Analysis of wave processes using beam-driven Langmuir/$\mathcal{Z}$-mode waveforms generated in Particle-In-Cell simulations

TL;DR

The paper addresses how Type III solar radio bursts arise from the competition between nonlinear electrostatic decay (ESD) of beam-driven Langmuir/$\mathcal{Z}$-mode waves and linear mode conversion (LMC) on random density fluctuations, which together produce fundamental and harmonic electromagnetic emissions. Using large-scale 2D PIC simulations with ensembles of virtual satellites, the authors statistically analyze localized wave processes and directly compare with in situ solar wind waveform observations, tracking how $\Delta N$ and the magnetization ratio $\omega_c/\omega_p$ shape wave dynamics. They find that density turbulence shifts the balance toward linear transformations (notably LMC) that can trigger EMD/ESD cascades earlier, while homogeneous plasmas show a higher occurrence of three-wave resonant ESD with substantial phase coherence; magnetization mostly modulates the small-$k$ LZ energy and magnetic signatures but does not erase ESD cascades. Overall, the work clarifies how linear and nonlinear pathways interact under realistic solar wind conditions, providing a robust framework to interpret spacecraft waveform data and guiding the analysis of future Type III radio burst observations.

Abstract

During Type III solar radio bursts, beam-driven upper-hybrid wave turbulence is converted into electromagnetic emissions at the fundamental plasma frequency and its harmonic, through a chain of various linear and nonlinear wave processes. In this work, we mainly investigate the relative roles and interplay of two key mechanisms: the nonlinear decay of Langmuir/$\mathcal Z$-mode waves and their linear transformations on random density fluctuations and, in particular, their mode conversion at constant frequency into electromagnetic waves. Using two-dimensional Particle-In-Cell simulations, we employ a diagnostic approach based on large ensembles of virtual satellites that record local waveforms, enabling detailed temporal and spatial characterization of wave processes in randomly inhomogeneous plasmas. This method allows robust statistical analysis and direct comparison with spacecraft observations. The study focuses on the dependence of wave dynamics on the average level of density fluctuations and the plasma magnetization. Our results quantify the occurrence rate of decay under varying physical conditions and demonstrate how developed plasma density turbulence can significantly alter the balance between nonlinear wave-wave interactions and linear wave transformations. These findings provide new insights into the mechanisms responsible for electromagnetic emissions during type III radio bursts and strengthen the connection between numerical simulations and in situ solar wind measurements, offering a valuable framework for the interpretation of future space-based waveform observations.

Figures (10)

• Figure 1: Waveforms in a homogeneous and unmagnetized plasma ($\Delta N=0$, $\omega_{c}=0$). (a) Time variations of the parallel (gray) and perpendicular (red) electric fields $E_{\parallel}(t)$ and $E_{\perp}(t)$. (b) Spectral electric field energy $|E|^{2}$ as a function of the normalized Doppler-shifted frequency $\omega^{D}/\omega_{p}$ and the time $\omega_{p}t.$ (c) Time variations of the ion density perturbation $\delta n_{i}(t)/n_{0}$. (d) Low-frequency spectral energy $|\delta n_{i}/n_{0}|^{2}$ in the map ($\omega^{D}/\omega_{p}$, $\omega_{p}t$). (e) High-frequency wave energy spectra $|E_{\parallel}|^{2}$ (black) and $|E_{\perp}|^{2}$ (red), calculated in the time interval $1000\lesssim\omega_{p}t\lesssim6000,$ as a function of $\omega^{D}/\omega_{p}$. (f) Low-frequency wave energy spectrum $|\delta n_{i}/n_{0}|^{2}$, in the same time interval and in linear scale, as a function of $\omega^{D}/\omega_{p}$. (g) Corresponding squared cross-bicoherence $b_{c}^{2}$ calculated for the triplet $(E_{\parallel},\delta n_{i},E_{\parallel})$, in the map ($\omega_{E_{\parallel}}^{D},\omega_{\delta n_{i}}^{D}$); the extrema $b_{c}\simeq0.72$ and $b_{c}\simeq0.68$, represented by stars, appear at $(\omega_{\mathcal{L}}^{D},\omega_{\mathcal{S}^{\prime}}^{D})=(0.978,0.0497)\omega_{p}$ (first cascade) and $(\omega _{\mathcal{L}^{\prime \prime }}^{D},\omega _{\mathcal{S}^{\prime \prime }}^{D})=(0.984,0.042)\omega_{p}$ (second cascade), respectively. All variables are in arbitrary units.
• Figure 2: High- and low-frequency energy spectra averaged over $N_s=256$ waveforms, as a function of the normalized Doppler-shifted frequency $\omega^{D}/\omega_{p}$. (a-b) Parallel (black) and perpendicular (red) electric field spectra $\left\langle |E_{\parallel}|^{2}\right\rangle$ and $\left\langle |E_{\perp}|^{2}\right\rangle$, in the time intervals $1000\lesssim\omega_{p}t\lesssim6000$ (a) and $6000\lesssim\omega_{p}t\lesssim15,000$ (b). (c-d) Low-frequency energy spectra $\left\langle |\delta n_{i}/n_{0}|^{2}\right\rangle$, in the same time intervals as (a) and (b), respectively. (a-b) : logarithmic scales. (c-d) : linear scales. All variables are in arbitrary units.
• Figure 3: (a) Wave distribution in the map ($\left\vert\Delta\omega_{\mathcal{LL}^{\prime}}^{D}\right\vert/\omega_p$, $\omega_{\mathcal{S}^{\prime}}^{D}/\omega_p$), obtained by using $N_s=150$ selected spectra consistent with ESD occurrence ($3$ peaks identified) out of a set of $256$; the dashed line represents the three-wave resonance condition $\omega_{\mathcal{L}}^{D}=\omega_{\mathcal{L}^{\prime}}^{D}-\omega_{\mathcal{S}^{\prime}}^{D}$. (b) Squared cross-bicoherence $\langle b_c\rangle^2$ of the triplet ($E_{\parallel},\delta n_{i},E_{\parallel}$), calculated in the time interval $1000\lesssim\omega_{p}t\lesssim6000,$ averaged over $N_{bc}=50$ selected waveforms satisfying at best the resonance condition and consistent with ESD occurrence, in a large frequency ($\omega_{\mathcal{L}}^{D},\omega_{\mathcal{S}^{\prime}}^{D}$) region. (c) Zoom of (b) in the region $0.95<\omega_{\mathcal{L}}^{D}/\omega_p<1.1$ and $0.03<\omega_{\mathcal{S}^{\prime}}^{D}/\omega_p<0.1$; the dashed line represents the theoretical curve $(\omega _{\mathcal{L}}^{D}(k),\omega _{\mathcal{S}^{\prime }}^{D}(2k-k_0))$ derived using the ESD resonance condition $\omega_{\mathcal{L}}^{D}=\omega_{\mathcal{L}^{\prime}}^{D}-\omega_{\mathcal{S}^{\prime}}^{D}$ and the wave dispersion relations. All variables are normalized.
• Figure 4: Waveforms of the parallel and perpendicular electric fields $E_{\parallel}(t)$ (gray) and $E_{\perp }(t)$ (red), to which the time variations of the normalized ion density perturbation $\delta n_{i}(t)/n_{0}$ are superimposed in (b)-(c) (green lines and right axes), for a plasma with different average levels of density fluctuations : $\Delta N=0$ (a), $\Delta N=0.025$ (b), and $\Delta N=0.05$ (c). Electric fields are in arbitrary units.
• Figure 5: Waveforms in a randomly inhomogeneous plasma with $\Delta N=0.025$ and $\omega_{c}=0$. (a) Time variations of the parallel (gray) and perpendicular (red) electric fields $E_{\parallel}(t)$ and $E_{\perp}(t)$ (left axis), as well as of the superimposed ion density perturbation $\delta n_i(t)/n_0$ (blue, right axis). (b) Spectral electric field energy $|E|^{2}$ in the map ($\omega_{p}t$, $\omega^{D}/\omega_{p}$). (c) Time variation of the induced ion density perturbation $\delta \tilde{n}_i(t)/n_{0}$ (applied density fluctuations $\delta n(t)$ have been removed from $\delta n_i(t)$ by filtering). (d) Low-frequency spectral energy $|\delta \tilde{n}_{i}/n_{0}|^{2}$ in the map ($\omega_{p}t$, $\omega^{D}/\omega_{p}$). (e) Spectral magnetic field energy $|B_{\perp 2}|^{2}$ in the map ($\omega_{p}t$, $\omega^{D}/\omega_{p}$). (f) High-frequency wave energy spectra $|E_{\parallel}|^{2}$ (black) and $|E_{\perp}|^{2}$ (red) versus $\omega^{D}/\omega_{p}$, calculated in the time interval $\Delta T=[500,6000]\omega_{p}^{-1}$; green labels and vertical lines indicate the excited Langmuir waves and their spectral peaks. (g) Low-frequency wave energy spectrum $|\delta \tilde{n}_{i}/n_{0}|^{2}$ versus $\omega^{D}/\omega_{p}$, calculated within $\Delta T$; green labels indicate the excited ion acoustic waves near their spectral peaks. (h) Magnetic wave energy spectrum $|B_{perp 2}|^2$ versus $\omega^{D}/\omega_{p}$, calculated within $\Delta T$. (i) Squared cross-bicoherence $b_{c}^{2}$ calculated within $\Delta T$ for the triplet $(E_{\parallel},\delta \tilde{n}_{i},E_{\parallel})$, in the map ($\omega_{E_{\parallel}}^{D},\omega_{\delta \tilde{n}_i}^{D}$)$/\omega_p$; $b_{c}\simeq0.78$ at $(\omega_{\mathcal{L}}^{D},\omega_{\mathcal{S}^{\prime}}^{D})=(0.975,0.058)\omega_{p}$ (first cascade); $b_{c}\simeq0.91$ at $(\omega_{\mathcal{L}^{\prime\prime}}^{D},\omega_{\mathcal{S}^{\prime\prime}}^{D})=(0.99,0.047)\omega_{p}$ (second cascade); $b_{c}\simeq0.67$ at $(\omega_{\mathcal{L}^{\prime\prime}}^{D},\omega_{\mathcal{S}^{(3)}}^{D})=(0.99,0.037)\omega_{p}$ (third cascade); positions in the frequency map are indicated by stars. (j) Squared cross-bicoherence $b_{c}^{2}$ calculated within $\Delta T$ for the triplet $(B_{\perp 2},\delta \tilde{n}_{i},E_{\parallel})$, in the map ($\omega_{B_{\perp 2}}^{D},\omega_{\delta \tilde{n}_i}^{D}$)$/\omega_p$; $b_{c}\simeq0.78$ at $(\omega_{\mathcal{F}}^{D},\omega_{\mathcal{S}_\mathcal{F}}^{D})=(1.01,0.04)\omega_{p}$, as indicated by stars. Parameters are the same as in Figure \ref{['fig1']}, but with $\Delta N=0.025$. All variables are in arbitrary units.