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The Cluster mission has produced a large data set of electron flux measurements in the Earth's magnetosphere since its launch in late 2000. Electron fluxes are measured using Research with Adaptive Particle Imaging Detector (RAPID)/Imaging Electron Spectrometer (IES) detector as a function of energy, pitch angle, spacecraft position, and time. However, no adiabatic invariants have been calculated for Cluster so far. In this paper we present a step-by-step guide to calculations of adiabatic invariants and conversion of the electron flux to phase space density (PSD) in these coordinates. The electron flux is measured in two RAPID/IES energy channels providing pitch angle distribution at energies 39.2-50.5 and 68.1-94.5 keV in nominal mode since 2004. A fitting method allows to expand the conversion of the differential fluxes to the range from 40 to 150 keV. Best data coverage for phase space density in adiabatic invariant coordinates can be obtained for values of second adiabatic invariant, K, similar to 10(2), and values of the first adiabatic invariant mu in the range approximate to 5-20 MeV/G. Furthermore, we describe the production of a new data product "LSTAR," equivalent to the third adiabatic invariant, available through the Cluster Science Archive for years 2001-2018 with 1-min resolution. The produced data set adds to the availability of observations in Earth's radiation belts region and can be used for long-term statistical purposes.
The Cluster mission has produced a large data set of electron flux measurements in the Earth's magnetosphere since its launch in late 2000. Electron fluxes are measured using Research with Adaptive Particle Imaging Detector (RAPID)/Imaging Electron Spectrometer (IES) detector as a function of energy, pitch angle, spacecraft position, and time. However, no adiabatic invariants have been calculated for Cluster so far. In this paper we present a step-by-step guide to calculations of adiabatic invariants and conversion of the electron flux to phase space density (PSD) in these coordinates. The electron flux is measured in two RAPID/IES energy channels providing pitch angle distribution at energies 39.2-50.5 and 68.1-94.5 keV in nominal mode since 2004. A fitting method allows to expand the conversion of the differential fluxes to the range from 40 to 150 keV. Best data coverage for phase space density in adiabatic invariant coordinates can be obtained for values of second adiabatic invariant, K, similar to 10(2), and values of the first adiabatic invariant mu in the range approximate to 5-20 MeV/G. Furthermore, we describe the production of a new data product "LSTAR," equivalent to the third adiabatic invariant, available through the Cluster Science Archive for years 2001-2018 with 1-min resolution. The produced data set adds to the availability of observations in Earth's radiation belts region and can be used for long-term statistical purposes.
Chorus waves play an important role in the dynamic evolution of energetic electrons in the Earth's radiation belts and ring current. Using more than 5 years of Van Allen Probe data, we developed a new analytical model for upper‐band chorus (UBC; 0.5fce < f < fce) and lower‐band chorus (LBC; 0.05fce < f < 0.5fce) waves, where fce is the equatorial electron gyrofrequency. By applying polynomial fits to chorus wave root mean square amplitudes, we developed regression models for LBC and UBC as a function of geomagnetic activity (Kp), L, magnetic latitude (λ), and magnetic local time (MLT). Dependence on Kp is separated from the dependence on λ, L, and MLT as Kp‐scaling law to simplify the calculation of diffusion coefficients and inclusion into particle tracing codes. Frequency models for UBC and LBC are also developed, which depends on MLT and magnetic latitude. This empirical model is valid in all MLTs, magnetic latitude up to 20°, Kp ≤ 6, L‐shell range from 3.5 to 6 for LBC and from 4 to 6 for UBC. The dependence of root mean square amplitudes on L are different for different bands, which implies different energy sources for different wave bands. This analytical chorus wave model is convenient for inclusion in quasi‐linear diffusion calculations of electron scattering rates and particle simulations in the inner magnetosphere, especially for the newly developed four‐dimensional codes, which require significantly improved wave parameterizations.
Plain Language Summary Radiation belts of the Earth, which are the zones of charged energetic particles trapped by the geomagnetic field, comprise enormous and dynamic systems. While the inner radiation belt, composed mainly of high-energy protons, is relatively stable, the outer belt, filled with energetic electrons, is highly variable and depends substantially on solar activity. Hence, extended reliable observations and the improved models of the electron intensities in the outer belt depending on solar wind parameters are necessary for prediction of their dynamics. The Cluster mission has been measuring electron flux intensities in the radiation belts since its launch in 2000, thus providing a huge dataset that can be used for radiation belts analysis. Using 16 years of electron measurements by the Cluster mission corrected for background contamination, we derived a uniform linear-logarithmic dependence of electron fluxes in the outer belt on the solar wind dynamic pressure.
Electromagnetic ion cyclotron (EMIC) waves play an important role in the dynamics of ultrarelativistic electron population in the radiation belts. However, as EMIC waves are very sporadic, developing a parameterization of such wave properties is a challenging task. Currently, there are no dynamic, activity-dependent models of EMIC waves that can be used in the long-term (several months) simulations, which makes the quantitative modeling of the radiation belt dynamics incomplete. In this study, we investigate Kp, Dst, and AE indices, solar wind speed, and dynamic pressure as possible parameters of EMIC wave presence. The EMIC waves are included in the long-term simulations (1year, including different geomagnetic activity) performed with the Versatile Electron Radiation Belt code, and we compare results of the simulation with the Van Allen Probes observations. The comparison shows that modeling with EMIC waves, parameterized by solar wind dynamic pressure, provides a better agreement with the observations among considered parameterizations. The simulation with EMIC waves improves the dynamics of ultrarelativistic fluxes and reproduces the formation of the local minimum in the phase space density profiles.
Data assimilation aims to blend incomplete and inaccurate data with physics-based dynamical models. In the Earth's radiation belts, it is used to reconstruct electron phase space density, and it has become an increasingly important tool in validating our current understanding of radiation belt dynamics, identifying new physical processes, and predicting the near-Earth hazardous radiation environment. In this study, we perform reanalysis of the sparse measurements from four spacecraft using the three-dimensional Versatile Electron Radiation Belt diffusion model and a split-operator Kalman filter over a 6-month period from 1 October 2012 to 1 April 2013. In comparison to previous works, our 3-D model accounts for more physical processes, namely, mixed pitch angle-energy diffusion, scattering by Electromagnetic Ion Cyclotron waves, and magnetopause shadowing. We describe how data assimilation, by means of the innovation vector, can be used to account for missing physics in the model. We use this method to identify the radial distances from the Earth and the geomagnetic conditions where our model is inconsistent with the measured phase space density for different values of the invariants mu and K. As a result, the Kalman filter adjusts the predictions in order to match the observations, and we interpret this as evidence of where and when additional source or loss processes are active. The current work demonstrates that 3-D data assimilation provides a comprehensive picture of the radiation belt electrons and is a crucial step toward performing reanalysis using measurements from ongoing and future missions.
The radiation belts of the Earth, filled with energetic electrons, comprise complex and dynamic systems that pose a significant threat to satellite operation. While various models of electron flux both for low and relativistic energies have been developed, the behavior of medium energy (120-600 keV) electrons, especially in the MEO region, remains poorly quantified. At these energies, electrons are driven by both convective and diffusive transport, and their prediction usually requires sophisticated 4D modeling codes. In this paper, we present an alternative approach using the Light Gradient Boosting (LightGBM) machine learning algorithm. The Medium Energy electRon fLux In Earth's outer radiatioN belt (MERLIN) model takes as input the satellite position, a combination of geomagnetic indices and solar wind parameters including the time history of velocity, and does not use persistence. MERLIN is trained on >15 years of the GPS electron flux data and tested on more than 1.5 years of measurements. Tenfold cross validation yields that the model predicts the MEO radiation environment well, both in terms of dynamics and amplitudes o f flux. Evaluation on the test set shows high correlation between the predicted and observed electron flux (0.8) and low values of absolute error. The MERLIN model can have wide space weather applications, providing information for the scientific community in the form of radiation belts reconstructions, as well as industry for satellite mission design, nowcast of the MEO environment, and surface charging analysis.
The Earth's inner magnetosphere is a very dynamic system, mostly driven by the external solar wind forcing exerted upon the magnetic field of our planet. Disturbances in the solar wind, such as coronal mass ejections and co-rotating interaction regions, cause geomagnetic storms, which lead to prominent changes in charged particle populations of the inner magnetosphere - the plasmasphere, ring current, and radiation belts. Satellites operating in the regions of elevated energetic and relativistic electron fluxes can be damaged by deep dielectric or surface charging during severe space weather events. Predicting the dynamics of the charged particles and mitigating their effects on the infrastructure is of particular importance, due to our increasing reliance on space technologies.
The dynamics of particles in the plasmasphere, ring current, and radiation belts are strongly coupled by means of collisions and collisionless interactions with electromagnetic fields induced by the motion of charged particles. Multidimensional numerical models simplify the treatment of transport, acceleration, and loss processes of these particles, and allow us to predict how the near-Earth space environment responds to solar storms. The models inevitably rely on a number of simplifications and assumptions that affect model accuracy and complicate the interpretation of the results. In this dissertation, we quantify the processes that control electron dynamics in the inner magnetosphere, paying particular attention to the uncertainties of the employed numerical codes and tools.
We use a set of convenient analytical solutions for advection and diffusion equations to test the accuracy and stability of the four-dimensional Versatile Electron Radiation Belt (VERB-4D) code. We show that numerical schemes implemented in the code converge to the analytical solutions and that the VERB-4D code demonstrates stable behavior independent of the assumed time step. The order of the numerical scheme for the convection equation is demonstrated to affect results of ring current and radiation belt simulations, and it is crucially important to use high-order numerical schemes to decrease numerical errors in the model.
Using the thoroughly tested VERB-4D code, we model the dynamics of the ring current electrons during the 17 March 2013 storm. The discrepancies between the model and observations above 4.5 Earth's radii can be explained by uncertainties in the outer boundary conditions. Simulation results indicate that the electrons were transported from the geostationary orbit towards the Earth by the global-scale electric and magnetic fields.
We investigate how simulation results depend on the input models and parameters. The model is shown to be particularly sensitive to the global electric field and electron lifetimes below 4.5 Earth's radii. The effects of radial diffusion and subauroral polarization streams are also quantified.
We developed a data-assimilative code that blends together a convection model of energetic electron transport and loss and Van Allen Probes satellite data by means of the Kalman filter. We show that the Kalman filter can correct model uncertainties in the convection electric field, electron lifetimes, and boundary conditions. It is also demonstrated how the innovation vector - the difference between observations and model prediction - can be used to identify physical processes missing in the model of energetic electron dynamics.
We computed radial profiles of phase space density of ultrarelativistic electrons, using Van Allen Probes measurements. We analyze the shape of the profiles during geomagnetically quiet and disturbed times and show that the formation of new local minimums in the radial profiles coincides with the ground observations of electromagnetic ion-cyclotron (EMIC) waves. This correlation indicates that EMIC waves are responsible for the loss of ultrarelativistic electrons from the heart of the outer radiation belt into the Earth's atmosphere.
Significant progress has been made in recent years in understanding acceleration mechanisms in the Earth's radiation belts. In particular, a number of studies demonstrated the importance of the local acceleration by analyzing the radial profiles of phase space density (PSD) and observing building up peaks in PSD. In this study, we focus on understanding of the local loss using very similar tools. The profiles of PSD for various values of the first adiabatic invariants during the previously studied 17 January 2013 storm are presented and discussed. The profiles of PSD show clear deepening minimums consistent with the scattering by electromagnetic ion cyclotron waves. Long-term evolution shows that local minimums in PSD can persist for relatively long times. During considered interval of time the deepening minimums were observed around L* = 4 during 17 January 2013 storm and around L* = 3.5 during 1 March 2013 storm. This study shows a new method that can help identify the location, magnitude, and time of the local loss and will help quantify local loss in the future. This study also provides additional clear and definitive evidence that local loss plays a major role for the dynamics of the multi-MeV electrons.
In this study we present analytical solutions for convection and diffusion equations. We gather here the analytical solutions for the one-dimensional convection equation, the two-dimensional convection problem, and the one- and two-dimensional diffusion equations. Using obtained analytical solutions, we test the four-dimensional Versatile Electron Radiation Belt code (the VERB-4D code), which solves the modified Fokker-Planck equation with additional convection terms. The ninth-order upwind numerical scheme for the one-dimensional convection equation shows much more accurate results than the results obtained with the third-order scheme. The universal limiter eliminates unphysical oscillations generated by high-order linear upwind schemes. Decrease in the space step leads to convergence of a numerical solution of the two-dimensional diffusion equation with mixed terms to the analytical solution. We compare the results of the third- and ninth-order schemes applied to magnetospheric convection modeling. The results show significant differences in electron fluxes near geostationary orbit when different numerical schemes are used.