Mechanisms of Fundamental Electromagnetic Wave Radiation in the Solar Wind

Large-scale and long-term two-dimensional particle-in-cell simulations performed for parameters relevant to type III solar radio bursts have provided new results on the generation mechanisms of fundamental electromagnetic waves radiated at the plasma frequency ω p . The paper first considers the nonlinear wave interaction process of electromagnetic decay (EMD) in a homogeneous solar wind plasma with an electron-to-ion temperature ratio T e /T i > 1. The dynamics of ion-acoustic waves (dispersion, spectra, growth/damping) is studied, and signatures confirming the three-wave interactions (cross-bicoherence, correlations between waves’ phases and between waves' growths, resonance conditions) are provided. The decisive role played in EMD by the backscattered Langmuir waves coming from the electrostatic decay (ESD) is demonstrated. EMD can be triggered by ion acoustic waves coming from the two cascades of the faster and more intense ESD. The same study is then performed in a solar wind plasma with random density fluctuations. In this case, EMD is not suppressed but develops only within plasma regions of reduced or quasi-uniform density. It coexists with linear mode conversion (LMC) of Langmuir waves into electromagnetic radiation, which is the fastest and most prominent process, as well as with ESD. LMC can lead to enhanced occurrence of EMD in the early stage. Moreover, the impact of T e /T i on electromagnetic energy growth and saturation is shown to be rather weak. Ion-acoustic waves are heavily damped at T e ∼ T i , so that EMD is overcome by nonlinear induced scattering on thermal ions. In actual solar wind plasmas, EMD should be more easily observed in plasma regions weakly perturbed by the background density turbulence and where ion temperature is decreased.


Introduction
The exceptional capabilities of recent ground-based and space-born missions have resulted in an abundant harvest of observations on type III solar radio bursts (Reid & Ratcliffe 2014;Reid 2020, and references therein), such as their localization in conjunction with solar flares (Krupar et al. 2024) using multispacecraft data provided by Parker Solar Probe, Solar Orbiter, Wind and STEREO-A (Russell 1995;Kaiser et al. 2008;Fox et al. 2016;Müller et al. 2020); the tracking of electron beams from corona into interplanetary space (Badman et al. 2022) with satellites and the Low Frequency Array (LOFAR; van Haarlem et al. 2013); the study of fine bursts' structures revealed by LOFAR (Reid & Kontar 2021); the evidence of electrostatic decay (ESD) in a complex type III burst (Thejappa & MacDowall 2021); the detection and detailed analysis of fundamental-harmonic pairs and interplanetary type IIIb radio bursts (Jebaraj et al. 2023;Chen et al. 2021); or the study of electron fluxes, spectra, and pitch-angle distributions associated with type III bursts (Lorfing et al. 2023).
Fundamental electromagnetic emissions radiated by type III solar radio bursts at plasma frequency ω p were first attributed to scattering of Langmuir waves generated by electron beams off thermal ions (Ginzburg & Zhelezniakov 1958).However, this process was shown not to account for the brightness temperatures of emissions observed in the solar wind.Therefore, it was argued that nonlinear wave processes such as decay or coalescence of Langmuir waves      (Tsytovich 1970), leading to the generation of the fundamental electromagnetic mode  and ion-acoustic waves , could be responsible for such emissions (Melrose 1980).On the other hand, in inhomogeneous plasmas, the linear conversion mechanism of Langmuir waves on density gradients or random fluctuations (Hinkel-Lipsker et al. 1989, 1991;Kim et al. 2008;Volokitin & Krafft 2018;Krasnoselskikh et al. 2019;Volokitin & Krafft 2020;Tkachenko et al. 2021;Krafft & Volokitin 2021;Krafft & Savoini 2021, 2022a, 2022b) and the so-called antenna mechanism, where Langmuir eigenmodes are trapped in density cavities (Malaspina et al. 2012), were shown to lead to fundamental electromagnetic waves' radiation.
The dominant nonlinear wave interaction process in the solar wind is believed to be the ESD of Langmuir waves,  ¢ + ¢    (e.g., Tsytovich 1970;Cairns & Robinson 1995;Graham & Cairns 2013;Krafft & Savoini 2024), where ¢  and ¢  are backscattered Langmuir and ion-acoustic waves, respectively.It has a significantly larger growth rate than electromagnetic decay (EMD)  +   .However, Lin et al. (1986) observed low-frequency noise in close time coincidence with Langmuir waves and concluded in favor of EMD rather than ESD.On the other hand, using joint observations of Langmuir and  -mode waves, it was concluded that strong turbulence and linear conversion phenomena could both be viable (Thejappa et al. 1993).However, conditions for strong turbulence are rarely satisfied in the solar wind (e.g., Graham & Cairns 2013).Finally, it appears to date that, in homogeneous Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
plasmas, EMD should be responsible for the generation of  waves, whereas the conversion mechanism of Langmuir waves on density inhomogeneities into  waves should play the main role in nonuniform plasmas (Krasnoselskikh et al. 2019;Volokitin & Krafft 2018, 2020;Krafft & Volokitin 2021;Tkachenko et al. 2021;Krafft & Savoini 2022a, 2022b).In the solar wind and corona, density fluctuations are ubiquitous (Celnikier et al. 1987;Krupar et al. 2020), and their influence on the mechanisms of  wave generation is crucial.
Theoretical and numerical studies on the kinematics of EMD and its radiation properties were performed in the framework of weak turbulence theory (e.g., Tsytovich 1970;Melrose 1980;Cairns 1987;Edney & Robinson 1999;Li et al. 2005;Ziebell et al. 2015).However, this nonlinear generation mechanism was never studied using particle-in-cell (PIC) simulations-mostly because of its high numerical requirements-even if  -mode energy spectra were shown with more or less resolution for homogeneous and randomly inhomogeneous plasmas (Rhee et al. 2009;Lee et al. 2019;Krafft & Savoini 2022a).It was shown that, in plasmas with random density fluctuations,  waves are radiated at the early stage of the beam-plasma interaction, with electromagnetic energies up to two orders of magnitude larger than in homogeneous plasmas (Krafft & Savoini 2022a).In this view, the question arises whether EMD, which requires resonance conditions to be fulfilled, can occur in such plasmas.
By analyzing large-scale and long-term two-dimensional (2D) PIC simulations performed for typical parameters of type III solar radio bursts near 1 au (Reid & Ratcliffe 2014), several objectives are reached.First, we highlight the EMD mechanism in a homogeneous plasma, explain the 2D dynamics of ionacoustic waves (spectra, dispersion, growth, and damping), and demonstrate that such a three-wave interaction process can actually take place under type III radio bursts' conditions, by presenting signatures of their nonlinear interactions (crossbicoherence, correlations between waves' phases and between waves' growths, resonance conditions).In inhomogeneous plasmas, signatures can be most easily detected in regions of weaker density turbulence, but also using ion density distributions and low-frequency wave spectra and dispersion.Second, we explain the role played by ESD, linear mode conversion, and coalescence processes in the occurrence of EMD and determine under what conditions each process can be prominent.Third, we study the impact of random plasma density fluctuations and electron-to-ion temperature ratios T e /T i on the 2D dynamics of EMD, in particular for plasmas with T e /T i ∼ 1, where ionacoustic waves are heavily damped.The final objective is to state under what conditions EMD can occur in actual solar wind plasmas, where random density fluctuations are ubiquitous and temperatures satisfy 1  T e /T i  10.To be fulfilled, the study requires performing challenging large-scale and long-term 2D PIC simulations, in order to simultaneously follow high-and low-frequency waves' dynamics and electrostatic and electromagnetic wavevector scales.
Note finally that, given the large group velocity of electromagnetic waves radiated at ω p in the solar wind, PIC simulations performed with periodic boundary conditions (as in this paper) cannot take into account their rapid escape from their source.When this plasma region is optically thin, the electromagnetic wave energy calculated by the simulations should be considered as an upper limit of the energy that would actually be radiated.Nevertheless, the study of the growth and the development of nonlinear wave instabilities such as EMD and nonlinear induced scattering (NLIS) on thermal ions can be achieved by analyzing their signatures in high-frequency Langmuir and low-frequency ion-acoustic waves' or quasimodes' dynamics, as well as the qualitative properties of these mechanisms, as done in this paper.When the plasma source is randomly inhomogeneous, with density fluctuations of average levels of a few percent of the ambient plasma density, it can be considered as an optically thick medium, as discussed, for example, by Li et al. (2008aLi et al. ( , 2008b)).Then, PIC simulations with periodic boundary conditions can describe such solar wind situations, as well as the amount of electromagnetic waves' energy radiated at ω p .

Numerical Simulations
Simulations are performed using the 2D3V version of the relativistic full electromagnetic PIC code SMILEI (Derouillat et al. 2017) with periodic boundary conditions.They involve three particle species, i.e., beam electrons, plasma electrons, and ions.In order to reduce the numerical noise well below the average level ) ) N n n 0 2 1 2 of applied density fluctuations δn, which reaches a few percent of the background plasma density n 0 (Krafft & Savoini 2021, 2022a), 5400 macroparticles per cell are used.Two simulation boxes of sizes L x × L y = 1448λ D × 1448λ D and L x × L y = 5792λ D × 2896λ D are used, where λ D is the electron Debye length.The ion-to-electron mass ratio is m i /m e = 1836, where m i and m e are the proton and electron masses.
Random density fluctuations are applied initially and evolve self-consistently, with typical wavelengths around 10 times larger than those of the Langmuir waves.A type III electron beam is injected along the x-axis, with a 2D Maxwellian velocity distribution (Krafft & Savoini 2023); drift and thermal velocities are v b = 0.25c and , where v T is the thermal velocity of the plasma electrons (v b /v T = 12.7).Its weak relative density n b /n 0 = 5 × 10 −4 allows us to reach the kinetic regime.The characteristic wavenumber  k of Langmuir waves excited by the beam along the x-axis is l  k 0.1 D .Simulations are presented for conditions relevant to type III solar bursts near 1 au (Reid & Ratcliffe 2014), for homogeneous (ΔN ; 0) and randomly inhomogeneous plasmas (ΔN = 0.025, 0.05) and temperature ratios T e /T i = 1 and 10.We consider here, as a first step, the case of an unmagnetized plasma close to the weakly magnetized solar wind.Long-term simulations are performed, up to 15,000w - p 1 , in order to clearly highlight ion-acoustic waves.Moreover, we need to follow simultaneously the dynamics of high-and lowfrequency waves, whose frequencies differ by around three orders of magnitude, and the spatial scales of electrostatic and electromagnetic waves, whose wavenumbers extend over two decades.The total energy W tot is well conserved along the whole simulation, as the relative error is ΔW tot /W tot ; 0.0003, where ΔW tot is the total energy variation up to ω p t = 15,000; with respect to the energy ΔW b deposited to the system by the beam within the same time period, we get that ΔW tot /ΔW b ; 0.08.


, as well as k and ω refer to wavevectors and frequencies, respectively.They imply that In a homogeneous and unmagnetized plasma, the Langmuir, ion-acoustic, and electromagnetic wave dispersion relations can be written as w w l 1 2 .Another nonlinear wave interaction process, ESD, plays an important role in the development of the EMD.For the parameters used here, two cascades of decay can be identified, i.e.,  → ¢  + ¢  and ¢  →   +   , where ¢  and   are backscattered and forward-propagating Langmuir waves, and ¢  and   are forward-and backward-propagating ion-acoustic modes, respectively (Krafft & Savoini 2024).The three-wave resonance conditions are written as w In the one-dimensional approximation, they lead, together with the dispersion relations, to the relations

Electromagnetic Decay in a Homogeneous Plasma
In order to determine hereafter the influence of T e /T i and ΔN on EMD, let us first study the case of a homogeneous plasma where ion-acoustic waves are weakly damped (T e /T i = 10 and ΔN = 0).Figure 1 shows the dynamics of ion-acoustic spectra from the time ω p t ; 5500 (Figure 1(a)) at which the waves  generated by the EMD  +    reach amplitudes discernible on the spectrum of the waves ¢  and   , which are the forward-and backward-propagating ion-acoustic waves coming from the first and second ESD cascades,  , and, later, shows the concomitant dynamics of four processes of wave-wave interaction.
Note that the low-frequency excitation appearing at k x ∼ 0 in Figure 1, also visible in 1D simulations with modified Zakharov equations (Krafft et al. 2015) or studies performed in the framework of weak turbulence theory (Ziebell et al. 2015), corresponds to ion-acoustic waves of large wavelengths that do not play a role in the processes studied here.This structure appears likely due to linear beatings between close (in the frequency and wavelength domains) beam-driven Langmuir waves generated during the beam deceleration (see also Krafft & Savoini 2024).
The scenario of  waves' generation revealed by the spectra (and confirmed below) is the following.First, backscattered Langmuir waves  .Indeed, this three-wave interaction can occur despite its small growth rate predicted by theory, because the spectral domain of   waves produced by is satisfied, where D  k is the width of the beam-driven Langmuir waves' spectrum; see also Robinson et al. 1994); consequently,   waves stimulate the growth of  b waves, whose amplitude becomes sufficiently large to trigger the EMD ¢  +    b .Moreover, at Note that panels (a)-(f) are the continuation from ω p t ; 5500 to ω p t ; 13,500 of the left panels of Figure 2 in Krafft & Savoini (2024).All variables are normalized.
, and later at ω p t ; 13,500 (Figure 1 . This scenario shows that the presence of ESD is essential for the generation of electromagnetic waves by EMD.Note also that their thresholds are defined by e  ( ) th , where g w g w = ( )( ) for ESD (Shukla et al. 1983);  E is the beam-driven Langmuir wave electric field; and g  , g  , g ¢  , and g ¢  are the absolute values of the damping rates of  ,  , ¢  and ¢  waves.Using the same normalization by the initial beam energy as in the figures, the EMD threshold, of the order of 10 −11 , is overcome as e  ( ) (see, e.g., Figure 3(a)), as is the ESD threshold, which is around 10 −7 .All waves  b ,  f , ¢  , and   lie on the theoretical ionacoustic dispersion laws (Figure 2), at times corresponding to Figure 1.In Figures 2(a  and   exhibit damping.Meanwhile, Figures 2(g)-(i) present energy spectra of waves  , at ω p t ; 4000, 5500, and 7500.They exhibit a dipolar energy distribution, as expected by theory for homogeneous plasmas.Such high-resolution spectra, performed with the large box L x × L y = 5792λ D × 2896λ D , were never obtained before in homogeneous plasmas, but only for randomly inhomogeneous ones, by Krafft & Savoini (2022a).
Note that our results are consistent with those obtained by Li et al. (2005) and Ziebell et al. (2015).The former studied electromagnetic emission in a homogeneous plasma, taking into account its propagation away from its source and calculating emission rates with an approximate analytic formula; they show that beam-driven and backscattered Langmuir waves produced by ESD can radiate electromagnetic waves through EMD when stimulated by ion-acoustic waves.The latter determined numerically the complete solutions of the entire set of electromagnetic weak turbulence theory equations and showed evidence of Langmuir, ion-acoustic, and electromagnetic waves' spectra consistent with ours.
Figures 3(a Other diagnostics can confirm that  waves are produced by a three-wave interaction process.Indeed, space and time crossbicoherences are calculated using the electric field E x , the induced ion density perturbations δn i , and the magnetic field B z of waves participating in the EMD, where the brackets denote ensemble averaging; (X 1 , X 2 ) are the parallel wavenumbers (k x1 , k x2 ) or frequencies (ω 1 , ω 2 ).At and (e)-(g) are in logarithmic scales.All variables are dimensionless.Note that, as the plasma is homogeneous (ΔN = 0), the energy of electromagnetic waves should be considered as an upper limit of the actual energy radiated in an optically thin region of the solar wind, due to the periodic boundary conditions used, which do not allow electromagnetic waves to escape from the simulated system.ω p t ; 8000, when  W starts to saturate, the maximum cross-bicoherence is b c (k x1 , k x2 ) = 0.9 (Figure 3 2), so that, when resonance conditions W ). Ionacoustic waves are weakly damped (T e /T i = 10), and the method employed in our previous works to confirm three-wave interactions can be used (Krafft & Savoini 2021, 2022b).One observes that µ during the growth of  and  b waves (4000  ω p t  8000; Figure 3(e)).Similar features can also be seen at later times.This shows that the EMD ¢  +    b actually takes place.This procedure is also applied to the time variations of for ω p t  7000, highlighting that the EMD  +    f occurs in this time interval, as revealed by Figures 1-2.Nevertheless, the proportionality between energies' and energy products' time variations is not always very good (compare with Figures 3(c)-(d) of Krafft & Savoini (2024), where it is much better, the ESD being the prominent nonlinear wave process), and the relations µ could not be evidenced (and thus are not shown in Figures 3(e) and (g)).
This can be due to several reasons.For example, the coalescence process +     f , which absorbs energy of ion-acoustic waves but does not lead to exponential growth of  waves, indeed participates in the waves' interactions, as confirmed by bicoherence studies (not shown here; see also Figure 1(f), where  f waves are partially damped near k x λ D ; 0.15).Moreover, the two ESD cascades, which are prominent compared to EMD, occur during the same periods.Thus, each kind of Langmuir wave is involved in several processes simultaneously.

Influence of Density Fluctuations
Let us now study EMD in a randomly inhomogeneous plasma where, as shown in Krafft & Savoini (2022a), the generation process of  waves by linear mode conversion of Langmuir waves on density fluctuations takes place.As resonance conditions between waves required for EMD should be a priori hardly fulfilled, the question rises of whether EMD can occur in such plasmas.Moreover, Langmuir wave energy lost during conversion cannot be used further by other processes, not to mention that Langmuir wave turbulence itself also decays owing to the appearance of accelerated beam electrons resulting from wave scattering on density fluctuations (Krafft et al. 2013;Krafft & Volokitin 2016, 2017;Krafft & Savoini 2023).
Figure 4(a) shows the time variations of  W up to ω p t = 15,000, for ΔN = 0, 0.025, and 0.05.For ΔN > 0,  W reaches its maximum at early times around ω p t ; 2000 (linear mode conversion process), where it exceeds by almost two orders of magnitude the energy radiated in the homogeneous plasma, which increases slowly (nonlinear EMD) and stabilizes asymptotically at one order of magnitude above the energy reached in the inhomogeneous plasmas, due to the decay of Langmuir wave turbulence at ΔN > 0 (Figure 4(b)).At ω p t  11,000, Langmuir wave excitation by the beam can no longer overcome this decrease and generate electromagnetic waves so that  W stabilizes.Note that we recover similar results to those in Figure 5(b) of Krafft & Savoini (2022a), where initial parameters are slightly different but close and simulations are twice as short; we note that, in this previous figure, only magnetic energy was shown, which is around a factor 20 smaller than the corresponding electromagnetic energy.
Note that the growth of beam-driven Langmuir waves (Figure 4(b)) is not significantly different for ΔN = 0 and ΔN > 0 (even if it is noticeably smaller for ΔN = 0.05), contrary to what is stated in the framework of 1D geometry, where density fluctuations lead to a much slower growth of Langmuir waves (e.g., Krafft et al. 2013;Voshchepynets et al. 2015).Indeed, in 2D geometry, the resonance condition between the beam particles of velocity v(v x , v y ) and the Langmuir waves of wavevectors k(k x , k y ) involves a second degree of freedom, so that the probability to find particles that can resonate with a given wave is larger than in a 1D plasma, where the resonance condition involves only the parallel direction (Krafft & Savoini 2023).
Figure 4(c) shows the spatial distribution of the normalized ion density perturbation δn i (x, y)/n 0 , at ω p t = 1700, filtered around l ~- , on which the corresponding density fluctuations' isocontours δn(x, y) = 0 (red lines) are superimposed.One observes several enhancements of ion-acoustic waves' intensity in localized regions with δn(x, y)  0, where propagation structures appear with wavelengths p p . This indicates the occurrence of EMD, at times when the linear conversion of Langmuir waves into fundamental radiation operates.At such early time (ω p t = 1700), Langmuir and  -mode waves are strongly excited (Figures 4(a)-(b)) and thus can nonlinearly interact and trigger three-wave interaction processes as EMD, and consequently generate ion-acoustic waves.
In Figures 4(d As a conclusion, EMD coexists with linear mode conversion of Langmuir waves and ESD and can be persistent up to large times, but only in specific plasma regions where density is quasi-uniform or depleted. Ion-acoustic energy spectra are strongly broadened and scattered as a result of Langmuir waves' transformations on density fluctuations (not shown here; see also Krafft & Savoini 2024); due to this fact, the small amplitudes of the low-frequency waves, and the coexistence of ESD, it is difficult to provide reliable evidence of EMD using spectra.Nevertheless, Figure 4(f) shows, for ΔN = 0.05, the dispersion curves of ion-acoustic waves at ω p t = 5000, where  b waves are visible near l - and persist up to ω p t ; 8000.Note finally that our results are in some aspects different from those obtained by Ratcliffe & Kontar (2014), who developed a model in an inhomogeneous plasma with a decreasing gradient from the source to the observer, involving effects of beam-plasma interaction, wave-wave nonlinear interaction, electromagnetic radiation, and escape out of the source, and describing the scattering of Langmuir waves on  small-scales density fluctuations by a diffusion equation in wavenumber; they found that density inhomogeneities can suppress the production of plasma radio emission.We show that density fluctuations enhance electromagnetic radiation by linear mode conversion, whereas EMD becomes a localized process occurring in regions of weak density turbulence.

Influence of the Electron-to-ion Temperature Ratio
In the solar wind, the electron-to-ion temperature ratio can vary from T e /T i > 1 to T e /T i = 1 and even reach values T e /T i < 1 (Gurnett et al. 1979;Newbury et al. 1998).In a homogeneous plasma with T i ∼ T e , ion-acoustic waves are strongly damped and the EMD  +    is overcome by NLIS, where thermal ions interact with the low-frequency beating of Langmuir and electromagnetic waves (Tsytovich 1970).Then, ion density perturbations are related to ionacoustic quasi-modes.
In a homogeneous plasma with T e /T i = 1, Figures 5(a 5(e) shows the corresponding dispersion at ω p t = 12,500, where these quasi-modes can be clearly distinguished, at ω ; 0.002ω p and ω ; 0.004ω p , respectively; note that they do not follow ion-acoustic waves' dispersion.
Figure 5(f) shows the time variations of  W for T e /T i = 1 (dashed) and T e /T i = 10 (solid), for ΔN = 0 and ΔN = 0.05.The influence of T e /T i on  W is rather weak, for both ΔN; decreasing T e /T i from 10 to 1 does not significantly modify the growth rates of  W .However, for ΔN = 0 (see also Figures 5(a)-(d)),  W is significantly decreased during a long time (4000  ω p t  9000), where energies  W of quasi-modes  b qm and  f qm are slowly growing owing to NLIS (inset of Figure 5(f)).The impact of T e /T i on  W is significant; compared to the case with T e /T i = 10 (Figure 3(b)),  W is reduced by almost one order of magnitude.
Finally, the important question rises of whether the excitation of  waves can be observed in actual solar wind plasmas, where T e /T i > 1 and ΔN ∼ 0.01-0.05. Figure 5(g) shows, in such conditions, the low-frequency spectrum at ω p t = 3500, which exhibits the excitation of quasi-modes ¢  qm ,   qm ,  f qm , and  ; b qm the most excited ones are produced by NLIS via ¢  +    b qm .They are, however, significantly damped and distorted compared to the case with ΔN = 0 and T e /T i = 10, where the ion-acoustic waves ¢  ,   ,  f , and  b are clearly visible (Figure 1).Note the large increase of the central spectral excitation, due to scattering phenomena; its origin and growth are beyond the scope of this paper, as they do not play a noticeable role here up to ω p t = 15,000 (see also our remark in Section 2.1 when discussing Figure 1).Moreover, at the same time, Figure 5(h) shows the spatial distribution of δn i (x, y)/n 0 , filtered around l , where enhanced ion-acoustic mode structures with wavelengths p p ~ x are visible in localized regions with δn(x, y)  0 only, for the same reasons as discussed above (Figure 5(c)).Then, NLIS occurs, with ion-acoustic quasi-modes of small intensities and in plasma regions where density turbulence is weak.

Conclusion
Large-scale and long-term 2D PIC simulations performed for parameters typical of type III solar radio bursts near 1 au have provided new results on the generation mechanisms of fundamental electromagnetic waves.
First, the occurrence of EMD has been demonstrated in a homogeneous plasma with T e /T i = 10.The 2D dynamics of ion-acoustic waves has shown the decisive role played in EMD by the backscattered Langmuir waves coming from the ESD.The electromagnetic waves, which are radiated at w , p are slowly growing, with quasi-dipolar spectra.Moreover, it is shown that EMD can be stimulated and triggered by ionacoustic waves coming from the faster and more intense ESD.This is due to the overlap of two spectral domains of backwardpropagating ion-acoustic waves (i) produced by the EMD of backscattered Langmuir waves, i.e., ¢  +    b , and (ii) coming from the second ESD cascade, ¢   +     .Similarly, and for the same reasons, the EMD of forwardpropagating Langmuir waves,  +    f and   +    f , is stimulated by the ion-acoustic waves coming from the first ESD cascade  ¢ + ¢    .All these nonlinear processes coexist over long times, even if EMD is slower.The coalescence process ¢ +     f likely occurs but is significantly less prominent, leading to ion-acoustic waves' absorption in the same spectral region as beam-driven Langmuir waves.The occurrence of all these decay channels suggests that power spectra of magnetic fields' waveforms should present fine peaks around the frequency ω p .
In randomly inhomogeneous plasmas, EMD is not suppressed but develops only within confined plasma regions of reduced or quasi-uniform density.It coexists with linear mode conversion of Langmuir waves into electromagnetic emission at constant frequency, which is the fastest and most prominent process.The energy growth of electromagnetic waves radiated at w p is fast, and their spectra are quasi-isotropic.The mode conversion process, by producing at early times fundamental electromagnetic waves of large amplitudes (up to two orders of magnitude larger than in a homogeneous plasma) that can nonlinearly interact with simultaneously intense Langmuir waves, can therefore trigger EMD.Moreover, ESD and EMD evolve in competition, the linear mode conversion process reducing eventually the energy carried by Langmuir waves and thus by wave products of ESD and EMD.
In a homogeneous plasma, the impact of the electron-to-ion temperature ratio on fundamental electromagnetic energy growth and saturation is rather weak.At T e ∼ T i , ion-acoustic waves are heavily damped and EMD is overcome by NLIS on thermal ions; ion-acoustic quasi-modes appear.
In actual solar wind conditions, where random density fluctuations are ubiquitous, EMD should become rare but is not suppressed.Indeed, it can occur in specific plasma regions only, if T e /T i  3; otherwise, NLIS takes place.Then, due to the smallness of ion-acoustic waves' or quasi-modes' amplitudes and their localization, these nonlinear processes should not easily be detected by satellites.It should be more probable to observe them in plasma regions weakly perturbed by the background density turbulence and where ion temperature is decreased.
Note finally that our PIC simulations, which involve periodic boundary conditions, present some limitations, as they do not allow us to determine quantitatively, in an optically thin solar wind plasma, the energy of electromagnetic waves radiated at ω p , but only its upper limit.Nevertheless, analytical and numerical studies (Li et al. 2008a(Li et al. , 2008b) ) report that, in a solar wind plasma with ΔN = 0.01 (0.07), the fraction of electromagnetic emission at ω p escaping from its source is around 10 −3 to 10 −2 (less than 5 × 10 −4 ), so that, even for very low levels of inhomogeneities, solar wind plasmas should not be optically thin but thick.However, such models include many assumptions, and the question of solar wind transparency remains to date still open, so that future PIC simulations will perhaps require considering new kinds of boundary conditions.
leads to the generation of fundamental electromagnetic waves  and ion-acoustic waves .The resonance conditions are w w respectively, so that resonance conditions on wavenumbers k x are readily fulfilled (see Krafft & Savoini 2024 for more details).Moreover, ionacoustic waves  b (the subscript "b" indicates backward propagation) appear with a small spectral intensity at the right border of the spectral domain of   waves (centered around l grows further with time (Figure 1(b)).Later, at ω p t = 7500 (Figure 1(c)), a similar phenomenon manifests at the left side of the spectral domain of ¢  waves, and forward-propagating ion-acoustic waves  f rise with an intensity growing around l ω p t ; 10,500 and ω p t ; 13,500 (Figures 1(e)-(f)), the spectra exhibit four well-discernible regions, corresponding to waves ¢  and   coming from the two ESD cascades, as well as to waves  f and  b (Figure 1(f)) generated via the EMD of Langmuir waves , ¢  , and   , with the conditions
ω p t ; 7500 (Figure 1(c)), the left border of the ¢  waves' spectral domain reveals the excitation of S f waves, stimulated by ¢  waves; they trigger the EMD of beam-driven Langmuir waves

Figure 2 .
Figure 2. (a-f) Dispersion curves in the plane (ω/ω p , k x λ D ), corresponding to spectra of Figure 1, in logarithmic scales, for the box l ´= Ĺ L 1448 1448 x y D 2 .Dashed black lines represent the theoretical dispersion law of ion-acoustic waves.(g-i) Magnetic energy spectra of  waves, filtered around ω p , in the plane (k x λ D , k y λ D ), at ω p t ; 4000, 5500, and 7500, in logarithmic scales, for the large box l ´= Ĺ L 5792 2896 x y D 2 .All variables are normalized.

Figure 3 .
Figure 3. Variations with time of wave energy (integrated on the 2D simulation box and normalized by the initial beam energy W b ) of (a) Langmuir waves , ¢  , and   involved in the two ESD cascades  ¢ + ¢    and ¢   +     , electromagnetic waves  , and (b) backward-and forward-propagating ion-acoustic waves  b and  f . (a-b) The energies of waves radiated directly or indirectly by the beam are several orders of magnitude above the noise, as shown by the analysis of a "thermal" simulation with no beam injection.(c) Cross-bicoherence b c (k x1 , k x2 ) at ω p t ; 8000; k x1 and k x2 represent the parallel wavenumbers of waves ¢  and  ; b the maximum is b c = 0.9, for l = ---( ) ( ) k k , 0.117, 0.121 x x 1 2 D 1 .(d) Cross-bicoherence b c (ω 1 , ω 2 ) at ω p t ; 8000, where ω 1 and ω 2 represent the frequencies of waves  and  ; b the maximum is b c = 0.85, for (ω 1 , ω 2 ) = (1.020,0.0042)ω p .(e) Time variations of energies  W (blue) and  W b (red) and of energy products ¢   W W b (dotted light blue, * ¢   b theoretical range of values of  k x .In the frequency plane (Figure3(d)), the maximum cross-bicoherence is b c (ω 1 , ω 2 ) = 0.85, for w )-(e), the profiles along x (at fixed y) of δn i /n 0 (applied density fluctuations δn are subtracted by filtering) show localized sequences of wavelengths corresponding to ionacoustic when Langmuir wave energy is decreased by one (ω p t = 4700; Figure4(d)) and two (ω p t = 13,400; Figure4(e)) orders of magnitude compared to its maximum (Figure4(b)).These sequences occur around 180  x/λ D  300 (Figure 4(d)) and 300  x/λ D  600 (Figure 4(e)).Moreover, wavelengths of ion-acoustic waves produced by the ESD  ¢ + ¢    , which are roughly two times smaller (∼30λ D ), as their wavevectors satisfy Section 2 above and Krafft & Savoini 2024), are also visible (0  x/λ D  200, Figure 4(e)).

Figure 5 .
Figure 5. (a-d) Ion-acoustic energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 4500, 7500, 9500, and 12,500, for ΔN = 0 and T e /T i = 1 ; the ion-acoustic quasimodes are indicated on each panel.(e) Corresponding dispersion (ω/ω p , k x λ D ) at ω p t = 12,500.(f) Time variations of  W (integrated on the 2D simulation box and normalized by the initial beam energy) for T e /T i = 1 (dashed) and T e /T i = 10 (solid), for ΔN = 0 (black) and ΔN = 0.05 (blue), in logarithmic scales.Inset shows the time variation of energies  W of quasi-modes  b qm and  f qm (red), for ΔN = 0, in logarithmic scales.(g) Low-frequency energy spectrum at ω p t = 3500, where the quasi-modes ¢  qm ,   , qm  f qm , and  b qm are visible, for T e /T i = 1 and ΔN = 0.05, in logarithmic scale.(h) Spatial distribution of δn i (x, y)/n 0 filtered around  k , x at ω p t = 3500, showing enhanced ion-acoustic quasi-modes' structures corresponding to l ~~~-| | | |    k k k 0.1 x x x D 1