Publisher Correction: Wave energy budget analysis in the Earths radiation belts uncovers a missing energy.

This corrects the article DOI: 10.1038/ncomms8143

Wave-particle interactions in the outer radiation belts

Data from the Van Allen Probes have provided the first extensive evidence of non-linear (as opposed to quasi-linear) wave-particle interactions in space with the associated rapid (fraction of a bounce period) electron acceleration to hundreds of keV by Landau resonance in the parallel electric fields of time domain structures (TDS) and very oblique chorus waves. The experimental evidence, simulations, and theories of these processes are discussed. {\bf Key words:} the radiation belts, wave-particle interaction, plasma waves and instabilitiesComment: 9 pages, 2 figure

Acceleration of radiation belts electrons by oblique chorus waves

International audience[1] The redistribution of energy during the recovery phase of geomagnetic storms related to the acceleration of electrons in the Earth's outer radiation belt by cyclotron-resonant chorus waves is an important and challenging topic of magnetospheric plasma physics. An approximate analytical formulation of energy diffusion coefficients is derived in this paper, on the basis of a quasi-linear formalism valid for large enough bandwidths or for successive random scatter by uncorrelated waves of different frequencies and moderate average amplitudes. We make use of chorus wave parameterizations derived from CLUSTER measurements to show that oblique whistler waves can significantly increase the energy diffusion rate of small pitch angle electrons on the dayside. On the other hand, the energization rate of the more numerous high pitch angle electrons is typically reduced by a factor of 2 on the dayside, while it remains nearly unchanged on the nightside where high-intensity waves are less oblique. Besides, lifetimes are strongly reduced on the dayside, which could also impact the long-term time-integrated acceleration rates of injected electrons. Comparison between the analytical formulas and full numerical results demonstrates a good agreement and provides new scaling laws as a function of whistler mean frequency, plasma density and particle energy. It is also suggested that the enhancement of energy diffusion of low energy electrons (<100 keV) at small pitch angles with oblique waves could result in an intensification of wave growth at latitudes higher than 15. This might contribute to explain high chorus intensities measured by CLUSTER on the dayside at high latitudes. Citation: Mourenas, D., A. Artemyev, O. Agapitov, and V. Krasnoselskikh (2012), Acceleration of radiation belts electrons by oblique chorus waves

Chorus wave-normal statistics in the Earth's radiation belts from ray tracing technique

Discrete ELF/VLF (Extremely Low Frequency/Very Low Frequency) chorus emissions are one of the most intense electromagnetic plasma waves observed in radiation belts and in the outer terrestrial magnetosphere. These waves play a crucial role in the dynamics of radiation belts, and are responsible for the loss and the acceleration of energetic electrons. The objective of our study is to reconstruct the realistic distribution of chorus wave-normals in radiation belts for all magnetic latitudes. To achieve this aim, the data from the electric and magnetic field measurements onboard Cluster satellite are used to determine the wave-vector distribution of the chorus signal around the equator region. Then the propagation of such a wave packet is modeled using three-dimensional ray tracing technique, which employs K. Rönnmark's WHAMP to solve hot plasma dispersion relation along the wave packet trajectory. The observed chorus wave distributions close to waves source are first fitted to form the initial conditions which then propagate numerically through the inner magnetosphere in the frame of the WKB approximation. Ray tracing technique allows one to reconstruct wave packet properties (electric and magnetic fields, width of the wave packet in k-space, etc.) along the propagation path. The calculations show the spatial spreading of the signal energy due to propagation in the inhomogeneous and anisotropic magnetized plasma. Comparison of wave-normal distribution obtained from ray tracing technique with Cluster observations up to 40&deg; latitude demonstrates the reliability of our approach and applied numerical schemes

Trapped fast MGD waves in dayside magnetosphere

We studied the dynamics of the magnetosphere response to the solar wind parameters changes using the Alfvén velocity distribution in the Earth's magnetosphere obtained from the IGRF and T89 Earth's magnetic field models and plasma density diflusive equilibrium model. We used the solution of the eigenvalue problem for trapped waves in the dayside magnetosphere cavity as the initial condition for simulation. Numerical simulation of the fast MHD wave packet propagation in 3D magnetosphere cavity shows the occurrence of two global quasi-periodic modes of the dayside magnetosphere: cavity modes at the sub-solar region and waveguide modes at the magnetosphere flanks. The periods obtained in the numerical simulation are consistent with the theoretical predictions and the THEMIS measurements

Non-diffusive resonant acceleration of electrons in the radiation belts

International audienceWe describe a mechanism of resonant electron acceleration by oblique high-amplitude whistlerwaves under conditions typical for the Earth radiation belts. We use statistics of spacecraftobservations of whistlers in the Earth radiation belts to obtain the dependence of the angle hbetween the wave-normal and the background magnetic field on magnetic latitude k. According tothis statistics, the angle h already approaches the resonance cone at k 15 and remains close to itup to k 30–40 on the dayside. The parallel component of the electrostatic field of whistlerwaves often increases around k 15 up to one hundred of mV/m. We show that due to thisincrease of the electric field, the whistler waves can trap electrons into the potential well via waveparticle resonant interaction corresponding to Landau resonance. Trapped electrons then move withthe wave to higher latitudes where they escape from the resonance. Strong acceleration is favoredby adiabatic invariance along the increasing magnetic field, which continuously transfers theparallel energy gained to perpendicular energy, allowing resonance to be reached and maintained.The concomitant increase of the wave phase velocity allows for even stronger relative accelerationat low energy <50 keV. Each trapping-escape event of electrons of 10 keV to 100 keV results inan energy gain of up to 100 keV in the inhomogeneous magnetic field of the Earth dipole. Forelectrons with initial energy below 100 keV, such rapid acceleration should hasten their drop intothe loss-cone and their precipitation into the atmosphere. We discuss the role of the consideredmechanism in the eventual formation of a trapped distribution of relativistic electrons for initialenergies larger than 100 keV and in microbursts precipitations of lower energy particles

Very oblique whistler generation by low-energy electron streams

International audienceWhistler mode chorus waves are present throughout the Earth's outer radiation belt as well as at larger distances from our planet. While the generation mechanisms of parallel lower band chorus waves and oblique upper band chorus waves have been identified and checked in various instances, the statistically significant presence in recent satellite observations of very oblique lower band chorus waves near the resonance cone angle remains to be explained. Here we discuss two possible generation mechanisms for such waves. The first one is based on Landau resonance with sporadic very low energy (<4 keV) electron beams either injected from the plasma sheet or produced in situ. The second one relies on cyclotron resonance with low-energy electron streams, such that their velocity distribution possesses both a significant temperature anisotropy above 3–4 keV and a plateau or heavy tail in parallel velocities at lower energies encompassing simultaneous Landau resonance with the same waves. The corresponding frequency and wave normal angle distributions of the generated very oblique lower band chorus waves, as well as their frequency sweep rate, are evaluated analytically and compared with satellite observations, showing a reasonable agreement

Fast transport of resonant electrons in phase space due to nonlinear trapping by whistler waves

International audienceWe present an analytical, simplified formulation accounting for the fast transport of relativistic electrons in phase space due to wave-particle resonant interactions in the inhomogeneous magnetic field of Earth's radiation belts. We show that the usual description of the evolution of the particle velocity distribution based on the Fokker-Planck equation can be modified to incorporate nonlinear processes of wave-particle interaction, including particle trapping. Such a modification consists in one additional operator describing fast particle jumps in phase space. The proposed, general approach is used to describe the acceleration of relativistic electrons by oblique whistler waves in the radiation belts. We demonstrate that for a wave power distribution with a hard enough power law tail inline image such that η < 5/2, the efficiency of nonlinear acceleration could be more effective than the conventional quasi-linear acceleration for 100 keV electrons

Outer radiation belt electron lifetime model based on combined Van Allen Probes and Cluster VLF measurements

The flux of energetic electrons in the outer radiation belt shows a high variability. The interactions of electrons with very low frequency (VLF) chorus waves play a significant role in controlling the flux variation of these particles. Quantifying the effects of these interactions is crucially important for accurately modeling the global dynamics of the outer radiation belt and to provide a comprehensive description of electron flux variations over a wide energy range (from the source population of 30 keV electrons up to the relativistic core population of the outer radiation belt). Here, we use a synthetic chorus wave model based on a combined database compiled from the Van Allen Probes and Cluster spacecraft VLF measurements to develop a comprehensive parametric model of electron lifetimes as a function of L‐shell, electron energy, and geomagnetic activity. The wave model takes into account the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. We provide general analytical formulas to estimate electron lifetimes as a function of L‐shell (for L = 3.0 to L = 6.5), electron energy (from 30 keV to 2 MeV), and geomagnetic activity parameterized by the AE index. The present model lifetimes are compared to previous studies and analytical results and also show a good agreement with measured lifetimes of 30 to 300 keV electrons at geosynchronous orbit

Model of the propagation of very low-frequency beams in the Earth&amp;#8211;ionosphere waveguide: principles of the tensor impedance method in multi-layered gyrotropic waveguides

The modeling of very low-frequency (VLF) electromagnetic (EM) beam propagation in the Earth–ionosphere waveguide (WGEI) is considered. A new tensor impedance method for modeling the propagation of electromagnetic beams in a multi-layered and inhomogeneous waveguide is presented. The waveguide is assumed to possess the gyrotropy and inhomogeneity with a thick cover layer placed above the waveguide. The influence of geomagnetic field inclination and carrier beam frequency on the characteristics of the polarization transformation in the Earth–ionosphere waveguide is determined. The new method for modeling the propagation of electromagnetic beams allows us to study the (i) propagation of the very low-frequency modes in the Earth–ionosphere waveguide and, in perspective, their excitation by the typical Earth–ionosphere waveguide sources, such as radio wave transmitters and lightning discharges, and (ii) leakage of Earth–ionosphere waveguide waves into the upper ionosphere and magnetosphere. The proposed approach can be applied to the variety of problems related to the analysis of the propagation of electromagnetic waves in layered gyrotropic and anisotropic active media in a wide frequency range, e.g., from the Earth–ionosphere waveguide to the optical waveband, for artificial signal propagation such as metamaterial microwave or optical waveguides