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 radial diffusion coefficients
(

The outer radiation belt consists of electrons at energies from a few hundred kiloelectronvolts (keV) to several megaelectronvolts (MeV)

Ultra-low-frequency (ULF) waves in the Pc4–5 band (1–25 mHz) can violate the third adiabatic invariant

Currently there are two widely used formalisms in order to derive radial diffusion coefficients.

Specifically,

These two components of the radial diffusion coefficients are given by

It is clear, from the aforementioned formulation, that in order to have accurate calculations of the radial diffusion coefficients, we need accurate magnetic and electric field measurements, which, of course, are not always available. To that end, efforts have been devoted to provide empirical relationships of

Widely used semi-empirical models for the estimation and prediction of the radial diffusion coefficients, their mathematical formulation, trained datasets and limitations.

In addition, the observed

In this work we present a new database of ULF power spectral density (PSD) and the derived radial diffusion coefficients, which has been developed in the framework of the “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 research to operations. To that end, a database of radial diffusion coefficients derived from in situ magnetic and electric field measurements, coupled with solar wind and geomagnetic parameters, as well as the accompanied analysis, is of outmost importance, not only for statistical purposes, but also for any future efforts to develop accurate models for nowcasting and forecasting the

The radial diffusion coefficients were calculated directly from
in situ measurements using the approach based on the

Figure

Work logic towards the creation of the SafeSpace radial diffusion coefficient database.

The next step was to estimate the power spectral density (PSD) of the
waves in the 2–25 mHz frequency range for the two time series,
which corresponds to the drift periods of near-equatorial mirroring
electrons roughly in the 0.4–13 MeV. For the spectral analysis of the
electric and magnetic field measurements we made use of the continuous
wavelet transform

Finally, using the estimated PSDs, the

The PSDs of both the toroidal electric and the compressional magnetic
field as a function of time,

The wave power included in Eqs. (

Even though we have followed a well-established methodology in order
to calculate – as accurately as possible – the ULF PSD and the corresponding

As already discussed, important differences can exist between the two approaches by

In addition, the theoretical approach of

Finally, we have to mention the uncertainties introduced by the use of the Olson–Pfitzer quiet model for the estimation of the magnetic ephemeris data, as well as the uncertainties of the instruments used

Equations (

In order to discuss the possible uncertainties generated by the limited azimuthal coverage, we calculate a 1 min resolution proxy of the

Logarithms of the mean

As shown, there are obvious differences between the two
components. During quiet times, the

The aforementioned feature of the

Pearson correlation coefficients between the natural
logarithms of the hourly mean values of

Our

Figure

Generally, the CCs of the magnetic component exhibit greater values
than the ones of the electric component with maxima at

Furthermore, the CC between solar wind speed and

It is worth mentioning that the only parameter which exhibits an
anti-correlation with the

Figure

Pearson correlation coefficients between the natural
logarithms of the hourly mean values of

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 in the generation of ULF waves and, consequently, in the evolution of radial diffusion coefficients

Figure

Superposed epoch analysis of the 25 ICME-driven (left column panels) and 46 SIR-driven (right column panels) geospace disturbances. Top to bottom: median (black line), 25th and 75th quantiles (red lines) of the magnetopause location predicted by the

Finally, a very important feature is exhibited by the ratio of the
electric over the magnetic component, which generally spans the
0.1–100 range. As shown in the bottom panels of Fig.

We must emphasize the fact that this feature introduces a significant energy dependence on the

As already discussed in the Introduction, even though the semi-empirical Kp-parameterized models have the advantage of providing estimations and predictions of the

We note that the comparison of radial diffusion coefficients among multiple methods is anything but a straightforward process since the details of each method are different (e.g. different datasets, different time periods in a solar cycle or different theoretical approaches). In addition, here we attempt a comparison of a dataset inferred from in situ measurements with the estimations of semi-empirical models, which are by definition two different things. Nevertheless, we attempt a comparison in order to show a clearer picture of the statistical behaviour of each model compared to our

Figure

Comparison of the SafeSpace

It is worth mentioning that the

Concerning the

To the same extent, the

The

Finally, we have to consider the uncertainties in the SafeSpace-calculated

In the previous section, we presented an extended comparison of the various semi-empirical models with the calculated

Simulation of the outer radiation belt dynamics during the St. Patrick's event of 2015 using the Salammbô model for two electron energies: (left column panels) 500 keV and (right column panels) 1.5 MeV. From bottom to top the Kp index, the AL index, the electron flux measured by the MagEIS instrument on board the Van Allen Probes, the simulation results using the

As shown in the 500 keV electron energy, simulation results exhibit more intense radial transport at the outer edge of the outer radiation belt (

We must emphasize the fact that the aforementioned comparison is performed between the calculated

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 ULF power spectral density and the estimated radial diffusion coefficients. We have further exploited this database in order to investigate the dependence of these calculated

The results of this analysis can be summarized as follows:

Both

The superposed epoch analysis reveals important differences between the evolution of

Furthermore, the comparison of the semi-empirical models with the

We believe that these results may offer significant insight for future modelling efforts in order to develop an accurate nowcasting and forecasting model for radial diffusion coefficients.

The scientific products of the SafeSpace radial diffusion coefficient database can be found at

CK drafted and wrote the paper with participation of all co-authors. CP contributed to the software development, AN to the development of the database and SAG to 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.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has received funding from the European Union's Horizon 2020
Research and Innovation programme “SafeSpace” under grant agreement
no. 870437. The authors acknowledge the THEMIS/FGM and EFI teams for
the use of the corresponding datasets which can be found online at

This research has been supported by the European Commission, Horizon 2020 (SafeSpace (grant no. 870437)).

This paper was edited by Elias Roussos and reviewed by two anonymous referees.