,947 1 h data points in the correlation.

In Figure 6 the F_e1.2 index is plotted in black for 90 days in the year 2005, when there are regularly occurring high-speed streams. Each black data point is the F_e1.2 index for an hour of universal time. The blue, green, and red curves in Figure 6 are the right-hand sides of expression 8 (blue), expression 9 (green), and expression 10 (red). Note the difference between the green curve and the blue curve in Figure 6 in relation to the black points: the green curve (which has the addition of the integral ∫ log(P_ips) dt to the integral ∫ F_e130 dt) does a much better job of describing the sudden drops in F_e1.2, which may account for some of the better correlation with F_e1.2. Drops in the relativistic electron flux at geosynchronous orbit associated with occurrences of superdense plasma sheet (i.e., large values of n_ips and P_ips) were reported by Borovsky and Denton (2009b), where because of the strong temporal connection they hypothesized (perhaps incorrectly) that EMIC waves driven by the superdense plasma sheet were the cause of radiation-belt dropouts.

Figure 6

For 90 days in 2005, hourly values of the multispacecraft-averaged 1.2 MeV flux of electrons F_e1.2 are plotted as the black points. The blue curve is the one-integral right-hand side of expression 8, the green curve is the two-integral expression on the right-hand side of expression 9, and the red curve is the three-integral expression on the right-hand side of expression 10.

4.4 Generating an Optimized Formula Comprised Only of Solar Wind Variables

The motivation to find a formula to describe the variance of the electron-radiation-belt flux that is comprised only of solar wind variables is the belief that the solar wind is the ultimate driver (controller) of the dynamics of the magnetosphere and the evolution of the radiation belt. The desire in this section is to find a collection of solar wind variables that yields the highest correlation with F_e1.2. The starting point is the best single-variable correlation (cf. Table 2) and progressively adding more solar wind variables that improve the correlation.

Adding a progression of solar wind variables into correlation with F_e1.2, the first solar wind variable (with the highest single variable correlation with F_e1.2) is v_sw, then the addition of B_mag gives the largest increase in r_corr, then the addition of sin²(θ_clock/2) gives the largest increase in r_corr, then the addition of log(n_sw) gives the largest increase in r_corr, and then the variable I₂₇₂ gives the largest increase in r_corr. Variables examined that did not yield the largest increases in r_corr during the progressive steps were log(v_sw/n_sw), log(S_p), log(P_ram), T_p, log(R_qo), R_quick, log(B_mag/n_sw), T_p/v_sw, B_z, B_south, and log(M_A). The progressive adding of five solar wind variables yields the correlation expression

(11)

with r_corr = 0.760. (The equivalent 5-variable equation using time-lagged variables instead of time-integrated variables has a correlation with F_e1.2 of r_corr = 0.666.) I₂₇₂ is unavailable in early solar wind data: the optimized 4-variable formula before the addition of I₂₇₂ is

(12)

with r_corr = 0.752.

Note that the optimized solar wind -variable equations (expressions 11 and 12) have correlations with F_e1.2 that are much less than the equation in expression 9 that uses only the two magnetospheric variables F_e130 and P_ips.

4.5 Adding Persistence (Autoregression)

Adding a prior value of F_e1.2 to the correlation formulas can improve the correlation coefficients r_corr if the prior value is recent enough (e.g., Boynton et al., 2013; Sakaguchi et al., 2013; Simms et al., 2016; Ukhorskiy et al., 2004). An example of this is the addition of a prior value of F_e1.2 to the right-hand side of the simple two-variable expression 9:

(13)

where t₀ is the time selected for the prior values. In Figure 7 the optimized correlation coefficient r_corr is plotted (blue points) as a function of the selected time t₀ for expression 13 with the coefficients c₀, c₁, and c₂ and the integration times t₁ and t₂ optimized by the evolutionary algorithm for every value of t₀. At t₀ = 0 the correlations is r_corr = 1 (because F_e1.2(t − 0) on the right-hand side has all the information for F_e1.2(t) on the left-hand side) and the correlation decays as t₀ increases. The autocorrelation function A(t₀) of F_e1.2 is given by

Figure 7

The correlation coefficient r_corr with F_e1.2 is plotted (blue points) as a function of the autoregression-term time lag t₀ for expression 13. The autocorrelation function of F_e1.2 is plotted as the red curve. A baseline value of r_corr (with no autoregression term in expression 13 is plotted as the green dashed line.

(14)

where ⟨F_e1.2⟩ is the average value of F_e1.2 and where the time integrations in the numerator and the denominator of expression 14 are over the entire time series. The autocorrelation function of F_e1.2 is plotted in red in Figure 7. For small values of t₀ the decay of r_corr for expression 14 (blue points) follows the autocorrelation function of F_e1.2 (red curve): in this regime the relevant information on the right-hand side of expression 13 is dominated by the F_e1.2(t − t₀) term. With further increasing t₀ the values of r_corr deviate from the autocorrelation function of F_e1.2, eventually asymptoting to a baseline value of r_corr (green dashed line) that is the correlation of the equation of expression 13 without the F_e1.2(t − t₀) term on the right-hand side (i.e., expression 9). Examining Figure 7 for expression 13, it is seen that persistence values t₀ less than about 75 h help the correlation, and longer persistence values do not help.

For a reference, expression 13 is optimized for t₀ = 24 h and the expression obtained for F_e1.2(t) is

(15)

with r_corr = 0.879.

5 A Physics-Based Picture of Radiation-Belt Control

In this section a correlation picture will be constructed based on physical insight underpinned by the current understanding of radiation-belt physics. The development will be organized by listing the physical processes that act on the electron radiation belt and the likely variables in the magnetosphere and in the solar wind that control or measure each process. Two correlation pictures will be developed organized on (1) what variables in the magnetosphere control the radiation-belt evolution and (2) what variables in the solar wind control those magnetospheric variables.

A list of eight physical processes that act on the electron radiation belt is contained in Table 9, along with educated guesses about the magnetospheric and solar wind variables that control or measure those processes and educated guesses about the expected sign of the correlation of each of those variables with F_e1.2. The eight processes are described in the following eight paragraphs.

1. The seed electrons for the electron radiation belt are believed to be substorm-injected energetic electrons (e.g., Cayton et al., 1989; Friedel et al., 2002; Hwang et al., 2004; McDiarmid & Burrows, 1965). Seed electrons are important for increasing the flux of the radiation belt. A measuring variable for the strength of the population of these injected electrons is the substorm-injected-electron flux index F_e130 (Borovsky & Yakymenko, 2017a). Since it could be expected that more seed electrons will lead to stronger radiation-belt fluxes, the expected sign of the correlation of F_e130 with F_e1.2 for this process is positive (cf. Table 9).
2. Whistler mode chorus waves in the magnetosphere energize the electrons of the radiation belt (e.g., Horne et al., 2005; Meredith et al., 2002; Thorne et al., 2013), and the chorus waves are driven by substorm-injected electrons (e.g., He et al., 2015; Meredith et al., 2001; Summers et al., 1998). It is expected that this mechanism acts to increase the electron flux of the radiation belt. A measuring variable for the strength of the driving of chorus waves should be F_e130, the strength of the population of substorm-injected electrons. The sign of the correlation of F_e130 with F_e1.2 for this physical process should be positive (cf. Table 9).
3. ULF wave-driven radial diffusion redistributes the electrons of the radiation belt (e.g., Falthammar, 1973; Lanzerotti et al., 1978; Shprits, Elkington, et al., 2008) and undoubtedly affects the electron fluxes at geosynchronous orbit. A measure of the strength of this process is S_geo, which is a measure of the amplitude of ULF fluctuations at geosynchronous orbit. The sign of the correlation of S_geo with F_e1.2 for this physical process is not known (cf. Table 9); common sense would say that the sign of the effect at geosynchronous orbit depends on the sign of the radial gradient of the radiation-belt intensity at geosynchronous orbit.
4. Radiation-belt electrons can be lost into the magnetosheath via the magnetopause-shadowing process (e.g., Desorgher et al., 2000; West, Buck, & Walton, 1972) when the solar wind ram pressure moves the magnetosheath inward and when increased dayside reconnection moves the magnetosphere inward. Magnetopause shadowing is an important mechanism for decreasing the flux of the radiation belt. This process is controlled by the magnetopause standoff distance d_mp. The sign of the correlation of d_mp with F_e1.2 for this physical process should be positive, with more radiation-belt loss when d_mp is smaller (cf. Table 9).
5. Enhanced ULF wave-driven radial diffusion is thought to act in concert with magnetopause shadowing to transport radiation-belt electrons outward to the magnetopause to be lost (e.g., Ozeke et al., 2014; Shprits et al., 2006; Yu, Koller, & Morley, 2013). A controlling variable for this processes is S_geo, the amplitude of ULF fluctuations at geosynchronous orbit. The sign of the correlation of S_geo with F_e1.2 for this physical process should be negative (cf. Table 9).
6. Electromagnetic ion cyclotron (EMIC) waves in the magnetosphere can pitch angle scatter radiation-belt electrons into the atmospheric loss cone whereupon they are lost (e.g., Horne & Thorne, 1998; Jordanova et al., 2008; Thorne et al., 2006). EMIC scattering is an important mechanism for decreasing the radiation-belt flux. EMIC waves in the magnetosphere are believed to be driven by resonances with ion-plasma sheet ions with temperature anisotropies (e.g., Cornwall, 1977; Fraser & Nguyen, 2001; Meredith, Thorne, et al., 2003). The strength of the scattering and the rate of loss of radiation-belt electrons should increase with increasing amplitude of the EMIC waves. In plasma simulations, the saturation amplitude of hot-ion-driven EMIC waves was found to increase when the hot-ion density increased and when the hot-ion temperature anisotropy A_ips = T_⊥i/T_||i increased (Bortnik et al., 2011). More specifically, plasma simulations have found the saturation amplitude of EMIC waves to be proportional to the β = 8πnk_BT/B² of the plasma and to the ion anisotropy A_ips (Fu et al., 2016; Gary et al., 2017); for the magnetosphere with fixed magnetic field strength B, the plasma β depends on the ion-plasma sheet pressure P_ips = n_ipsT_ips. The temperature anisotropy of the ion plasma sheet A_ips is anticorrelated with the Kp index (cf. Figure 2 of Denton et al., 2005). Hence, for this physical processes two controlling variables are P_ips and Kp. The sign of the correlation of P_ips with F_e1.2 for this physical process should be negative, and since the sign for A_ips should be negative, the sign for Kp should be positive (cf. Table 9).
7. During quiet geomagnetic times the plasmasphere fills out to large radii and whistler mode hiss in the plasmasphere acts to pitch angle scatter radiation-belt electrons into the atmospheric loss cone (e.g., Borovsky & Denton, 2009a; Lam et al., 2007; Meredith et al., 2006). Scattering by hiss is an important mechanism to produce decreases in the radiation-belt flux. The location of hiss is restricted to the plasmasphere, the plasmaspheric drainage plume, and to detached plasma regions (e.g., Chan & Holzer, 1976; Hayakawa et al., 1986; Summers et al., 2008); hence, the quiet time pitch angle scattering of the radiation belt is dependent on the location of the plasmapause r_pp. The radius of the plasmasphere r_pp increases as the level of magnetospheric convection decreases; hence, r_pp increases as Kp decreases (Carpenter & Anderson, 1992; Heilig & Luhr, 2013). The amplitude of the hiss increases with increasing plasmaspheric number density n_psp (e.g., Chan et al., 1974; Chen et al., 2012; Cornilleau-Wehrlin et al., 1978). The number density of the plasmasphere increases with time, so as long as Kp remains low, so n_psp is inversely proportional to Kp or to the time integral of Kp. A controlling magnetospheric factor for this physical process is the Kp index, and the sign of the correlation of Kp with F_e1.2 for this physical process should be positive; the radiation belt should decay when Kp is low (cf. Table 9).
8. Via the Dst effect (Dessler & Karplus, 1961; Kim & Chan, 1997), the increase of diamagnetism in the dipolar magnetosphere (associated with the plasma thermal pressure producing the storm time Dst perturbation) moves magnetic flux outward to larger L shells. The radiation-belt electrons move outward to larger L-shells with the outward migration of the magnetic flux and are adiabatically deenergized; for typical radiation-belt radial profiles this results in a significant lowering of the electron flux at geosynchronous orbit as the magnitude of Dst or magnitude of SYM-H increases. As the plasma pressure subsides and the Dst and SYM-H indices return to nonactive levels, the reduction of plasma diamagnetism in the magnetosphere (associated with a decay in the strength of the storm time Dst perturbation) moves magnetic flux radially inward to lower L-shells. During this process radiation-belt electrons move inward to lower L-shells along with the magnetic flux and are adiabatically energized. Depending on the radial profile of the radiation belt before the magnetic flux migrates inward, the radiation belt at geosynchronous orbit can be enhanced as the magnitudes of Dst and SYM-H decrease. Dst and SYM-H are both good measures of the magnitude of the Dst effect: SYM-H will be chosen because it utilizes a more-even distribution of ground-based magnetometers. Because the radiation-belt flux decreases as Dst and SYM-H become negative, and because the radiation-belt flux increases as Dst and SYM-H become less negative, the expected correlation between F_e1.2 and SYM-H is positive.

Table 9. Physical Mechanisms Affecting the Electron Radiation Belt, the Magnetospheric Controlling Variables for Those Mechanisms, and the Solar Wind Controlling Variables for Those Mechanisms

	Physical process affecting Fe1.2	Expected effect on Fe1.2	Magnetospheric controlling variable	Expected sign of correlation with Fe1.2	Solar wind controlling variable or measuring variable	Expected sign of correlation with Fe1.2
1	Substorm-injected seed population	Increase	Fe130	+	vsw, Rquick	+, +
2	Chorus driving by substorm injected electrons	Increase	Fe130	+	vsw, Rquick	+, +
3	ULF radial diffusion distributing radiation-belt electrons	Increase or Decrease	Sgeo	?	Rquick, Pram	?, ?
4	Radiation-belt losses via magnetopause shadowing	Decrease	dmp	+	Pram, Rquick	−, −
5	ULF radial diffusion enhancing loss to magnetopause	Decrease	Sgeo	−	Rquick, Pram	−, −
6	Precipitation losses caused by EMIC and magnetosonic waves driven by the ion plasma sheet	Decrease	Pips, Kp	−, +	Bmag, Pram	−, −
7	Decay of radiation belts from hiss scattering in quiet-time plasmasphere	Decrease	Kp	+	Rquick, vsw	+, +
8	Dst effect	Decrease or Increase	SYM-H	+	Rquick, Bsouth	−, −

For the eight physical mechanisms described above and in Table 9, six magnetospheric variables are anticipated to be controlling factors for the radiation belt: F_e130, S_geo, d_mp, P_ips, Kp, and SYM-H. Values for d_mp are not available. Constructing a correlation for F_e1.2 versus the five magnetospheric variables F_e130, S_geo, P_ips, Kp, and SYM-H yields

(16)

with r_corr = 0.830. Removing each of the five variables on the right-hand side of expression 16 one at a time reduces r_corr by 0.105 for the removal of F_e130, by 0.060 for the removal of P_ips, by 0.007 by the removal of Kp, by 0.006 by the removal of SYM-H, and by 0.001 for the removal of S_geo. F_e130 and P_ips by far dominate the correlation of expression 16 (cf. expression 9).

In this physics-based picture of radiation-belt evolution the variables in the solar wind that control the radiation belt should be the solar wind variables that control the eight physical processes listed in Table 9. In the top two rows of Figure 8 the control of F_e1.2 (red) by the six magnetospheric variables F_e130, S_geo, d_mp, P_ips, Kp, and SYM-H (blue) that control the eight physical processes is sketched. In the bottom two rows the control of the magnetospheric variables (blue) by solar wind variables (green) is sketched. To determine the solar wind variables and the connections denoted by the green arrows, the two leading solar wind variables that control each of the five magnetospheric variables are determined with optimized time-lagged linear correlations. First, the leading single-variable correlation is found from all of the available solar wind variables (variables 2–5, 7, 11–14, 19, 22–24, 26, and 28–30 in Table 2) and then all of the available solar wind variables are examined to find the second variable that produces the most improved correlation. For F_e130, Kp, S_geo, log(P_ips), and SYM-H the results are

Figure 8

A sketch of the driving of the radiation-belt index F_e1.2 (red) by the magnetospheric variables in Table 9 (blue), which are in turn driven by the solar wind variables in Table 9 (green).

(17a)

(17b)

(17c)

(17d)

(17e)

These connections are sketched with the green arrows in Figure 8. No values are available for the magnetopause standoff distance d_mp, so no correlations with the solar wind variables could be examined. It is reasonable to expect that d_mp will depend strongly on the dynamic pressure of the solar wind P_ram (e.g., Schield, 1969) and on the rate of dayside reconnection (e.g., Shue et al., 1997), with the reconnection rate parameterized by R_quick (Borovsky & Birn, 2014). In expressions 17a–17e and the educated guess for the solar wind control of d_mp, five solar wind variables are involved: v_sw, R_quick, P_ram, B_mag, and B_south. Constructing a time-integral correlation of these four solar wind variables with F_e1.2 yields

(18)

with r_corr = 0.736. Removing each of the five variables on the right-hand side of expression 18 one at a time reduces r_corr by 0.063 for the removal of v_sw, by 0.045 for the removal of B_mag, by 0.011 for the removal of R_quick, by 0.025 for the removal of log(P_ram), and by 0.005 for the removal of B_south. Comparing expression 18 with the optimized four-variable solar wind correlation of expression 4, the correlation coefficient r_corr of expression 18 is slightly lower (0.736 compared with 0.752).

For this physical picture the expected signs of the correlations collected in Table 9 can be compared with the single-variable correlations in Table 4. The first two physical mechanisms of Table 9 are the substorm-injected seed electrons and the substorm-electron-driven chorus waves, which are both expected to be important during intervals when the radiation-belt electron flux is increasing. For both of these mechanisms the magnetospheric control variable is F_e130 and F_e130 is indeed important during intervals of increasing F_e1.2 (cf. Table 4). The expected positive correlation in Table 9 for F_e130 is positive in Table 4. For the first two mechanisms of Table 9 the solar wind variables associated with F_e130 are v_sw and R_quick: in Table 4 both v_sw and R_quick are dominant in increasing-F_e1.2 intervals as anticipated in Table 9. And as anticipated in Table 9 the signs of the correlations of v_sw and R_quick with F_e1.2 are both positive. For the radiation-belt loss mechanisms 4–7 of Table 9 there are three controlling magnetospheric variables: S_geo, P_ips, and Kp. These three variables should all be important in correlations during F_e1.2-decreasing intervals: in Table 4 S_geo and P_ips are, but contrary to the Table 9 expectation Kp is more important during F_e1.2-increasing intervals. In Table 9 the expected signs of the correlations of S_geo and P_ips are negative and for Kp it is positive: Table 4 confirms the expected signs of the correlations for all three of these magnetospheric variables. For the loss mechanisms 4–7 of Table T20, the four solar wind variables P_ram, R_quick, v_sw, and B_mag control the magnetospheric variables. For the loss mechanisms 4–7 it is therefore expected that P_ram, R_quick, v_sw, and B_mag will be more important during F_e1.2-decreasing intervals: Table 4 finds that P_ram and B_mag are more important during F_e1.2-decreasing intervals, but v_sw and R_quick are more important during F_e1.2-increasing intervals. For the loss mechanisms 4–7 of Table 9 the expected signs of the correlations with F_e1.2 are negative for P_ram, negative for B_mag, positive for v_sw, and either negative (mechanisms 4 and 5) or positive (mechanism 7) for R_quick; Table 4 finds the signs of the correlations to be negative for P_ram and B_mag (both as predicted in Table 9), positive for v_sw (as predicted in Table 9), and positive for R_quick (agreeing with the expectation for mechanism 7 but disagreeing with the expectations for mechanisms 4 and 5). For the Dst effect (mechanism 8 in Table 9) it is expected that SYM-H should be positively correlated with F_e1.2: in Table 4 this is confirmed. For this mechanism it is expected in Table 9 that R_quick and B_south should be negatively correlated with F_e1.2: in Table 4 they are both positively correlated. (As for R_quick, this variable acts in Figure 8 in both positive and negative manners.) Overall, the observations in Table 4 of the signs of the correlations and whether they are dominant in F_e1.2-increasing or F_e1.2-decreasing intervals agree with the expected behaviors in the physical picture organized in Table 9. The deviations from the physical picture of Table 9 are in the variables R_quick, Kp, and (partially in) v_sw: these three variables could also be playing roles in the F_e1.2-increasing mechanisms that compete with their roles in the F_e1.2-decreasing mechanisms.

There are difficulties with determining physical cause and effect between the solar wind variables and the radiation belt caused by the intercorrelations of the solar wind variables. This leads to a complicated “web of correlation” that is difficult to untangle. For the five solar wind variables of Figure 8, this web of correlation with F_e1.2 is sketched in Figure 9. The red node in the figure is F_e1.2, and the five green nodes are the five solar wind variables. The red arrows are sketched from the green nodes to the red node, and the red labels denote the single-variable correlation with F_e1.2 with the optimal integration time denoted in parenthesis. The green arrows between green nodes are labeled with the correlation coefficient r_corr between the pairs of solar wind variables with the optimal time lag in parenthesis. The direction of the arrow indicates the direction of the lag with the variable at the base of the arrow leading in time the variable at the tip of the arrow. As can be seen, there is a web of connections and differing time lags. Note of course that Figure 9 is an oversimplification of the set of solar wind variables that act on the radiation belt.

Figure 9

For the solar wind variables of Table 9 (green nodes) driving the radiation-belt flux index F_e1.2 (red node), the web of correlations is sketched. On each arrow the large number is the two-variable correlation coefficient (in per cent) and the small number in parenthesis is the optimal lag time between the two variables. The direction of each arrow is the direction of the time lag with the variable at the base of the arrow moving first (leading in time).

For the Earth driven by the solar wind, a particular difficulty in interpreting cause and effect using correlative methods is that multiple solar wind variables carry the same information about the temporal pattern of change of the solar wind that drives the temporal pattern of change of the magnetosphere and its subsystems. This is demonstrated in Figure 10, where the hourly values of F_e1.2 are plotted (black dots) for 100 days in 2005; also plotted are the hourly values of the 62 h integral of F_e130 (red), the 97 h integral of v_sw (orange), the 38 h integral of -log(n_sw) (green), and the 138 h integral of the Kp index (blue). Note the similarities in the shapes of all of the colored curves representing solar wind variables and geomagnetic activity. All of these variables are carrying similar information about the time sequence of solar wind variations that drive magnetospheric and radiation-belt variations. A separate study (Borovsky, 2017a) examines the origins at the Sun of the intercorrelations between solar wind variables and why they carry the same information.

Figure 10

For 100 days in 2005 hourly values of the multispacecraft radiation-belt flux index F_e1.2 are plotted as the black dots. Plotted in color are the time integrals of F_e130 (red), v_sw (orange), −log(n_sw) (green), and Kp (blue) with their integration times optimized (cf. Table 2) to maximize the single-variable correlation with F_e1.2.

Another difficulty in interpreting cause and effect of the solar wind variables on the radiation belt is the fact that a single solar wind variable can influence the radiation belt via several of the physical mechanisms acting on the radiation belt and by influencing several of the magnetospheric control variables in different manners. This is illustrated in Figure 8 by the different pathways information flows from a solar wind variable to F_e1.2. There are confusing pathways acting with different signs of correlation with differing timescales. Note again that Figure 8 is a great oversimplification of the system.

6 Discussion

Two topics are discussed: (1) the interpretation of the optimal integration times found by the evolutionary algorithm and (2) why the Pearson correlation coefficients found in this study are never higher than r_corr ≈ 0.88.

6.1 Interpreting the Optimal Integration Times

The optimal times to time integrate variables into the past to produce the strongest correlations with F_e1.2 (at the present time) were calculated. It is of interest to know what the values of those optimal times can tell us.

Based on the reasonable assumption that it does not make sense to integrate a variable longer than its autocorrelation time, one might guess that the optimal integration times of the individual variables are related to the autocorrelation times of those individual variables: but this is not so. Comparing the autocorrelation times of specific variables in Table 1 of Borovsky and Yakymenko (2017a) with the optimal integration times in Table 2 of the present paper finds some integration times that are longer than autocorrelation times and some that are shorter. Note that a complication here is the fact that a single variable tends to have multiple autocorrelation times related to short-term variability in the time series and long-term variability in the time series.

One might also guess that the optimal integration times are related to the autocorrelation time of F_e1.2: this may be the case. The autocorrelation time of F_e1.2 is 108 h (cf. Table 1 of Borovsky & Yakymenko, 2017a): for the rank-1 to rank-11 variables in Table 2 (which all have |r_corr| > 0.5), the mean value and standard deviation of the optimal integration times is 97 ± 33 h, which is in the ballpark of the 108 h autocorrelation time of F_e1.2.

Related to the autocorrelation time of F_e1.2, the obtained integration times could represent typical storm durations when for F_e1.2 is increasing and could represent typical F_e1.2 decay times before storms. In Table 4, looking at the rank-1 to rank-11 variables, the mean and standard deviation of the optimal integration times for F_e1.2-increasing intervals is 72 ± 26 h and the mean and standard deviation of the optimal integration times for F_e1.2-decreasing intervals is 138 ± 34 h. The 72 h = 3 day integration time for F_e1.2-increasing intervals agrees with previous observations of the timescales for increases in the ~1 MeV fluxes at geosynchronous orbit during storms (e.g., Figure 3b of Borovsky & Denton, 2010a), and the 138 h ~ 6 day integration time for F_e1.2-decreasing intervals might be in agreement with previous observations of a exp(−t/6 day) exponential decay of the ~1 MeV fluxes at geosynchronous orbit prior to storms (cf. black curve in Figure 3 of Borovsky & Denton, 2009a). Sudden dropouts and sudden recoveries of the ~1 MeV flux at geosynchronous orbit with timescales of 6–10 h (cf. Figure 6 of Borovsky & Denton, 2009b) may be represented in the shorter optimal integration times of variables like P_ips and P_ram in Table 4.

A complicating factor in interpreting the optimal integration times is proxy effects between various variables (e.g., Figure 9). The multiple intercorrelations between the variables of the solar wind and of the magnetosphere involve multiple lag times; as variables interfere with each other in the correlations, various lag times become involved and these diverse lag times could alter the optimal integration times.

6.2 Why Are the Correlations Not Higher Than 88%?

In correlating time-integrated variables with the F_e1.2 index, correlation coefficients r_corr below 0.88 are found. Several factors could be working to prevent higher correlation coefficients from being obtained. Five inter-related factors are discussed.

1. It may be that variables describing key physical processes are missing from this study.

In section 5, key physical processes that are believed to affect the evolution of the electron radiation belt were discussed and variables that describe those processes were sought. It is possible that our understanding of how those processes are controlled is lacking. It is further possible that there are additional key physical processes that we have not yet identified.

2. There is noise in the measurements that prevent perfect correlations.

Noise in the F_e1.2 index was quantified by looking at the first-point drop of the autocorrelation function of F_e1.2 and comparing that drop with the drops in the next few points of the autocorrelation function (cf. Figure 2b of Borovsky & Yakymenko, 2017a): noise values of less than 10% were found. Other measured values of course also have error.

Two well-known sources of error in the solar wind-at-Earth data sets are (a) the fact that the solar wind that is measured by an upstream monitor is not the solar wind that hits the Earth (see point 4 below) and (b) there are errors in the propagation of solar wind measurements between the upstream monitor and the Earth (Bargatze et al., 2005; Weimer & King, 2008).

3. The physical processes driving the radiation belt may be strong, but the variables used in the correlations are not exactly the appropriate variables to measure those physical processes.

For example, F_e1.2 is not the global intensity of the electron radiation belt; it is the peak flux in the 1.2 MeV channel at geosynchronous orbit as measured by multiple satellites. Many things affect the value of F_e1.2, including (a) compressions and decompressions of the magnetosphere that shift the radiation-belt electrons radially and (b) the chance locations in local time of the multiple satellites used to construct F_e1.2.

A second example is the ion-plasma sheet pressure P_ips, which is only measured at geosynchronous orbit, and it is an average value determined by multiple satellites in varying local times. Maybe geosynchronous orbit is not the optimal location to measure P_ips to gauge its effect on the radiation belt. Further, the value of P_ips used here is a partial pressure constructed from proton measurements between 0 and 40 keV, whereas much of the ion-plasma sheet pressure at geosynchronous orbit is contained in protons with kinetic energies greater than 40 keV (cf. section 2.1 of Borovsky et al., 1998). Note also that the oxygen content of the ion plasma sheet is not accounted for in the moments analysis of the energy-per-charge electrostatic analyzers of the MPA instruments (Thomsen et al., 1999), producing systematic errors in P_ips during storms and during solar maximum.

As other examples, geomagnetic indices measure different aspects of magnetospheric activity (strengths of various current systems, strength of convection, plasma pressure, etc). A pertinent question is what are the aspects of magnetospheric activity that physically affect the evolution of the electron radiation belt? And do any of the available geomagnetic indices accurately describe those relevant aspects?

4. There are inaccuracies in solar wind variables because the solar wind that is measured by an upstream monitor is not the solar wind that hits the Earth.

Upstream monitors located at L1 have severe shortcomings caused by the combination that (a) the monitors are orbiting about the Sun-Earth line and (b) the solar wind streamlines are highly time variable. The solar wind flow at Earth is systematically aberrated by the Earth's motion about the Sun, and the direction of the plasma flow vector is highly variable in time. An analysis that examined (a) characteristic scale sizes of the spatial structure of the solar wind plasma, (b) the amplitudes of the variations of the flow vector, and (c) the spacecraft orbits at L1 found strong differences between the solar wind variations hitting the Earth and the solar wind variations measured by the upstream monitors (Borovsky, 2017b).

5. Physically, there are a lot of factors acting in the system.

The magnetosphere-ionosphere-thermosphere system driven by the solar wind is not a very simple system: multiple physical processes are acting that couple the various subsystems to each other. F_e1.2 depends simultaneously on how strong the driving has been, on how strong the seed-particle population has been, and on how strong loss processes have been. The system has hysteresis effects (Valdivia et al., 2013), feedback loops (Borovsky, 2014b), and preconditioning (Lavraud et al., 2006): the time histories of the system and the driving are important.

Further, there are high-Reynolds-number aspects to the magnetosphere which forces one to replace an exact picture with a probabilistic picture (Borovsky et al., 1997; Stepanova et al., 2011; Voros et al., 2005). In such a system energy may go into one path during one event and go into a different path during another event. The magnitude of a result from a driving event may be a matter of chance.

7 Summary

Correlations are performed with F_e1.2, the multisatellite 1.2 MeV electron-flux hourly index at geosynchronous orbit derived in Borovsky and Yakymenko (2017a). The 93,947 hourly values from the years 1995–2006 are used.

Thirty one different variables are correlated with F_e1.2 of the electron radiation belt: 14 of them are magnetospheric variables and 17 of them are solar wind variables. These variables include substorm-injected-electron fluxes, substorm-occurrence rates, ion-plasma sheet properties, estimates of the dayside reconnection rate, magnetospheric ULF indices, geomagnetic indices, and the diverse properties (speed, density, temperature, magnetic field, entropy, ram pressure, Mach number, clock angle, etc.) of the solar wind at Earth including the intensity of the electron strahl. All variables have 1 h time resolution, and the years 1995–2006 are used.

The correlations are optimized with an evolutionary algorithm that randomly adjusts coefficients, integration times, and time lags to find formulas with the highest correlations.

Correlations with F_e1.2 are found to be better with time-integrated variables rather than time-lagged variables, where the time integrations are into the past. This improvement is probably because (1) the physical processes that act on the radiation belt act in a time-integrated fashion (i.e., there is cumulative energization and there is cumulative loss) and (2) the time integration introduces averaging into the variables that reduces noise, with the noise being uncorrelated variance.

The maximum single-variable correlation with F_e1.2 is with the time integral of F_e130, the multispacecraft average of the 130 keV substorm-injected electron flux at geosynchronous orbit. To ensure that this high correlation cannot be explained by the 130 keV electrons being part of the radiation belt, this correlation is checked with the multispacecraft 60 keV electron flux F_e60: the correlation with the time integral of F_e60 is nearly as high as the time integral of F_e130. The correlation with F_e130 is probably high because F_e130 is (1) a measure of the seed electron population for the electron radiation belt and (2) a measure of the electrons that drive whistler mode chorus.

Times when F_e1.2 is increasing are separated from times when F_e1.2 is decreasing, and correlations with F_e1.2 are performed separately for the two groups. Some variables are prominent during F_e1.2-increasing times, and some are prominent during F_e1.2-decreasing times. Increasing-time-dominant variables may be playing stronger roles in the energization of the radiation belt, and decreasing-time variables may be playing stronger roles in the loss of the radiation belt. The highest correlation is for the time integral of F_e130 during F_e1.2-increasing times with r_corr = 0.798.

Combinations of time-integrated variables are correlated with F_e1.2. The combinations include (1) the set of historical variables that are well known for their correlation with the radiation belt, (2) an optimized set of magnetospheric variables, (3) an optimized set of solar wind variables, (4) a physics-motivated set of magnetospheric variables, and (5) a physics-motivated set of solar wind variables.

It is found that F_e1.2 correlations with magnetospheric variables are superior to F_e1.2 correlations with solar wind variables. It is argued that the solar wind controls magnetospheric phenomena and that magnetospheric phenomena control the evolution of the radiation belt. In that chain of causality the magnetospheric variables are closer to the description of the radiation-belt evolution than are solar wind variables.

A standout correlation with F_e1.2 is the sum of the time integrals of F_e130 plus log(P_ips) with r_corr = 0.820, where P_ips is a multisatellite average of the ion-plasma sheet pressure at geosynchronous orbit. The most significant third variable to add was the integral of F_e60, the 60 keV electron flux, yielding r_corr = 0.838. Adding a fourth variable to this three-variable combination improves the correlation (to r_corr = 0.854 with the addition of SYM-H), but this is almost not a statistically significant increase. Examination of this correlation indicates that F_e130 tends to describe the energization of the radiation-belt electrons and log(P_ips) describes rapid losses of the radiation-belt electrons.

The persistence of F_e1.2 (autoregression) is added to the time-integral correlations with F_e1.2 by adding a prior value of F_e1.2. It is found that if the prior value is recent, then the prior value (the autoregression term) dominates the correlation and if the prior value is not recent enough, then the time-integrated variables dominate the correlation.

The web of correlations in the solar wind /magnetosphere/radiation-belt system is examined. It is demonstrated that most of the variables of the solar wind and of the magnetosphere carry the same information about the time sequence of the solar wind that drives the time sequence of the magnetosphere. The difficulties of discerning cause and effect in the web of correlations were argued to be (1) because the multiple intercorrelations lead to confounding and suppression and (2) because a single variable can act on the radiation belt simultaneously through multiple physical mechanisms (i.e., it can act to control or describe the energization of the radiation-belt electrons and also act to control or describe the loss of the radiation-belt electrons).

The source of much of the intercorrelation in the solar wind/magnetosphere/radiation-belt system is the intercorrelation of solar wind variables that drive the system. The variations of solar wind variables are not independent of each other and are not random. The origin of the intercorrelations of solar wind variables is examined, and two processes are demonstrated to introduce systematic correlations between solar wind variables: (1) plasma-type switching caused by solar rotation and (2) the dynamic interaction of the solar wind plasma (specifically compression and rarefaction) as the solar wind advects from the Sun to the Earth.

Acknowledgments

The author thanks Tom Cayton, Mick Denton, Xiangrong Fu, Peter Gary, Mike Henderson Ruth Skoug, John Steinberg, Michelle Thomsen, and Kateryna Yakymenko for data and for helpful conversations. This work was supported at the Space Science Institute by NASA Heliophysics LWS TRT program via grants NNX14AN90G and NNX16AB75G, by the NSF GEM Program via award AGS-1502947, by the NSF Solar-Terrestrial Program via grant AGS-12GG13659, and by the NASA Heliophysics Guest Investigator Program via grant NNX14AC15G. The 1 h resolution indices are available upon request from Joe Borovsky at [email protected].