Electrostatic Wave Decay in the Randomly Inhomogeneous Solar Wind

Despite a few space observations where Langmuir and ion acoustic waves are expected to participate in the mechanism of electrostatic decay, this is to date believed to be the main and fastest nonlinear wave process in the solar wind. However, in such a plasma where random density fluctuations are ubiquitous, the question of whether nonlinear wave processes play a significant role in Langmuir wave turbulence generated by electron beams associated with type III solar radio bursts remains still open. This paper provides several answers by studying, owing to two-dimensional challenging particle-in-cell simulations, the dynamics and the properties of the ion acoustic waves excited by such Langmuir wave turbulence and the role they play in the electrostatic decay. The impact on this process of plasma background density fluctuations and electron-to-ion temperature ratio is studied. Moreover, it is shown that, for a typical solar wind plasma with an average level of density fluctuations of a few percent of the ambient density and a temperature ratio of the order of 1, nonlinear induced scattering off ions occurs, with small intensity low-frequency quasi-modes and only in localized plasma regions where density is depleted or weakly perturbed by low-frequency turbulence.


Introduction
Waveforms containing the beating signature of two highfrequency waves were first observed upstream of the foreshock of Jupiter (Gurnett et al. 1981) and, later, of the Earth (e.g., Kellogg et al. 1999).In some cases, such observations were associated with type III solar radio bursts (Gurnett et al. 1993;Hospodarsky & Gurnett 1995).An interpretation of the beating waves as signatures of nonlinear three-wave decay was proposed by Cairns & Robinson (1992), arguing that the two high-frequency peaks correspond to the beam-driven Langmuir waves  and the backscattered ones ¢  , produced via the resonant electrostatic decay (ESD)  ¢ + ¢    , where ¢  are ion acoustic waves (e.g., Tsytovich 1970;Melrose 1986).More recently, such double-peak features were observed by several satellites together with low-frequency emissions, with the authors arguing for ESD (Henri et al. 2009;Graham & Cairns 2013;Kellogg et al. 2013).
However, another interpretation was also proposed, presenting the double peak as a beam-driven incident Langmuir wave and its reflection on a density gradient (Willes et al. 2002).The solar wind indeed involves random density fluctuations δn of a few percent of the ambient plasma density n 0 (Celnikier et al. 1987;Krasnoselskikh et al. 2007;Krupar et al. 2020), that have a strong impact on the propagation of Langmuir waves and can be the cause of their conversion into electromagnetic waves (Kellogg et al. 1999;Krasnoselskikh et al. 2011;Volokitin & Krafft 2018).Due to the difficulty to distinguish the origin of a double-peak structure (a three-wave interaction process or wave scattering on plasma density fluctuations), the question of whether nonlinear processes play a significant role in Langmuir wave turbulence generated by electron beams during type III radio bursts in the solar wind remains open and still actively debated (e.g., Soucek et al. 2005).Moreover, in the framework of such radio bursts, the generation of backscattered Langmuir waves is of highest importance, as they participate to wave coalescence processes leading to harmonic electromagnetic wave radiation at frequency 2ω p , whereas ion acoustic waves ¢  produced via ESD can stimulate the production of ion acoustic waves  (Robinson et al. 1994) involved in the electromagnetic decay  +    supposed to lead to the radiation of electromagnetic waves  at frequency ω p (e.g., Tsytovich 1970;Melrose 1986).
In this Letter we study the ESD  ¢ + ¢    in homogeneous and randomly inhomogeneous plasmas, with different electron-to-ion temperature ratios T e /T i and average levels of density fluctuations ñ , for conditions typical of type III solar radio bursts near 1 au (e.g., Reid &Ratcliffe 2014, andreferences therein, Krupar et al. 2015;Dakeyo et al. 2022;Wilson et al. 2023).For the first time, two objectives are reached owing to large-scale and long-term challenging 2D PIC simulations, i.e., (i) to show the occurrence of 2D ESD in a homogeneous plasma by presenting in detail the low-frequency waves' dynamics and characteristics (spectra, dispersion, energy growth, saturation, and damping), as well as relevant signatures of their nonlinear interactions with Langmuir waves (resonance conditions, waves' phase correlations, space and time cross-bicoherences), and (ii) to study the dependence of the decay process as a function of ΔN and T e /T i , in order to state if, in an actual solar wind plasma with ΔN ∼ 0.05 and T e /T i ∼ 1, ESD can be the dominant process, or if the linear conversion of Langmuir waves on density fluctuations is the main source of backscattered Langmuir waves.Whereas ESD is to date considered the main and fastest nonlinear wave interaction process in homogeneous plasmas with developed Langmuir wave turbulence, this statement has to be seriously reconsidered in actual solar wind plasmas where random density fluctuations are ubiquitous and ion acoustic damping is significant.

Simulation Results
A 2D3V version of the relativistic full electromagnetic PIC code SMILEI is used (Derouillat et al. 2017).Simulations involve 1800 macroparticles per cell and per each of the three species (plasma ions and electrons, beam electrons) in order to minimize the numerical noise below the average levels ΔN = 0.025 and 0.05 of the external density fluctuations, of the order of a few percent of the background plasma density n 0 (Krafft & Savoini 2021, 2022).These initially applied random inhomogeneities evolve self-consistently during the simulations.The drift and thermal velocities of the type III beam considered are v b ; 0.25c and v v T T b = , where v T is the electron plasma thermal velocity.The weak beam relative density, n b /n 0 = 5 × 10 −4 , enables us to work in the kinetic instability regime.The beam is injected parallel to the x-axis with a 2D Maxwellian velocity distribution function (Krafft & Savoini 2023).
PIC simulations are performed for conditions typical of type III solar radio source regions near 1 au (e.g., Reid & Ratcliffe 2014, and references therein), considering homogeneous (ΔN = 0) and randomly inhomogeneous plasmas (ΔN = 0.025, 0.05), with electron-to-ion temperature ratios T e /T i = 1 and 10, which delimit the range of actual solar wind temperature ratios near 1 au (Dakeyo et al. 2022;Wilson et al. 2023).The lengths of the simulation box are L x = L y = 1448λ D ; λ D is the electron Debye length.The mass ratio is m i /m e = 1836; m i and m e are the proton and electron masses.The weak solar wind magnetic field is neglected.
The characteristic wavenumber k  of the beam-driven Langmuir waves along the beam direction is k 0.1 where (Robinson et al. 1993).Depending on the value of k 0 , the second decay cascade  in 1D geometry (e.g., Li et al. 2003;Krafft et al. 2015).These wavevector values can be used as estimates for the case of 2D geometry.

Homogeneous Plasma
, involved in the same decay process, reach their maxima roughly at the same time ω p t ∼ 5000 (ω p t ∼ 10,000).Note the very large time domain (up to 15,000 p 1 w -) allowing us to observe simultaneously the growth and the saturation stages of all the highand low-frequency waves involved in the first and the second decay cascades.The Langmuir wave spectrum is shown at time ).The spatial distribution of the normalized ion density perturbations δn i /n 0 can be observed at the same time in Figure 1(c).Their wavelength  occurs in the whole simulation box.Moreover, Figure 1(d) shows the spatiotemporal cross-distribution (at fixed y) of δn i /n 0 up to ω p t ; 15,000, exhibiting the presence of ¢  and   waves revealed by descending and ascending lines, spaced by distances equal to the ion acoustic wavelengths ), and propagating with opposite group velocities c s /v T and −c s /v T .
Figure 2 shows the energy spectra of ion acoustic waves at three times before the saturation of their energy near ω p t ; 6000, together with their dispersion curves (ω, k x ), to which the theoretical ones are superimposed.At ω p t = 1500, forward propagating waves ¢  produced by the decay  ¢ + ¢    appear near k x λ D ; 0.18, as expected by theory, as well as, starting from ω p t = 2500, backward propagating waves   produced by the second decay cascade ¢   +     .Dispersion curves show, at k x > 0, the excitation of ¢  waves near ω/ω p ; 0.005 (maximum of intensity), corresponding to the expected value k x λ D ; 0.18.Finally, at ω p t = 4500, the two spectral domains corresponding to ion acoustic waves produced by the first and second decay cascades are clearly visible, as well as on the dispersion curves.Note the low-frequency excitation near k x ∼ 0, also visible in 1D studies (Krafft et al. 2015) or weak turbulence studies (Ziebell et al. 2015), concerning waves of large wavelengths that do not play a role in the processes studied here.
In order to demonstrate that three-wave interaction actually takes place, we present in Figures 3(a)-(b) the space and time cross-bicoherences b c (Kim & Powers 1979) calculated using the electric fields' components E x and E y as well as the ion density perturbation δn i associated with the waves participating in the three-wave process 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 the time . However, as propagation is not strictly parallel, i.e., k k 0.06  which is satisfied with high accuracy.This demonstrates that the resonance conditions w  are fulfilled when the nonlinear interaction between the waves takes place.
Moreover, Figure 3(c) shows the time variations of energies W  , W ¢  , and W ¢  , as well as of energy products W W ¢   and W W ¢   .For the ratio T e /T i = 10 considered here, ion acoustic waves are weakly damped and we can use the method employed in our previous works (Krafft & Savoini 2021, 2022) that, on the basis of energy conservation of a three-wave system, allows us to find a signature of this nonlinear interaction by showing that the time  and ¢  waves (ω p t  5000).Such result, together with the waves' phase correlations highlighted by the cross-bicoherence, confirms the occurrence of a three-wave nonlinear interaction.Finally, let us apply this procedure to the time variations of energies W ¢  , W   , and W   and energy products  (pink dotted line follows red line) during the growth of   waves (ω p t  5000), confirming that the process observed is the second decay cascade In conclusion, both decay cascades are shown to be three-wave interaction mechanisms, in conditions relevant to type III solar radio bursts.
In 3D geometry, at the saturation of the ESD, the occupation numbers N  and N ¢  of  and ¢  waves (Tsytovich 1970)  where w  and w ¢  are the energy densities of the  and ¢  waves.Considering that solid angles satisfy  DW DW ¢   (Robinson et al. 1993), one can write at saturation that W W (depending on the time chosen in the saturation stage), in good agreement with W W th ( ) ¢   , showing that the saturation condition expected for ESD is recovered by the simulations and that the decay process identified behaves as expected by theory.

Impact of Random Density Fluctuations
Let us now study the impact of applied random density fluctuations of average level ΔN on the decay process.For an inhomogeneous plasma with ΔN = 0.05, Figures 4(a)-(d) show, at ω p t = 2500 and ω p t = 4500, the energy spectra and the dispersion curves of the ion acoustic waves excited.Despite significant wave scattering and broadening, one can identify these modes on dispersion curves (and wave spectra) even if, compared to the homogeneous plasma case, the latter are no more well shaped but strongly scattered, with amplitudes at least one decade less.At ω p t = 2500, dispersion curves show the excitation of ¢  waves around k k 0.2 2 and ω/ω p ; 0.006; at ω p t = 4500,   waves appear, indicating the occurrence of the second decay cascade.At larger times (not shown here), low-frequency spectral energy mostly gathers toward smaller wavenumbers below 0.05; however, dispersion curves continue to exhibit the excitation of ion acoustic waves along the theoretical dispersion laws, but with no specific amplitude enhancements near k ¢  and k   .For a smaller ΔN = 0.025, ¢  and   waves can be more clearly identified  are no more visible.All these features are in agreement with Figure 4(e), which shows the time variations of energies of ion acoustic and Langmuir waves, for ΔN = 0.025 and ΔN = 0.05.After reaching a maximum near ω p t ; 1000, the energy W  slowly decreases, due to the formation of a tail of accelerated beam electrons resulting from Langmuir wave scattering on random density fluctuations (Krafft et al. 2013;Krafft & Savoini 2023).This effect is more prominent for larger ΔN than for smaller ones.The ion acoustic energy W ¢  , which is around 1 order of magnitude less than in the homogeneous plasma case, reaches its maximum at roughly the same time as the Langmuir backscattered waves' energy W ¢  , showing a correlation between them.During the saturation stage, W ¢  and W ¢  decrease with nearly the same rate, for both ΔN.One can conclude that ESD can occur and persist in a randomly homogeneous plasma, depending on the value of ΔN, and with significantly reduced energies of participating waves and scattering of their spectral and dispersion properties, compared to the homogeneous plasma case.Spatial distributions of wave fields show that ESD only occurs in regions where the plasma density is quasi-uniform or depleted and the resonance conditions necessary for three-wave interaction can be met.Indeed, density fluctuations scatter randomly the waves out of resonance one with another.During time evolution, Langmuir wave energy mostly self-organizes outside regions of density humps (Krafft & Volokitin 2021), where interactions between incident and reflected (on density fluctuations) Langmuir waves can occur.Figures 4(h)-(i) show the joint time evolution of profiles (cross sections at fixed y) of the parallel electric field E x and δn i /n 0 , for 1300  ω p t  3970.The latter exhibits localized oscillations with wavelengths around 35-45λ D (corresponding ) propagating with group velocities around c s /v T and −c s /v T , i.e., to forward and backward ion acoustic waves able to interact resonantly with Langmuir waves via ESD, whereas the former one shows, within the same time interval, the interaction of beamdriven and backscattered Langmuir waves, manifesting exactly in the plasma region where ion acoustic waves appear.This illustrates a decay process occurring in a localized region of the 2D simulation box, where plasma density is weakly perturbed by random fluctuations.To conclude, the plasma density turbulence has a strong impact on the dynamics of ESD, limiting its occurrence to regions of depleted or quasi-uniform plasma, where correlations between waves' phases and resonance conditions can be realized.

Influence of the Electron-to-ion Temperature Ratio
Let us now study the influence of the electron-to-ion temperature ratio T e /T i on ESD, by presenting firstly results obtained for the above homogeneous plasma, with T e /T i = 1 instead of T e /T i = 10.
The ratio g w ¢ ¢   of the damping rate to the frequency of the ion acoustic waves ¢  (with k 0.2 ) can be estimated at T e /T i = 1 as (Gary 1993) 0.48 ) and at T e /T i = 10 as 0.03 ).Thus, ion acoustic waves are heavily damped at T e = T i and nonlinear induced scattering (NLIS) of Langmuir waves off ions, where thermal ions interact with the beat between a beam-driven and a backscattered wave, becomes dominant over ESD (Tsytovich 1970).An ion acoustic quasi-mode qm ¢  appears, which is a low-frequency beating between  and ¢  waves, with a phase velocity v p satisfying v k  , that interacts nonlinearly with thermal ions.Further work on the NLIS at T e  T i will be done in a forthcoming paper.


( l -- ), which is connected with another wave interaction process that will be studied in a companion paper.Moreover, the time variations of high-and low-frequency energies show that (Figure 5(f)), whereas the waves  and ¢  present roughly the same behavior than at T e /T i = 10, the waves   exhibit a significantly larger saturation amplitude at late times (compare with Figure 1(a)), indicating that another process than ESD is responsible for Langmuir wave generation.The energies of the ion acoustic quasi-modes qm ¢  and qm   are reduced by more than 1 order of magnitude compared to the case with T e /T i = 10, and their growth rates are significantly decreased (compare the solid and dashed red lines in Figure 5(f)).
Finally, Figures 5(g)-(i) show the time variation of lowfrequency energy spectra in a plasma with ΔN = 0.05 and T e /T i = 1.Quasi-modes qm ¢  and qm   appear near ω p t ; 1500, reach a maximum intensity at ω p t ; 3000, and are almost indiscernible from the ambient noise at ω p t  4000 (not shown here).Then, signatures of NLIS off ions can be observed in inhomogeneous solar wind regions with T e /T i ∼ 1, but rarely, due to the modes' localization in plasma regions with quasi-uniform or reduced density only, and to the small intensity of the lowfrequency quasi-modes. and   in the plane (k x λ D , k y λ D ), at ω p t = 2500 and 4500. (b), (d) Dispersion curves, at the same times, in the plane (ω/ω p , k x λ D ); the black dashed lines represent the theoretical dispersion law; ΔN = 0.05.(e) Variations with time of Langmuir and ion acoustic wave energies (integrated on the 2D simulation box and normalized by the initial beam energy), for ΔN = 0.05 (solid lines) and ΔN = 0.025 (dashed lines), in logarithmic scale. : black; ¢  : blue; ¢  ,   : red. (f), (g) Ion acoustic energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 1500 and 11,500, for ΔN = 0.025. (h) Time evolution (1300  ω p t  3970) of profiles along x of the parallel electric field E x .(i) Corresponding profiles of δn i /n 0 , in the same time interval as (h); ion acoustic waves propagate in the forward and backward directions (oblique lines with slopes c s /v T and −c s /v T ).All variables are normalized.

Conclusion
The dynamics of nonlinear electrostatic wave-wave interactions, studied first in a homogeneous plasma where Langmuir wave turbulence is generated by a type III electron beam and ion acoustic waves are weakly damped (T e /T i ∼ 10), show that the dominant nonlinear three-wave process is ESD, which is also the source of backscattered Langmuir waves.This conclusion is also valid for solar wind regions with T T 3 10 -where ion acoustic waves are moderately damped-and with very low levels of density turbulence (quasi-homogeneous plasmas).This result was obtained owing to challenging 2D PIC simulations, which allowed us for the first time to study in detail the 2D dynamics of low-frequency waves and to demonstrate the occurrence of ESD generated by type III beams in the solar wind.Furthermore, in solar wind plasmas where the electron-to-ion temperature ratio reaches values down to , with heavily damped ion acoustic waves ( 0.17 0.48 ), simulations show that NLIS off ions leads to Langmuir waves' growth and amplitudes close to those of ESD-with some small differences however-and to small amplitude ion acoustic quasi-modes.
For inhomogeneous plasmas with random density fluctuations of average levels of several percent of the ambient density, both ESD and NLIS processes can only arise in localized plasma regions of reduced or quasi-uniform density, where Langmuir waves' energy accumulates during their scattering on the density humps.Moreover, Langmuir wave turbulence, which generates at early times emission of fundamental electromagnetic waves via linear conversion, is decaying with time due to the formation of a tail of accelerated beam electrons caused by Langmuir waves' scattering on density fluctuations; then, the energy available for further ESD or NLIS, which due to their nonlinear character arise later, is reduced.
Due to the combined action of heavy ion acoustic waves' damping at T T 1 3 and Langmuir wave scattering on random density fluctuations, the occurrence rate of ESD can be significantly reduced in favor of NLIS, so that, in certain conditions, it should not be easy to detect it by satellites in the solar wind near 1 au.Nevertheless, one can expect to observe it in regions where the plasma density is weakly perturbed by lowfrequency turbulence and the temperature ratio satisfies T e  3T i .Finally, the Doppler-shifted double-peak structures commonly observed by spacecraft in the solar wind and exhibiting two Langmuir waves with close frequencies, usually attributed to ESD, could also result from interactions of Langmuir waves with density fluctuations, as reflections off density gradients or scattering at a range of angles, for example, or to NLIS on ions.

Figure 1
Figure 1(a) presents the time variations of the energies (i) W ,  W ¢  , and W   of the Langmuir waves , ¢  , and   , and (ii) W ¢  and W   of the ion acoustic waves ¢  and   .WhereasW  increases due to beam instability, the energies W ¢  (W   ) and W ¢ ω p t = 4500 corresponding to Langmuir (Figure 1(b)) and ion acoustic waves' spectra (Figure 2).In Figure 3(b), the highest level of cross-bicoherence is b c (k x1 , k x2 ) = 0.86, at k x1 = k 0.225 growth of the energy of one wave is proportional to the growth of the product of the two other waves energies.One observes that W dotted line follows red line) during the growth stages of ¢

Figure 1 .
Figure 1.Homogeneous plasma with T e /T i = 10.(a) Time variations of energies (integrated on the 2D simulation box and normalized by the beam initial energy) of Langmuir waves , ¢  , and   , and ion acoustic waves ¢  and   , in logarithmic scale. (b) Langmuir wave energy spectrum in the plane (k x λ D , k y λ D ), at ω p t = 4500, in logarithmic scale.(c) Space distribution of δn i /n 0 , at ω p t = 4500, in the plane (x/λ D , y/λ D ).(d) Spatiotemporal variation of the cross-distribution along x (at fixed y) of δn i /n 0 , with descending and ascending lines exhibiting the propagation of waves ¢  and   , with group velocities −c s /v T and c s /v T .All variables are normalized.

Figure 2 .
Figure 2. Homogeneous plasma with T e /T i = 10.(Left column) Ion acoustic energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 1500, 2500, and 4500, in logarithmic scale.(Right column) Ion acoustic dispersion curves, at the same times, in the plane (ω/ω p , k x λ D ); the black dashed lines represent the theoretical dispersion law.All variables are normalized.Labels indicate the ion acoustic waves ¢  and   .

Figure 3 .
Figure 3. Homogeneous plasma with T e /T i = 10.(a) Cross-bicoherence b c of the triplet (δn i , E x , E y ) in the plane (ω 1 /ω p , ω 2 /ω p ), in logarithmic scales; the maximum is b c (ω 1 , ω 2 ) = 0.5, at ω 1 /ω p = 0.006 and ω 2 /ω p = 1.014.(b) Cross-bicoherence b c of the triplet (δn i , E x , E y ) in the plane (k x1 λ D , k x2 λ D ); the maximum is b c (k x1 , k x2 ) = 0.86, at k x1 λ D = 0.225 and k x2 λ D = -0.096. (c) Time variations of energies of Langmuir waves  and ¢  (black and blue) and ion acoustic modes ¢  (red), as well as of the energy products W W ¢   (dotted pink, ¢ *   ) and W W ¢   (dotted light blue, ¢ *   ). (d) Time variations of energies of Langmuir waves ¢  and   (black and blue) and ion acoustic modes   (red), as well as of the energy products W W     (dotted green,   *   ) and W W  ¢   (dotted pink,  ¢ *   ).(c)-(d) Energies, in logarithmic scales, are integrated on the 2D simulation box and normalized by the initial beam energy.All variables are dimensionless.

Figure 4 .
Figure 4. (For a better representation of (h) and (i), please zoom or print.)Randomly inhomogeneous plasma with ΔN > 0 and T e /T i = 10.(a), (c) Energy spectra of ion acoustic waves ¢ and   in the plane (k x λ D , k y λ D ), at ω p t = 2500 and 4500. (b), (d) Dispersion curves, at the same times, in the plane (ω/ω p , k x λ D ); the black dashed lines represent the theoretical dispersion law; ΔN = 0.05.(e) Variations with time of Langmuir and ion acoustic wave energies (integrated on the 2D simulation box and normalized by the initial beam energy), for ΔN = 0.05 (solid lines) and ΔN = 0.025 (dashed lines), in logarithmic scale. : black; ¢  : blue; ¢  ,   : red. (f), (g) Ion acoustic energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 1500 and 11,500, for ΔN = 0.025. (h) Time evolution (1300  ω p t  3970) of profiles along x of the parallel electric field E x .(i) Corresponding profiles of δn i /n 0 , in the same time interval as (h); ion acoustic waves propagate in the forward and backward directions (oblique lines with slopes c s /v T and −c s /v T ).All variables are normalized.

Figure 5 .
Figure 5. (a) Space distribution of δn i /n 0 , at ω p t = 4500, for a homogeneous plasma with T e /T i = 1.(b)-(c) Ion acoustic quasi-modes' qm ¢  and qm   energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 2500 and 4500, for ΔN = 0 and T e /T i = 1, in logarithmic scales.(d)-(e) Dispersion curves corresponding to (b)-(c).(f) Time variations of energies (integrated on the 2D simulation box and normalized by the initial beam energy) of Langmuir waves , ¢  , and   (black) and ion acoustic quasi-modes qm ¢  and qm   (solid red), in logarithmic scale, for ΔN = 0 and T e /T i = 1; for comparison, the dashed red line is the energy variation of the ¢  mode for ΔN = 0 and T e /T i = 10.(g)-(i) Ion acoustic quasi-modes' energy spectra in the plane (k x λ D , k y λ D ), at ω p t = 1500, 2500, and 4500, for ΔN = 0.05 and T e /T i = 1.All variables are normalized.