# Decay of Turbulent Upper-hybrid Waves in Weakly Magnetized Solar Wind Plasmas

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

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F. Polanco-Rodríguez, C. Krafft, P. Savoini. Decay of Turbulent Upper-hybrid Waves in Weakly Magnetized Solar Wind Plasmas. The Astrophysical Journal Letters, 2025, 982 (1), pp.L24. 10.3847/2041-8213/ adba64 . hal-05156506

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# Decay of Turbulent Upper-hybrid Waves in Weakly Magnetized Solar Wind Plasmas

F. J. Polanco-Rodríguez¹, C. Krafft¹,², and P. Savoini¹

¹Laboratoire 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

²Institut Universitaire de France (IUF), France

Received 2025 February 4; revised 2025 February 24; accepted 2025 February 26; published 2025 March 19

## Abstract

Large-scale and long-term two-dimensional particle-in-cell simulations of high resolution are performed for the first time to study the dynamics of electrostatic decay of upper-hybrid wave turbulence generated by electron beams into Langmuir/Z-mode ($LZ$) waves in weakly to moderately magnetized plasmas, in conditions relevant to type III solar radio bursts. Simulations use parameters characteristic of beam-plasma interactions between ~0.1 and 1 au. The impact of plasma magnetic field on decay is shown, and magnetic properties of $LZ$ waves are determined. During their energy transport through $k$ wavevector scales, waves undergo several decay cascades, acquiring increasing magnetic energy until they reach electromagnetic Z-mode dispersion below the plasma frequency. Whereas the impact of magnetic field on decaying waves of large $k = |k|$ is weak, important differences with respect to the unmagnetized plasma case manifest at small $k$-scales, where a boundary layer delimiting a spectral domain free of $LZ$ energy is revealed. It prevents decayed waves from reaching the Z-mode cutoff frequency and a high level of left-handed polarization, and it modifies the conditions for the appearance of modulational instabilities and strong turbulence phenomena at $k \sim 0$. Ordinary O-mode waves are generated jointly with Z-mode waves at comparable energy levels, via electromagnetic decay, whereas X-mode emissions are much weaker in most cases. These results provide support for the interpretation of observations by satellites such as Parker Solar Probe and Solar Orbiter, and they supply a solid basis for tackling the more complex problem of dynamics of upper-hybrid wave turbulence in magnetized plasmas where random density fluctuations cannot be neglected.

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

## 1. Introduction

Electrostatic decay (V. N. Tsytovich 1970; D. B. Melrose 1980) is considered the most important nonlinear wave process in the solar wind, and in particular, it contributes to wave energy transport from larger to smaller wavevectors' scales. Many authors have reported observations in the solar wind of double-peaked wave spectra near the plasma frequency $\omega_p$, sometimes accompanied by low-frequency waves, which have been interpreted as signatures of three-wave decay of upper-hybrid waves, also known as Langmuir/Z-mode waves and hereinafter referred to as $LZ$ waves (D. A. Gurnett et al. 1981; I. H. Cairns & P. A. Robinson 1992, 1995; G. B. Hospodarsky & D. A. Gurnett 1995; S. D. Bale et al. 1996; G. Thejappa et al. 2003; J. Soucek et al. 2005; P. Henri et al. 2009; D. M. Malaspina et al. 2011; D. B. Graham & I. H. Cairns 2013, 2014; P. J. Kellogg et al. 2013; G. Thejappa 2022, and references therein). Today, interest in processes involving $LZ$ waves with detectable magnetic signatures is growing while satellites such as Parker Solar Probe and Solar Orbiter (N. J. Fox et al. 2016; D. Müller et al. 2020), which approach the Sun at distances never reached before, can observe magnetized radio sources up close.

Decay of electrostatic waves was studied theoretically as well via numerical simulations, and in particular, for solar wind conditions (e.g., V. N. Tsytovich 1970; D. B. Melrose 1986;

I. H. Cairns 1987; E. P. Kontar & H. L. Pécseli 2002; B. Li et al. 2003; T. Umeda & T. Ito 2008; C. Krafft et al. 2015, 2019; C. Krafft & P. Savoini 2024). However, most works have neglected the weak magnetization of the solar wind. More recently, some authors studied the kinematics of decay in one- and two-dimensional magnetized plasmas, determining solutions in $k$-space by using three-wave resonance conditions between monochromatic $LZ$ and ion acoustic waves (A. Layden et al. 2013; I. H. Cairns & A. Layden 2018). In particular, it was shown that $LZ$ wave decay can persist down to very small $k$-scales near the transition between the Langmuir-like electrostatic and the Z-mode-like electromagnetic wave dispersive behaviors of $LZ$ waves, and that magnetization effects induce significant changes in wave kinematics at small wavenumbers only.

Furthermore, the magnetic signature of $LZ$ waves was observed recently for the first time by the Parker Solar Probe mission (A. Larosa et al. 2022), even if the role of such waves was already discussed in previous observations (e.g., F. L. Scarf et al. 1970; S. D. Bale et al. 1998; D. M. Malaspina et al. 2011; D. B. Graham & I. H. Cairns 2013; P. J. Kellogg et al. 2013, and references therein). Great interest is now focused on resolving open questions about their generation and transformation processes in the solar wind, and in particular, on their role in electromagnetic wave radiation by upper-hybrid wave turbulence excited by electron beams during type III solar radio bursts. In this context, it is very timely to understand the impact of magnetization on the dynamics of nonlinear wave processes involving $LZ$ waves.

<figure><img src="image_3.png" /></figure>

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.
Owing to large-scale and long-term particle-in-cell (PIC) simulations, we study the 2D dynamics of decay of upper-hybrid wave turbulence into *LZ* wavepackets in weakly magnetized solar wind plasmas. Beyond recovering wave kinematic features (I. H. Cairns & A. Layden 2018), we address several open questions. First, can the decay of turbulent *LZ* waves lead to the generation of small-*k* daughter waves with significant electromagnetic character, i.e., with frequencies below the electron plasma frequency $\omega_p$ down to the *Z*-mode cutoff frequency, and what are the consequences of this energy transport in terms of electromagnetic radiation? What are the characteristics of *LZ* waves' magnetic signatures? Are there any other electromagnetic modes excited jointly by upper-hybrid wave turbulence generated by beams? How and to what extent does magnetization modify decay dynamics in the high- and low-frequency range, and can it have an impact on ultimate energy deposition at small *k*-scales ($k = |k|$) and on the subsequent triggering of strong turbulence phenomena?

The characterization of *LZ* waves' magnetic signatures and the determination of electromagnetic modes emitted during beam-plasma interactions will be a useful support for the interpretation of observations by satellites such as Parker Solar Probe and Solar Orbiter. Beyond that major aspect, this work also supplies a solid and indispensable basis for tackling the more complex problem of nonlinear decay processes competing with mode conversion and transformations of *LZ* waves in magnetized solar wind plasmas where random density fluctuations cannot be neglected.

## 2. Nonlinear Decay of *LZ* Waves

The long-term 2D dynamics of *LZ* wavepackets' decay is studied using 2D/3V PIC simulations performed with the SMILEI code (J. Derouillat et al. 2018). Large boxes of sizes up to $L_x \times L_y = 5792 \times 2896\lambda_D^2$ are used ($\lambda_D$ is the electron Debye length), with high parallel and perpendicular wavenumber resolutions reaching $\delta k_\parallel \simeq 0.001\lambda_D^{-1}$ and $\delta k_\perp \simeq 0.002\lambda_D^{-1}$, respectively. Simulations allow electrostatic and electromagnetic scales as well as high- and low-frequency dynamics to be followed simultaneously. The plasma is initially homogeneous and weakly magnetized, with ratios of electron cyclotron to plasma frequencies $0.02 \le \omega_c/\omega_p \le 0.14$. A weak electron beam of relative density $n_b/n_0 = 5 \cdot 10^{-4}$ is injected in the plasma of average density $n_0$ along the ambient magnetic field $B_0$ (directed along the x-axis), with velocities $v_b \simeq 0.08c$ and $v_b \simeq 0.25c$, both corresponding to $v_b/v_T \simeq 12.7$ ($v_T$ is the electron thermal velocity), i.e., to electron temperatures $T_e = 20$ eV and $T_e = 200$ eV, respectively. The electron-to-ion temperature ratios are $T_e/T_i = 3$ and $T_e/T_i = 10$, and the ion-to-electron mass ratio is $m_i/m_e = 1836$. Simulations are performed until $\omega_p t \simeq 25,000$, in order to follow simultaneously the whole high- and low-frequency decay dynamics. Numerical parameters can be found in previous works (C. Krafft & P. Savoini 2021, 2022, 2023, 2024; C. Krafft et al. 2024).

Parameters are relevant to type III radio bursts' conditions from ~0.1 au up to ~1 au. Indeed, the first set of parameters used, i.e., $v_b \simeq 0.25c$, $T_e = 200$ eV, $T_e/T_i = 10$ (higher than the realistic value to evidence ion acoustic waves), and $0.02 \le \omega_c/\omega_p \le 0.14$ can reasonably describe type III beam-plasma interactions near 0.1–1 au, whereas the second one, i.e.,

$v_b \simeq 0.08c$, $T_e = 20$ eV, $T_e/T_i = 3$, and $\omega_c/\omega_p = 0.02$, can account for such interactions near 0.5–1 au.

According to the kinematics of electrostatic decay in a 1D unmagnetized plasma (e.g., I. H. Cairns 1987; C. Krafft et al. 2015; I. H. Cairns & A. Layden 2018), four decay cascades of beam-driven Langmuir waves *L* are expected to occur for our simulation parameters, i.e., $L \to L' + S'$, $L' \to L'' + S''$, $L'' \to L^{(3)} + S^{(3)}$ and $L^{(3)} \to L^{(4)} + S^{(4)}$, where $S^{(n)}$ are the ion acoustic waves produced by the $n^{th}$ cascade (note that $S' = S^{(1)}$ and $S'' = S^{(2)}$). The wavenumbers of wave products $L^{(n)}$ are $k^{(n)} \simeq (-1)^n(k_L - nk_0)$, where $k_0\lambda_D = 2c_s/3v_T$ ($c_s$ is the ion acoustic velocity) and $k_L \sim \omega_p/v_b$; for our first set of parameters (used in all figures except Figures 5(g)–(h)), $k_L\lambda_D \sim 0.1$ and $k_0\lambda_D \simeq 0.017$.

However, in a 2D plasma with wave turbulence generated by an electron beam, the number of cascades can actually be higher for some waves, due to beam deceleration and the finite spectral widths of the wavepackets. In addition, if the plasma is weakly magnetized, the 2D dynamics of *LZ* wave decay is slightly modified at large *k*-scales (compared to the unmagnetized case), and more significantly at small ones, as shown below. Note that we can take the values $k^{(n)}$ as reasonable approximations for wavenumbers along the parallel direction *x*.

### 2.1. The 2D Dynamics of Decay at Large Wavenumber Scales

Let us first study the dynamics of decay at large $k = |k|$, when *LZ* waves are quasi-electrostatic, with dispersion $\omega/\omega_p \simeq 1 + 3k^2\lambda_D^2/2 + (\omega_c^2/2\omega_p^2)\sin^2\theta \gtrsim 1$, where $\theta$ is the propagation angle between *k* and $B_0$. Figures 1(a)–(h) present the electric field spectra $|E_k^2|$ of *L* and *LZ* waves excited around $\omega_p$ (see caption), at different times, in an unmagnetized (top row) and a weakly magnetized plasma with $\omega_c/\omega_p = 0.07$ (bottom row), respectively. At $\omega_p t = 5400$, one observes in both cases three spectral domains of wave excitation, corresponding to (i) beam-driven *L* and *LZ* waves (centered around $k_x\lambda_D \simeq 0.12 > k_L\lambda_D$, due to beam deceleration), (ii) backscattered waves *L'* and *LZ'* coming from the first decay cascade (with $k_x' < 0$), and (iii) *L''* and *LZ''* waves produced by the second cascade, with $0.02 \lesssim k_x''\lambda_D \lesssim 0.05$ (see also C. Krafft & P. Savoini 2024). One observes that the first stages of this dynamics are not significantly modified by the magnetization. Further, at $\omega_p t = 9900$, for both $\omega_c = 0$ and $\omega_c > 0$, the energy of beam-driven waves is decreased by around an order of magnitude, due mostly to wave decay, beam relaxation (with small instability growth rates at $k \gtrsim k_L$), and Landau damping; note that these wavepackets exhibit significant propagation angles (up to $\theta \simeq \pi/4$) and that their decay products, as *LZ'* and *LZ''* (*L'* and *L''*), are angularly scattered. For both cases $\omega_c = 0$ and $\omega_c > 0$, the amplitudes of *L*^{(3)} and *LZ*^{(3)} (*L*^{(4)} and *LZ*^{(4)}) wavepackets generated by the third (fourth) decay cascade and centered around $k_x^{(3)}\lambda_D \simeq -0.04$ ($k_x^{(4)}\lambda_D \simeq 0.03$) become significant at $\omega_p t = 14,800$. At the asymptotic stage ($\omega_p t = 25,000$), beam-driven waves are reduced to a thin crescent-shaped spectral domain at large *k*. The spectral energy of wave products forms in both cases a filled surface with the highest intensity reached at $k \lesssim 2k_0$ (see the dashed black lines representing $k = 2k_0$), where discrepancies between unmagnetized and magnetized spectra become important. In particular, for $\omega_c \ne 0$ only, one observes around $k \sim 0$ a spectral region free of *LZ* wave energy, with a characteristic size significantly smaller than $k_0$, whereas for the unmagnetized case, this region forms a condensate where wave
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Spectral electric field energy $|E_k|^2$ around the frequency $\omega_p$ (in logarithmic scales and arbitrary units) in the map $(k_x\lambda_D, k_y\lambda_D)$, at different times $\omega_p t = 5400, 9900, 14,800$, and $25,000$ corresponding to the four columns, respectively, for an unmagnetized plasma (upper row (a)–(d)) and a weakly magnetized plasma with $\omega_c/\omega_p = 0.07$ (bottom row (e)–(h)). Filtering in the frequency domain $0.9\omega_p \le \omega \le 1.1\omega_p$ is performed. The circles $k = k_0$ and $k = 2k_0$ are represented by black dashed lines. The lengths of simulation boxes are $L_x = L_y = 1448\lambda_D$, with grid numbers $N_x = N_y = 1024$. All variables are normalized. The same colorbars are used for all panels, to enable relevant comparisons. The first set of physical parameters is used (see above).</figcaption></figure>

energy accumulates at $k \lesssim k_0$ (see the dashed circles representing $k = k_0$). Note that this small-k spectral domain will be studied in detail in the next section, by using high k-space resolution.

To conclude, differences between the magnetized and unmagnetized cases are not significant for $k \gtrsim 2k_0$, in agreement with kinematic studies (I. H. Cairns & A. Layden 2018), even if some noticeable differences should be mentioned—for example, the reduction of the spectral domain along the perpendicular direction, due to the Lorentz force, a tendency that is accentuated with increasing magnetization. However, for $k \lesssim 2k_0$, the differences between both cases are important and are discussed in detail in the next section.

Figure 2(a) shows the time variations of the energy density $W$ carried by the beam-driven $\mathcal{L}$ and $\mathcal{L}\mathcal{Z}$ waves and their decay products, for the unmagnetized case as well as for plasmas with $\omega_c/\omega_p = 0.07$ and $0.14$. For a given $\omega_c/\omega_p$, the growth rates and the saturation levels of $W$ decrease with the number of cascade $n$ ($n \le 4$). The excitation of $\mathcal{L}$ and $\mathcal{L}\mathcal{Z}$ waves by the beam weakly depends on $\omega_c$ ($\omega_p t \lesssim 1000$), in agreement with D. Krauss-Varban (1989); this is also the case for waves produced by the first and second decay cascades, whose wavenumbers satisfy $k \gtrsim 2k_0$, but not for waves corresponding to $n=3$ and $n=4$ (which lie mostly in the small k-range), whose growth rates noticeably decrease with increasing $\omega_c$. This is likely due to the fact that energy carried by $\mathcal{L}\mathcal{Z}$ wavepackets involved in the last decay cascades, with broad spectral width, is shared between their decay products (shifted in wavenumber by $k_0$) and the lowest attainable k-layer during the ultimate stage of decay (with shift smaller than $k_0$). Such an effect, described in detail in the commentary on Figure 3, cannot manifest for monochromatic waves.

Figure 2(b) shows the time variations of wave energy densities $W_>$ and $W_<$ (within the spectral domains $k > 2k_0$ and

$k < 2k_0$, respectively) for $\omega_c/\omega_p = 0, 0.02, 0.07$, and $0.014$. One observes that $W_>$ decreases (at $\omega_p t \gtrsim 3000$) due to energy transport to smaller k-scales during decay cascades, as confirmed by the corresponding growth of $W_<$, which reaches at asymptotic time values up to an order of magnitude less than $W_>$ and whose growth decreases with increasing $\omega_c/\omega_p$ so that, for weak magnetization, the higher the ambient magnetic field intensity, the slower the energy transport to small k-scales via decay. So far, one can state that the four cascades predicted by a simple 1D kinematic study involving three-wave resonance conditions and waves' dispersion in an unmagnetized plasma have been identified on energy spectra at expected k-regions, showing the transport of wave energy over k-scales during successive cascades, with a dynamics at late times in small k-ranges that strongly differs between magnetized and unmagnetized plasma cases.

Low-frequency waves generated by decay are visible almost ubiquitously on the spacetime variations of the ion density perturbations $\delta n_i(x, t) = (n_i(x, t) - n_0)/n_0$—at fixed $y$—(Figures 2(c)–(d)), filtered in the wavelength ranges of $S'$ and $S''$ waves (c) as well as of $S^{(3)}$ and $S^{(4)}$ waves (d), respectively, in a plasma with $\omega_c/\omega_p = 0.07$. At $\omega_p t \gtrsim 1400$, one observes (see, e.g., orange-colored regions at $600 \lesssim x/\lambda_D \lesssim 1000$) the forward propagation of $S'$ waves, with wavelengths $\lambda_{S'} \simeq 30\lambda_D$, appearing as straight ascending lines with slopes near the normalized group velocity $c_s/v_T$ of ion acoustic waves. At $\omega_p t \gtrsim 10,000$, the backward propagation of $S''$ waves manifests as descending lines, with slopes $-c_s/v_T$ and wavelengths $\lambda_{S''} \simeq 38\lambda_D$ (see, e.g., $0 \lesssim x/\lambda_D \lesssim 200$). In the spectral domain of $S^{(3)}$ and $S^{(4)}$ waves, other ion acoustic waves generated by electromagnetic decay (EMD) of $\mathcal{L}\mathcal{Z}$ waves can additionally appear (C. Krafft et al. 2024). However, at $\omega_p t \gtrsim 17,000$ (see, e.g., $600 \lesssim x/\lambda_D \lesssim 1000$), forward-propagating waves appear with $\lambda_{S^{(3)}} \simeq 76\lambda_D$, which is, at our parameters, too large of an ion acoustic wavelength to
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (a) Variations with time of the energy densities $W$ carried by the beam-driven $\mathcal{L}$ and $\mathcal{L}\mathcal{Z}$ waves and their decay products $\mathcal{L}^{(n)}$ and $\mathcal{L}\mathcal{Z}^{(n)}$, respectively, with $n \le 4$, for an unmagnetized plasma and weakly magnetized plasmas with $\omega_c/\omega_p = 0.07$ and $\omega_c/\omega_p = 0.14$. $\mathcal{L}\mathcal{Z}$ and $\mathcal{L}$ are indicated by solid lines, $\mathcal{L}\mathcal{Z}'$ and $\mathcal{L}'$ by dashed lines, $\mathcal{L}\mathcal{Z}''$ and $\mathcal{L}''$ by dashed-dotted lines, $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}^{(3)}$ by dotted lines, and $\mathcal{L}\mathcal{Z}^{(4)}$ and $\mathcal{L}^{(4)}$ by solid lines (see also legend). (b) Variations with time of wave energy densities $W_>$ (solid lines) and $W_<$ (dashed lines) within the spectral domains $k > 2k_0$ and $k < 2k_0$, respectively, for $\omega_c/\omega_p = 0, 0.02, 0.07$, and $0.14$ (see also legend). Panels (a) and (b): Energies are normalized by the initial beam energy and expressed in logarithmic scales. All energies are several orders of magnitude higher than numerical noise. Panels (c) and (d): Ion density profiles $\delta n_i(x, t)/n_0 = (n_i(x, t) - n_0)/n_0$ at fixed $y = 33\lambda_D$, filtered in wavelength ranges corresponding to $S'$ and $S''$ waves (c) and $S^{(3)}$ and $S^{(4)}$ waves (d), as a function of $x/\lambda_D$ and $\omega_p t$. Energies of $S^{(n)}$ waves propagate with group velocities $c_i/v_T$ for $n=1$ (c), $n=3$ (d) and $-c_i/v_T$ for $n=2$ (c), $n=4$ (d), as highlighted by the four orange-colored regions (see also the text). For comparison, green arrows with slopes $\pm c_s/v_T = \pm 0.026$ are represented. All variables are normalized. Parameters are the same as in Figure 1.</figcaption></figure>

correspond to low-frequency products generated by EMD. Then, at $\omega_p t \simeq 19,000$ (see, e.g., $100 \lesssim x/\lambda_D \lesssim 500$), the backward propagation of $S^{(4)}$ waves appears with $\lambda_S^{(4)} \simeq 100\lambda_D$. The low-frequency waves' behavior remains quasi-similar to that in an unmagnetized plasma (C. Krafft & P. Savoini 2024). Note that, in such weakly magnetized plasmas with $\omega_c/\omega_p \lesssim 0.14$, ion acoustic waves dominate the low-frequency range and the ion response is not significantly changed compared to the unmagnetized case. The coordinated appearance of $\mathcal{L}\mathcal{Z}^{(n)}$ wave products (Figure 1) confirms the occurrence of the electrostatic decay process of $\mathcal{L}\mathcal{Z}$ waves.

## 2.2. The 2D Dynamics of Decay at Small Wavenumber Scales

As revealed by Figure 1, the decay dynamics of $\mathcal{L}\mathcal{Z}$ waves is very sensitive to the plasma magnetic field at small $k$-scales. This point is considered in detail in the present section. Therefore, we first recall results obtained in previous works on the kinematics of 1D decay. Then, on the basis of our 2D PIC simulations, we describe the final stages of decay at small $k$-scales, when frequencies of $\mathcal{L}\mathcal{Z}$ waves drop below $\omega_p$ and these waves transition to electromagnetic $Z$-mode waves. Note that the decay dynamics of upper-hybrid wave turbulence is much

more complex than that of monochromatic waves, due to the finite widths of spectral domains generated by the beam and structured by successive decay cascades during its relaxation. The existence and the role of a boundary layer forming in the small-$k$ space due to plasma magnetization and surrounding a surface free of $\mathcal{L}\mathcal{Z}$ wave energy are evidenced. The ultimate stage of energy transport and deposition is discussed in comparison with the unmagnetized plasma case.

Kinematic studies were performed in an unmagnetized 1D plasma (A. Layden et al. 2013) with monochromatic Langmuir ($\mathcal{L}$ and $\mathcal{L}'$) and ion acoustic waves $S'$ interacting via three-wave decay $\mathcal{L} \rightarrow \mathcal{L}' + S'$ and satisfying the resonance condition $\omega_{\mathcal{L}}(\mathbf{k}) = \omega_{\mathcal{L}'}(\mathbf{k}') + \omega_{S'}(\mathbf{k} - \mathbf{k}')$. It was shown that, for $\mathcal{L}$ waves with $k > k_0$ ($k = \pm |\mathbf{k}|$), the decay produces backscattered $\mathcal{L}'$ waves with $k' = k_0 - k < 0$, whereas for $k_0/2 < k < k_0$, forward-propagating Langmuir waves are generated with $k' = k - k_0 > 0$. For $k < k_0/2$, no solution exists. The same methodology was used for $\mathcal{L}\mathcal{Z}$ waves propagating along $B_0$ (I. H. Cairns & A. Layden 2018). Using the wavenumber $k_*\lambda_D = (v_T/c)(1 + \omega_p/\omega_c)^{-1/2}$ at which the $\mathcal{L}\mathcal{Z}$ wave frequency $\omega_{\mathcal{L}\mathcal{Z}}$ is equal to $\omega_p$ at $\theta = 0$, it was found that (i) for $k > k_0 + k_*$, $k' = k_0 - k$, which is the backscattered wave
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a)–(h): Spectral electric field energy $|E_k|^2$ in the map $(k_x, k_y, \lambda_D)$, filtered around $\omega_p$, at times $\omega_p t = 3100, 4400, 7200$, and $14,800$, for an unmagnetized plasma (upper row (a)–(d)) and a magnetized plasma with $\omega_c/\omega_p = 0.07$ (middle row (e)–(h)); superimposed black dashed lines represent the circles $k = k_0/2$ and $k = k_0$ (upper row) and the curves $k_*(\theta)$ and $k_0 + k_*(\theta)$ (middle row); note that $|k_*(\theta)| \lesssim 0.01 \lambda_D^{-1}$. Black dots in panels (a), (b), (e), and (f) indicate the central wavenumbers of $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ broad spectral domains, estimated in 1D geometry; black and white arrows (in (b), (c), (f), and (g)) indicate forward or backward scatter and energy transport, respectively. Spectral domains of $\mathcal{L}^{(3)}$ and $\mathcal{L}^{(4)}$ ($\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$) waves are delimited by black solid lines (in (a), (b), and (e)–(f)). (f): The region of $\mathcal{O}$-mode wave excitation is indicated by an arrow. Colorbars are the same for panels (a)–(h). (i) Dispersion relation of $|E_k|^2$ integrated over propagation angles $\theta$, in the plane $(\text{sign}(k_x)k\lambda_D, \omega/\omega_p)$, for the magnetized plasma case, at late time $\omega_p t = 13,500$; superimposed white lines represent the dispersion curves of $\mathcal{L}\mathcal{Z}$ waves as well as of electromagnetic $\mathcal{O}$-, $\mathcal{X}$- and $\mathcal{Z}$-mode waves, calculated via the magnetоionic theory at $\theta = 0$ (solid lines) and $\theta = \pi/2$ (dashed lines). The inset presents the theoretical dispersion curves $(k, \omega)$ of these modes calculated in a thermal plasma without beam and for monochromatic waves; the wavenumbers $k_{*\parallel}$ and $k_{*\perp}$ mentioned in the text are indicated by red dots. In panels (a)–(i), spectral energies are shown in arbitrary units and logarithmic scales. The lengths of simulation boxes are $L_x \times L_y = 5792 \times 2896 \lambda_D^2$, with grid numbers $N_x = 4096$ and $N_y = 2048$. All variables are normalized. The first set of physical parameters is used (see above).</figcaption></figure>

solution found in unmagnetized plasmas, (ii) for $k_* < k < k_0 + k_*$, $k' = \pm k_*$, and (iii) for $k \le k_*$, $k' = -k$, which is the only solution in this $k$-range, as well as the only one satisfying $|k'| \le |k_*|$. We do not list here all possible solutions, but only those that can be useful hereafter. Note that

cases with $k_* > k_0$, which are not realistic for most solar wind plasmas, are not considered here.

In order to highlight important features of decay dynamics at small $k$-scales, large simulation boxes with very high spectral resolutions (for 2D PIC simulations) are used, i.e.,
$\delta k_{\parallel} = 10^{-3} \lambda_D^{-1}$ and $\delta k_{\perp} = 2 \cdot 10^{-3} \lambda_D^{-1}$, together with a frequency precision reaching $\delta\omega \simeq 2 \cdot 10^{-3} \omega_p$. Figures 3(a)-(h) show the evolution, at different times and within the range $k \lesssim 2k_0 \simeq 0.035 \lambda_D^{-1}$, of the electric field spectra $|E_k|^2$ generated around $\omega_p$ in an unmagnetized and a weakly magnetized plasmas with $\omega_c/\omega_p = 0.07$, respectively. Note that, for our first set of physical parameters (see above), the theoretical wavenumbers of parallel and perpendicular propagating $\mathcal{L}\mathcal{Z}$ waves are, at frequency $\omega_p$, given by $k_{*\parallel}|\lambda_D = (v_T/c)(1 + \omega_p/\omega_c)^{-1/2} \simeq 0.005$ and $k_{*\perp}\lambda_D = v_T/c \simeq 0.02$, respectively (see also the red dots in the inset of Figure 3(i)).

Figure 1 has shown the existence, in a magnetized plasma, of a spectral domain at small $k$ free of $\mathcal{L}\mathcal{Z}$ wave energy. Isocontours at frequencies $\omega < \omega_p$ can be determined in the map $(k, \theta)$ by numerically solving the full dispersion relation $\omega_{\mathcal{L}\mathcal{Z}}(k, \theta) = \omega$ of $\mathcal{L}\mathcal{Z}$ waves (without any approximation and with thermal effects that are however negligible at the small $k$ considered here). On this basis, we find that the isocontour $\omega_{\mathcal{L}\mathcal{Z}}(k, \theta) = \omega_{\min} \lesssim \omega_p$, where $\omega_{\min}$ is the lowest frequency reached asymptotically by $\mathcal{L}\mathcal{Z}$ waves via decay, corresponds to a boundary $k_*(\theta)$ delimiting a surface free of $\mathcal{L}\mathcal{Z}$ wave energy, as shown in Figures 3(e)-(h) by black dashed lines at $k\lambda_D \lesssim 0.005 - 0.01$ (see also Figures 5(a)-(c) for more details). The dispersion plot presented in Figure 3(i) provides that $\omega_{\min} \simeq 0.994\omega_p$. Then $k_*(\theta)$ represents the set of smallest wavevectors $k(k, \theta)$ toward which $\mathcal{L}\mathcal{Z}$ wave energy is transported during decay; no $\mathcal{L}\mathcal{Z}$ wave energy can penetrate via decay into the region $k < k_*(\theta)$.

Let us now evidence the dynamics of decay and the formation of the boundary $k_*(\theta)$. At $\omega_p t = 3100$ (Figures 3(a) and (e)), $\mathcal{L}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(3)}$ waves are generated around $k_x\lambda_D \simeq -0.035$ (near the 1D estimate $|k_{\mathcal{L}^{(3)}}| \simeq k_{\mathcal{L}} - 3k_0$), indicated by black dots surrounded by circles delimiting roughly the spectral domains of $\mathcal{L}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(3)}$ wave energies. Later, at $\omega_p t = 4400$, $\mathcal{L}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(3)}$ waves decay into $\mathcal{L}^{(4)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ ones, respectively, which appear around $k_x\lambda_D \simeq 0.015$, i.e., in the ranges $k_0/2 < k < k_0$ for $\omega_c = 0$ and $k_*(\theta) < k < k_0 + k_*(\theta)$ for $\omega_c/\omega_p = 0.07$ (see black solid lines in Figures 3(b) and (f), respectively); black dashed lines represent the circles $k = k_0/2$ and $k = k_0$ for $\omega_c = 0$, and the curves $k_*(\theta)$ and $k_0 + k_*(\theta)$ for $\omega_c > 0$. Meanwhile, the spectral domain of $\mathcal{L}\mathcal{Z}^{(3)}$ waves extends with time down to $k \lesssim k_0$ ($\omega_c = 0$) and to $k \lesssim k_0 + k_*(\theta)$ ($\omega_c > 0$) (see the white arrows in Figures 3(b) and (f)). Up to this stage, decay dynamics in both unmagnetized and weakly magnetized plasmas are quite similar, but at $\omega_p t \gtrsim 7000$, the dynamics becomes fundamentally different (Figures 3(c)-(g)). For $\omega_c = 0$, $\mathcal{L}^{(3)}$ waves with $k_0/2 \lesssim k < k_0$ (and $k_x < 0$) and most of $\mathcal{L}^{(4)}$ waves lying in the same spectral range (but with $k_x > 0$) do a last forward scatter down to the spectral domain $0 < k \lesssim k_0/2$, reaching asymptotically the condensate near $k \sim 0$ (Figure 3(c)). For $\omega_c/\omega_p = 0.07$, the wavenumbers of most $\mathcal{L}\mathcal{Z}^{(4)}$ waves lie in the range $k_*(\theta) < k < k_0 + k_*(\theta)$ and perform a scatter down to $k' \simeq \pm k_*(\theta)$ (Figures 3(g) and (h); see the black curved arrow in (g)). Meanwhile, the fraction of $\mathcal{L}\mathcal{Z}^{(3)}$ waves with $k_*(\theta) < k < k_0 + k_*(\theta)$ performs a scatter down to $k' \simeq \pm k_*(\theta)$, and the rest does a backscatter to $\mathcal{L}\mathcal{Z}^{(4)}$. Note that the important differences existing between the dynamics of energy transport down to $k \sim 0$ and to $k \sim k_*(\theta)$ shed light on the discrepancies already observed between the unmagnetized and the magnetized plasma cases in

energy variations with time of $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ waves (Figure 2(a)).

In the unmagnetized plasma, the spectral region around $k \sim 0$ is partially filled up at $\omega_p t = 14800$ with Langmuir wave energy, which is finally deposited in a condensate at $k \lesssim k_0/2 \simeq 0.008 \lambda_D^{-1}$, as shown in Figure 3(d) (see also Figure 1(d), where it appears more clearly at a significantly later time $\omega_p t = 25,000$). On the contrary, for $\omega_c > 0$, $\mathcal{L}\mathcal{Z}$ wave energy is transported to small wavenumbers and accumulates ultimately on a layer of thickness $\delta k(\theta)$, i.e., at $k_*(\theta) \lesssim k \lesssim k_*(\theta) + \delta k(\theta)$, without penetrating the internal surface bounded by $k_*(\theta)$ (see Figures 3(h) and 1(h)). This can be explained as $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ waves with $k_*(\theta) \lesssim k \lesssim k_0 + k_*(\theta)$ can only decay into waves with $k' = \pm k_*(\theta)$ (see above) and those can only decay in turn into waves with $k' = -k$, without reaching lower wavenumbers. The surface bounded by $k_*(\theta)$ remains already free of $\mathcal{L}\mathcal{Z}$ wave energy until at least $\omega_p t = 25,000$, as shown in Figure 1(h). Note the important fact that wave energy visible at $k \lesssim k_*(\theta)$ is not carried by $\mathcal{L}\mathcal{Z}$ waves (with large magnetic component), but by electromagnetic O-mode waves, with wavenumbers $k_O\lambda_D \simeq 0.005$ and frequencies $\omega_p \lesssim \omega_O \lesssim 1.05\omega_p$. More precisely, Figure 3(i) shows that, at $\omega_p t = 13,500$, wave energy at $k\lambda_D \lesssim 0.005$ covers the $(\omega, k)$-surface between the perpendicular and the parallel dispersion curves of the O-mode waves (black dashed and solid lines, respectively). Those are generated via EMD of $\mathcal{L}\mathcal{Z}$ waves of large or moderately small $k \gtrsim 2k_0$ (see also C. Krafft et al. 2024). In addition, no X-mode waves with significant amplitudes are observed; their energy is smaller by at least an order of magnitude compared to O- and Z-mode waves, with $\omega_O \gtrsim \omega_p$ and $\omega_Z \lesssim \omega_p$, respectively.

Figure 3(i) shows that the spectral energy $|E_k^2|$ lying on the theoretical $\mathcal{L}\mathcal{Z}$ dispersion curve extends from large $k$ down to $k \simeq k_{*\parallel} = 0.005 \lambda_D^{-1}$, reaching ultimately the frequency $\omega_{\min} \simeq 0.994\omega_p$ via decay. However, no significant energy is lying on the Z-mode dispersion curve between its cutoff frequency $\omega_{cZ} \simeq \omega_p - \omega_c/2 \simeq 0.965\omega_p$ and $\omega_{\min}$. This is true at all times and is verified for other weak magnetization intensities (not shown here). Then, $\mathcal{L}\mathcal{Z}$ wave decay cannot transport energy to smaller scales than $k_*(\theta)$ and $\omega_{\min}$, and only leads to Z-mode waves with partial left-handed polarization (~60%). This limiting frequency $\omega_{\min}$, which depends on the kinematics of three-wave decay and on the dispersive properties of $\mathcal{L}\mathcal{Z}$ waves, prevents the generation of Z-mode waves near their cutoff frequency. The only way to excite such waves in a weakly magnetized plasma is to increase the average level $\Delta N = \langle (\delta n/n_0)^2 \rangle^{1/2}$ of random density fluctuations $\delta n$ up to a few percent of the ambient plasma density (C. Krafft et al. 2013), as will be shown in a forthcoming paper by the authors.

In the small-$k$ region where $\omega < \omega_p$, the decay cascade $\mathcal{L}\mathcal{Z}^{(3)} \rightarrow \mathcal{L}\mathcal{Z}^{(4)} + S^{(4)}$ requires that the difference $\Delta\omega$ between the frequencies of $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ waves be equal to the ion acoustic frequency $\omega_S^{(4)}$, so that $|\Delta\omega/\omega_p| = (c_s/v_T)|k_S^{(4)}|\lambda_D \simeq 2(c_s/v_T)|v_T/v_b - 7c_s/3v_T|$ (rough estimate based on monochromatic waves in an unmagnetized plasma). Then the relation $|\Delta\omega/\omega_p|/|\Delta k\lambda_D| \ll 1$ holds for the usual solar wind parameters, taking into account that $\mathcal{L}\mathcal{Z}^{(3)}$ waves perform a scatter with the wavenumber shift $|\Delta k| \simeq k_0$. This estimate has to be compared with the group velocities $|v_g|/v_T \sim |\Delta\omega/\omega_p|/|\Delta k\lambda_D|$ of Z-mode wave dispersion curves, which significantly increase (see white lines at $\omega < \omega_p$ in Figure 3(i)), for any propagation angle $\theta$, in the $(\omega, k)$ region where $\mathcal{L}\mathcal{Z}$ wave decay stops. This
growth of the Z-mode group velocity explains why decay cannot go on further.

Note finally that the boundary layer surrounding $k_*(\theta)$ and delimiting the surface free of $\mathcal{L}\mathcal{Z}$ wave energy (but containing electromagnetic O-mode energy) is not only of interest for electrostatic wave decay. Indeed, in unmagnetized plasmas with developed Langmuir wave turbulence, the spectral region at $k\lambda_D \lesssim (m_e/m_i)^{1/2}/3 \simeq 0.008$ is well known to trigger modulational instabilities when energy threshold conditions are fulfilled (V. E. Zakharov et al. 1985). The fact that the surface free of $\mathcal{L}\mathcal{Z}$ wave energy and bounded by $k_*(\theta)$ extends toward larger $k$ with increasing ambient magnetic field should significantly modify the conditions for a possible onset of strong turbulence phenomena (V. E. Zakharov 1975), compared to the unmagnetized plasma case where a Langmuir condensate is formed at small $k$. However, such a study is beyond the scope of this Letter and should be investigated in forthcoming works.

## 2.3. Magnetic Signatures of $\mathcal{L}\mathcal{Z}$ Waves

Let us now take a look at the characteristics of $\mathcal{L}\mathcal{Z}$ waves' magnetic signatures, which are useful for analyzing and interpreting observations made in the solar wind. The dispersion and spectra of magnetic energies along the three directions x, y, and z are shown in Figures 4(a)–(f), at $\omega_p t = 13,500$, for $\omega_c/\omega_p = 0.07$. Bear in mind that the simulation map is on the plane (x,y), with the constant magnetic field $B_0$ directed along the x-axis. The spectral energies $|B_{xk}|^2$ and $|B_{yk}|^2$ contribute mostly to the magnetic signatures of $\mathcal{L}\mathcal{Z}$ waves, as one can see on the dispersion curves (sign($k_x$)$k_{\lambda_D}$, $\omega/\omega_p$) in Figures 4(a)–(c), integrated on the propagation angle $\theta$. $\mathcal{L}\mathcal{Z}$ waves are excited down to $\omega \simeq \omega_p$ until their transition to Z-mode waves with frequencies $\omega_c Z < \omega \lesssim \omega_p$ and $0 \lesssim \theta \lesssim \pi/2$ (see also Figure 3(i) for $|E_k^2|$). On the other hand, the dispersion of $|B_{xk}|^2$ exhibits quasi-parallel propagating Z-mode waves of weaker amplitudes, but is mainly dominated by O-mode waves propagating mostly perpendicular and oblique to $B_0$ (compare Figures 4(c) and (f)), which are also slightly visible in the small-k regions of $|B_{yk}|^2$ and $|B_{yk}|^2$ dispersion curves and spectra (Figures 4(d)–(e)). Note that, in an unmagnetized plasma, $|B_{xk}|^2 \ll |B_{yk}|^2$ and $|B_{yk}|^2 \ll |B_{\bar{z}k}|^2$ at small $k$.

The spectrum of the magnetic energy $|B_{\bar{z}k}|^2 = |B_{xk}|^2 + |B_{yk}|^2$ carried around frequency $\omega_p$ by $\mathcal{L}\mathcal{Z}$ and Z-mode waves (with $\omega \gtrsim \omega_p$ and $\omega \lesssim \omega_p$, respectively) is presented in Figure 4(g) up to large $k$, showing that most energy is gathered at $k\lambda_D \lesssim 0.05$; a small but noticeable anisotropy can be observed, favoring excitation of $\mathcal{L}\mathcal{Z}$ waves with $k_x < 0$, due to damping of waves with larger $k$ and $k_x > 0$. Magnetic energy is negligible along the parallel axis x, i.e., $\mathcal{L}\mathcal{Z}$ waves behave along this direction as Langmuir waves in unmagnetized plasmas. The spectra of $|B_{xk}|^2$ and $|B_{yk}|^2$ at small $k$ (Figures 4(d) and (e)) show that $\mathcal{L}\mathcal{Z}$ waves exhibit large propagation angles up to $\theta \simeq \pi/2$ and $\theta \simeq \pi/4$, respectively; the surface bounded by $k_*(\theta)$ remains free of $\mathcal{L}\mathcal{Z}$ wave energy, but with O-mode energy visible on its right and left borders. The spectrum of $|B_{\bar{z}k}|^2$ exhibits O-mode energy elongated along the magnetic field direction, due to the Lorentz force (Figure 4(f)), and is significantly different from the circular shape observed when $\omega_c = 0$. These waves are produced by the EMD process $\mathcal{L}\mathcal{Z} \rightarrow O + S$, where S is an ion acoustic wave,

as will be shown in a forthcoming paper (see also C. Krafft et al. 2024).

Figure 4(h) shows the time variations of $\mathcal{L}\mathcal{Z}$ waves' magnetic energies $W_{\mathcal{L}\mathcal{Z}}^M$, integrated on the box volume and normalized by the initial electron beam energy, for $\omega_c/\omega_p = 0, 0.02, 0.07$, and 0.14 (including all k-scales). As a reminder, the total energy carried by these waves during time and for each decay cascade is shown in Figure 2(a). For all $\omega_c > 0$, the growth of $W_{\mathcal{L}\mathcal{Z}}^M$ exhibits two main stages. At early times ($\omega_p t \lesssim 1000$), its fast increase results from $\mathcal{L}\mathcal{Z}$ wave excitation by the electron beam. Then, after a short stabilization ($1000 \lesssim \omega_p t \lesssim 4000$), a much slower growth occurs, due to energy transport via electrostatic decay toward smaller-k $\mathcal{L}\mathcal{Z}$ waves with larger magnetic components (4 to 6 orders of magnitude above the unmagnetized case). The final stage takes place at $\omega_p t \sim 25,000$ when decay goes to its end; the saturation energies reached at late times are proportional to $\omega_c^2$. For $\omega_c/\omega_p \gtrsim 0.07$, $W_{\mathcal{L}\mathcal{Z}}^M$ reaches a level $(10^{-6}-10^{-5})$ comparable to the magnetic energy carried by ordinary electromagnetic waves O produced via EMD in homogeneous unmagnetized plasmas (C. Krafft & P. Savoini 2022). In addition, Figure 4(i) shows the time variations of the electromagnetic energies $W_Z$, $W_O$, and $W_X$ carried by the Z-, O-, and X-mode waves in a magnetized plasma with $\omega_c/\omega_p = 0.07$. Note that $W_Z$ takes into account only waves with frequencies $\omega \lesssim \omega_p$ and wavenumbers $k_{*\parallel} \ll k < k_{*\perp}$ (see inset of Figure 3(i)). One observes that $W_Z$ is quite comparable to $W_O$, whereas $W_X$ is 1–2 orders of magnitude smaller. This confirms the importance of $\mathcal{L}\mathcal{Z}$ wave decay to radiate electromagnetic waves at $\omega \lesssim \omega_p$, but with only partial left-handed polarization, as the cutoff frequency is not reached, competing with O-mode radiation produced at $\omega \gtrsim \omega_p$ by EMD. Such results should help identify Z-mode magnetic components in the solar wind (A. Larosa et al. 2022).

Figures 5(a)–(c) show the time evolution of the spectral energy $|B_{\mathcal{L}\mathcal{Z}k}|^2$ at small scales, for $\omega_p t = 5800, 7200$, and 14,000, in a magnetized plasma with $\omega_c/\omega_p = 0.07$. Isocontours $k'_*(\theta, \omega, t)$ (black dashed lines) of constant frequency $\omega$ at time $t$ are superimposed to the spectra, together with the boundary curve $k_*(\theta)$ at $\omega = \omega_{min}$ (black solid lines); the former are calculated by numerically solving the equation $\omega_{\mathcal{L}\mathcal{Z}}(k, \theta) = \omega$ for $\omega/\omega_p = 0.996, 0.998$, and 0.9995, at $\omega_p t = 5800$ (a), 7200 (b), and 14,000 (c), respectively. At each time $t$, energy is mostly distributed around $k'_*(\theta, \omega, t)$, as a result of decay toward smaller $k$ of $\mathcal{L}\mathcal{Z}$ wavepackets of significant spectral widths, as shown in Figures 1 and 3 at $\omega_p t = 14,800$, where $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ waves continue to decay as well as to simultaneously get energy from $\mathcal{L}\mathcal{Z}^{(2)}$ and $\mathcal{L}\mathcal{Z}^{(3)}$ waves, respectively. Indeed, whereas some $\mathcal{L}\mathcal{Z}^{(3)}$ and $\mathcal{L}\mathcal{Z}^{(4)}$ waves reach the boundary $k_*(\theta)$ during their last scatter, others appear due to transport of energy from larger to smaller $k$, generating waves with $\omega \lesssim \omega_p$ in a finite range of small wavenumbers (and even a fifth decay cascade).

This prompts us to consider the case of very low magnetization with $\omega_c/\omega_p \lesssim 0.02$, for which the wavenumber and frequency resolutions required to discriminate the O-, Z-, and X-modes from one another in dispersion curves and spectra are not reached in this work. However, in actual randomly inhomogeneous solar wind plasmas where density turbulence cannot be neglected (i.e., for $\Delta N \gtrsim 3(v_T/v_b)^2$; see D. D. Ryutov & R. Z. Sagdeev 1970; C. Krafft et al. 2013), density fluctuations $\delta n$ of average levels $\Delta N \sim 0.01$ at least exist, and they are responsible for a frequency broadening
<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a)–(c): Dispersion of the spectral magnetic energies $|B_{xk}|^2$, $|B_{yk}|^2$, and $|B_{zk}|^2$ in the map (sign($k_x$)k$\lambda_D$, $\omega/\omega_p$), filtered around $\omega_p$, at $\omega_p t = 13,500$, in a magnetized plasma with $\omega_c/\omega_p = 0.07$; the dispersion curves of O- and X-modes (black lines) and Z-mode (white lines) calculated with the magnetоionic theory are superimposed for parallel (solid lines) and perpendicular (dashed lines) wave propagation (see also the inset of Figure 3(i)). (d)–(f): Corresponding energy spectra of $|B_{xk}|^2$, $|B_{yk}|^2$, and $|B_{zk}|^2$ around frequency $\omega_p$, in the map ($k_x\lambda_D$, $k_y\lambda_D$). The boundary curve $k_*(\theta)$ is represented in panels (d) and (e) by a black dashed contour; note that a small part of the O-mode energy is visible on its right and left borders. (a)–(f): arbitrary units and logarithmic scales. The colorbar is the same for panels (a)–(c) and for panels (d)–(f). (g) Spectrum of $|B_{LZk}|^2 = |B_{xk}|^2 + |B_{yk}|^2$, filtered around $\omega_p$, at $\omega_p t = 13,500$, for $\omega_c/\omega_p = 0.07$, up to large k. (h) Evolution with time of the spectral magnetic energy $W_{LZ}^M$, for different values of $\omega_c/\omega_p$ (see legend), normalized by the initial beam energy, in logarithmic scales. (i) Variations with time of the electromagnetic energies carried by the Z-, O- and X-mode waves (with $\omega \lesssim \omega_p$ and $\omega \gtrsim \omega_p$, respectively) near $\omega_p$ (see legend), normalized by the initial beam energy, in logarithmic scales, for $\omega_c/\omega_p = 0.07$. (a)–(g) and (i): The lengths of simulation boxes are $L_x \times L_y = 5792 \times 2896\lambda_D^2$, with grid numbers $N_x = 4096$ and $N_y = 2048$; (h): $L_x \times L_y = 1448 \times 1448\lambda_D^2$, $N_x = N_y = 1024$. The frequency resolution in (a)–(c) is $\delta\omega \simeq 2 \cdot 10^{-3}\omega_p$. The first set of physical parameters is used (see above).</figcaption></figure>

$\Delta\omega \simeq \omega_p\Delta N \sim 0.01\omega_p$; in this case, one can no longer distinguish one mode from the others and Z-mode generation near the cutoff frequency $\omega_cZ$ is possible (due to wave scattering on $\delta n$) if $\omega_p - \omega_cZ \simeq \omega_c/2 \lesssim \omega_p\Delta N$, i.e., if $\omega_c/\omega_p \lesssim 2\Delta N \sim 0.02$. However, in homogeneous plasmas where induced density fluctuations are much smaller ($\Delta N \ll 0.01$), our study suggests that Z-mode waves with $\omega \lesssim \omega_p$ cannot be excited down to their cutoff frequency as a

result of decay, and thus cannot reach 100% left-handed polarization.

Figures 5(d)–(f) show spectra of the magnetic energy $|B_{LZk}|^2$ at $\omega_p t = 14,000$, in magnetized plasmas with $\omega_c/\omega_p = 0.02, 0.07$, and 0.14, to which the curve $k_*(\theta)$ is superimposed (black dashed lines). The spectral domains with significant magnetic intensity extend and their energy amplifies with increasing $\omega_c$ (see also Figure 4(h)). As expected, no magnetic energy is found
<figure><img src="image_8.png" /><figcaption>**Figure 5.** (a)–(c): Spectra at small k-scales of $\mathcal{L}\mathcal{Z}$ wave energy $|B_{\mathcal{L}\mathcal{Z}\mathbf{k}}|^2 = |B_{\mathbf{k}}|^2 + |B_{\mathbf{\bar{k}}}|^2$, filtered around $\omega_p$, for $\omega_p t = 5800, 7200$, and 14,000, in a plasma with $\omega_c/\omega_p = 0.07$. The boundary curve $k = k_*(\theta)$ at $\omega = \omega_{\min}$ and the isocontours $k = k'_*(\theta, \omega, t)$ at frequency $\omega$ and time $t$ are represented by black solid and dashed lines, respectively; the latter are calculated by solving the equation $\omega_{\mathcal{L}\mathcal{Z}}(k, \theta) = \omega$, and are represented for $\omega/\omega_p = 0.996, 0.998$, and 0.9995 in panels (a), (b), and (c), respectively. (d)–(f): Spectra of $|B_{\mathcal{L}\mathcal{Z}\mathbf{k}}|^2$ radiated at $\omega_p$, for $\omega_p t = 14000$, in plasmas with $\omega_c/\omega_p = 0.02, 0.07$, and 0.14; the dashed line represents the curve $k = k_*(\theta)$ calculated in all cases by solving the equation $\omega_{\mathcal{L}\mathcal{Z}}(k, \theta) = \omega_{\min}$. (g) Spectral electric energy $|E_{\mathbf{k}}|^2$, radiated at $\omega_p$, at $\omega_p t = 9300$, for $T_e = 20\,\text{eV}$, $T_e/T_i = 3$, $\omega_c/\omega_p = 0.02$, $v_b \simeq 0.08c$, and $n_b/n_0 = 5 \cdot 10^{-4}$. (h): Magnetic energy $|B_{\mathcal{L}\mathcal{Z}\mathbf{k}}|^2$ corresponding to (g). (a)–(h): Arbitrary units and logarithmic scales. The first set of physical parameters is used for (a)–(f) and the second one for (g)–(h) (see above). In panels (a)–(c), the lengths of the simulation boxes are $L_x \times L_y = 5792 \times 2896\lambda_D^2$, with $N_x = 4096$ and $N_y = 2048$; in panels (d)–(h), they are $L_x \times L_y = 1448 \times 1448\lambda_D^2$, with $N_x = N_y = 1024$.</figcaption></figure>

at $\theta = 0$. Wave energy does not penetrate the energy-free region $k < k_*(\theta)$. Note that the same conclusion can be stated for plasmas with $\Delta N < 3(v_T/v_b)^2$, where scattering of $\mathcal{L}\mathcal{Z}$ waves on density fluctuations $\delta n$ is not effective.

The simulations presented above have been performed using the first set of physical parameters (see above), which allowed us to clearly identify the energy-free surface bounded by $k_*(\theta)$. Here, we present a simulation made with the second set of parameters closer to solar wind ones near 1 au, i.e., $T_e = 20\,\text{eV}$, $T_e/T_i = 3$, $\omega_c/\omega_p = 0.02$, $v_b \simeq 0.08c$ (V. Krupar et al. 2015), and $n_b/n_0 = 5 \cdot 10^{-4}$ (which is similar to that of the first set). Figures 5(g) and (h) show the electric and magnetic spectral energies $|E_{\mathbf{k}}|^2$ and $|B_{\mathbf{k}}|^2$ at $\omega_p t = 9300$, which can be compared, at close evolution stages, with Figures 1 and 5(d)–(f) (note that the

beam–plasma system and decay dynamics are almost two times slower for the second set than the first one). As in previous figures, one observes backscattered and forward-scattered $\mathcal{L}\mathcal{Z}'$ and $\mathcal{L}\mathcal{Z}''$ waves (Figure 5(g)) with magnetic signatures shown in Figure 5(h) (where the resolution is not sufficient to observe the energy-free zone near $k \sim 0$—which is around 10 times smaller than that for the first set of parameters), as $L_x \times L_y = 1448 \times 1448\lambda_D^2$, with $N_x = N_y = 1024$. The decay process works in a manner qualitatively similar to that of a hotter and more strongly magnetized plasma with a faster beam; however, $\mathcal{L}\mathcal{Z}$ waves exhibit much smaller energies. Thus, it should be easier to measure magnetic signatures of waves with satellites approaching as close to the Sun as Parker Solar Probe or Solar Orbiter (see also A. Larosa et al. 2022).
### 3. Conclusion

For the first time, owing to large-scale and long-term 2D PIC simulations with high space and time resolutions, the dynamics of three-wave decay of electrostatic upper-hybrid wave turbulence generated by electron beams into so-called Langmuir/Z-mode ($\mathcal{LZ}$) wavepackets is studied in weakly and moderately magnetized plasmas, in conditions relevant to type III solar radio bursts between 0.1 and 1 au. Studying wave energy transport down to small k-scales, we show the impact of plasma magnetic field on 2D decay and determine the magnetic properties of $\mathcal{LZ}$ waves.

During decay, the energy of $\mathcal{LZ}$ waves is transported through a few k-scales (here from $k\lambda_D \sim 0.1$ to $k\lambda_D \sim 0.001$) and from frequencies above ($\omega > \omega_p$) to below the plasma frequency ($\omega \lesssim \omega_p$). $\mathcal{LZ}$ waves undergo several decay cascades, during which they acquire increasing magnetic energy, until they reach electromagnetic Z-mode dispersion at $\omega \lesssim \omega_p$. Whereas the impact of plasma magnetic field on waves of large $k \gtrsim 2k_0 \simeq 4c_x\omega_p/3v_T^2$ is weak, strong differences with respect to the unmagnetized plasma case manifest at smaller scales $k \lesssim 2k_0$. The plasma magnetic field is responsible for the formation of a boundary $k_*(\theta) \ll 2k_0$ that delimits a surface free of $\mathcal{LZ}$ wave energy, which extends in k-space with increasing $\omega_c/\omega_p$. This impenetrable border around which energy accumulates over time prevents decayed waves from reaching the Z-mode cutoff frequency $\omega_{cZ}$ and a high level of left-handed polarization ($\gtrsim 60\%$). On the contrary, in an unmagnetized plasma, an energy condensate of Langmuir waves is formed ultimately at $k \sim 0$. The absence of $\mathcal{LZ}$ wave energy at very small scales can prevent the onset, in magnetized plasmas, of modulational instabilities and strong turbulence phenomena. Note finally that upper-hybrid wave turbulence generated by beam-plasma interactions radiates electromagnetic modes other than the Z-mode waves excited at $\omega_{cZ} < \omega \lesssim \omega_p$. Indeed, O-mode waves of comparable energy are jointly produced at $\omega \gtrsim \omega_p$ via EMD through the channel $\mathcal{LZ} \to O + S$, whereas X-mode waves are emitted with significantly weaker energy.

Results obtained on the dynamics of beam-generated upper-hybrid wave turbulence and its implications also provide a support for the interpretation of observations by satellites such as Parker Solar Probe and Solar Orbiter, and they supply a solid basis for tackling the more complex problem of its evolution in magnetized plasmas where background density turbulence cannot be neglected.

### Acknowledgments

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 authors' manuscript accepted for publication resulting from this submission.

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).

### 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

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