The "SafeSpace" Radial Diffusion Coefﬁcients Database: Dependencies and application to simulations

. Radial diffusion has been established as one of the most important mechanisms contributing to both the acceleration and loss of relativistic electrons in the outer radiation belt, as well as to the supply of particles to the inner radiation belt. In the framework of the SafeSpace project we have used 9 years (2011–2019) of multi-point magnetic and electric ﬁeld measurements from THEMIS A, D and E satellites to create a database of accurately calculated radial diffusion coefﬁcients (D LL ) spanning an L* range from 3 to 8. In this work we investigate the dependence of the D LL on the various solar wind parameters, geomagnetic 5 indices and coupling functions, and moreover, on the spatial parameters L* and Magnetic Local Time (MLT), during the solar cycle 24. The spatial distribution of the D LL reveals important MLT dependence rising from the various Ultra Low Frequency (ULF) wave generation mechanisms. Furthermore, we investigate via a superposed analysis, the dependence of the D LL on solar wind drivers. We show, for the ﬁrst time to our knowledge, that the Interplanetary Coronal Mass Ejections (ICME) driven disturbances accompanied by high solar wind pressure values combined with intense magnetospheric compression can produce 10 D BLL values comparable or even greater than the ones of D ELL . This feature cannot be captured by semi-empirical models and introduces a signiﬁcant energy dependence on the D LL . Finally, we show the advantages of the use of accurately calculated D LL by means of numerical of relativistic ﬂuxes several on the of here we show that it can similarly affect both D LL components. Another interesting feature is exhibited by the correlation between the D LL and solar wind dynamic pressure even


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The dynamics of the outer radiation belt are driven by a complex interplay between acceleration and loss mechanisms (Reeves et al., 2003;Reeves and Daglis, 2016;Daglis et al., 2019) leading to a broad energy range of energetic electrons (a few hundreds of keV to several MeV). Even though the relative contribution of each mechanism is still under debate, radial diffusion has been established as one of the most important ones since it can contribute to both energization (Jaynes et al., 2015;Li et al., 2016;Katsavrias et al., 2019a;Nasi et al., 2020) and loss of relativistic electrons (Morley et al., 2010;Turner et al., 2012; 20 Katsavrias et al., 2015Katsavrias et al., , 2019b. Radial diffusion due to drift-resonance is driven by Pc4-5 Ultra-Low Frequency (ULF) waves with frequencies between 1 and 22 mHz. ULF waves at these frequencies can violate the third adiabatic invariant L* of the energetic electrons allowing radial diffusion conserving the first two adiabatic invariants under the drift resonance condition ω = mω d , where ω is the wave frequency, m is the azimuthal wave mode number and ω d is the electron drift frequency (Elkington et al., 2003). Most often 25 radial transport is described as a stochastic process; the result of incoherent transport of particles by electromagnetic fields that vary irregularly on time scales of the drift period of radiation belt electrons (of the order of minutes). The radial diffusion coefficient, D LL , has been defined to represent the mean square change of L* for a large number of particles over time.
Currently there are two widely used formalisms in order to derive radial diffusion coefficients. Falthammar (1965) distinguished the contribution of single-mode fluctuations in Earth's magnetic field and induced electric fields (D M LL ) and per-30 turbations in convection electric fields (D E LL ) to derive a mathematical formulation for D LL . However he indicated that this formulation is valid for sub-relativistic particles, only. On the other hand, Fei et al. (2006) included the contributions from all azimuthal wave modes, thus including relativistic particles as well. Nevertheless, the latter authors, made the additional assumption that the magnetic field perturbations and the inductive electric field perturbations are independent, something that runs counter to basic physical concepts of electromagnetism. 35 Specifically, Fei et al. (2006) assumed radial diffusion coefficients as the sum of the effects of perturbations in the azimuthal electric field and the parallel magnetic field: These two components of the radial diffusion coefficients are given by: where µ is the first adiabatic invariant, L is the Roederer's L*, q is the charge of the diffused electrons, γ is the Lorentz D LL without the limitations of in-situ measurements. Nevertheless, it is also obvious (see also table 1) that the use of a single input parameter is an over-simplification for a complex process such as the radial diffusion of electrons. Moreover, Kp is a global geomagnetic index, which is a proxy for the global changes in the geomagnetic field (Mayaud , 1980). On the other hand, two of the most important (external) sources for ULF waves are a) solar wind pressure pulses and b) Kelvin-Helmholtz instabilities powered by the increased solar wind speed (Claudepierre et al., 2008). Since the Kp index does not 55 present significant correlation with either of these two solar wind parameters, it cannot account for the mechanism of radial diffusion that enhance or deplete the electron population in the outer radiation belt.
Moreover, the observed D LL have been shown to be highly event-specific (Jaynes et al., 2018) and physics-based models, such as the Versatile Electron Radiation Belt, cannot simulate the dynamics of the outer radiation belt observed during every storm using these empirically estimated coefficients (Drozdov et al., 2021). Moreover, several case studies have demonstrated 60 deviations of the event-specific diffusion coefficients from the Kp-parameterized models. The recent study of Liu et al. (2018) suggests that the difference between the various models is negligible for low levels of geomagnetic activity at an equatorial distance of L-shell = 7.5 R E but can be orders of magnitude different at high levels of geomagnetic activity. At the same extent, Moreover, the magnitude of mis-estimation varied throughout the event and, at times, the difference between empirically modelled values and event-specific diffusion coefficients was multiple orders of magnitude.
In this work we present a new database of radial diffusion coefficients, which has been developed in the framework of SafeSpace project funded by Horizon 2020. The SafeSpace project aims at advancing space weather nowcasting and forecasting capabilities and, consequently, at contributing to the safety of space assets through the transition of powerful tools from 70 research to operations. To that end, a database of accurately calculated radial diffusion coefficients coupled with solar wind and geomagnetic parameters, as well as the accompanied analysis, is of outmost importance, not only for the accurate quantification of radial diffusion but also, for any future efforts to develop accurate models for nowcasting/forecasting the D LL . The rest of this paper is organized as follows: section 2 describes the datasets used as input in the D LL database as well as the roadmap towards its creation, section 3 reports statistics which are important for future modelling efforts and, finally, section 4 presents examples of the importance of the use of event-specific D LL in radiation belt simulations.

Data and methods
We use 4-sec resolution measurements of the magnetic field vector from the THEMIS A, D and E fluxgate magnetometers (Auster et al., 2008) as well as electric field measurements from the EFI instrument (Bonnell et al., 2008) (Bourdarie and O'Brien, 2009) and the Olson-Pfitzer 1977 (Olson andPfitzer , 1977) external magnetic field model.
For the spectral analysis of the electric and magnetic field measurements we make use of the Continuous Wavelet Transform 85 (CWT-see also Torrence and Compo (1998)) with the Morlet wavelet as the wavelet basis function (Morlet et al., 1983). Figure 1 shows the steps followed in order to create the D LL database, from the collection of the input data to the final scientific products. In detail, THEMIS magnetic and electric field data were pre-processed by transforming them into a Mean Field Aligned (MFA) coordinate system, similar to Balasis et al. (2013). Furthermore, the transformed time-series were de-90 trended using a 20-min moving average, which is quite similar with a high-pass filtering with cutoff frequency at ≈0.83 mHz.

D LL database
Then, the wavelet transform is used to calculate the Power Spectral Density (PSD) in the 2 -25 mHz frequency range for Pc4-5 waves, which in turn is used for the calculation of the radial diffusion coefficients. Finally, the calculated D LL along with the PSD and the weighted averaged power, as a function of time, L* and Magnetic Local Time (MLT), are coupled with OMNIWeb data and stored in daily CDF files.

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For the calculation of the D LL , the Fei et al. (2006) approach is followed, which considers the compressional component of the magnetic field (parallel to the background magnetic field) and the toroidal component of electric field (perpendicular to the background magnetic field and in the east-west direction) for the calculation of the D B LL and D E LL , respectively, following the equations 2 and 3. As mentioned earlier, the wave power included in these equations, corresponds to the power at a specific drift frequency for all m values, which essentially means that particles are radially transported via stochastic acceleration with 100 various frequency waves (main frequency and harmonics). Nevertheless, to calculate the power at various m values, one would need at least 2m observations simultaneous in time, which is not trivial. To address this issue, it is often assumed that power at high m values is consistently lower than power at m = 1 and subsequently, that all power is contained in the lowest m = 1 wave mode of ULF waves driving diffusion (Ozeke et al., 2014). Nevertheless, this assumption denounces the very concept of stochastic acceleration restricting the process to a resonant interaction. More importantly, such an assumption can lead to 105 underestimation of the radial diffusion coefficient, since higher m values are shown to be often significant (e.g. m=2 up to m=5 at recovery phase of storms (see also Sarris et al. (2013)). To address this issue, we have opted to use, in the place of power at a specific frequency, the weighted averaged power over the whole frequency range under study (in our case Pc4 and Pc5 frequency range) calculated as follows: where Cdelta is a smoothing factor equal to 0.76 and dj = − log2 f min f min Scalemax (see also Torrence and Compo (1998)). Finally, as already mentioned, important differences can exist between the two approaches and it is indicated that the approach followed by Fei et al. (2006) can lead to an underestimation of the total D LL by a factor of 2 (Lejosne et al., 2019).
Nevertheless, this approach is the more widely used and it has been shown that this discrepancy is comparatively minor relative to the large variability in the observed values (Sandhu et al., 2021).  geomagnetic indices SYM-H, AE and Kp, the parameter (Akasofu, 1981), the southward solar wind field (here we show the exponential of Bs), the Half-Wave Rectifier (Burton et al., 1975) and Newell's function (Newell et al., 2007). Furthermore, the CC between solar wind speed and D LL is at ≈0.4 and ≈0.5 for the electric and magnetic component, respectively, but both at the 4.5-6.5 L* range. The importance of magnetopause instabilities-induced by the increased solar wind velocity-has been well established before (Bentley et al., 2018) but here we show that it can similarly affect both D LL components. Another interesting feature is exhibited by the correlation between the D LL and solar wind dynamic pressure even  4). This is expected since these parameters are known to be well correlated with substorm activity (Katsavrias et al., 2021).  Figure 4 shows the spatial distribution of D B LL and D E LL , as well as their ratio for three levels of geomagnetic activity: Kp< 3 (left column panels), 3 <Kp< 5 (middle column panels) and Kp> 5 (right column panels). D LL values with 1-min resolution are binned in L* and MLT with dL*=0.1 and dMLT=1 hour and the logarithm of the mean value of each bin is color-coded.

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As shown, there are significant differences at the distribution of the two components. During quiet times, the D E LL (top left panel) exceeds the value of 10 outside the geosynchronous orbit and is approximately equal to 1 at the 4.5-6 L* range, while there is a significant MLT asymmetry. More specifically, D E LL appears more intense at the dawn-noon and dusk-midnight sectors. As we move to higher geomagnetic activity levels (3 <Kp< 5-top middle panel), D E LL intensifies and, in addition, this asymmetry becomes stronger at L*> 5. During intense geomagnetic activity levels (top right panel), D E LL values range 160 between 10 and 100 at L*> 5 and they reach approximately the value of 1 even down to at L*= 3.5, while the MLT asymmetry becomes quite noisy.
On the other hand, the D B LL distribution exhibits a very different behaviour. During quiet times, the D B LL (middle left panel) values reach 1 at L*> 7 and only at the dayside sector (approximately in the 9-15 MLT range). As we move to higher geomagnetic activity levels, the D B LL exceeds the value of 10 even inside the geosynchronous orbit L*< 6. Furthermore, the MLT 165 asymmetry becomes more intense and wide (approximately in the 5-18 MLT range during Kp> 5 periods). It is worth mentioning that, during such intense geomagnetic activity levels, the D B LL becomes comparable with the D E LL -or even higher-as shown in the bottom right panel. The aforementioned feature of the spatial distribution of the D B LL component is in agreement with the correlation results shown in figure 2 and indicates that the magnetic component is linked with ULF waves generated through solar wind pressure pulses (Kepko et al., 2002). On the other hand, the observed asymmetry in the electric component 170 indicates that D E LL is not only linked with solar wind speed but with internal mechanisms such as substorm activity, something that is also in agreement with the results of figure 2. Moreover, we note a remarkable agreement of the D E LL MLT distribution (top row panels of figure 4) with Nosé et al. (1998), who stated that substorms generate azimuthal ULF fluctuations at the nightside which peak at 1-2 MLT.
All of the above suggest that, even though the radial diffusion coefficient is calculated with the drift-averaging assumption, 175 the MLT dependence of the D LL accounts for the coupling of external and internal ULF generation mechanisms and may be quite important for future modelling efforts. Finally, we emphasize the fact that our results on the MLT asymmetry are in good agreement with Sandhu et al. (2021) who used Van Allen probes data (different magnetic latitude) to infer the radial diffusion coefficients.

ICME vs SIR driven geospace disturbances 180
The role of solar wind drivers (e.g. Interplanetary Coronal Mass Ejections-ICMEs and Stream Interaction Regions-SIRs) has been suggested to play an important role to the generation of ULF waves and, consquently, to the evolution of radial diffusion coefficients (Simms et al., 2010;Kilpua et al., 2015). In order to investigate the dependence of the D E LL and D B LL on the solar wind driver we have selected 25 ICME-and 46 SIR-driven geospace disturbances (71 events in total) in the 2011-2019 time period, following the criteria of Katsavrias et al. (2019b). More specifically, we have chosen events that include a single driver 185 and have no pre-conditioning in solar wind parameters for at least 12 hours before the arrival of the ICME or SIR. Since we have applied no criteria depending on the Dst index (non-storm events are also included), we have used as zero-epoch time (t 0 ) the time of the maximum compression of the magnetopause (Lmp min ) as it is given by the empirical model of Shue et al. (1998). Figure 5 shows the results of the superposed epoch analysis. As shown, both groups exhibit several differences. During 190 ICME driven disturbances the maximum increase in D E LL takes place on t 0 at all L*> 4 and reaches a median value of 1000 at L*> 5, while significant activity reaches down to L≈ 3.5 up to 12 hours. After these 12 hours and the activity is still significant at L*> 5 and lasts up to 96 hours (4 days). During SIR driven disturbances, the D E LL exhibits a quite similar trend (it lasts up to 4 days after t 0 ) but both its magnitude and the penetration to inner L* are lower compared to the ICME driven disturbances.
On the other hand, the D B LL exhibits much more pronounced differences. During ICME driven disturbances the maximum increase in D B LL takes place on t 0 and the penetration of the activity reaches down to L*≈4. The overall enhancement occurs on -8< t 0 <12 hours. During SIR driven disturbances, the D B LL hardly reaches L*≈4 and the maximum increase reaches a value of 10. Nevertheless, the overall activity lasts up to approximately 30 hours after t 0 . Furthermore, the enhancement as well as the penetration of D B LL to low L*, is very well correlated with the enhancement in both solar wind dynamic pressure and Kp index and, consequently, is in agreement with the findings of figure 2. This result is also in agreement with Simms et al. 200 (2010) who indicated that ground Pc5 power was greater during CME storms, especially during the main and recovery phase.
One step further, Kalliokoski et al. (2020) studied 37 ICME-driven sheath regions in the Van Allen Probes era and linked the increased Pc5-ULF activity at GEO with the increased pressure during the sheath.
Finally, a very important feature is exhibited by the ratio of the electric over the magnetic component. As shown in the bottom panels of figure 5, the electric component is mostly dominant-up to two orders of magnitude compared with the magnetic 205 component. This feature changes dramatically during ICME driven disturbances and around the maximum compression of the magnetetopause (t 0 ) where the D B LL becomes equally (or even more) important than D E LL at all L*. Furthermore, at L*> 6, the D B LL is comparable to the D E LL up to approximately 12 hours after t 0 . The relative strength of the two D LL has been discussed before by Olifer et al. (2019) who studied the components ratio during the St. Patricks event of 2015. These authors indicated that during the main phase of this ICME driven storm, the magnetic component exceeded the electric by approximately one 210 order of magnitude, something that semi-empirical models cannot reproduce. Here we replicate this result using a statistical sample of 25 ICME driven disturbances independent of the magnitude of Dst index. Also note that this feature present during SIR disturbances as well. Nevertheless, it is less pronounced both in magnitude and L*. Finally, we must emphasize the fact that this feature introduces a significant energy dependence on the D LL , since the magnetic component is energy dependent, that may be of great importance to radiation belt simulations.  (Li et al., 2016) but also to further acceleration to ultra-relativistic energies (Jaynes et al., 2018).
We must emphasize the fact that the aforementioned comparison is performed between the calculated µ-dependent D LL from the SafeSpace database and the Boscher et al. (2018) model, only. This is done in accordance to the results discussed 260 in the previous section (see also figure 6) where we showed that the Boscher model exhibited the best comparison with the case-specific diffusion coefficients.

Conclusions
In the framework of the SafeSpace project we have used 9 years (2011 -2019) of multi-point magnetic and electric field measurements from THEMIS A, D and E satellites to create a database of accurately calculated radial diffusion coefficients. 265 We have further exploited this database in order to investigate the dependence of these calculated D LL to several solar wind and geomagnetic parameters, to solar wind drivers (ICMEs and SIRs), as well as to spatial parameters (MLT and L*).
The results of this analysis can be summarized as follows: 1. Both D LL components (magnetic and electric) exhibit good correlation with Kp and AE index. Furthermore, D E LL exhibits good correlation with solar wind speed, while D B LL exhibits good correlation with both solar wind speed and 270 pressure with zero time-lag.
2. MLT plays a significant role in the spatial distribution of both the components of D LL which exhibit asymmetries due to the coupling of external and internal ULF wave generation mechanisms.
3. The superposed epoch analysis reveals significant differences between the evolution of D LL during ICME-and SIRdriven disturbances. During the former, the high solar wind pressure values combined with the intense magnetospheric 275 compression produce D B LL values comparable or even greater than the ones of D E LL . This feature cannot be captured by semi-empirical models and introduces a significant energy dependence on the D LL .
Furthermore, the comparison of the semi-empirical models with the D LL from the SafeSpace database reveals significant deviations depending on the level of geomagnetic activity and the drift shell. Generally, all models underestimate the D LL during quiet times and at low L* values, while they overestimate the D LL during high levels of geomagnetic activity and 280 modules/document/?course=PHYS120.
Author contributions. CK drafted and wrote the paper with participation of all coauthors. CP contributed in the software development, AN in the development of the database and SAG in the statistical study. IAD and MG were consulted regarding the interpretation of the results.
ND, AB and SB contributed to the radiation belt simulations with the Salammbô model and were also consulted regarding the interpretation of the results .