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We conduct electromagnetic particle simulations in a uniform magnetic environment to verify the nonlinear wave growth process of plasmaspheric hiss in the equatorial plasmasphere. The satisfaction of the separability criterion for coexisting multiple frequency waves in the initial stage of wavenumber-time evolution declares that wave packets are coherent and capable of growing nonlinearly. Spatial and temporal evolutions of two typical modes located in wavenumber-time evolution demonstrate the consistency among wave growths, frequency variations, and inhomogeneity factor S in coherent wave packets, showing that rising and falling tones occur at negative and positive S values, respectively, and an obvious wave growth happens in a reasonable range of S satisfying the second-order resonance condition. Wave packets extracted from wave fields in space and time by a frequency band-pass filter confirm a good agreement between the nonlinear theory and simulation results. The nonlinear growth rates of the extracted wave packets possess similar magnitudes to the growth rates of wave packets in the simulation, and they are much greater than the theoretical linear growth rate, indicating that the nonlinear process is essential in the generation of plasmaspheric hiss.

Nonlinear processes associated with the generation process of whistler-mode chorus emissions are summarized. The nonlinear dynamics of energetic electrons interacting with a coherent whistler-mode wave and the formation of electromagnetic electron holes or hills in the velocity phase space are described. The condition for resonant electrons to be free from the anomalous trapping at low pitch angles is obtained. In the presence of the inhomogeneity due to the frequency variation and the gradient of the magnetic field, the electron holes or hills result in resonant currents generating rising-tone emissions or falling-tone emissions, respectively. After formation of a coherent wave at a frequency of the maximum linear growth rate, triggering of the nonlinear wave growth takes place when the wave amplitude is above the threshold amplitude. The wave grows to a level close to the optimum wave amplitude as an absolute instability near the magnetic equator. The nonlinear growth rate at a position away from the equator is derived for a subtracted Maxwellian momentum distribution function with correction to the formulas in the past publications. The triggering process is repeated sequentially at progressively higher frequencies in the case of a rising-tone emission, generating subpackets forming a chorus element. With a higher plasma density as in the plasmasphere, the triggering of subpackets takes place concurrently over a wide range of frequency forming discrete hiss elements with varying frequencies. The mechanism of nonlinear wave damping due to quasi-parallel propagation from the equator is presented, which results in the formation of a gap at half the electron cyclotron frequency, separating a long rising-tone chorus emission into the upper-band and lower-band chorus emissions. The theoretical formulation of an oblique whistler mode wave and its interaction with energetic electrons at the n-th resonance is also presented along with derivation of the inhomogeneity facto

We have conducted a one‐dimensional electromagnetic particle simulation with a parabolic magnetic field to reproduce whistler‐mode hiss emissions in the plasmasphere. We assume a bi‐Maxwellian distribution with temperature anisotropy for energetic electrons injected into the plasmasphere and find that hiss emissions are generated with spectrum characteristics typical of those observed by spacecraft near the magnetic equator. The hiss emissions contain fine structures involving rising tone and falling tone elements with variation in frequencies. The amplitude profile of the spectra agrees with the optimum wave amplitude derived from the nonlinear wave growth theory. The simulation demonstrates that hiss emissions are generated locally near the magnetic equator through linear and nonlinear interactions with energetic electrons with temperature anisotropy. The coherent hiss emissions efficiently scatter resonant electrons of 2.5–80 keV into the loss cone.

We perform a one‐dimensional electromagnetic full particle simulation for triggered falling‐tone emissions in the Earth's magnetosphere. The equatorial region of the magnetosphere is modeled with a parabolic magnetic field approximation. The short whistler‐mode waves with a large amplitude are excited and propagate poleward from an artificial current oscillating with a constant frequency and amplitude. Following the excited waves, clear emissions are triggered with a falling frequency. Without the inhomogeneity of the background magnetic field, no triggered emission appears. The falling tone has several subpackets of amplitude and decreases the frequency in a stepwise manner. The positive resonant current formed by resonant electrons in the direction of the wave magnetic field clearly shows that an electron hill is formed in the phase space and causes the frequency decrease. The entrapping of the resonant electrons at the front of the packets and the decrease of the amplitude at the end of packets are essential for the generation of falling‐tone emissions. Each wavefront of the emission has a strongly negative resonant current −JE, which results in the wave growth. In the formation process of the resonant currents, we investigate the inhomogeneous factor S, which controls the nonlinear motion of the resonant electrons interacting with waves. The factor S consists of two terms, a frequency sweep rate and a gradient of the background magnetic field. The resonant current JE in the wave packet changes its sign from negative to positive as the packet moves away from the equator, terminating the wave growth.

We present a statistical study of electromagnetic ion cyclotron (EMIC) waves observed at Antarctic station (geographic 70.7° S, 11.8° E, L=5) on quiet and disturbed days during 2011–2017. The data span a fairly good period of both ascending and descending phases of the solar cycle 24, which has witnessed extremely low activity. We noted EMIC wave occurrence by examining wave power in different frequency ranges in the spectrogram. EMIC wave occurrence during ascending and descending phases of solar cycle 24, its local time, seasonal dependence and durations have been examined. There are total 2367 days for which data are available. Overall, EMIC waves are observed for 3166.5 h (≈5.57% of total duration) which has contributions from 1263 days. We find a significantly higher EMIC wave occurrence during the descending phase (≈ 6.83%) as compared to the ascending phase (≈ 4.08%) of the solar cycle, which implies nearly a twofold increase in EMIC wave occurrence. This feature is attributed to the higher solar wind dynamic pressure during descending phase of solar activity. There is no evident difference in the percentage occurrence of EMIC waves on magnetically disturbed and quiet days. On ground, EMIC waves show marginally higher occurrence during winter as compared to summer. This seasonal tendency is attributed to lower electron densities and conductivities in the ionosphere, which can affect the propagation of EMIC waves through ionospheric ducts. In local time, the probability distribution function of EMIC wave occurrence shows enhancement during 11.7–20.7 LT (i.e., afternoon–dusk sector). Daily durations of EMIC waves are in the range of 5–1015 min and it is noted that the longer duration (240–1015 min) events are prevalent on quiet days and are mostly seen during the descending phase of solar cycle.

In this study we present an inversion method which provides thermal plasma population parameters from characteristics of chorus emissions only. Our ultimate goal is to apply this method to ground‐based data in order to derive the lower‐energy boundary condition for many radiation belt models. The first step is to test the chorus inversion method on in situ data of the Van Allen Probes in the generation region. The density and thermal velocity of energetic electrons (few kiloelectron volts to 100 keV) are derived from frequency sweep rate and starting frequencies of chorus emissions through analysis of wave data from the Electric and Magnetic Field Instrument Suite and Integrated Science on board the Van Allen Probes. The nonlinear wave growth theory of Omura and Nunn (2011, https://doi.org/10.1029/2010JA016280) serves as the basis for our inversion method, assuming that the triggering wave is originated by the linear cyclotron instability. We present 16 consecutive rising‐tone emissions recorded in the generation region between 11 and 12 UT on 14 November 2012. The results of the inversion are compared with density and thermal velocities (parallel and perpendicular) of energetic electrons derived from the unidirectional flux data of the Helium, Oxygen, Proton, and Electron instrument, showing a good agreement: The normalized root‐mean‐square deviation between the measured and predicted values are less than ∼15%. We found that the theoretical amplitudes are consistent with the measured ones. The relation between linear and nonlinear wave growth agrees with our basic assumption; namely, linear growth is a preceding process of nonlinear wave growth. We analyze electron distributions at the relativistic resonant energy ranges.

Efficient acceleration of relativistic electrons at Landau resonance with obliquely propagating whistler‐mode chorus emissions is confirmed by theory, simulation, and observation. The acceleration is due to the perpendicular component of the wave electric field. We first review theoretical analysis of nonlinear motion of resonant electrons interacting with obliquely propagating whistler‐mode chorus. We have derived formulae of inhomogeneity factors for Landau and cyclotron resonances to analyze nonlinear wave trapping of energetic electrons by an obliquely propagating chorus element. We performed test particle simulations to confirm that nonlinear wave trapping by both Landau and cyclotron resonances can take place for a wide range of energies. For an element of large amplitude chorus waves observed by the Van Allen Probes, we have performed detailed analyses of the wave form data based on theoretical framework of nonlinear trapping of resonant electrons. We compare the efficiencies of accelerations by cyclotron and Landau resonances. We find significant acceleration can take place both in Landau and cyclotron resonances. What controls the dynamics of relativistic electrons in the Landau resonance is the perpendicular field components rather than the parallel electric field of the oblique chorus wave. In evaluating the efficiency of nonlinear trapping, we have taken into account variation of the wave trapping potential structure controlled by the inhomogeneity factors.

The concept of metamaterials has led to extraordinary schemes of wave propagation, which has been verified using various meta-atoms, constituent units of metamaterials, as well as applied to a number of categories in physics. Although its definition clarifies the maximum size of meta-atoms as a fraction of one wavelength, the size may vary by several orders, like from millimeters ('macroscopic level') to atomic scales ('microscopic level') for microwaves. This review surveys several patterns of parameter combinations, like permittivity and permeability in electromagnetic metamaterials, which have been achieved at either macroscopic or microscopic levels, with the similar analogy under the concept of metamaterials. Various experimental and theoretical efforts reported so far and shown here verify that the parameter identification of these values (permittivity, permeability, and refractive index) is independent of meta-atom sizes, with importance of spatial integration procedure on the order of a wavelength.

Understanding of underlying mechanisms of drastic variations of the near-Earth space (geospace) is one of the current focuses of the magnetospheric physics. The science target of the geospace research project Exploration of energization and Radiation in Geospace (ERG) is to understand the geospace variations with a focus on the relativistic electron acceleration and loss processes. In order to achieve the goal, the ERG project consists of the three parts: the Arase (ERG) satellite, ground-based observations, and theory/modeling/integrated studies. The role of theory/modeling/integrated studies part is to promote relevant theoretical and simulation studies as well as integrated data analysis to combine different kinds of observations and modeling. Here we provide technical reports on simulation and empirical models related to the ERG project together with their roles in the integrated studies of dynamic geospace variations. The simulation and empirical models covered include the radial diffusion model of the radiation belt electrons, GEMSIS-RB and RBW models, CIMI model with global MHD simulation REPPU, GEMSIS-RC model, plasmasphere thermosphere model, self-consistent wave–particle interaction simulations (electron hybrid code and ion hybrid code), the ionospheric electric potential (GEMSIS-POT) model, and SuperDARN electric field models with data assimilation. ERG (Arase) science center tools to support integrated studies with various kinds of data are also briefly introduced.

We carry out a self-consistent simulation of the generation process of whistler-mode chorus by a spatially one-dimensional electron hybrid code, by assuming the magnetic field inhomogeneity corresponding to L = 4 of the dipole field. Chorus emissions with rising tones are reproduced in the simulation result, while the frequency range, sweep rate, and the amplitude profiles in the spectra of the reproduced elements are consistently explained by the nonlinear wave growth theory. We compare the simulation results with the observation by the Cluster spacecraft (Santolik et al. in J Geophys Res 108:1278, 2003, doi:10.1029/2002JA009791; Santolik in Nonlinear Process Geophys 15:621–630, 2008) and reveal similarities of the spectral fine structure of reproduced chorus elements with the observation. On the other hand, there is no gap at half the gyrofrequency in the spectra of the reproduced chorus elements, which is evident in the observation. This difference implies that the mechanism of a gap at half the gyrofrequency is governed by the process that is not described by the spatially one-dimensional simulation treating purely parallel propagating electromagnetic waves.

Recent observations of plasmaspheric hiss emissions by the Van Allen Probes show that broadband hiss emissions in the plasmasphere comprise short-time coherent elements with rising and falling tone frequencies. Based on nonlinear wave growth theory of whistler mode chorus emissions, we have examined the applicability of the nonlinear theory to the coherent hiss emissions. We have generalized the derivation of the optimum wave amplitude for triggering rising tone chorus emissions to the cases of both rising and falling tone hiss elements. The amplitude profiles of the hiss emissions are well approximated by the optimum wave amplitudes for triggering rising or falling tones. Through the formation of electron holes for rising tones and electron hills for falling tones, the coherent waves evolve to attain the optimum amplitudes. An electromagnetic particle simulation confirms the nonlinear wave growth mechanism as the initial phase of the hiss generation process. We find very good agreement between the theoretical optimum amplitudes and the observed amplitudes as a function of instantaneous frequency. We calculate nonlinear growth rates at the equator and find that nonlinear growth rates for rising tone emissions are much larger than the linear growth rates. The time scales of observed hiss emissions also agree with those predicted by the nonlinear theory. Based on the theory, we can infer properties of energetic electrons generating hiss emissions in the equatorial region of the plasmasphere.

We show that the anomalous cyclotron resonance between relativistic electrons and electromagnetic ion cyclotron (EMIC) triggered emissions takes place very effectively near the magnetic equator because of the variation of the ambient magnetic field. Efficient precipitations are caused by nonlinear trapping of relativistic electrons by electromagnetic wave potentials formed by EMIC triggered emissions. We derive the necessary conditions of the wave amplitude, kinetic energies, and pitch angles that must be satisfied for the nonlinear wave trapping. We have conducted test particle simulations with a large number of relativistic electrons trapped by a parabolic magnetic field near the magnetic equator. In the presence of coherent EMIC-triggered emissions with increasing frequencies, a substantial amount of relativistic electrons is trapped by the wave, and the relativistic electrons at high pitch angles are guided to lower pitch angles within a short time scale much less than a second, resulting in rapid precipitation of relativistic electrons or relativistic electron microbursts.

We perform one-dimensional fluid simulation of ion acoustic (IA) solitons propagating parallel to the magnetic field in electron-ion plasmas by assuming a large system length. To model the initial density perturbations (IDP), we employ a KdV soliton type solution. Our simulation demonstrates that the generation mechanism of IA solitons depends on the wavelength of the IDP. The short wavelength IDP evolve into two oppositely propagating identical IA solitons, whereas the long wavelength IDP develop into two indistinguishable chains of multiple IA solitons through a wave breaking process. The wave breaking occurs close to the time when electrostatic energy exceeds half of the kinetic energy of the electron fluid. The wave breaking amplitude and time of its initiation are found to be dependent on characteristics of the IDP. The strength of the IDP controls the number of IA solitons in the solitary chains. The speed, width, and amplitude of IA solitons estimated during their stable propagation in the simulation are in good agreement with the nonlinear fluid theory. This fluid simulation is the first to confirm the validity of the general nonlinear fluid theory, which is widely used in the study of solitary waves in laboratory and space plasmas.

We derive the second-order resonance condition for interaction between a relativistic electron and a coherent Electromagnetic Ion Cyclotron (EMIC) wave with a variable frequency. We perform test particle simulations of relativistic electrons interacting with EMIC waves with a fixed frequency and a rising-tone frequency such as EMIC triggered emissions observed in the inner magnetosphere. Trapping of resonant electrons leads to rapid and efficient pitch angle scattering of relativistic electrons, resulting in bursty precipitation of relativistic electrons. The efficiency of the pitch angle scattering depends on the gradient of the magnetic field, the frequency sweep rate, and the wave amplitude. The effective wave trapping occurs for a wide range of pitch angles from 10 to 60 degrees. The most effective pitch angle scattering takes place for the case of a rising-tone emission with an enhanced magnetic field gradient. Since the efficiency of pitch angle scattering also depends on the wave amplitude, resonant electrons may not be scattered into the loss cone in a single passage through the wave packet. However, repeated interactions with a series of wave packets result in scattering of relativistic electrons into the loss cone.

A decomposition scheme was applied to ALOS/PALSAR data obtained from a fast-growing tree plantation in Sumatra, Indonesia to extract tree stem information and then estimate the forest stand volume. The scattering power decomposition of the polarimetric SAR data was performed both with and without a rotation matrix and compared to the following field-measured forest biometric parameters: tree diameter, tree height and stand volume. The analytical results involving the rotation matrix correlated better than those without the rotation matrix even for natural scattering surfaces within the forests. Our primary finding was that all of the decomposition powers from the rotated matrix correlated significantly to the forest biometric parameters when divided by the total power. The surface scattering ratio of the total power markedly decreased with the forest growth, whereas the canopy and double-bounce scattering ratios increased. The observations of the decomposition powers were consistent with the tree growth characteristics. Consequently, we found a significant logarithmic relationship between the decomposition powers and the forest biometric parameters that can potentially be used to estimate the forest stand volume.

In the equatorial region of the Earth's inner magnetosphere, the electromagnetic ion cyclotron (EMIC) triggered emissions are generated through interaction with energetic protons. We investigate the generation process of the EMIC triggered emissions in the He+branch and associated precipitation of the energetic protons using a one-dimensional hybrid simulation with a cylindrical parabolic magnetic geometry. The simulation results show a good agreement with the nonlinear wave growth theory. As the electron density becomes higher as in the plasmasphere or the plasmaplume, the wave amplitude thresholds for both H+ and He+ band triggered emissions become lower and their nonlinear growth rates become higher. The higher hot proton density also makes the thresholds lower. While the H+ branch triggered emissions interact with a few keV protons, the He+ branch triggered emissions interact with more energetic protons of a few hundred keV with a larger nonlinear growth rate.

Recently much progress has been made in the simulation and theoretical understanding of rising frequency triggered emissions and rising chorus. Both PIC and Vlasov VHS codes produce risers in the region downstream from the equator toward which the VLF waves are traveling. The VHS code only produces fallers or downward hooks with difficulty due to the coherent nature of wave particle interaction across the equator. With the VHS code we now confine the interaction region to be the region upstream from the equator, where inhomogeneity factor S is positive. This suppresses correlated wave particle interaction effects across the equator and the tendency of the code to trigger risers, and permits the formation of a proper falling tone generation region. The VHS code now easily and reproducibly triggers falling tones. The evolution of resonant particle current JE in space and time shows a generation point at −5224 km and the wavefield undergoes amplification of some 25 dB in traversing the nonlinear generation region. The current component parallel to wave magnetic field (JB) is positive, whereas it is negative for risers. The resonant particle trap shows an enhanced distribution function or 'hill', whereas risers have a 'hole'. According to recent theory (Omura et al., 2008, 2009) sweeping frequency is due primarily to the advective term. The nonlinear frequency shift term is now negative (∼−12 Hz) and the sweep rate of −800 Hz/s is approximately nonlinear frequency shift divided by TN, the transition time, of the order of a trapping time.

We analyze the nonlinear evolution of whistler mode chorus waves propagating along a magnetic field line from their equatorial source. We solve wave evolution equations off the equator for the wave magnetic field amplitude and wave frequency, subject to boundary conditions at the equator comprising model "chorus equations" that describe the generation of a seed chorus element. The electron distribution function is assumed to evolve adiabatically along a field line. The wave profiles exhibit nonlinear convective growth followed by saturation. Convective growth is due to nonlinear wave trapping, and the saturation process is partly due to a combination of adiabatic effects and a decreasing resonant current with latitude. Notwithstanding computationally expensive full-scale kinetic simulations, our study appears to be the first to analyze the nonlinear evolution and saturation of whistler mode waves off the equator.

We perform self-consistent electromagnetic particle simulations to analyze whistler-mode triggered emissions in the magnetosphere. The whistler-mode triggering waves with different wave amplitudes and a constant frequency are injected at the equator. With triggering wave amplitudes greater than the threshold for the nonlinear wave growth, rising-tone emissions are successfully excited near the equator. The detailed time evolutions of amplitudes and frequencies of the rising-tone emissions show similar development. A recent theoretical study found the optimum amplitude of triggering waves for rising-tone emissions. The optimum amplitude condition is confirmed by the simulations. A triggering wave with an amplitude much greater than the optimum amplitude cannot trigger a rising emission. In a process where triggered emissions develop, phase-organized resonant electrons clearly appear in the velocity-phase space, contributing to the nonlinear wave growth. The simulation study shows that amplitudes and frequency sweep rates of triggered emissions do not depend on the amplitude of a triggering wave.

Electromagnetic ion cyclotron (EMIC) triggered emissions with rising tones between the H+ and He+ cyclotron frequencies were found in the inner magnetosphere by the recent Cluster observations. Another type of EMIC wave with a constant frequency is occasionally observed below the He+ cyclotron frequency after the multiple EMIC triggered emissions. We performed a self-consistent hybrid simulation with a one-dimensional cylindrical magnetic flux model approximating the dipole magnetic field of the Earth's inner magnetosphere. In the presence of energetic protons with a sufficient density and temperature anisotropy, multiple EMIC triggered emissions are reproduced due to the nonlinear wave growth mechanism of rising-tone chorus emissions, and a constant frequency wave in the He+ EMIC branch is subsequently generated. Through interaction with the multiple EMIC rising-tone emissions, the velocity distribution function of the energetic protons is strongly modified. Because of the pitch angle scattering of the protons, the gradient of the distribution in velocity phase space is enhanced along the diffusion curve of the He+ branch wave, resulting in the linear growth of the EMIC wave in the He+ branch.

In a recent observation by the Cluster spacecraft, emissions triggered by electromagnetic ion cyclotron (EMIC) waves were discovered in the inner magnetosphere. We perform hybrid simulations to reproduce the EMIC triggered emissions. We develop a self-consistent one-dimensional hybrid code with a cylindrical geometry of the background magnetic field. We assume a parabolic magnetic field to model the dipole magnetic field in the equatorial region of the inner magnetosphere. Triggering EMIC waves are driven by a left-handed polarized external current assumed at the magnetic equator in the simulation model. Cold proton, helium, and oxygen ions, which form branches of the dispersion relation of the EMIC waves, are uniformly distributed in the simulation space. Energetic protons with a loss cone distribution function are also assumed as resonant particles. We reproduce rising tone emissions in the simulation space, finding a good agreement with the nonlinear wave growth theory. In the energetic proton velocity distribution we find formation of a proton hole, which is assumed in the nonlinear wave growth theory. A substantial amount of the energetic protons are scattered into the loss cone, while some of the resonant protons are accelerated to higher pitch angles, forming a pancake velocity distribution.

The frequency and amplitude characteristics of chorus emissions are studied by performing electron hybrid code simulations with different initial number densities of energetic electrons. Chorus emissions with rising tones are generated in all simulation runs except for the simulation assuming the lowest number density. The frequency sweep rates of reproduced chorus vary depending on the variation of the wave amplitude of respective chorus elements. We find that the theoretically estimated frequency sweep rates are consistent with the simulation results. The simulation results reveal that the characteristic frequency variation of chorus elements showing rising tones are formed at the region very close to the magnetic equator while the wave amplitudes of chorus elements are significantly amplified through their propagation away from the equator. The spatial scale of the region where the observed explosive wave growth varies in each simulation run, corresponding to the difference of the wave amplitudes of reproduced chorus elements. We estimate the spatial scale hc, a measure of the spatial extent of the generation region derived from the nonlinear wave growth theory, using parameters of the simulation runs and find its consistency with the simulation results. We also analyze the energy spectra of reproduced chorus elements in the simulation results and find that the spectra of chorus are essentially different from those predicted by the linear growth rates in the frequency domain. These results clearly demonstrate that the nonlinear wave-particle interaction governs the chorus generation mechanism.

Self-consistent electromagnetic particle simulations were performed to analyze whistler mode triggered emissions in the magnetosphere. Triggering whistler mode waves injected at the magnetic equator induces a nonlinear absolute instability that results in rising tone emissions similar to natural whistler mode chorus emissions. The triggering wave causes a depletion of resonant electrons at the resonance velocity. The phase-organized resonant electrons released by the triggering waves generate coherent waves that undergo nonlinear growth with increasing frequencies. The essential mechanism is the same as that for the generation of chorus emissions. Saturation of the nonlinear wave growth is caused by enhancement of resonant electrons at high pitch angles that have been trapped and guided along the resonance velocity by the triggered emissions. Because of the decreasing resonance velocity resulting from the increasing frequency, trapped electrons are accelerated to higher pitch angles. Saturation of the nonlinear wave growth thereby results in electron acceleration.