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.
Variable | Definition | Source and reference |
---|---|---|
F_{e1.2} | Radiation-belt 1.2 MeV electron flux index | SOPA, Ref. 1 |
n_{erb} | Radiation-belt number density | SOPA, Ref. 2 |
T_{erb} | Radiation-belt temperature (hardness) | SOPA, Ref. 2 |
F_{e130} | Substorm-injected 130 keV electron flux index | SOPA, Ref. 1 |
F_{e60} | Substorm-injected 60 keV electron flux index | SOPA, Sect. 2 |
S_{rate} | Rate of substorm occurrence | SuperMag, Ref. 1 |
n_{ips} | Number density of ion plasma sheet | MPA, Sect. 2 |
T_{ips} | Temperature of ion plasma sheet | MPA, Sect. 2 |
P_{ips} | Pressure of the ion plasma sheet n_{ips}T_{ips} | 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 |
S_{grd} | ULF index, ground based | Ref. 5 |
S_{geo} | ULF index, geosynchronous based | Ref. 5 |
v_{sw} | Solar wind speed | OMNI2, Ref. 3 |
n_{sw} | Solar wind number density | OMNI2, Ref. 3 |
B_{mag} | Solar wind magnetic field strength | OMNI2, Ref. 3 |
T_{p} | Solar wind proton temperature | OMNI2, Ref. 3 |
sin^{2}(θ_{clock}/2) | Solar wind magnetic clock angle function | OMNI2, Ref. 6 |
S_{p} | Proton specific entropy of solar wind | OMNI2, Ref. 7 |
v_{A} | Alfven speed in the solar wind plasma | OMNI2, Ref. 3 |
MA | Alfven Mach number of solar wind | OMNI2, Ref. 3 |
P_{ram} | Dynamic pressure of solar wind | OMNI2, Ref. 3 |
R_{quick} | Dayside reconnection function | OMNI2, Ref. 8 |
R_{qo} | Logarithm of R_{quick} without clock-angle term | OMNI2, Ref. 9 |
I_{272} | Intensity of 272 eV strahl in solar wind | ACE, Sect. 2 |
d_{mp} | Magnetopause standoff distance | No values |
A_{ips} | Temperature anisotropy T_{⊥i}/T_{||i} of ion plasma sheet | No values |
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_{ips}T_{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_{y}^{2} + B_{z}^{2})^{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_{p}n_{sw})^{1/2}, with m_{p} being the proton mass and B_{mag} = (B_{x}^{2} + B_{y}^{2} + B_{z}^{2})^{1/2}. The Alfven Mach number of the solar wind is MA = v_{sw}/v_{A}. The dynamic pressure is P_{ram} = 0.5m_{p}n_{sw}v_{sw}^{2}.
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
Finally in Table 1 an index I_{272} 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 f_{272} at 272 eV are utilized. The hourly average of log_{10}(f_{272}) in the two angular bins in the direction parallel to the measured magnetic field direction B and the average of log_{10}(f_{272}) 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
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
Variable | r_{corr} for integral | Integration time | Rank | r_{corr} for lagged variable |
---|---|---|---|---|
F_{e130} | 0.736 | 62 h | 1a | 0.617 |
F_{e60} | 0.699 | 85 h | 1b | 0.531 |
v_{sw} | 0.647 | 98 h | 2 | 0.602 |
log(v_{sw}/n_{sw}) | 0.627 | 51 h | 3 | 0.568 |
log(S_{p}) | 0.620 | 91 h | 4 | 0.502 |
log(n_{sw}) | −0.579 | 38 h | 5 | −0.518 |
Kp | 0.562 | 138 h | 6 | 0.413 |
T_{p} | 0.554 | 121 h | 7 | 0.377 |
AE | 0.544 | 140 h | 8 | 0.327 |
S_{grd} | 0.535 | 123 h | 9 | 0.344 |
S_{rate} | 0.509 | 118 h | 10 | 0.415 |
log(R_{qo}) | 0.501 | 96 h | 11 | 0.281 |
R_{quick} | 0.493 | 152 h | 12 | 0.267 |
log(B_{mag}/n_{sw}) | 0.466 | 81 h | 13 | 0.389 |
T_{p}/v_{sw} | 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 |
I_{272} | −0.358 | 13 h | 19 | −0.168 |
log(P_{ips}) | −0.356 | 6 h | 20 | −0.347 |
T_{ips} | −0.322 | 4 h | 21 | −0.336 |
v_{A} | 0.309 | 133 h | 22 | 0.232 |
B_{mag} | −0.306 | 10 h | 23 | −0.291 |
log(P_{ram}) | −0.286 | 16 h | 24 | −0.265 |
log(n_{ips}) | −0.264 | 9 h | 25 | −0.247 |
sin^{2}(θ_{clock}/2) | 0.252 | 86 h | 26 | 0.103 |
S_{geo} | 0.251 | 138 h | 27 | 0.207 |
B_{z} | −0.23 | 116 h | 28 | −0.093 |
B_{south} | 0.143 | 192 h | 29 | 0.086 |
log(MA) | 0.101 | 4 h | 30 | 0.101 |
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.
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 I_{272} 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_{p}n_{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_{sw}v_{sw}^{2} 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 sin^{2}(θ_{clock}/2) of the solar wind magnetic field ranks 26, with ∫ sin^{2}(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 sin^{2}(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 sin^{2}(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 I_{272}, 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).
Integrated variable | r_{corr} with F_{e1.2} | Integration time with F_{e1.2} | r_{corr} with log(n_{hi}) | Integration time with log(n_{hi}) | r_{corr} with T_{hi} | Integration time with T_{hi} |
---|---|---|---|---|---|---|
F_{e130} | 0.736 | 62 h | 0.675 | 12 h | 0.552 | 99 h |
F_{e60} | 0.699 | 85 h | 0.594 | 35 h | 0.506 | 117 h |
v_{sw} | 0.647 | 98 h | 0.537 | 39 h | 0.522 | 127 h |
log(v_{sw}/n_{sw}) | 0.627 | 51 h | 0.608 | 11 h | 0.532 | 80 h |
log(S_{p}) | 0.620 | 91 h | 0.538 | 35 h | 0.515 | 116 h |
log(n_{sw}) | −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 |
T_{p} | 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 |
S_{grd} | 0.535 | 123 h | 0.488 | 65 h | 0.366 | 147 h |
S_{rate} | 0.509 | 118 h | 0.459 | 63 h | 0.349 | 144 h |
log(R_{qo}) | 0.501 | 96 h | 0.495 | 76 h | 0.312 | 159 h |
R_{quick} | 0.493 | 152 h | 0.445 | 87 h | 0.332 | 173 h |
Log(B_{mag}/n_{sw}) | 0.466 | 81 h | 0.492 | 26 h | 0.347 | 103 h |
T_{p}/v_{sw} | 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 |
I_{272} | −0.358 | 1 h | −0.224 | 24 h | −0.404 | 24 h |
P_{ips} | −0.356 | 6 h | 0.270 | 89 h | −0.479 | 12 h |
T_{ips} | −0.322 | 4 h | 0.284 | 61 h | −0.517 | 8 h |
v_{A} | 0.309 | 133 h | 0.337 | 71 h | 0.206 | 161 h |
B_{mag} | −0.306 | 10 h | −0.244 | 1 h | −0.378 | 19 h |
log(P_{ram}) | −0.286 | 16 h | −0.337 | 3 h | −0.330 | 23 h |
log(n_{ips}) | −0.264 | 9 h | −0.202 | 1 h | −0.316 | 17 h |
sin^{2}(θ_{clock}/2) | 0.252 | 86 h | 0.202 | 50 h | 0.168 | 127 h |
S_{geo} | 0.251 | 138 h | −0.322 | 81 h | −0.237 | 14 h |
B_{z} | −0.228 | 116 h | −0.191 | 68 h | −0.155 | 150 h |
B_{south} | 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 I_{272}; the ion-plasma sheet temperature T_{ips}; and the clock-angle function sin^{2}(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.
Variable | Rank | r_{corr} for all intervals | r_{corr} for intervals of increasing F_{e1.2} | r_{corr} for intervals of decreasing F_{e1.2} | Dominance |
---|---|---|---|---|---|
F_{e130} | 1a | 0.736 (62 h) | 0.798 (49 h) | 0.645 (107 h) | F_{e1.2} Increasing |
F_{e60} | 1b | 0.699 (85 h) | 0.761 (56 h) | 0.594 (129 h) | F_{e1.2} Increasing |
v_{sw} | 2 | 0.647 (98 h) | 0.641 (62 h) | 0.567 (139 h) | F_{e1.2} Increasing |
log(v_{sw}/n_{sw}) | 3 | 0.627 (51 h) | 0.593 (34 h) | 0.552 (98 h) | |
log(S_{p}) | 4 | 0.620 (91 h) | 0.584 (77 h) | 0.552 (128 h) | |
log(n_{sw}) | 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) | F_{e1.2} Increasing |
T_{p} | 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) | F_{e1.2} Increasing |
S_{grd} | 9 | 0.535 (123 h) | 0.510 (96 h) | 0.437 (149 h) | |
S_{rate} | 10 | 0.509 (118 h) | 0.558 (78 h) | 0.396 (153 h) | F_{e1.2} Increasing |
log(R_{qo}) | 11 | 0.501 (96 h) | 0.529 (92 h) | 0.370 (164 h) | F_{e1.2} Increasing |
R_{quick} | 12 | 0.493 (152 h) | 0.507 (106 h) | 0.378 (188 h) | F_{e1.2} Increasing |
log(B_{mag}/n_{sw}) | 13 | 0.466 (81 h) | 0.435 (60 h) | 0.365 (115 h) | |
T_{p}/v_{sw} | 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) | F_{e1.2} 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) | |
I_{272} | 19 | −0.358 (13 h) | −0.397 (9 h) | −0.327 (22 h) | F_{e1.2} Increasing |
log(P_{ips}) | 20 | −0.356 (6 h) | −0.435 (17 h) | −0.488 (7 h) | F_{e1.2} Decreasing |
T_{ips} | 21 | −0.322 (4 h) | −0.476 (4 h) | −0.345 (33 h) | F_{e1.2} Increasing |
v_{A} | 22 | 0.309 (133 h) | 0.272 (94 h) | 0.229 (163 h) | |
B_{mag} | 23 | −0.306 (10 h) | −0.243 (10 h) | −0.374 (24 h) | F_{e1.2} Decreasing |
log(P_{ram}) | 24 | −0.286 (16 h) | −0.158 (12 h) | −0.338 (28 h) | F_{e1.2} Decreasing |
log(n_{ips}) | 25 | −0.264 (9 h) | −0.316 (31 h) | −0.441 (7 h) | F_{e1.2} Decreasing |
sin^{2}(θ_{clock}/2) | 26 | 0.252 (86 h) | 0.277 (87 h) | 0.204 (136 h) | F_{e1.2} Increasing |
S_{geo} | 27 | 0.251 (138 h) | 0.198 (106 h) | −0.262 (15 h) | F_{e1.2} Decreasing |
B_{z} | 28 | −0.228 (116 h) | −0.233 (97 h) | −0.181 (142 h) | |
B_{south} | 29 | 0.143 (192 h) | 0.122 (115 h) | −0.263 (31 h) | F_{e1.2} Decreasing |
log(MA) | 30 | 0.101 (4 h) | 0.110 (15 h) | 0.192 (9 h) |
Term | Coefficient | Integration time | Added correlation |
---|---|---|---|
v_{sw} | 0.294 | 116 h | 0.009 |
log_{10}n_{sw} | −0.165 | 14 h | 0.006 |
S_{grd} | −0.226 | 172 h | 0.006 |
Kp | −0.496 | 31 h | 0.017 |
F_{e130} | 0.912 | 30 h | 0.092 |
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
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 log_{10}(n_{sw}), and the blue curve (left axis) is −log_{10}(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.
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
No. of inputs | v_{sw} | log_{10}n_{sw} | S_{grd} | Kp | F_{e130} | r_{corr} | |
---|---|---|---|---|---|---|---|
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}.
∫^{98} v_{sw} dt | ∫^{38} log(n_{sw}) dt | ∫^{123} S_{grd} dt | ∫^{138} Kp dt | ∫^{62} F_{e130} dt | |
---|---|---|---|---|---|
∫^{98} v_{sw} dt | 1 | −0.672 | 0.719 | 0.806 | 0.766 |
∫^{38} log(n_{sw}) dt | −0.672 | 1 | −0.373 | −0.458 | −0.567 |
∫^{123} S_{grd} dt | 0.719 | −0.373 | 1 | 0.887 | 0.718 |
∫^{138} Kp dt | 0.806 | −0.458 | 0.887 | 1 | 0.777 |
∫^{62} F_{e130} 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
If the five-variable correlation of expression 6 is constructed with time-lagged variables rather than time-integrated variables, the optimized result is
4.3 Creating an Optimized Correlation Formula With F_{e1.2}
In Table 2 the strongest single-variable correlation with F_{e1.2} is
Added variable | Addition to r_{corr} for the addition of the variable to ∫ F_{e130} dt | Addition to r_{corr} for the addition of the variable to ∫ F_{e130} dt + ∫ log(P_{ips}) dt |
---|---|---|
F_{e60} | 0.001 | 0.017 |
v_{sw} | 0.011 | 0.001 |
log(v_{sw}/n_{sw}) | 0.025 | 0.001 |
log(S_{p}) | 0.024 | 0.005 |
log(n_{sw}) | 0.028 | 0.005 |
Kp | 0.024 | 0.002 |
T_{p} | 0.017 | 0.005 |
AE | 0.017 | 0.001 |
S_{grd} | 0.010 | 0.004 |
S_{rate} | 0.002 | 0.005 |
log(R_{qo}) | 0.011 | 0.002 |
R_{quick} | 0.021 | 0.000 |
log(B_{mag}/n_{sw}) | 0.000 | 0.001 |
T_{p}/v_{sw} | 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 |
I_{272} | 0.023 | 0.000 |
log(P_{ips}) | 0.084 | |
T_{ips} | 0.057 | 0.010 |
v_{A} | 0.012 | 0.002 |
B_{mag} | 0.054 | 0.005 |
log(P_{ram}) | 0.028 | 0.004 |
log(n_{ips}) | 0.055 | 0.006 |
sin^{2}(θ_{clock}/2) | 0.002 | 0.007 |
S_{geo} | 0.029 | 0.000 |
B_{z} | 0.000 | 0.003 |
B_{south} | 0.022 | 0.001 |
log(MA) | 0.013 | 0.003 |
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
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.
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 sin^{2}(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 I_{272} 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
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:
For a reference, expression 13 is optimized for t_{0} = 24 h and the expression obtained for F_{e1.2}(t) is
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.
- 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).
- 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).
- 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.
- 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).
- 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).
- 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_{B}T/B^{2} 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_{ips}T_{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).
- 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).
- 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.
Physical process affecting F_{e1.2} | Expected effect on F_{e1.2} | Magnetospheric controlling variable | Expected sign of correlation with F_{e1.2} | Solar wind controlling variable or measuring variable | Expected sign of correlation with F_{e1.2} | |
---|---|---|---|---|---|---|
1 | Substorm-injected seed population | Increase | F_{e130} | + | v_{sw}, R_{quick} | +, + |
2 | Chorus driving by substorm injected electrons | Increase | F_{e130} | + | v_{sw}, R_{quick} | +, + |
3 | ULF radial diffusion distributing radiation-belt electrons | Increase or Decrease | S_{geo} | ? | R_{quick}, P_{ram} | ?, ? |
4 | Radiation-belt losses via magnetopause shadowing | Decrease | d_{mp} | + | P_{ram}, R_{quick} | −, − |
5 | ULF radial diffusion enhancing loss to magnetopause | Decrease | S_{geo} | − | R_{quick}, P_{ram} | −, − |
6 | Precipitation losses caused by EMIC and magnetosonic waves driven by the ion plasma sheet | Decrease | P_{ips}, Kp | −, + | B_{mag}, P_{ram} | −, − |
7 | Decay of radiation belts from hiss scattering in quiet-time plasmasphere | Decrease | Kp | + | R_{quick}, v_{sw} | +, + |
8 | Dst effect | Decrease or Increase | SYM-H | + | R_{quick}, B_{south} | −, − |
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
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
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
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.
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.
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%?
- It may be that variables describing key physical processes are missing from this study.
- 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.
- 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.
- 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.
- 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].