Nonlinear Wave Growth Theory Research Articles

Observations of co-existing rising and falling tone emissions of electromagnetic ion cyclotron waves

We report observations of co-existing rising and falling tone emissions of electromagnetic ion cyclotron (EMIC) waves by THEMIS E spacecraft. The investigation of these fine structures of the EMIC waves is essential from the point of view of understanding the connection between the proton holes and the proton hills in velocity phase-space. The wave packets of rising and falling tones are tracked by Poynting vector analysis, where we observe that the rising tones are propagating northward and the falling tones are propagating southward. The nonlinear wave growth theory supports our observations. We propose a model where the proton velocity distribution function evolves through the formation of proton holes on the negative side of the distribution function and mirrored resonant protons forming proton hills on the positive side of the distribution function, allowing us to observe the co-existing rising and falling tone EMIC waves.Graphical abstract

Direct observations of energy transfer from resonant electrons to whistler-mode waves in magnetosheath of Earth

Electromagnetic whistler-mode waves in space plasmas play critical roles in collisionless energy transfer between the electrons and the electromagnetic field. Although resonant interactions have been considered as the likely generation process of the waves, observational identification has been extremely difficult due to the short time scale of resonant electron dynamics. Here we show strong nongyrotropy, which rotate with the wave, of cyclotron resonant electrons as direct evidence for the locally ongoing secular energy transfer from the resonant electrons to the whistler-mode waves using ultra-high temporal resolution data obtained by NASA’s Magnetospheric Multiscale (MMS) mission in the magnetosheath. The nongyrotropic electrons carry a resonant current, which is the energy source of the wave as predicted by the nonlinear wave growth theory. This result proves the nonlinear wave growth theory, and furthermore demonstrates that the degree of nongyrotropy, which cannot be predicted even by that nonlinear theory, can be studied by observations.

Nonlinear Wave Growth Analysis of Chorus Emissions Modulated by ULF Waves

AbstractWe present Van Allen Probes observations of periodic chorus wave emissions in the troughs of compressional ultralow frequency (ULF) waves. During this event, the spectral gap of chorus waves gradually widens as the spacecraft moves from the equatorial source region towards higher latitudes. Moreover, chorus wave intensity increases and frequency range widens after a substorm injection. We show that the periodic occurrence of chorus waves is attributed to the modulation of threshold amplitude for nonlinear growth of chorus waves by the second spatial derivative of ULF compressional magnetic field. The widening gap can be interpreted in terms of the nonlinear damping mechanism. A good agreement is also found between the nonlinear wave growth theory and the observations regarding the influence of substorm injection on the chorus. These findings support the applicability of the nonlinear theory in describing the chorus wave generation and damping, together with their modulations by ULF waves.

Simulation study on parametric dependence of whistler-mode hiss generation in the plasmasphere

We conduct electromagnetic particle simulations to examine the applicability of nonlinear wave growth theory to the generation process of plasmaspheric hiss. We firstly vary the gradient of the background magnetic field from a realistic model to a rather steep gradient model. Under such variation, the threshold amplitude in the nonlinear theory increases quickly and the overlap between threshold and optimum amplitude disappears correspondingly, the nonlinear process is suppressed. In the simulations, as we enlarge the gradient coefficient of the background magnetic field, waves generated near the equator do not grow through propagation. By examining the range of suitable values of inhomogeneity factor S (i.e., |S|<2), we find the generation of wave packets is limited to the equatorial region when the background field is steep, showing a good agreement with what is indicated by critical distance in the theory. We then check the dependence of generation of hiss emissions on different hot electron densities. Since the overlap between threshold and optimum amplitude vanishes, the nonlinear process is weakened when hot electron density becomes smaller. In the simulation results, we find similar wave structures in all density cases, yet with different magnitudes. The existence of suitable S values implies that the nonlinear process occurs even at a low level of hot electron density. However, by examining J_E that closely relates to the wave growth, we find energy conveyed from particles to waves is much limited in small density cases. Therefore, the nonlinear process is suppressed when hot electron density is small, which agrees with the theoretical analysis.Graphical

Multipoint Analysis of Source Regions of EMIC Waves and Rapid Growth of Subpackets

AbstractElectromagnetic Ion Cyclotron (EMIC) rising tone emissions are important in understanding the nonlinear wave evolution and interaction with the energetic particles. We present observations of rising tone emissions of EMIC waves by THEMIS A, D, and E spacecraft in the outer magnetosphere. These emissions with subpacket structures in the proton band were observed during the time interval 14:20 UT to 14:30 UT on September 9, 2010. The observed waves are left‐handed polarized with wave normal angles less than . THEMIS A was closest to the equator and was in a higher L‐shell than THEMIS E and D. The smallest radial separation is 2,000 km between THEMIS E and D spacecraft. This configuration of THEMIS allows us to investigate the subpackets of rising tone EMIC waves observed simultaneously at 14:23 UT by three spacecraft. Hilbert Huang Transformation (HHT) is applied to show the variations of the instantaneous frequency and the observed wave amplitude. The direction of energy flow is determined from the analysis of the Poynting flux. There is a rapid nonlinear growth of the EMIC subpackets within one wavelength. Subpackets are dynamic in nature as their structure changes within one wave period, which is further supported by the nonlinear wave growth theory. Optimum and threshold amplitudes for the EMIC wave growth are calculated beside the nonlinear transition time (). Observed ion energies and pitch angle spectra of the ion fluxes are consistent with the energy associated with the Landau and cyclotron resonance conditions.

Particle Simulation of the Generation of Plasmaspheric Hiss

AbstractWe 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.

Evaluation of Plasma Properties From Chorus Waves Observed at the Generation Region

AbstractIn 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.

Generation Process of Large‐Amplitude Upper‐Band Chorus Emissions Observed by Van Allen Probes

AbstractWe analyze large‐amplitude upper‐band chorus emissions measured near the magnetic equator by the Electric and Magnetic Field Instrument Suite and Integrated Science instrument package on board the Van Allen Probes. In setting up the parameters of source electrons exciting the emissions based on theoretical analyses and observational results measured by the Helium Oxygen Proton Electron instrument, we calculate threshold and optimum amplitudes with the nonlinear wave growth theory. We find that the optimum amplitude is larger than the threshold amplitude obtained in the frequency range of the chorus emissions and that the wave amplitudes grow between the threshold and optimum amplitudes. In the frame of the wave growth process, the nonlinear growth rates are much greater than the linear growth rates.

Dependence of Generation of Whistler Mode Chorus Emissions on the Temperature Anisotropy and Density of Energetic Electrons in the Earth's Inner Magnetosphere

AbstractWe carry out a series of self‐consistent electron hybrid code simulations for the dependence of chorus generation process on the temperature anisotropy and density of energetic electrons in the Earth's inner magnetosphere. We use the same magnetic field gradient in the simulation system and different temperature anisotropy AT for the initial distribution of energetic electrons at the magnetic equator. We conduct 6 sets of simulations for different AT from 4 to 9, changing the initial number density Nh of energetic electrons at the equator in each set of simulations. By analyzing the spectra obtained in the simulation results, we identify chorus elements with rising tones in the results for higher Nh but no distinct chorus in smaller Nh. We compare the simulation results with estimations of the threshold and optimum amplitude proposed by the nonlinear wave growth theory. We find that the chorus generation processes reproduced in the simulation results are consistently explained by the theoretical estimates. We also compare the simulation results with linear growth rates for all simulation runs. We find clear disagreement between the spectral characteristics of reproduced chorus and the predictions by the linear theory. The present study clarifies that the spectra of chorus are essentially different from those predicted by the linear theory and are determined fully by nonlinear processes of wave‐particle interactions in the chorus generation region.

Electron hybrid code simulation of whistler-mode chorus generation with real parameters in the Earth’s inner magnetosphere

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.

Observational evidence of the nonlinear wave growth theory of plasmaspheric hiss

AbstractWe test the recently developed nonlinear wave growth theory of plasmaspheric hiss against discrete rising tone elements of hiss emissions observed by the Van Allen Probes. From the phase variation of the waveforms processed by bandpass filters, we calculate the instantaneous frequencies and wave amplitudes. We obtain the theoretical relation between the wave amplitude and frequency sweep rates at the observation point by applying the convective growth rates and dispersion factors to the known relation at the equator. By plotting the theoretical relation over scatterplots of the wave amplitudes and the frequency sweep rates for rising tone elements, we find good agreement between the hiss observations and the nonlinear theory. We also find that the duration periods of the hiss elements are in good agreement with the nonlinear transition time necessary for the formation of a resonant current through coherent nonlinear wave‐particle interactions.

A statistical study of EMIC rising and falling tone emissions observed by THEMIS

AbstractElectromagnetic ion cyclotron (EMIC) waves with rising or falling frequency variations have been studied intensively because of their effects on energetic particles in the Earth's magnetosphere. We develop an automated classification method of EMIC events based on the characteristics of frequency variations. We report some basic statistical properties of frequency variations in EMIC waves observed over 5–10 RE by three Time History of Events and Macroscale Interactions during Substorms probes from January 2012 to December 2014. We clarify whether rising tones or falling tones are observed in each chosen 20 min time segment. In the present analysis, we find that the occurrence rate of EMIC rising or falling tone events is more than 30% of the total EMIC wave events. The dayside magnetosphere is a preferential region for the EMIC frequency variations. The occurrence rate of rising tone events is slightly greater than that of falling tone events. We examine the relation between the frequency characteristics and the magnetospheric conditions. The solar wind pressure strongly controls the occurrence rates of frequency variations. We also calculate ranges of frequency variations. Large‐amplitude EMIC waves tend to have wider frequency variations, and the range of frequency variation is largest around the prenoon region. In addition, rapid variations in wave amplitudes called “subpacket structures” are found in 70% of the EMIC rising or falling tone events in the dayside region. Subpacket structures appear mainly in large‐amplitude EMIC rising or falling tones. These features are consistent with nonlinear wave growth theory.

A direct link between chorus emissions and pulsating aurora on timescales from milliseconds to minutes: A case study at subauroral latitudes

AbstractA correlation was observed between chorus emissions and pulsating aurora (PA) from observations at Athabasca (L≈4.3) in Canada at 9:00–9:20 UT on 7 February 2013, using an electron multiplying charge‐coupled device camera and a VLF loop antenna with sampling rates of 110 Hz and 100 kHz, respectively. Pulsating aurora having a quasiperiodic variation in luminosity and a few hertz modulation was observed together with chorus emissions consisting of a group of successive rising‐tone elements. The repetition period and modulation frequency of the PA are in good agreement with those of the modulated chorus. After 9:11 UT, the temporal features of the aurora became aperiodic PA of indistinct modulation. Simultaneously, the rising‐tone chorus turned into chorus emissions consisting of numerous rising‐tone elements. The equatorial geomagnetic field inhomogeneity calculated using the Tsyganenko 2002 model shows a decreasing trend during the period. This result is consistent with nonlinear wave growth theory having a small geomagnetic field inhomogeneity, which contributes to a decrease in the threshold amplitude to trigger discrete chorus elements. These observations show a close connection between chorus emissions and PA on timescales from milliseconds for generation of discrete chorus elements on the microphysics of wave‐particle interaction to minutes for the variations of the geomagnetic field inhomogeneity related with the substorm activity.

Nonlinear wave growth theory of coherent hiss emissions in the plasmasphere

AbstractRecent 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.

Multiscale temporal variations of pulsating auroras: On‐off pulsation and a few Hz modulation

AbstractA statistical study on the cross‐scale property on the temporal variations of pulsating aurora intensity was conducted on 53 events observed at the Poker Flat Research Range during the period from 1 December 2011 to 1 March 2012. The observed modulation frequency ranged from 1.5 to 3.3 Hz, and strong modulations were not seen in the frequency range higher than about 3 Hz. This suggests that the time of flight of electrons has a time‐smoothing effect on the more rapid variations above 3 Hz. Furthermore, the frequency of modulation showed relatively strong correlation to auroral intensity (correlation coefficient of 0.58), and it can be explained with nonlinear wave growth theory, in which the modulation frequency increases with the wave amplitude of the whistler mode chorus. In contrast, the on‐off pulsations showed no significant correlations with auroral intensity. This result probably implies that several different plasma processes with different time scales from nonlinear wave growth should be taken into account when determining the on‐off periods. In particular, we suggest that long‐term variations in the cold plasma density play a dominant role in controlling the conditions of wave‐particle interactions that have temporal scale of the on‐off pulsation periods.

Spectrum characteristics of electromagnetic ion cyclotron triggered emissions and associated energetic proton dynamics

AbstractWe perform parametric analyses of electromagnetic ion cyclotron (EMIC) triggered emissions with a gradient of the nonuniform ambient magnetic field using a hybrid simulation. According to nonlinear wave growth theory, as the gradient of the ambient magnetic field becomes larger, the theoretical threshold of the wave amplitude becomes larger, although the optimum wave amplitude for nonlinear wave growth does not change. With a larger magnetic field gradient, we obtain coherent rising‐tone spectra because the triggering process of the EMIC triggered emission takes place only under a limited condition on the wave amplitude. On the other hand, with a smaller magnetic field gradient, triggering of the emissions can be caused with various wave amplitudes, and then the subpackets are generated at various locations at the same time. The concurrent triggering of emissions results in incoherent waves, observed as “broadband” EMIC bursts. Broadband emissions induce rapid precipitation of energetic protons into the loss cone since the scattering by the concurrent triggering takes place faster than that of the coherent emissions. The coherent triggered emission causes efficient proton acceleration around the equator because of the stable particle trapping by the coherent rising‐tone emission.

Electromagnetic ion cyclotron rising tone emissions observed by THEMIS probes outside the plasmapause

We report observations of electromagnetic ion cyclotron (EMIC) triggered emissions observed by the Time History of Events and Macroscale Interactions during Substorms (THEMIS) probes outside the plasmasphere. Although these phenomena have recently received much attention because of the possibility of strong interaction with particles, only a few events of EMIC triggered emissions have been reported near the equatorial plasmapause. We performed a survey of the THEMIS probe data and found various types of emissions mainly on the dayside at radial distances of 6–10 RE. We study three distinctive events in detail. The first is a typical event with an obvious rising tone emission in the afternoon sector. The emissions in the second event are simultaneously excited in different frequency bands separated by the cyclotron frequency of helium ions. In the third event, which occurred near local noon, rising tone emissions were excited in an extended region near the equator where the field‐aligned B gradient was much reduced because of compression of the magnetosphere by the solar wind. We compare these events with the nonlinear wave growth theory developed by Omura et al. (2010). In all events, it is found that the observed relationship between the amplitudes and frequencies of the emissions are in good agreement with the theory.

Frequency drift of Saturn chorus emission compared to nonlinear theory

The fine structure of nonlinear drifting‐frequency chorus is observed at Saturn by the Cassini Radio and Plasma Wave Investigation. During a high‐inclination orbit in which Cassini is at near‐constant L‐shell within about 10° of the magnetic equator, moderately intense nonlinear chorus is observed. Cassini observed a region of intense chorus and large bandwidth a few degrees on either side of the magnetic equator, with lower intensities and bandwidths observed nearest the magnetic equator. Using the observed plasma wave spectra and electron phase space distribution, we have measured plasma parameters within or near the chorus generation region and evaluated the theoretical value of the frequency sweep rate, ∂f/∂t, based on the nonlinear wave growth theory of Omura et al. and the backward wave oscillator theory of Trakhtengerts. Both theories produce rates that are within a factor of 2 of the observed values, but the nonlinear wave growth theory values are closer to the observations for the cases examined. The work presented is consistent with nonlinear theory in the generation of chorus, but also reveals a distinct region of weaker or linear chorus growth nearest to the magnetic equator at Saturn.

Precipitation of highly energetic protons by helium branch electromagnetic ion cyclotron triggered emissions

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.

THEMIS observation of chorus elements without a gap at half the gyrofrequency

Using waveform data obtained by one of the THEMIS satellites, we report properties of rising tone chorus elements without a gap at half the gyrofrequency in a region close to the magnetic equator. The wave normal angle of the chorus elements is typically field‐aligned in the entire frequency range of both upper‐band and lower‐band chorus emissions. We find that the observed frequency sweep rates are consistent with the estimation based on the nonlinear wave growth theory of Omura et al. (2008). In addition, we compare the frequency profiles of the chorus wave amplitudes with those of the optimum and threshold wave amplitudes derived from the nonlinear wave growth theory for triggering rising tone chorus emissions. The results of the comparison show a reasonable agreement, indicating that rising tone chorus elements are continually generated through a triggering process which generates elements with the optimum amplitudes for nonlinear growth.