Evidence of Langmuir/$\mathcal{Z}$-mode Wave Decay into $\mathcal{Z}$-mode Electromagnetic Radiation in the Solar Wind

TL;DR

This study provides the first observational evidence that beam-driven Langmuir/$\mathcal{Z}$-mode waves nonlinearly decay into electromagnetic $\mathcal{Z}$-mode radiation at the plasma frequency $f_p$ in the solar wind. Using high-resolution electric and magnetic field data from Solar Orbiter's RPW, the authors identify three-wave resonance conditions, strong phase coherence via cross-bicoherence, and tight temporal correlation between interacting waves, with results fully consistent with theoretical predictions. A second, independent event corroborates the decay, and 2D/3V PIC simulations reproduce the observed waveforms and dynamics, including wave trapping in broad density wells that facilitates decay. Collectively, these findings clarify the generation mechanism for $f_p$-localized radiation during Type III bursts and bridge spacecraft measurements with kinetic simulations, enhancing our understanding of beam-plasma interactions in the solar wind.

Abstract

The nonlinear decay of Langmuir/$\mathcal{Z}$-mode waves into electromagnetic $\mathcal{Z}$-mode wave radiation at the plasma frequency is observed for the first time in the solar wind. This finding was enabled by the unprecedented high-resolution electric and magnetic field measurements provided by the Radio Plasma Waves (RPW) instrument aboard the Solar Orbiter spacecraft, which encountered an electron beam associated with a Type III radio burst. The decay process is definitively identified through multiple lines of evidence: satisfaction of frequency and wavevector resonance conditions, strong phase coherence and temporal coincidence between the interacting waves, exclusion of competing mechanisms, and full agreement with theoretical predictions. Particle-in-cell simulations, conducted under close beam-plasma conditions, successfully reproduce the key features of the observations. Notably, they suggest that the wave packet observed by Solar Orbiter may be trapped within an extended, nearly flat-bottomed density well, where the decay process is not overcome by wave scattering on random density fluctuations and subsequent mode conversion effects.

Figures (5)

• Figure 1: Snapshot captured in the survey mode by the Solar Orbiter instrument RPW, on 22 September 2022, at 13:52:28 UT. (a) Waveform of the two electric field components $E_z$ (blue) and $E_y$ (red), in the SRF frame. (b) Hodograms $E_y(E_z)$ calculated within equidistant time windows of $0.7$ ms. (c-d) Wavelet spectrograms of the electric energy $|E(f,t)|^2=|E_y(f,t)|^2+|E_z(f,t)|^2$ in the high- and low-frequency ranges, i.e. 45 $\leq f\leq48$ kHz and $f \leq 1$ kHz, respectively. (e) Wavelet spectrogram of the magnetic energy $|B(f,t)|^2=|B_x(f,t)|^2$, in the same frequency range as (c). (c-e) : the color bars are in logarithmic scales. (f-i) Power spectra of the field components $E_z$ (blue), $E_y$ (red) and $cB_x$ (green), respectively. Zoomed-in views are presented, in logarithmics and linear scales, in the ranges $44\leq f\leq48$ kHz (g-h) and $f\leq1$ kHz (i); the black curve in (i) represents the spectrum of $|E|^2$. The dashed vertical lines correspond to peaks at $f_{3}\simeq45.74$ kHz, and $f_{1}\simeq46.04$ kHz (g-h), and at $f_{1}'\simeq0.3$ kHz (i). Spectra in (f-i) are calculated in the time interval 2 ms $\leq t\leq22$ ms.
• Figure 2: Determination of phase coherence between waves using cross-bicoherence. (a) Square of the cross-bicoherence $b_c^2(f_{B_x},f_{E_z})$ computed with the field triad $(B_x,E_z,B_x)$ over the time interval $0 \leq t \leq 62$ ms, with $\Delta T=15$ ms and $\Delta t=1$ ms (see equation \ref{['bico']} and text), in the frequency ranges $f_{B_x}\leq 60$ kHz and $f_{E_z} \leq 60$ kHz. (b) Zoom-in of (a) in the map region 44 kHz $\leq f_{B_x} \leq 48$ kHz and $f_{E_z}\leq 1$ kHz, with $b_c\simeq0.7$ at $(f_{B_x},f_{E_z})=(f_\mathcal{Z},f_\mathcal{S})\simeq(45.74,0.3)$ kHz (black point). (c) Square of the cross-bicoherence averaged on 18 different triads with $E_z$ or $E_y$ in the second position, $\langle b_c(f_1,f_2)\rangle^2$, in the same frequency range as (a). (d) Zoom-in of (c) in the same frequency range as in (b), with $b_c\simeq0.7$ at $(f_{B_x},f_{E_z})\simeq(45.74,0.3)$ kHz (black point). (b,d) : The white vertical lines indicate the local plasma frequency. The magenta curves represented on the borders of the panels refer to the high- and low-frequency spectra used to calculate $b_c$ (see Figures \ref{['fig:snapshot111']}(h,i)). (e) Variations with time of the $\mathcal{LZ}$, $\mathcal{Z}$ and $\mathcal{S}$ wave energies obtained by integration of the wavelet spectrograms of $E$, $B$ and $E$ fields, respectively, in the ranges $45.2\textnormal{ kHz}\leq f\leq 47.8\textnormal{ kHz}$ for $\mathcal{LZ}$ and $\mathcal{Z}$-mode waves, and $f\leq0.4\textnormal{ kHz}$ for $\mathcal{S}$ waves.
• Figure 3: Snapshot captured in the survey mode by RPW on 22 September 2022, at 14:05:04 UT. (a) Waveform of the two electric field components $E_z$ (blue) and $E_y$ (red), in the SRF frame. Hodograms $E_y(E_z)$ calculated within equidistant time windows of $0.7$ ms. (c-d) Wavelet spectrograms of the electric energy $|E(f,t)|^2$ in the high- and low-frequency ranges, i.e. 45 kHz $\leq f\leq48$ kHz and $f \leq 0.7$ kHz, respectively. (e) Wavelet spectrogram of the magnetic energy $|B(f,t)|^2$, in the same frequency range as (c). (c-e) : the color bars are in logarithmic scales. (f-i) Power spectra of the field components $E_z$ (blue), $E_y$ (red) and $cB_x$ (green), respectively. Zoomed-in view are presented, in linear scales, in the ranges 45 kHz $\leq f\leq48$ kHz (g-h) and at $f\leq1.7$ kHz (i). (h-i) : linear scales; the black curve represents the spectrum of $|E|^2$. The vertical lines correspond to peaks at $f_{3}\simeq52.71$ kHz and $f_{1}\simeq53.05$ kHz (g-h), and at $f_{1}'\simeq0.18$ kHz (i). Spectra in (f-i) are calculated in the time interval $0\leq t\leq62$ ms.
• Figure 4: Determination of the phase coherence between waves using cross-bicoherence. (a) Square of the cross-bicoherence $b_c^2(f_{B_x},f_{E_y})$ computed with the field triad $(B_x,E_y,E_y)$ over the time interval $0 \leq t \leq 62$ ms, with $\Delta T=15$ ms and $\Delta t=1$ ms (see equation \ref{['bico']} and text), in the frequency range $f_{B_x},f_{E_y} \leq 60$ kHz. (b) Zoom-in of (a) in the domain $52\textnormal{ kHz} \leq f_{B_x} \leq 53.7$ kHz and $f_{E_y} \leq 1$ kHz, with $b_c\simeq0.65$ at $(f_{B_x},f_{E_y})=(f_\mathcal{Z},f_\mathcal{S})\simeq(52.86, 0.18)$ kHz (black point). (c) Square of the cross-bicoherence averaged over the 18 triads with $E_y$ or $E_z$ in the second position, $\langle b_c(f_1,f_2)\rangle^2$, in the same frequency range as (a). (d) Zoom-in of (c), in the same frequency domain as in (b), with $b_c\simeq0.55$ at $(f_{B_x},f_{E})=(f_\mathcal{Z},f_\mathcal{S})$ kHz (black point). (b,d) : The white vertical line indicates the local plasma frequency. The pink curves represented on the borders of the panel are the high- and the low-frequency spectra used to calculate $b_c$ (see Figures \ref{['fig:snapshot113']}(h,i)).
• Figure 5: Waveform recorded by a virtual satellite moving into a 2D PIC simulation plane with the velocity $|v_{s}|=0.1v_T$, in the direction opposite to $\mathbf{B}_0$. Plasma and beam parameters are $\Delta N=\langle (\delta n/n_0)^2\rangle^{1/2}=0.025$, $f_c/f_p=0.02$, and $v_b=0.25c$. (a) Waveforms of the parallel $E_x=E_\parallel$ (blue) and perpendicular $E_{\perp}\simeq E_y$ (red) electric fields ($E_z$ is negligibly small). The green line represents the normalized ion density $n_i/n_0$. (b) Spectrograms of the electric energy $|E(f,t)|^2=|E_x(f,t)|^2+|E_y(f,t)|^2+|E_z(f,t)|^2$ in the frequency range $0.97\leq f/f_p\leq1.08$; the white line represents the local plasma frequency $f_{pl}$ normalized to the average plasma frequency $f_p$. (c) Spectrogram of $|\delta n_i(f,t)/n_0|^2$ in the range $0.005\leq f/f_p\leq0.035$. (d) Spectrogram of $|B(f,t)|^2$ in the same frequency range as (b). (e) Power spectra of the electric field energy $|E(f)|^2$ (black), in the range $0.97\leq f/f_p\leq1.06$; parallel and perpendicular energies are shown in blue and red, respectively. (f) Power spectra of the magnetic field energy $|B(f)|^2$ (black), in the same frequency range as (e); the three field components are indicated in color. (e-f) : spectra are calculated in the time interval $500\leq f_pt\leq4000$. (g) Low-frequency power spectrum $|\delta n_i(f)/n_0|^2$ in the range $f/f_p\leq0.035$. (e-g) : vertical dashed lines indicate the frequencies $f_3\simeq0.999f_p$, $f_2\simeq1.004f_p$ and $f_1\simeq1.012f_p$ (e,f) as well as $f_3'\simeq0.004f_p$, $f_2'\simeq0.009f_p$ and $f_1'\simeq0.0125f_p$ (g). (h-i) Square of the cross-bicoherence averaged on the 4 triads $(B_x, \delta n_i,E_x)$, $(B_y,\delta n_i,E_x)$, $(B_x,\delta n_i,E_y)$ and $(B_y,\delta n_i,E_y)$, in the ranges $f/f_p\leq1.1$ (h), as well as $0.97\leq f_1/f_p\leq1.06$ and $f_2/f_p\leq0.06$ (i). Sliding windows of duration $\Delta T=1300f_p^{-1}$, separated by $\Delta t=800f_p^{-1}$, are employed to calculate $b_c$ over the time range $f_pt\leq5000$.