Nonlinear dynamics of resonant electrons interacting with coherent Langmuir waves

Tobita, M.; Omura, Yoshiharu

doi:10.1063/1.5018084

Cited by 14 publications

(10 citation statements)

References 11 publications

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“…We consider a factor

F false(S_{n} false)

evaluating the normalized size of trapping potential in the velocity phase space (Omura et al, ; Tobita & Omura, ). The

F false(S_{n} false)

is treated as a proportion of trapped particles with the same

K

and

α

and is given by

F (S_{n}) = \{\begin{array}{ll} \sqrt{1 - S_{n}^{2}} + ({, \sin^{- 1} (||, S_{n}) - \frac{π}{2}) (||, S_{n}), & if false| S_{n} false| < 1 \\ 0, & otherwise. \end{array}

Multiplying the acceleration rate

W_{n}

, the trapping time

normalΔ T_{n}

, and the trapped proportion

F false(S_{n} false)

, we obtain the energy gain

normalΔ H_{n}

of trapped electrons within a subpacket period

δ t

= 0.02 s, which is a typical value of a subpacket.…”

Section: Results Of Test Particle Simulationsmentioning

confidence: 99%

“…According to (45) and (57) of Omura et al () larger wave amplitude leads to smaller inhomogeneity factor

S_{n}

. In other words, if the amplitude is larger, the distance along a field line with

false| S_{n} false| < 1

is longer (Omura et al, ), and the wave potential to entrap electron is wider (Tobita & Omura, ), resulting in more electrons undergoing nonlinear trapping. Equations and indicate that larger wave amplitude leads to greater acceleration rate

W_{n}

.…”

Section: Results Of Test Particle Simulationsmentioning

confidence: 99%

“…We consider a factor F(S n ) evaluating the normalized size of trapping potential in the velocity phase space Tobita & Omura, 2018). The F(S n ) is treated as a proportion of trapped particles with the same K and and is given by…”

Section: Acceleration Via Landau and Cyclotron Resonancesmentioning

confidence: 99%

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Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions

2020

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Nonlinear whistler mode wave‐particle interaction is one of the processes to generate relativistic electrons in the Earth's outer radiation belt. Applying test particle simulations with a pair of whistler mode chorus emissions, we traced a large number of electrons in various initial conditions along an L=4.5 magnetic field line to build a set of Green's functions for analyzing evolution of the electron distribution under the chorus emissions. Employing convolution integral for the Green's functions, we tracked the formation of relativistic electron fluxes from an injected energetic electron through interaction with consecutive chorus emissions. We compared the simulation results among a purely parallel wave model and two oblique wave models with wave normal angles gradually increased from 0° to 20° and 60°. Multiple resonances result in a higher probability of nonlinear trapping. Hence, the trapped electrons for the oblique cases are scattered to a wider range of pitch angles than those for the parallel case. Oblique chorus can accelerate 10–30 keV electrons to MeV level rapidly in several wave emission cycles (about 10 s), which is much faster than that of the parallel wave model. The study shows comprehensive evolutions of electron fluxes interacting with oblique chorus emissions through cyclotron and Landau resonances in the outer radiation belt.

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“…We consider a factor

F false(S_{n} false)

evaluating the normalized size of trapping potential in the velocity phase space (Omura et al, ; Tobita & Omura, ). The

F false(S_{n} false)

is treated as a proportion of trapped particles with the same

K

and

α

and is given by

F (S_{n}) = \{\begin{array}{ll} \sqrt{1 - S_{n}^{2}} + ({, \sin^{- 1} (||, S_{n}) - \frac{π}{2}) (||, S_{n}), & if false| S_{n} false| < 1 \\ 0, & otherwise. \end{array}

Multiplying the acceleration rate

W_{n}

, the trapping time

normalΔ T_{n}

, and the trapped proportion

F false(S_{n} false)

, we obtain the energy gain

normalΔ H_{n}

of trapped electrons within a subpacket period

δ t

= 0.02 s, which is a typical value of a subpacket.…”

Section: Results Of Test Particle Simulationsmentioning

confidence: 99%

“…According to (45) and (57) of Omura et al () larger wave amplitude leads to smaller inhomogeneity factor

S_{n}

. In other words, if the amplitude is larger, the distance along a field line with

false| S_{n} false| < 1

W_{n}

.…”

Section: Results Of Test Particle Simulationsmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions

2020

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“…Given a finite radial scale of ULF waves, our scenario may require adjustments if particle's radial displacement is large compared to half width of the wave excitation (Wang et al, ). The azimuthal scale of ULF waves, as discussed in Degeling et al (), could provide an inhomogeneity factor for the pendulum equation to enable significant particle scattering (Tobita & Omura, ). Similar effects may also appear if a convective electric field is introduced to the model.…”

Section: Summary and Discussionmentioning

confidence: 99%

Nonlinear Drift Resonance Between Charged Particles and Ultralow Frequency Waves: Theory and Observations

Zhou

Omura

et al. 2018

Geophysical Research Letters

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In Earth's inner magnetosphere, electromagnetic waves in the ultralow frequency (ULF) range play an important role in accelerating and diffusing charged particles via drift resonance. In conventional drift resonance theory, linearization is applied under the assumption of weak wave‐particle energy exchange so particle trajectories are unperturbed. For ULF waves with larger amplitudes and/or durations, however, the conventional theory becomes inaccurate since particle trajectories are strongly perturbed. Here we extend the drift resonance theory into a nonlinear regime, to formulate nonlinear trapping of particles in a wave‐carried potential well, and predict the corresponding observable signatures such as rolled‐up structures in particle energy spectrum. After considering how this manifests in particle data with finite energy resolution, we compare the predicted signatures with Van Allen Probes observations. Their good agreement provides the first observational evidence for the occurrence of nonlinear drift resonance, highlighting the importance of nonlinear effects in magnetospheric particle dynamics under ULF waves.

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“…The size of the trapping potential in the velocity phase space is given by

\frac{normalΔ V}{V_{tr}} = \sqrt{\frac{\cos ζ_{1} - \cos ζ_{0} + false(ζ_{0} - ζ_{1} false) S}{2},}

where

\cos ζ_{0}

and

\cos ζ_{1}

are solutions of the second‐order resonance conditions:

\sin ζ + S = 0

, and V tr is the trapping velocity, that is, the size of the potential around the resonance velocity for S = 0. The phases ζ 0 and ζ 1 are stable equilibrium point for trapped electrons and the saddle point for the separatrix of the trapping potential (Tobita & Omura, ). Since the trapping potential is two‐dimensional in the velocity phase space, we can evaluate the number of trapped electrons by a function F t ( S ) = (Δ V / V tr ) 2 .…”

Section: Evaluation Of Electron Acceleration Based On Wave Observationsmentioning

confidence: 99%

Cyclotron Acceleration of Relativistic Electrons Through Landau Resonance With Obliquely Propagating Whistler‐Mode Chorus Emissions

et al. 2019

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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. Key Points: • Oblique chorus waves can accelerate relativistic electrons effectively through Landau resonance • The acceleration is due to the perpendicular component of the wave electric field • Nonlinear trapping conditions and efficiency of electron acceleration are confirmed by spacecraft observation

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Nonlinear dynamics of resonant electrons interacting with coherent Langmuir waves

Cited by 14 publications

References 11 publications

Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions

Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions

Nonlinear Drift Resonance Between Charged Particles and Ultralow Frequency Waves: Theory and Observations

Cyclotron Acceleration of Relativistic Electrons Through Landau Resonance With Obliquely Propagating Whistler‐Mode Chorus Emissions

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