Unified View of Nonlinear Wave Structures Associated with Whistler-Mode Chorus

Xin An, Jinxing Li, Jacob Bortnik, Viktor Decyk, Craig Kletzing, and George Hospodarsky

Phys. Rev. Lett. 122, 045101 – Published 30 January 2019

Abstract

A range of nonlinear wave structures, including Langmuir waves, unipolar electric fields, and bipolar electric fields, are often observed in association with whistler-mode chorus waves in near-Earth space. We demonstrate that the three seemingly different nonlinear wave structures originate from the same nonlinear electron trapping process by whistler-mode chorus waves. The ratio of the Landau resonant velocity to the electron thermal velocity controls the type of nonlinear wave structures that will be generated.

Received 6 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.045101

Physics Subject Headings (PhySH)

Electrostatic waves & oscillations Nonlinear phenomena in plasmas Plasma microinstabilities Whistler plasma waves

Space & astrophysical plasma

Particle-in-cell methods

Plasma Physics

Authors & Affiliations

Xin An^1,*, Jinxing Li¹, Jacob Bortnik¹, Viktor Decyk², Craig Kletzing³, and George Hospodarsky³

¹Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, California 90095, USA
²Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
³Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA

^*xinan@atmos.ucla.edu

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Vol. 122, Iss. 4 — 1 February 2019

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Figure 1
Typical nonlinear wave structures associated with whistler-mode chorus waves from spacecraft observations. (a)–(f) The parallel (red) and perpendicular (blue) electric waveforms measured by the EMFISIS instrument aboard Van Allen Probes. (a),(c),(e) Example with chorus propagation parallel to the background magnetic field. (b),(d),(f) Example with chorus propagation antiparallel to the background magnetic field. (a),(b) Chorus-related Langmuir waves. (c),(d) Chorus-related unipolar electric fields. (e),(f) Chorus-related bipolar electric fields.
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Figure 2
Langmuir waves driven by whistler-mode chorus in simulation 1 with $v_{r} / v_{th} = 3.2$ . (a) The temporal evolution of the wave number spectrum of the longitudinal electric field $δ E_{L}$ (along the direction of wave propagation). The horizontal axis is shown in units of time normalized to the plasma frequency $ω_{p e}$ . The pump field is turned off at $t_{off} = 1100 ω_{p e}^{- 1}$ . The mode number of whistler-mode chorus is 10, which represents the number of wavelengths in the system. Two harmonics of whistler occur consecutively in time at mode numbers 20 and 30. Langmuir waves are located mainly in the mode number range 300–400. (b) The dispersion diagram, obtained by Fourier transforming the wave number spectrum from the time domain to the frequency domain. The whistler frequency is $0.0215 ω_{p e}$ . Beam-mode Langmuir waves are found just below $ω_{p e}$ . The reciprocal of the slope of the dash-dotted line represents the phase velocity of whistler, which is the same as that of the whistler harmonics, but is smaller than that of Langmuir waves. (c) The waveforms of the electric fields parallel (red) and perpendicular (blue) to $B_{0}$ at $t_{0} = 1540 ω_{p e}^{- 1}$ . The horizontal axis is the spatial coordinate $x$ normalized by grid size $Δ_{x}$ . Langmuir waves are seen in $δ E_{∥}$ . (d) The phase space portrait at $t_{0} = 1540 ω_{p e}^{- 1}$ . The magnitude of phase space density $f$ is coded in color. The separatrix (black line) encircles the resonant island, dividing trapped and untrapped particles. Trapped particles are bounded in phase in the wave frame of the propagating whistler waves, whereas untrapped particles are able to freely oscillate in phase.
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Figure 3
The wave number spectrum and dispersion diagram in simulation 2 with $v_{r} / v_{th} = 2.1$ . (a) The temporal evolution of the wave number spectrum of the longitudinal electric field. The pump field is turned off at $t_{off} = 750 ω_{p e}^{- 1}$ . The mode number of the whistler wave is seven. Electron acoustic waves start to be excited in the interval $1000 < t ω_{p e} < 2000$ . (b) The dispersion diagram for $0 < t ω_{p e} \leq 2000$ . The phase velocity of electron acoustic waves is slightly larger than that of the whistler waves (dash-dotted line). (c) The dispersion diagram for $2000 < t ω_{p e} \leq 4000$ , showing that all the whistler harmonics propagate at the same phase speed as the fundamental whistler wave itself. Beam-mode waves with phase velocities smaller than that of the whistler wave propagate in both forward and backward directions.
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Figure 4
The distortion of whistler wave electric field into unipolar electric field in simulation 2 with $v_{r} / v_{th} = 2.1$ . (a) The waveforms of the electric fields parallel (red) and perpendicular (blue) to $B_{0}$ at $t_{1} = 1300 ω_{p e}^{- 1}$ . Electron acoustic waves are seen in $δ E_{∥}$ . (b) The phase space portrait at $t_{1} = 1300 ω_{p e}^{- 1}$ . A spatially modulated beam is formed within the separatrix (black line). (c) The waveforms of the electric fields parallel (red) and perpendicular (blue) to $B_{0}$ at $t_{2} = 2500 ω_{p e}^{- 1}$ . The whistler electric field is distorted into the unipolar electric field shown in $δ E_{∥}$ . (d),(e) The phase space portrait at $t_{2} = 2500 ω_{p e}^{- 1}$ , color coded by phase space density (d) and perturbed phase space density (e). The signed logarithm of the perturbed phase space density ${snglog}_{10} (δ f)$ is defined as $s n g (δ f) \cdot {l o g}_{10} (| δ f |)$ . The electrons are accumulated in phases where the separatrix (black line) is narrowed. An example of electron accumulation is indicated by the green box. Several small beams are seen in the velocity range $- 2 < v_{∥} / v_{th} < 1$ .
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Figure 5
Bipolar electric field structures driven by whistler-mode chorus waves in simulation 3 with $v_{r} / v_{th} = 1.0$ . (a) The temporal evolution of the wave number spectrum of the longitudinal electric field. The pump field is turned off at $t_{off} = 500 ω_{p e}^{- 1}$ . The mode number of the whistler wave is seven. The transient broadband spectrum with mode numbers ranging between 20 and 100 corresponds to the bipolar field structures. (b) The dispersion diagram showing that the phase velocity of bipolar structures is around that of whistler (dash-dotted line). (c) The waveforms of the electric fields parallel (red) and perpendicular (blue) to $B_{0}$ at $t_{3} = 1200 ω_{p e}^{- 1}$ . Bipolar electric fields are seen in $δ E_{∥}$ . (d) The phase space portrait at $t_{3} = 1200 ω_{p e}^{- 1}$ . Each phase space hole is a region of electron deficit inside the separatrix (black line), generating a bipolar electric field structure. An example of a phase space hole is indicated by the green arrow.
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