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# Polarization Ratios of Turbulent Langmuir/Z-mode Waves Generated by Electron Beams in Magnetized Solar Wind Plasmas

F J Polanco-Rodríguez, C. Krafft, P. Savoini

► To cite this version:

F J Polanco-Rodríguez, C. Krafft, P. Savoini. Polarization Ratios of Turbulent Langmuir/Z-mode Waves Generated by Electron Beams in Magnetized Solar Wind Plasmas. The Astrophysical Journal Letters, 2025, 989 (2), pp.L38. 10.3847/2041-8213/adf4ce. hal-05263639

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Submitted on 16 Sep 2025

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# Polarization Ratios of Turbulent Langmuir/Z-mode Waves Generated by Electron Beams in Magnetized Solar Wind Plasmas

F. J. Polanco-Rodríguez<fnref n="1" /> ID, C. Krafft<fnref n="1" />,<fnref n="2" /> ID, and P. Savoini<fnref n="1" /> ID

## Abstract

The polarization ratios $F = |E_{\perp}|^2/|E|^2$ of beam-generated turbulent Langmuir/Z-mode ($\mathcal{L}Z$) waves and electromagnetic emissions radiated at plasma frequency $\omega_p$ from such sources are studied in weakly magnetized and randomly inhomogeneous plasmas. Large-scale and long-term 2D/3V particle-in-cell simulations with parameters relevant to type III solar radio bursts are performed. Statistical studies using waveforms recorded by virtual satellites are carried out to determine the distributions of polarization ratios as a function of beam and plasma parameters. This efficient method, which mimics waveform recording by spacecraft in the solar wind, leads to results consistent with observations. Moreover, plasma random density fluctuations $\delta n$ turn out to be the key factor responsible for the increase in polarization ratios up to $F \sim 1$. Indeed, it is demonstrated that linear mode conversion at constant frequency near $\omega_p$ of $\mathcal{L}Z$ waves scattering on $\delta n$ is the most efficient and fastest process to produce large polarization ratios in randomly inhomogeneous plasmas. This is due to electromagnetic slow extraordinary Z-mode wave emission by $\mathcal{L}Z$ wave turbulence. The results provide guidance to theoretical studies and useful support to estimate the average level of density fluctuations in solar wind plasmas.

*Unified Astronomy Thesaurus concepts: Solar wind (1534); Radio bursts (1339); Solar electromagnetic emission (1490); Solar flares (1496)*

## 1. Introduction

For decades, a large number of high- and low-frequency waveforms have been observed at different solar wind conditions by various spacecraft (e.g., G. B. Hospodarsky & D. A. Gurnett 1995; P. J. Kellogg et al. 1999; D. M. Malaspina & R. E. Ergun 2008; V. V. Krasnoselskikh et al. 2011; D. M. Malaspina et al. 2011; D. B. Graham & I. H. Cairns 2013, 2014; D. B. Graham et al. 2021; D. Přša et al. 2021; J. Soucek et al. 2021; C. Y. Lorfing et al. 2023; N. E. Raouafi et al. 2023 and references therein) and, more recently, by the Solar Orbiter and Parker Solar Probe satellites (N. J. Fox et al. 2016; D. Müller et al. 2020). Their analysis has shed light on dispersion, polarization, and excitation mechanisms of various types of waves.

In particular, electric field waveforms of Langmuir/Z-mode waves (hereinafter referred to as $\mathcal{L}Z$ waves) excited by electron beams propagating along open magnetic field lines during type III solar radio bursts are commonly observed. Most of them reveal a predominant parallel (to the ambient magnetic field $B_0$) electric field component $E_{\|}$. However, a significant proportion of waves with large perpendicular electric fields $E_{\perp}$ and high polarization ratios $F = \sum_t E_{\perp}(t)^2/\sum_t (E_{\perp}(t)^2 + E_{\|}(t)^2)$ are also reported (S. D. Bale et al. 1996; D. M. Malaspina et al. 2011; D. B. Graham & I. H. Cairns 2013, 2014; P. J. Kellogg et al. 2013). Their origin, which remains unclear, is linked to the major problem of electromagnetic radiation by electrostatic wave turbulence generated by beams. Therefore, the question of which wave processes

and beam-plasma parameters are responsible for such large polarization ratios is addressed in this work.

Several hypotheses have been proposed to explain why polarization ratios calculated using measured $\mathcal{L}Z$ waveforms can reach large values up to $F \sim 0.4$ and even more (D. M. Malaspina et al. 2011; D. B. Graham et al. 2012; D. B. Graham & I. H. Cairns 2013, 2014). Electrostatic wave decay was invoked, as it transports the energy of $\mathcal{L}Z$ waves toward smaller k-scales while enhancing their electromagnetic character (e.g., A. J. Willes & I. H. Cairns 2000; D. B. Graham & I. H. Cairns 2013; A. Layden et al. 2013; F. J. Polanco-Rodríguez et al. 2025), as well as 3D Langmuir eigenmodes (R. E. Ergun et al. 2008; D. M. Malaspina & R. E. Ergun 2008) or linear transformations of $\mathcal{L}Z$ waves on density gradients (e.g., D. Krauss-Varban 1989; S. D. Bale et al. 1998; P. J. Kellogg et al. 1999; D. M. Malaspina et al. 2011).

Random density fluctuations $\delta n$ exist in the solar wind with average levels up to $\Delta N = \langle (\delta n/n_0)^2 \rangle^{1/2} \simeq 0.07$, where $n_0$ is the ambient plasma density (e.g., L. M. Celnikier et al. 1987; V. Krupar et al. 2020). Recently, the magnetic signature of an electromagnetic slow extraordinary Z-mode wave was observed for the first time in a solar wind plasma with strong density fluctuations (A. Larosa et al. 2022) and is thought to be produced via linear mode conversion (LMC) at constant frequency of $\mathcal{L}Z$ waves scattering on $\delta n$ (e.g., D. E. Hinkel-Lipsker et al. 1989; V. V. Krasnoselskikh et al. 2019; C. Krafft & P. Savoini 2022a). Recently, this process has been studied analytically and numerically in randomly inhomogeneous and weakly magnetized 2D and 3D plasmas (C. Krafft et al. 2025), demonstrating the predominance of Z-mode wave radiation by beam-generated radio sources. In this view, it is now timely to explain the link between Z-mode waves and large polarization ratios.

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Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ Laboratoire de Physique des Plasmas (LPP), CNRS, Sorbonne Université, Observatoire de Paris, Université Paris-Saclay, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France; catherine.krafft@universite-paris-saclay.fr</footnote>

<footnote n="2">² Institut Universitaire de France (IUF), France

*Received 2025 June 27; revised 2025 July 28; accepted 2025 July 28; published 2025 August 13*</footnote>
The principal questions raised in this work are the following. What is the main process responsible for large wave polarization ratios *F* in solar wind plasmas? How are they distributed for a statistical ensemble of *LZ* and electromagnetic wavepackets? What is the link between large polarization ratios and the generation of electromagnetic *Z*-mode waves? What is the impact of beam-plasma parameters (magnetization, electron and ion temperatures, average level of density fluctuations, beam velocity) on *F* distributions? How can the properties of polarization ratios be used to diagnose solar wind plasmas?

These issues are addressed by using large-scale and long-
term 2D/3V particle-in-cell (PIC) simulations with parameters
relevant to type III solar radio bursts. In this context, we also
demonstrate the effectiveness of the technique used to analyze
the simulations. It consists of the generation of a large number
of virtual satellites measuring waveforms (C. Krafft et al.
2014) during their motion through a large 2D simulation box.
To our knowledge, such an approach is used for the first time
in the framework of PIC simulations and is accompanied here
by statistical studies using several thousand waveforms. This
method, which mimics wave measurements by spaceborne
satellites, aims to support the analysis and interpretation of
solar wind observations. Indeed, long-term and high-resolution
waveforms can be obtained simultaneously for the six fields'
components as well as for the ion and electron densities. This
considerably increases the possibility of identifying ongoing
processes and wave characteristics, compared to the situation
in space. Moreover, it is shown that this local approach is
particularly useful for randomly inhomogeneous plasmas
where waves are strongly scattered and for which the global
approach previously used (e.g., C. Krafft & P. Savoini 2024)
provides, for some diagnostics, less clear and interpretable
results.

## 2. Methods

2.1. 2D/3V PIC Simulations

Large-scale and long-term 2D/3V PIC simulations are
performed using the SMILEI code (J. Derouillat et al. 2018),
with beam and plasma parameters relevant to type III solar
radio bursts at distances of ~0.1–1 au from the Sun.
Randomly inhomogeneous and weakly-to-moderately magne-
tized plasmas are considered, with ΔN ≪ 0.05 and ratios of
cyclotron-to-plasma frequency in the range ω_c/ω_p ≪ 0.5. An
electron beam with drift velocity v_b = 12.7v_T ≃ 0.25c (where
v_T is the electron thermal velocity) and weak relative density
n_b = 5 · 10⁻⁴n₀ is injected in the 2D plasma along the ambient
magnetic field B₀ directed along the x-axis. The electron-to-ion
mass and temperature ratios are m_e/m_i = 1/1836 and
T_e/T_i = 10.

Two other sets of parameters are used to understand the
dependence of polarization ratios on ΔN, ω_c/ω_p, T_e, T_e/T_i, and
v_b. The first one, with v_b = 12.7v_T ≃ 0.25c like mentioned
above, is characterized by a low electron temperature
T_e = 20 eV, with T_e/T_i = 3, ω_c/ω_p = 0.02, and ΔN = 0,
0.025. The second one includes slower beams with
6v_T ≪ v_b ≪ 12.7v_T (i.e., 0.12c ≪ v_b ≪ 0.25c) and plasma para-
meters 0.025 ≪ ΔN ≪ 0.05, T_e/T_i = 10, and ω_c/ω_p = 0.07.

Simulation planes with sizes Lₓ × Lᵧ = 1448 × 1448λ<sup>2</sup><sub>D</sub> (λ<sub>D</sub> is the electron Debye length) are used to perform exceptionally long-term simulations up to t = 30,000ω<sub>p</sub><sup>-1</sup>,

which can describe the full dynamics of electrostatic decay in
the presence of a weak beam. Note that the large number of
particles per cell used (N<sub>c</sub> = 5400) reduces the total energy
relative error to ~10<sup>-4</sup> at t = 30,000ω<sub>p</sub><sup>-1</sup>. More details can be
found in previous articles by the authors (C. Krafft &
P. Savoini 2021, 2022a, 2022b, 2023, 2024; C. Krafft et al.
2024, 2025; F. J. Polanco-Rodríguez et al. 2025).

## 2.2. Virtual Satellites

Hereafter, we simulate thousands of virtual satellites
moving in a 2D simulation box and recording at each time *t*
and position **x**(*t*) the electric and magnetic wave fields **E**(**x** (*t*), *t*) and **B**(**x**(*t*), *t*), as well as the ion and electron densities
*n*<sub>i</sub>(**x**(*t*), *t*) and *n*<sub>e</sub>(**x**(*t*), *t*), with a sampling time of 0.18**ω**<sub>p</sub><sup>-1</sup>
(0.06**ω**<sub>p</sub><sup>-1</sup> for the low-temperature simulation with *T*<sub>e</sub> = 20 eV).
For purposes of simplicity and without loss of generality, the
velocity **v**<sub>s</sub> of each satellite with respect to the plasma
considered as immobile is chosen constant and directed along
the *x*-axis, i.e., along **v**<sub>b</sub> and **B**<sub>0</sub>, in order to simplify the
determination of the Doppler shift. Its modulus *v*<sub>s</sub> = 0.3*v*<sub>T</sub> is a
nominal value for spacecraft moving relative to the slow solar
wind (e.g., D. B. Graham & I. H. Cairns 2013).

The duration of the simulated waveforms is
T = 15,000 – 30,000ω<sub>p</sub><sup>-1</sup>, which is roughly equivalent to
0.5–1 s in the solar wind near 1 au. Wave frequencies
measured by the virtual satellites are Doppler-shifted, i.e.,
ω<sup>D</sup> = ω – **k** · **v**<sub>s</sub>, where ω and **k** are the frequency and the
wavevector in the plasma (laboratory) frame, respectively.
Note that **v**<sub>s</sub> can be adjusted to increase or decrease the
Doppler shift. This can be useful for distinguishing between
forward- and backward-propagating LZ waves. For ordinary
and extraordinary electromagnetic waves radiated at ω<sub>p</sub> (i.e.,
O-, X-, and Z-mode waves), the Doppler shift term **k** · **v**<sub>s</sub> is
negligible (for the satellite velocity chosen) compared to their
frequencies ω<sub>em</sub> in the motionless plasma frame, as their
wavenumbers satisfy k<sub>em</sub>λ<sub>D</sub> ≲ 0.01, so that ω<sub>em</sub><sup>D</sup> ≃ ω<sub>em</sub>.
Frequencies ω<sub>S</sub> of ion acoustic waves in this frame can be
neglected compared to those in the satellite frame (ω<sub>S</sub><sup>D</sup>) and
the Doppler shift term, so that ω<sub>S</sub><sup>D</sup> ≃ ±k<sub>S</sub> · **v**<sub>s</sub>. Finally, the
frequencies of the LZ waves are Doppler-shifted below or
above ω<sub>p</sub> depending on their forward or backward direction of
propagation, respectively. For example, for beam-driven LZ
waves with k<sub>LZ</sub>λ<sub>D</sub> ~ 0.1, we get ω<sub>LZ</sub><sup>D</sup> ≃ ω<sub>p</sub>(1 + 3k<sub>LZ</sub><sup>2</sup>λ<sub>D</sub><sup>2</sup>/2 +
ω<sub>c</sub><sup>2</sup> sin<sup>2</sup>θ/2ω<sub>p</sub><sup>2</sup>) – v<sub>s</sub>k<sub>LZ</sub> ≃ ω<sub>p</sub>(1 + 3k<sub>LZ</sub><sup>2</sup>λ<sub>D</sub><sup>2</sup>/2) – v<sub>s</sub>k<sub>LZ</sub> ≃ 0.984
for ω<sub>c</sub>/ω<sub>p</sub> = 0.07 ≪ 1, where k<sub>LZ</sub> = k<sub>LZ</sub> cos θ =
· k<sub>LZ</sub> · **B**<sub>0</sub>/B<sub>0</sub>.

2.3. Tools for Waveform Analysis

We present here the methodology used to analyze the
numerous waveforms recorded by virtual satellites, based on
averaging methods and calculation of probability density
functions (PDFs). The polarization ratio of a single waveform
s at time *t* and frequency *ω* ≃ *ω*<sub>*p*</sub> is designated by

$$
F_s(t) = \frac{|E_{\perp}^s(\omega_p, t)|^2}{|E^s(\omega_p, t)|^2}, \quad (1)
$$

where |E<sup>s</sup><sub>⊥</sub>(ω<sub>p</sub>, t)|<sup>2</sup> and |E<sup>s</sup>(ω<sub>p</sub>, t)|<sup>2</sup> are the spectral energies in
the interval 0.9ω<sub>p</sub> ≪ ω<sup>D</sup> ≪ 1.1ω<sub>p</sub>, calculated using complex
Morlet wavelet transforms of the perpendicular and total
electric fields of the waveform s. Note that |E<sup>s</sup><sub>⊥</sub>(ω<sub>p</sub>, t)|<sup>2</sup> =
<figure><img src="image_5.png" /><figcaption>Figure 1. Typical example of waveforms, together with hodograms, polarization ratios, and spectral wave energies, in a weakly magnetized homogeneous plasma with $\omega_c/\omega_p = 0.07$ and $\Delta N = 0$, for $0 \le \omega_p t \le 30,000$. (a) Waveform of electric fields $E_{\parallel}(t) = E_x(t)$ (gray), $E_{\perp 1}(t) = E_y(t)$ (red), and $E_{\perp 2}(t) = E_z(t)$ (blue), filtered in the Doppler-shifted frequency interval [0.9, 1.1]$\omega_p$. (b) Hodograms $E_{\parallel}(E_{\perp 1})$ calculated in equidistant time windows of $60\omega_p^{-1}$. (c) Time variation of the polarization ratio $F_s(t)$ of the waveform. (d) Spectral electric energy $|E(\omega^D, t)|^2$ in the map $(\omega_p t, \omega^D/\omega_p)$; the beam-driven $\mathcal{L}Z$ and backscattered $\mathcal{L}Z'$ waves are indicated by labels. (e) Spectral magnetic energy $|B(\omega^D, t)|^2$ in the map $(\omega_p t, \omega^D/\omega_p)$; O- and Z-modes are indicated by labels. Both energies are calculated using wavelet transforms. (f) Variation of the ion density perturbation $\delta n_i(t)/n_0 = (n_i(t) - n_0)/n_0$ as a function of the time $\omega_p t$. (g) Spectral energy $|\delta n_i(\omega^D, t)|^2$ in the map $(\omega_p t, \omega^D/\omega_p)$, calculated using wavelet transform. All fields and energies are presented in arbitrary units. Color bars are in logarithmic scales.</figcaption></figure>

$|E_{\perp 1}^s(\omega_p, t)|^2 + |E_{\perp 2}^s(\omega_p, t)|^2$ and $(E_x^s, E_y^s, E_z^s) = (E_x^s, E_y^s, E_z^s)$. We define the average of $F_s(t)$ over $N_s$ waveforms as

$$ \langle F(t) \rangle_s = \frac{1}{N_s} \sum_s F_s(t), \quad (2) $$

whereas the average of $F_s(t)$ over the time interval $t_0 \le t \le t_0 + \Delta T$ is

$$ \langle F_s \rangle_t = \frac{1}{\Delta T} \int_{t_0}^{t_0 + \Delta T} F_s(t) dt. \quad (3) $$

The polarization ratio averaged over time and waveforms is designated by $\langle F \rangle_{s,t}$. The PDF of $F_s(t)$ for a single waveform is $f_s(F)$, with $0 \le F \le 1$ representing $F_s(t)$ at a given time, whereas the PDF of $F_s(t)$ for $N_s$ waveforms is defined as $\langle f \rangle_s$. The distribution $\langle f(F, X_0) \rangle_s$ represents the PDF of $N_s$ waveforms with polarization ratio $F$ at a given physical variable $X = X_0$. Finally, since $F_s(t)$ is a ratio of energies, one can

define a similar quantity in the $(k, \omega)$-space as

$$ F_{k,\omega} = \frac{|E_{\perp k,\omega}|^2}{|E_{k,\omega}|^2}, \quad (4) $$

where $E_{\perp k,\omega}$ and $E_{k,\omega}$ are the Fourier components of $E_{\perp}(\mathbf{x}, t)$ and $E(\mathbf{x}, t)$, respectively. $F_{k,\omega}$ represents the polarization ratio of a wave with wavevector $k$ and frequency $\omega$. Some correspondence between $\langle F \rangle_{s,t}$ and $F_{k,\omega}$ can be established, since $\langle F \rangle_{s,t}$ is averaged over time and space (each waveform depends on $(x(t), y(t))$).

### 3. Waveforms and Related Diagnostics

Before performing statistical studies, let us analyze the typical waveforms obtained by simulations in a homogeneous and a randomly inhomogeneous plasma.
### 3.1. Homogeneous Magnetized Plasma

Figure 1 shows a typical waveform in a homogeneous magnetized plasma with $\omega_c/\omega_p = 0.07$. It presents the electric fields $E_\parallel(t)$, $E_{\perp 1}(t)$, and $E_{\perp 2}(t)$ filtered in the Doppler-shifted frequency range [0.9, 1.1]$\omega_p$, together with the hodograms $E_\parallel(E_{\perp 1})$ calculated within time periods of $60\omega_p^{-1}$, the polarization ratio $F_s(t)$, and the ion density perturbation $\delta n_i(t)/n_0 = (n_i(t) - n_0)/n_0$ (Figures 1(a)-(c), (f)). The electric, magnetic, and low-frequency spectral energies $|E(\omega^D, t)|^2$, $|B(\omega^D, t)|^2$, and $|\delta n_i(\omega^D, t)|^2$ are shown in Figures 1(d), (e), and (g), respectively. The field $E_\parallel(t)$ remains the largest one during most of time until $E_{\perp 1}(t)$ reaches comparable amplitudes at large $\omega_p t \gtrsim 15,000$ (Figure 1(a)), with a constantly negligible contribution of $E_{\perp 2}(t)$. In Figure 1(d), $|E(\omega^D, t)|^2$ reaches large intensities at early times around $\omega^D \approx 0.98\omega_p$, corresponding to beam-driven $\mathcal{L}\mathcal{Z}$ waves. Later, at $\omega_p t \gtrsim 4000$, backward-propagating waves $\mathcal{L}\mathcal{Z}'$ produced by the first cascade $\mathcal{L}\mathcal{Z} \rightarrow \mathcal{L}\mathcal{Z}' + S'$ of electrostatic decay (e.g., C. Krafft & P. Savoini 2024) appear in the frequency range $1.01 \lesssim \omega^D/\omega_p \lesssim 1.04$ simultaneously with ion acoustic wavepackets of frequencies $0.04 \lesssim \omega^D/\omega_p \lesssim 0.06$ (Figures 1(f) and (g)). $\mathcal{L}\mathcal{Z}$-wave beatings correlated with ion acoustic oscillations are signatures of this process (e.g., in the range $3500 \lesssim \omega_p t \lesssim 5000$; see Figures 1(a), (d), (f), and (g)) and are clearly visible up to $\omega_p t \approx 15,000$. Later, such oscillations become irregular; indeed, forward- and backward-propagating waves coming from the successive decay cascades, i.e., $\mathcal{L}\mathcal{Z}^{(2)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ (with $\omega^D \lesssim \omega_p$) and $\mathcal{L}\mathcal{Z}'$ and $\mathcal{L}\mathcal{Z}^{(3)}$ (with $\omega^D \gtrsim \omega_p$) appear (F. J. Polanco-Rodríguez et al. 2025). Their frequencies approach $\omega_p$ as time increases, whereas at earlier times, $\omega_p t \lesssim 10,000$, no energy is visible at $\omega_p$ (Figure 1(d)). Most of the electric energy is concentrated around $\omega_p$ at $\omega_p t \gtrsim 20,000$ as a result of the ultimate stage of $\mathcal{L}\mathcal{Z}$ wave decay.

In Figure 1(b), the hodograms are mainly linearly polarized and elongated along the magnetic field direction, as expected for quasi-parallel propagating electrostatic $\mathcal{L}\mathcal{Z}$ waves. However, they present more complex structures as time increases, due to the occurrence of nonlinear wave processes. The polarization ratio $F_s(t)$ oscillates with time around the mean $\langle F_s \rangle_t \approx 0.26$ (averaged over $0 \lesssim \omega_p t \lesssim 30,000$) while slowly growing (Figure 1(c)). Enhancements up to $F_s(t) \approx 0.6$ are observed at $\omega_p t \gtrsim 10,000$, in conjunction with the radiation of $\mathcal{Z}$-mode wave magnetic energy (Figure 1(e)), resulting from the late stage of electrostatic decay when $\mathcal{L}\mathcal{Z}$ wave energy has been transported down to electromagnetic $k$-scales (F. J. Polanco-Rodríguez et al. 2025). Indeed, the time variation of $|B(\omega^D, t)|^2$ exhibits a gradual increase of $\mathcal{Z}$-mode energy with frequencies $\omega^D \lesssim \omega_p$. At $\omega_p t \gtrsim 10,000$, magnetic signatures of $\mathcal{L}\mathcal{Z}$ waves with $\omega^D \gtrsim \omega_p$ become visible in Figure 1(e); meanwhile, the energy of $\mathcal{Z}$-mode waves increases continuously, simultaneously with $F_s(t)$. Note also the presence of $\mathcal{O}$-mode waves with $\omega^D > \omega_p$, which are generated by electromagnetic decay via $\mathcal{L}\mathcal{Z} \rightarrow \mathcal{O} + S$, where $S$ is an ion acoustic wave, as will be shown in a forthcoming paper (see also C. Krafft et al. 2024). The increase with time of $F_s(t)$ is mainly due to nonlinear decay that transports wave energy to smaller and more oblique wavevectors $k$ until the final generation of $\mathcal{Z}$-mode waves with frequencies $\omega_c\mathcal{Z} < \omega^D \lesssim \omega_p$, where $\omega_c\mathcal{Z}$ is the $\mathcal{Z}$-mode's cutoff frequency (C. Krafft et al. 2024; F. J. Polanco-Rodríguez et al. 2025).

<figure><img src="image_6.png" /></figure>

**Figure 2.** Typical example of waveforms, together with hodograms, polarization ratios, and spectral wave energies, in a randomly inhomogeneous and weakly magnetized plasma with $\Delta N = 0.05$ and $\omega_c/\omega_p = 0.07$, for $0 \le \omega_p t \le 10,000$. (a) Waveform of electric fields $E_\parallel(t)$ (gray), $E_{\perp 1}(t)$ (red), and $E_{\perp 2}(t)$ (blue), filtered in the Doppler-shifted frequency interval [0.9, 1.1]$\omega_p$; the applied density fluctuations $\delta n(t)/n_0$ are superimposed (green lines), and a reflection region is indicated by a black star, as it is also in panels (d) and (e). (b) Hodograms $E_\parallel(E_{\perp 1})$ calculated in equidistant time windows of $60\omega_p^{-1}$. (c) Time variation of the polarization ratio $F_s(t)$ of the waveform. (d) Spectral electric energy $|E(\omega^D, t)|^2$ in the map $(\omega_p t, \omega^D/\omega_p)$. (e) Spectral magnetic energy $|B(\omega^D, t)|^2$ in the map $(\omega_p t, \omega^D/\omega_p)$. Both energies are calculated using wavelet transforms. Electromagnetic modes $\mathcal{O}$ and $\mathcal{Z}$ are indicated by labels. All fields and energies are presented in arbitrary units. Color bars are in logarithmic scales.

### 3.2. Randomly Inhomogeneous Magnetized Plasma

Figure 2 shows a typical waveform in a randomly inhomogeneous and magnetized plasma with $\Delta N = 0.05$ and $\omega_c/\omega_p = 0.07$. Contrary to the homogeneous plasma case, $E_\parallel(t)$ and $|E_{\perp 1}(t)| \gg |E_{\perp 2}(t)|$ reach similar intensities almost all the time (Figure 2(a)), whereas most intense wavepackets are trapped into density depletions $\delta n < 0$ (C. Krafft & A. S. Volokitin 2021). Note that the wave energy peaks when $\mathcal{L}\mathcal{Z}$ waves are reflected on gradients of $\delta n$, i.e., when their frequency reaches the local plasma frequency, $\omega_{\mathcal{L}\mathcal{Z}} \approx \omega_p(1 - |\delta n(t)|/2n_0) < \omega_p$. At this time, a part of the $\mathcal{L}\mathcal{Z}$ wave energy can also be transformed at constant frequency into electromagnetic $\mathcal{Z}$-mode wave energy via the linear mode conversion process (LMC). The polarization ratio $F_s(t)$ exhibits strong enhancements up to 0.85 (Figure 2(c)), with a mean $\langle F_s \rangle_t \approx 0.49$ (averaged over $0 \lesssim \omega_p \lesssim 10,000$) around 2 times larger than in a homogeneous plasma. Similarly, hodograms reveal more complex elliptical structures and rarely linear polarizations compared with the case $\Delta N = 0$ (Figure 2(b)). The spectral electric and magnetic energies show strong electromagnetic emissions at early times (Figures 2(d) and (e)) in the frequency range $0.96 \lesssim \omega^D/\omega_p \lesssim 0.98$. At $2000 \lesssim \omega_p t \lesssim 4000$ (corresponding to a density depletion, as
evidenced in Figure 2(a) by green lines), a wide frequency spectrum of quasi-electrostatic *LZ* waves of significant energy appears (Figure 2(d)). Those are scattered on density fluctuations and experience refraction, reflection, and linear mode conversion (LMC), while *Z*-mode emission at $\omega \lesssim \omega_p$ is enhanced.

Let us focus in more detail on the trapped *LZ* wavepacket observed within $500 \lesssim \omega_p t \lesssim 1500$ (Figure 2(a)). At $500 \lesssim \omega_p t \lesssim 1000$, its energy is mostly carried by the parallel wave electric field $E_\| (t)$. When it is reflected on the gradient of a density depletion at $\omega_p t \simeq 1500$ (see the black stars in Figures 2(a), (d), and (e) indicating the reflection region), a large perpendicular field $|E_{\perp 1}(t)| > |E_\| (t)|$ is generated. Meanwhile, the corresponding hodogram exhibits quasi-circular polarization (Figure 2(b)). Moreover, a strong increase in electric and magnetic energies occurs simultaneously around $\omega^D/\omega_p \simeq 0.965$, in conjunction with the reflection process (Figures 2(d) and (e)), which is the signature of electromagnetic *Z*-mode wave emission by linear conversion of *LZ* waves at constant frequency. Meanwhile, $F_s(t)$ reaches large values up to 0.85, confirming a correlation between the generation of *Z*-mode waves via LMC and the appearance of large polarization ratios.

To conclude, in a weakly magnetized homogeneous plasma where *LZ* wave turbulence is generated by an electron beam, polarization ratios grow slowly with time due to nonlinear decay of *LZ* waves. In randomly inhomogeneous plasmas, polarization ratios increase rapidly due to the fast linear interactions of wave turbulence with random density fluctuations, leading to scattering and conversion of *LZ* waves to *Z*-mode waves. Let us now confirm these observations with statistical studies.

# 4. Statistical Study of Polarization Ratios

## 4.1. Distributions and Their Dependencies on Beam-plasma Parameters

In Figures 3(a)–(c), the PDFs $\langle f \rangle_s$ (see Section 2.3) represent the histograms of polarization ratios $F$ calculated at any time along $N_s = 1000$ waveforms filtered in the Doppler-shifted frequency range [0.9, 1.1]$\omega_p$, where, in weakly magnetized plasmas with $\Delta N \lesssim 0.05$, all *LZ* and fundamental electromagnetic wave energy is gathered. Note that for all distributions and averaged values presented below, convergence is reached at $N_s \simeq 400$. Waveforms are recorded in the time intervals $3000 \le \omega_p t \le 6000$ ((a) and (c)) and $12,000 \le \omega_p t \le 15,000$ (b), in homogeneous ($\Delta N = 0$) and inhomogeneous ($\Delta N = 0.05$) plasmas with various $\omega_c/\omega_p$ (see caption). The first time interval covers the period when the process of LMC at constant frequency of *LZ* waves into electromagnetic waves is most intense (in randomly inhomogeneous plasmas with $\Delta N \gtrsim 3(v_T/v_b)^2$, as discussed below). The second one corresponds to larger times when nonlinear three-wave decay processes are active or close to being achieved (in quasi-homogeneous plasmas with $\Delta N \lesssim 3(v_T/v_b)^2$; see also hereafter).

Figure 3(a) shows that $\langle f \rangle_s$ peaks near $F_m \simeq 0.1$ (polarization ratio at the maximum of $\langle f \rangle_s$) for $\omega_c/\omega_p < 0.2$ and covers the range $F \lesssim 0.6$, with $\langle F \rangle_{s,t} \simeq 0.17$. At larger $\omega_c/\omega_p \gtrsim 0.2$, the maxima of $\langle f \rangle_s$ correspond to much smaller $F_m$, due to the Lorentz force. At later times (Figure 3(b)), $F_m$ is significantly larger for all $\omega_c/\omega_p$, due to the occurrence of nonlinear three-wave interactions (see discussion in

Section 4.2); the largest $F_m \simeq 0.3$ are reached for $\omega_c/\omega_p < 0.2$. The dependence of $\langle f \rangle_s$ on $\omega_c/\omega_p < 0.2$ is weak, but it is much more important for $\omega_c/\omega_p \gtrsim 0.2$ (moderately magnetized plasmas). For $\Delta N = 0.05$ (with $\Delta N \gtrsim 3(v_T/v_b)^2$; see Figure 3(c)), random density fluctuations are responsible for the significant increase of $F_m$, compared with the homogeneous plasma case (Figure 3(a)). For $\omega_c/\omega_p < 0.2$, all PDFs $\langle f \rangle_s$ extend up to $F \simeq 1$ and peak near $F_m \simeq 0.4$, whereas for $\omega_c/\omega_p \gtrsim 0.2$, $F_m$ is significantly smaller. Note that for such inhomogeneous plasmas, the distributions $\langle f \rangle_s$ change little over time (not shown here). It appears that $F_m$ mainly increases due to *LZ* wave interactions with density fluctuations, whereas it is weakly (significantly) impacted by plasma magnetization for $\omega_c/\omega_p < 0.2$ ($\omega_c/\omega_p \gtrsim 0.2$).

Figure 3(d) shows the decrease with $\omega_c/\omega_p$ of the mean $\langle F \rangle_{s,t}$ of all distributions presented in Figures 3(a)–(c). At early times, $\langle F \rangle_{s,t}$ reaches up to 3 times larger values for $\Delta N = 0.05$ than for $\Delta N = 0$. At late times, when electrostatic decay is completed, $\langle F \rangle_{s,t}$ remains at any time significantly smaller than in inhomogeneous plasmas, confirming that, whatever the plasma magnetization is, LMC is more efficient and faster than electrostatic decay at generating waves with large polarization ratios. Note that when the plasma is randomly inhomogeneous, $\langle F \rangle_{s,t}$ follows a dependence of the form $\cosh^{-2}(\omega_c/\omega_p)$ (the fit satisfies $R^2 > 0.99$), which can be approximated by a parabolic profile for $\omega_c/\omega_p \lesssim 0.2$, where the dependence of $\langle F \rangle_{s,t}$ with magnetization is weak. For homogeneous plasmas where nonlinear decay dominates, the tendency is linear at $\omega_c/\omega_p \le 0.2$.

Figure 3(e) shows the distributions $\langle f \rangle_s$ for the set of parameters corresponding to a significantly lower electron temperature $T_e = 20$ eV, with $T_e/T_i = 3$, $\omega_c/\omega_p = 0.02$, and $\Delta N = 0, 0.025$. The PDFs obtained for homogeneous and inhomogeneous plasmas correspond well to the distributions with $\omega_c/\omega_p < 0.2$ in Figures 3(a) and (c), respectively, showing that $T_e$ and $T_e/T_i$ weakly influence the polarization ratio.

Figure 3(f) shows the variation of the mean $\langle F \rangle_{s,t}$ (colored points) with its corresponding standard deviations $\sigma_F = \langle\langle \langle F \rangle_{s,t} - F_s(t) \rangle\rangle_{s,t}^{1/2}$ (gray vertical lines) as a function of $\Delta N v_b^2/3v_T^2$, i.e., for different $v_b/v_T$ and $\Delta N$. Whereas $\langle F \rangle_{s,t} \simeq 0.2$ for $\Delta N \lesssim 3(v_T/v_b)^2$, a sharp increase of $\langle F \rangle_{s,t}$ is observed at $\Delta N \simeq 3(v_T/v_b)^2$, reaching $\langle F \rangle_{s,t} \simeq 0.4$ for $\Delta N \gtrsim 3(v_T/v_b)^2$ (as found in Figure 3(c)). This is due to a significant increase of LMC efficiency, which can be used to estimate $\Delta N$ in the solar wind (see Section 4.2).

Figures 3(g)–(i) present the distributions $F_{k,\omega}$ in the (sign$(k_x)k\lambda_D$, $\omega/\omega_p$) map, calculated in the time range $3500 \le \omega_p t \le 6500$ using a high-resolution simulation plane with $\delta k_x = 0.001\lambda_D^{-1}$ and $\delta k_y = 0.002\lambda_D^{-1}$ for three propagation angles $\theta = 0^\circ, 45^\circ$, and $90^\circ$ ($\cos\theta = \mathbf{k} \cdot \mathbf{B}_0/kB_0$). For $\theta < 90^\circ$, one observes that the largest $F_{k,\omega} \simeq 1$ are reached near the cutoff frequency $\omega_c Z$ of *Z*-mode waves in the spectral domain where $k\lambda_D \lesssim 0.005-0.008$. At $\theta = 90^\circ$, $F_{k,\omega} \simeq 1$ for *Z*-mode waves with frequencies $\omega_c Z \lesssim \omega \lesssim \omega_p$ and $k\lambda_D \lesssim 0.02$. This confirms the link between large polarization ratios and *Z*-mode waves’ excitation. On the other hand, for *O*-mode waves, which are lying in the spectral range corresponding to $k\lambda_D \lesssim 0.005$ and $\omega \gtrsim \omega_p$, $F_{k,\omega}$ varies between 0 and 1 depending on $\theta$, contrary to *Z*-mode waves at $\omega_c Z$, with $F_{k,\omega} \simeq 1$ for any $\theta$. Moreover, waves with $k\lambda_D \gtrsim 0.02$ are quasi-electrostatic *LZ* waves with polarization
<figure><img src="image_7.png" /></figure>

**Figure 3.** Distributions of polarization ratios as a function of beam and plasma parameters. (a)–(c) Distributions $\langle f \rangle_s$ calculated using $N_s = 1000$ waveforms with frequencies $\omega^D$ filtered in the range [0.9, 1.1]$\omega_p$, in magnetized plasmas with $0 \le \omega_c/\omega_p \le 0.5$, in the time intervals $3000 \le \omega_p t \le 6000$ ((a) and (c)) and $12,000 \le \omega_p t \le 15,000$ (b), for $\Delta N = 0$ ((a) and (b)) and $\Delta N = 0.05$ (c). (d) Variations of $\langle f \rangle_{s,t}$ with $\omega_c/\omega_p$ for $\Delta N = 0$ (triangles) and $\Delta N = 0.05$ (stars), in the ranges $3000 \le \omega_p t \le 6000$ (black) and $12,000 \le \omega_p t \le 15,000$ (red); stars are fitted by an $\alpha + \beta \cosh^2(\gamma\omega_c/\omega_p)$ curve for $\omega_c/\omega_p \le 0.5$ (green). (e) Distribution $\langle f \rangle_s$ calculated similarly to (a) for simulations with $v_b = 12.7v_T$, $T_e = 20$ eV = $3T_i$, $\omega_c/\omega_p = 0.02$, $\Delta N = 0$ (blue) and $\Delta N = 0.025$ (red). (f) Variation of $\langle F \rangle_{s,t}$ with $\Delta N v_b^2/3v_T^2$ for $\Delta N = 0.025$ (red points) and $\Delta N = 0.05$ (blue points), with normalized beam velocities $v_b/v_T = 6, 7, 9, 11.5, 12.7$; the gray vertical lines indicate the standard deviations $\sigma_F$ of the distributions, and the curve fitting the color points is represented by a black dashed line. (a)–(f) In order to ignore noise, only times $t$ of wavepackets $E(t)$ corresponding to spectral energies $|E(\omega^D, t)|^2$ greater than $|E(\omega^D, t)|^2_{\text{max}}/1000$ are retained when building the distributions. (g)–(i) Polarization ratio $F_{k,\omega}$ (Equation (4)) calculated using the Fourier components $E_{k,\omega}$ in a randomly inhomogeneous and magnetized plasma with $\Delta N = 0.025$ and $\omega_c/\omega_p = 0.07$, in the time range $3500 \le \omega_p t \le 6500$. $F_{k,\omega}$ is presented in the map ($\text{sign}(k_x)k_{\lambda_D}, \omega/\omega_p$) for three different propagation angles $\theta = 0^\circ, 45^\circ$, and $90^\circ$; only colored regions, which correspond to spectral electric energies satisfying $|E_k|^2 > |E_{k_{\text{max}}}|^2/1000$, are shown (other regions are represented in light gray). O- and Z-mode waves are indicated by labels. The linear color bar holds for (g)–(i). A large simulation box with $L_x \times L_y = 5792 \times 2896\lambda_D^2$ ($N_x = 4096$ and $N_y = 2048$ grid points) is used. The theoretical dispersion curves of electromagnetic modes are superimposed with white and black lines for parallel (solid) and perpendicular (dashed) wave propagation.

$F_{k,\omega} \simeq 0$ at $\theta \simeq 0^\circ$, which satisfies $E_\parallel/E_\perp \simeq k_\parallel/k_\perp$ (i.e., $F(\theta) \simeq \sin^2\theta$, as we can observe). Note that X-mode waves are only weakly emitted (not shown here), so that they are of no interest in this study.

## 4.2. Discussion

At this stage, several conclusions can be stated. First, random density fluctuations $\delta n$ are primarily responsible for large polarization ratios due to their fast interactions with $\mathcal{L}\mathcal{Z}$ waves, which are scattered and linearly converted via LMC into electromagnetic Z-mode waves excited below $\omega_p$ down to their cutoff frequency $\omega_c\mathcal{Z}$ (C. Krafft et al. 2025), where polarization ratios can reach $F \simeq 1$. On the other hand, in homogeneous plasmas, $\mathcal{L}\mathcal{Z}$ wave energy can be transported toward smaller and more oblique wavevectors $k$ by electrostatic decay on much longer time scales, so that polarization

ratios increase slowly with time, ultimately reaching lower values than in inhomogeneous plasmas.

Second, the polarization ratios $F_m$ (at which the PDF $\langle f \rangle_s$ is maximum) and $\langle F \rangle_{s,t}$ (averaged over time and waveforms) decrease with increasing $\omega_c/\omega_p$ for $\Delta N = 0$ and $\Delta N > 0$, due to the Lorentz force and the fact that the Z-mode cutoff frequency $\omega_c\mathcal{Z} \simeq \omega_p - \omega_c/2$ decreases with increasing $\omega_c/\omega_p$. Indeed, density fluctuations cannot scatter $\mathcal{L}\mathcal{Z}$ waves down to frequencies smaller than $\omega \simeq (1 - \Delta N)\omega_p$, and electrostatic decay ultimately leads to waves with frequencies $\omega_c\mathcal{Z} < \omega_{\text{min}} \le \omega \lesssim \omega_p$ above a limit $\omega_{\text{min}}$ as well (F. J. Polanco-Rodríguez et al. 2025). Consequently, for all $\Delta N$, when $\omega_c/\omega_p$ increases, Z-mode waves are more difficult to generate at $\omega \simeq \omega_c\mathcal{Z}$ by beam-driven $\mathcal{L}\mathcal{Z}$ wave turbulence, so that $F_m$ decreases. More precisely, Z-mode waves can only be generated down to their cutoff frequency $\omega_c\mathcal{Z}$ via linear
transformations of $\mathcal{LZ}$ waves on density fluctuations if $\Delta N \gtrsim \omega_c/2\omega_p$. Therefore, solar wind observations of circularly polarized $Z$-mode waves can provide a low estimate of $\Delta N$, provided that the satellite measures $\omega_c/\omega_p$. Moreover, in plasmas with $\Delta N \lesssim \omega_c/2\omega_p$, polarization ratios should be lower compared to those with $\Delta N \gtrsim \omega_c/2\omega_p$, as the radiated $Z$-mode waves cannot be excited down to their cutoff frequency. This can be observed in Figure 3(d), where, for $\Delta N = 0.05$, $\langle F \rangle_{s,t}$ starts decreasing at $\omega_c/\omega_p \simeq 2\Delta N = 0.1$.

Third, for larger beam velocities $v_b$, beam-driven $\mathcal{LZ}$ waves exhibit smaller $k_b\lambda_D = v_T/v_b$ and are more affected by the presence of density fluctuations $\delta n$ (provided that the wavelengths of $\mathcal{LZ}$ waves remain significantly smaller than those of $\delta n$). Such a regime was studied by the authors (C. Krafft et al. 2013), who considered beam-driven electrostatic wave turbulence in 1D plasmas where $\Delta N \lesssim 3(v_T/v_b)^2$ and $\Delta N \gtrsim 3(v_T/v_b)^2$ (see also D. D. Ryutov 1970). In the former case, $\mathcal{LZ}$ waves are insensitive to density fluctuations but can experience electrostatic decay, and in the latter one, they are transformed on $\delta n$ via various linear effects (reflection, refraction, scattering, trapping, tunneling, etc.), reducing the occurrence of nonlinear decay to plasma regions where density turbulence is weak.

Moreover, space observations in type III solar wind regions showed that polarization ratios obtained from recorded waveforms increase sharply at $v_b \simeq v_0$ ($v_0 \simeq 0.08c$) (D. M. Malaspina et al. 2011; D. B. Graham et al. 2012; D. B. Graham & I. H. Cairns 2013, 2014) from polarization ratios $F \simeq 0.1$ at $v_b \lesssim v_0$ to $F \simeq 0.45$ at $v_b \gtrsim v_0$. These authors suggested that electrostatic decay or scattering on density fluctuations could explain such a dependence. In Figure 3(f), one observes that the boundary $v_b \simeq v_0$ is associated with the condition $\Delta N \simeq 3(v_T/v_b)^2$. At $v_b \lesssim v_0$ ($v_b \gtrsim v_0$), the inequality $\Delta N \lesssim 3(v_T/v_b)^2$ ($\Delta N \gtrsim 3(v_T/v_b)^2$) is satisfied, corresponding to lower (higher) values of $F$, i.e., to the prevalence of electrostatic decay (LMC). This result allows us to diagnose the average level of density fluctuations of solar wind plasmas by calculating the polarization ratios using recorded waveforms. For example, applying $\Delta N \simeq 3(v_T/v_b)^2$ at $v_b \simeq v_0 \simeq 0.08c$ in a plasma near 1 au ($T_e \simeq 10$ eV) as in D. M. Malaspina et al. (2011), we obtain $\Delta N \simeq 0.01$, which corresponds well to levels of density turbulence expected at such distance. Note that a rough estimate of $\Delta N$ was performed in the latter work (which does not contradict our result), using, however, a simplified model of monochromatic Langmuir-wave scattering on a density gradient. On the contrary, our simulations involve random density fluctuations and magnetization, and our method provides a more precise determination of $\Delta N$. Furthermore, D. B. Graham & I. H. Cairns (2013) classified the waveforms based on their polarization ratios $\langle F_s \rangle_t$, designating those with $\langle F_s \rangle_t > 0.2$ as corresponding to small-k-waves at the final stage of electrostatic decay. By comparing Figures 3(a) and (b) (where decay is predominant), we observe that polarization ratios grow from 0.1 at early times, when decay starts to develop, to 0.25, when decay has transported wave energy toward small k-scales. This trend supports, at least in part, the classification scheme of D. B. Graham & I. H. Cairns (2013).

We have shown that both LMC and electrostatic decay lead to the increase of polarization ratios but that LMC is clearly the fastest and most efficient one for beam-plasma systems with $\Delta N \gtrsim 3(v_T/v_b)^2$. Otherwise, when $\Delta N \lesssim 3(v_T/v_b)^2$, distributions of polarization ratios are closer to those obtained

in homogeneous plasmas where $F_m \sim 0.2-0.3$ at late times. Therefore, the increase of polarization ratios with $v_b$, as observed in the solar wind, is strongly related to the condition $\Delta N \gtrsim 3(v_T/v_b)^2$ and the efficiency of LMC, rather than to electrostatic decay of $\mathcal{LZ}$ waves. Furthermore, the sharp growth of polarization ratios occurring at $\Delta N \simeq 3(v_T/v_b)^2$ can be used to measure the average level of random density fluctuations or the beam drift velocity in beam-plasma systems, as we did above with space observations at 1 au. In addition, measurements of polarization ratios' distributions as presented in Figures 3(a)-(d) can also provide useful estimates of $\Delta N$ and $\omega_c/\omega_p$.

Finally, the electron temperature $T_e$ and the ratio $T_e/T_i$, when varied in the ranges $20 \text{ eV} \le T_e \le 200 \text{ eV}$ and $3 \le T_e/T_i \le 10$, weakly affect the polarization ratios in plasmas with parameters relevant to type III solar radio bursts. However, in a homogeneous plasma with $T_e = 20 \text{ eV}$, the beam-plasma instability growth rate and the energy saturation level of $\mathcal{LZ}$ waves are smaller than in a plasma with larger temperatures $T_e$, so that the dynamics of nonlinear wave processes is significantly slower.

### 4.3. Correlations between Polarization Ratios and $Z$-mode Waves

The preceding conclusions can be completed and reinforced by studying the time dependence of the averaged polarization ratio $\langle F(t) \rangle_s$ for various $\omega_c/\omega_p$ in plasmas with $\Delta N \lesssim 3(v_T/v_b)^2$ and $\Delta N \gtrsim 3(v_T/v_b)^2$. Figures 4(a) and (b) show the time variations of $\langle F(t) \rangle_s$, averaged over $N_s = 1000$ waveforms recorded in the time ranges $0 \le \omega_p t \le 15,000$ (a) and $0 \le \omega_p t \le 30,000$ (b), with $\Delta N = 0.05$ and $\Delta N = 0$, respectively, for $\omega_c/\omega_p \le 0.33$ (only simulations with $v_b = 12.7 v_T \simeq 0.25c$ are used). In inhomogeneous plasmas with $\Delta N \gtrsim 3(v_T/v_b)^2$ (Figure 4(a), dashed lines), $\langle F(t) \rangle_s$ increases rapidly within the time interval $\omega_p t \lesssim 2000$ for all $\omega_c/\omega_p$, and, for $\omega_c/\omega_p < 0.2$, it remains at a quasi-constant level around $\langle F(t) \rangle_s \simeq 0.45$ during a long period ($3000 \le \omega_p t \le 15,000$). However, the maxima and saturation levels reached by $\langle F(t) \rangle_s$ decrease with increasing $\omega_c/\omega_p \ge 0.2$. Indeed, linear interactions of $\mathcal{LZ}$ waves with density fluctuations rapidly transform and convert them to small-k electromagnetic waves via LMC, increasing their polarization ratios very fast. On the other hand, for $\Delta N \lesssim 3(v_T/v_b)^2$ (Figures 4(a) and (b), solid lines), $\langle F(t) \rangle_s$ slowly grows during the time interval $\omega_p t \lesssim 15,000$ for all $\omega_c/\omega_p$. Indeed, for $\omega_c/\omega_p < 0.2$, $\langle F(t) \rangle_s$ increases from 0.15 up to 0.3 within $5000 \le \omega_p t \le 20,000$ (Figure 4(b)), which is at least an order of magnitude longer than the time needed to reach saturation in a randomly inhomogeneous plasma with $\Delta N = 0.05$. This is because, when $\Delta N \lesssim 3(v_T/v_b)^2$, nonlinear electrostatic decay of $\mathcal{LZ}$ waves—much slower than LMC—is the dominant wave process. Likewise, at $\omega_p t \simeq 30,000$, $\langle F(t) \rangle_s$ saturates at noticeably smaller values than for $\Delta N = 0.05$ (for any $\omega_c/\omega_p$), as LMC can transfer wave energy to smaller wavevectors than electrostatic decay.

In Figures 4(c) and (d), we replicate the time variations of $\langle F(t) \rangle_s$ presented in (a) for $\Delta N = 0.05$ and (b) for $\Delta N = 0$ at $\omega_c/\omega_p = 0.07$, onto which we superimpose (black lines) the corresponding time variations of $W_Z$ and $W_{Z+\mathcal{LZ}}^S$. Those represent, respectively, (i) the electromagnetic energy of $Z$-mode waves (as calculated in C. Krafft et al. 2025) and (ii) the energy carried by magnetized small-k $\mathcal{LZ}$ waves (as $\mathcal{LZ}^{(3)}$ and $\mathcal{LZ}^{(4)}$; see Section 3.1) and by $Z$-mode waves
<figure><img src="image_8.png" /></figure>

**Figure 4.** (a)–(b) Variations with time of $\langle f(t) \rangle_s$ (Equation (2)) averaged over $N_s = 1000$ waveforms filtered in the Doppler-shifted frequency range [0.9, 1.1]$\omega_p$ in plasmas with various $\Delta N$ and $\omega_c/\omega_p$. (a) Homogeneous ($\Delta N = 0$; solid lines) and randomly inhomogeneous ($\Delta N = 0.05$; dashed lines) plasmas with different $\omega_c/\omega_p \le 0.33$ (see legend in panel (b)); the time range is $\omega_p t \le 15,000$. (b) Homogeneous plasmas with weak magnetization $\omega_c/\omega_p \le 0.14$ in the very long time range $\omega_p t \le 30,000$. (c) Variation with time of $\langle f(t) \rangle_s$ for $\omega_c/\omega_p = 0.07$ and $\Delta N = 0.05$ (dashed blue line replicating that in panel (a), right axis), to which the time variation of the $\mathcal{Z}$-mode wave energy $W_{\mathcal{Z}}$ is superimposed (black line, left axis). (d) Variation with time of $\langle f(t) \rangle_s$ for $\omega_c/\omega_p = 0.07$ and $\Delta N = 0$ (solid blue line as in panel (b), right axis), to which the time variation of the energy $W_{\mathcal{Z}}$ of small-k $\mathcal{L}\mathcal{Z}$- and $\mathcal{Z}$-mode waves is superimposed (black line, left axis). Energies are normalized by the initial beam energy.

<figure><img src="image_9.png" /></figure>

**Figure 5.** Distributions $\langle f(F, X) \rangle_s$ averaged over $N_s = 1000$ waveforms in the time interval $1000 \le \omega_p t \le 4000$ (with frequencies $\omega^D$ filtered in the range [0.9, 1.1]$\omega_p$), represented in the map $(F, X)$. (a) and (b) $X = |B_{\omega_p}|^2$, $\Delta N = 0.05$, $\omega_c/\omega_p = 0.07$ (a) and $\omega_c/\omega_p = 0.2$ (b). (c) $X = \delta n_i/n_0$, $\Delta N = 0.05$, and $\omega_c/\omega_p = 0.07$. The upper limit of the linear color bar represents, for each distribution, the maximum value of $\langle f(F, X) \rangle_s$, which is different for each panel. In order to ignore noise, only wavepackets whose spectral energy $|E(\omega_p, t)|^2$ is greater than $|E(\omega_p, t)|^2_{\text{max}}/100$ are considered when building distributions. Energies are in arbitrary units.

generated at the ultimate stage of decay (as calculated in F. J. Polanco-Rodríguez et al. 2025). One can observe that blue and black curves follow the same variations within the time range where LMC is dominant ($\omega_p t \lesssim 7000$ for (c)) or during the late stage of decay ($\omega_p t \gtrsim 7000$ for (d)). This shows the strong correlation between the growth of polarization ratios and wave energies $W_{\mathcal{Z}}$ and $W_{\mathcal{Z}}^{\mathcal{L}\mathcal{Z}}$.

Finally, let us study the dependence of the total magnetic energy carried by electromagnetic waves with polarization ratios in a plasma with $\Delta N \gtrsim 3(v_T/v_b)^2$. Figures 5(a) and (b) present the PDFs $\langle f(F, |B_{\omega_p}|^2) \rangle_s$, obtained using $N_s = 1000$ waveforms filtered in the Doppler-shifted frequency interval $[0.9, 1.1]\omega_p$, within the time range $1000 \le \omega_p t \le 4000$ (where LMC is most efficient) for $\Delta N = 0.05$ and $\omega_c/\omega_p = 0.07, 0.2$. We observe that the magnetic energy $|B_{\omega_p}|^2$ radiated at $\omega_p$ increases quasi-linearly with $F$. As it is mostly carried by $\mathcal{Z}$-mode waves (C. Krafft et al. 2025), a clear correlation exists between $\mathcal{Z}$-mode wave emission and the growth of polarization ratios. Note that, at later times, when $\mathcal{L}\mathcal{Z}$ wave turbulence starts to damp (see Figure 4(c)), due mainly to the formation of a tail of accelerated beam electrons (C. Krafft & P. Savoini 2023), the correlation between $|B_{\omega_p}|^2$ and $F$—which is strong during the period when LMC is most efficient—tends to disappear (not shown here).

Figure 5(c) shows the distribution $\langle f(F, \delta n_i/n_0) \rangle_s$ as a function of $F$ and the ion density perturbation $\delta n_i/n_0 = (n_i - n_0)/n_0$ in a plasma with $\Delta N = 0.05$ and $\omega_c/\omega_p = 0.07$. The distribution peaks at $-0.05 \lesssim \delta n_i/n_0 < 0$, with most of polarization ratios satisfying $0.2 \lesssim F \lesssim 0.7$ and a nonnegligible fraction reaching $F \gtrsim 0.9$. This observation can be done for all $\omega_c/\omega_p$ studied (not shown here; see also
Figure 2). Therefore, in weakly-to-moderately magnetized plasmas with $\Delta N \gtrsim 3(v_T/v_b)^2$, most of the $\mathcal{L}Z$ waves of significant energy are trapped in density wells, where their frequency is shifted below $\omega_p$, enabling them to convert at constant frequency to electromagnetic $\mathcal{Z}$-mode waves with $\omega \lesssim \omega_p$ (see also Figures 2(d) and (e)). This confirms the fact that LMC is indeed the most important process of $\mathcal{L}Z$ waves' transformation into $\mathcal{Z}$-mode waves.

## 5. Conclusion

Large-scale and long-term 2D/3V PIC simulations are performed with parameters relevant to type III radio bursts. $\mathcal{L}Z$ wave turbulence is generated by an electron beam in a weakly magnetized and randomly inhomogeneous solar wind plasma and radiates electromagnetic waves near the plasma frequency $\omega_p$. The polarization ratios $F = |E_{\perp}|^2/|E|^2$ of turbulent $\mathcal{L}\mathcal{Z}$ and electromagnetic wavepackets, which are studied in detail, are closely related to wave generation mechanisms and to beam and plasma properties. This work provides answers to long-standing questions and offers new insights into the role of random density fluctuations in weakly magnetized plasmas with developed wave turbulence.

Statistical studies using large numbers of waveforms recorded by virtual satellites moving in the simulation plane are performed to determine the distributions of polarization ratios $F$ and their variations with beam and plasma parameters (beam velocity as well as plasma magnetization, electron and ion temperatures, and average level of density fluctuations). This method, which mimics waveform recording by spacecraft in the solar wind, leads to results consistent with observations.

Plasma random density fluctuations turn out to be the key factor behind the increase in polarization ratios, whereas high plasma magnetization $\omega_c/\omega_p \gtrsim 0.2$ tends to reduce them. We demonstrate unambiguously that LMC at constant frequency of $\mathcal{L}Z$ waves scattering on $\delta n$ is the most efficient and fastest process to increase polarization ratios up to $F \sim 1$ in randomly inhomogeneous plasmas where $\Delta N \gtrsim 3(v_T/v_b)^2$, due to $\mathcal{Z}$-mode wave emission. On the other hand, the nonlinear electrostatic decay—which is a dominant process only in plasmas with weaker average levels of density fluctuations satisfying $\Delta N \lesssim 3(v_T/v_b)^2$—ultimately leads to lower values of $F$, reached after at least an order of magnitude longer time periods. Furthermore, it is shown that, if $\Delta N \gtrsim \omega_c/2\omega_p$, large polarization ratios up to $F \simeq 1$ can be reached by electromagnetic $\mathcal{Z}$-mode waves generated by LMC near their cutoff frequency, where they exhibit left-hand circular polarization.

Finally, our results provide support to estimate type III beam and plasma parameters in the solar wind. In particular, the average level of density fluctuations $\Delta N$ can be inferred from the dependence of polarization ratios—derived from in situ observed waveforms—on measured beam velocities by using the relation $\Delta N \simeq 3(v_T/v_b)^2$ at the beam velocity where the sharp growth of polarization ratios is observed. Likewise, observations in the solar wind of circularly polarized $\mathcal{Z}$-mode waves can supply a low estimate for $\Delta N$, provided that the magnetization ratio $\omega_c/\omega_p$ can be measured by the spacecraft, which is usually the case. Moreover, the dependence of the averaged polarization ratio on $\omega_c/\omega_p$, evidenced in this work, can provide guidance to theoretical studies.

## Acknowledgments

This work was granted access to the HPC computing and storage resources under the allocation 2023-A0130510106 and 2024-A017051010 made by GENCI. This research was also financed in part by the French National Research Agency (ANR) under the project ANR-23-CE30-0049-01. C.K. thanks the International Space Science Institute (ISSI) in Bern through ISSI International Team project No. 557, Beam-Plasma Interaction in the Solar Wind and the Generation of Type III Radio Bursts. C.K. thanks the Institut Universitaire de France (IUF).

For open-access purposes, a CC-BY license has been applied by the authors to this document and will be applied to any subsequent version up to the author's manuscript accepted for publication resulting from this submission.

## ORCID iDs

F. J. Polanco-Rodríguez https://orcid.org/0009-0005-2951-697X

C. Krafft https://orcid.org/0000-0002-8595-4772

P. Savoini https://orcid.org/0000-0002-2117-3803

## References

* Bale, S. D., Burgess, D., Kellogg, P. J., et al. 1996, *GeoRL*, 23, 109

* Bale, S. D., Kellogg, P. J., Goetz, K., & Monson, S. J. 1998, *GeoRL*, 25, 9

* Celnikier, L. M., Muschietti, L., & Goldman, M. V. 1987, *A&A*, 181, 138

* Derouillat, J., Beck, A., Pérez, F., et al. 2018, *CoPhC*, 222, 351

* Ergun, R. E., Malaspina, D. M., Cairns, I. H., et al. 2008, *PhRvL*, 101, 051101

* Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, *SSRv*, 204, 7

* Graham, D. B., & Cairns, I. H. 2013, *JGR*, 118, 3968

* Graham, D. B., & Cairns, I. H. 2014, *JGRA*, 119, 2430

* Graham, D. B., Cairns, I. H., Prabhakar, D. R., et al. 2012, *JGRA*, 117, A09107

* Graham, D. B., Khotyaintsev, Y. V., Vaivads, A., et al. 2021, *A&A*, 656, A23

* Hinkel-Lipsker, D. E., Fried, B. D., & Morales, G. J. 1989, *PhRvL*, 62, 2680

* Hospodarsky, G. B., & Gurnett, D. A. 1995, *GeoRL*, 22, 1161

* Kellogg, P. J., Goetz, K., Monson, S. J., & Bale, S. D. 1999, *JGR*, 104, 17069

* Kellogg, P. J., Goetz, K., Monson, S. J., & Opitz, A. 2013, *JGR*, 118, 4766

* Krafft, C., & Savoini, P. 2021, *ApJL*, 917, L23

* Krafft, C., & Savoini, P. 2022a, *ApJL*, 924, L24

* Krafft, C., & Savoini, P. 2022b, *ApJL*, 934, L28

* Krafft, C., & Savoini, P. 2023, *ApJ*, 949, 24

* Krafft, C., & Savoini, P. 2024, *ApJL*, 964, L30

* Krafft, C., Savoini, P., & Polanco-Rodríguez, F. J. 2024, *ApJL*, 967, L20

* Krafft, C., & Volokitin, A. S. 2021, *ApJ*, 923, 103

* Krafft, C., Volokitin, A. S., & Krasnoselskikh, V. V. 2013, *ApJ*, 778, 111

* Krafft, C., Volokitin, A. S., Krasnoselskikh, V. V., & de Wit, T. D. 2014, *JGRA*, 119, 9369

* Krafft, C., Volokitin, A. S., Polanco-Rodríguez, F. J., & Savoini, P. 2025, *NatAs*, 2397

* Krasnoselskikh, V., Voshchepynets, A., & Maksimovic, M. 2019, *ApJ*, 879, 51

* Krasnoselskikh, V. V., Dudok de Wit, T., & Bale, S. D. 2011, *AnGeo*, 29, 613

* Krauss-Varban, D. 1989, *JGR*, 94, 3527

* Krupar, V., Szabo, A., Maksimovic, M., et al. 2020, *ApJS*, 246, 57

* Larosa, A., de Wit, T. D., Krasnoselskikh, V., et al. 2022, *ApJ*, 927, 95

* Layden, A., Cairns, I. H., Li, B., & Robinson, P. A. 2013, *PhRvL*, 110, 185001

* Lorfing, C. Y., Reid, H. A. S., Gómez-Herrero, R., et al. 2023, *ApJ*, 959, 128

* Malaspina, D. M., Cairns, I. H., & Ergun, R. E. 2011, *GeoRL*, 38, L13101

* Malaspina, D. M., & Ergun, R. E. 2008, *JGRA*, 113, A12108

* Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, *A&A*, 642, A1

* Písa, D., Souček, J., Santolík, O., et al. 2021, *A&A*, 656, A14

* Polanco-Rodríguez, F. J., Krafft, C., & Savoini, P. 2025, *ApJL*, 982, L24

* Raouafi, N. E., Matteini, L., Squire, J., et al. 2023, *SSRv*, 219, 8

* Ryutov, D. D. 1970, *JETP*, 30, 131

* Soucek, J., Písa, D., Kolmasova, I., et al. 2021, *A&A*, 656, A26

* Willes, A. J., & Cairns, I. H. 2000, *PhI*, 7, 3167