Single-mode wave decay

Explosive events that energize and accelerate particles are ubiquitous in space and astrophysical environments. Of particular importance are phenomena that generate beams of energetic electrons that will propagate through a, usually denser, background plasma composed also of electrons and ions. This kind of setup is usually called a beam-plasma system and it has been observed in solar flares and coronal mass ejections,^1–9 the foreshock region of Earth's magnetosphere,¹⁰ and the auroral region,¹¹ among other space environments.

The excess of free energy contained in the electronic beam frequently converts to electrostatic and/or electromagnetic waves through a series of physical processes. In fact, in many regions of the heliosphere and planetary magnetospheres, the remote observation of free-propagating electromagnetic waves was the first and only possible evidence of the energetic beams. Such was the case of the solar corona. For decades, before in situ observations were possible, thanks to the recently launched Parker Solar Probe and Solar Orbiter, the properties of particle beams originating from the solar chromosphere and expanding into the solar corona were inferred by wave phenomena such as solar type III and x-ray emissions.^{1,2,4–8,12,13} The same happens with radio emissions from the auroral regions of other planets of the solar system, such as Jupiter and Saturn, which are believed to be caused by energetic electron fluxes accelerated along the magnetic field lines, sometimes in combination with other free-energy sources such as loss-cone or ring distributions.^14–19 

Regarding a pure beam-plasma system, one of the most accepted physical mechanisms whereby free energy from the electronic beam is ultimately converted to electromagnetic waves is the plasma emission process, which occurs in a series of steps, beginning by the direct, linear conversion of particle kinetic energy into longitudinal Langmuir and ion-sound waves, followed by nonlinear mechanisms that couple transverse waves with the longitudinal waves, allowing the former to grow from the enhanced level of the latter. A solid evidence for this mechanism is the frequent in situ observation of enhanced Langmuir waves associated with particle beams.^{7,8,10,12,20–22}

The subsequent nonlinear mechanisms that convert Langmuir waves into other normal modes are usually assumed to occur in a timescale larger than the quasilinear diffusion time. One recently proposed mechanism starts with the parametric decay of Langmuir waves into other longitudinal modes such as the electron- and ion-acoustic waves, followed by conversion to electromagnetic waves.^13,20,23

Another frequently proposed mechanism is based in the weak turbulence theory derived from a perturbative approach to the Klimontovich–Dupree formalism.^24,25 According to the theory, the particle velocity distribution functions and the spectral intensities of the existing normal modes of oscillation evolve in time according to kinetic equations that include the processes occurring during the quasilinear relaxation phase, followed by the nonlinear three-wave decay and nonlinear wave-particle scattering that occur in a longer timescale. The weak turbulence theory, in particular, and the plasma emission hypothesis, in general, have been developed and compared with numerical simulations by several contributions along the years, of which Refs. 26–44 is but a sample of recent contributions.

In the context of beam-plasma systems and weak turbulence theory, the self-consistent evolution of longitudinal modes in a field-free plasma is usually described in terms of two interplaying modes, namely, Langmuir and ion-sound modes.^29,31,45,46 Specifically, the wave–wave process accounted is the three-wave decay involving both modes. Despite that, three-wave decay involving ion-sound mode only and four-wave decay involving Langmuir mode only, for instance, are feasible.^47–49 

Wave decay processes play an important role in regulating wave spectra and particle velocity distributions in plasma systems. In particular, beam-plasma systems are shown to be significantly influenced by wave decay processes. The higher the energy carried by the beam, the more relevant the decay processes are shown to be, since such processes usually take place as higher order nonlinear contributions.^26,50–53 Henceforth, it is emphasized the meaning of wave decay as a wave–wave process, that is to say, a process involving waves only, not particles. This distinction is called for since waves can decay intermediated by particles, as in the so-called nonlinear wave scattering process.

The present work addresses a beam-plasma system, aiming to describe the system in terms of a single wave mode, taking into account the quasilinear interaction with particles and the nonlinear interaction among waves, through the three-wave process, and compare against the usual approach that accounts for Langmuir and ion-sound modes.^{11,21,25,36,39,40,42,54,55}

Although not usual in the study of the beam-plasma process (see, e.g., Ref. 25 for a list of references), a single-mode approach is by no means uncommon. For instance, the nonlinear decay of Alfvén waves into a pair of other Alfvén or related waves in conditions pertaining to the solar wind plasma has been studied within the frameworks of MHD^56,57 and hybrid⁵⁸ simulations. Three-wave decay processes among kinetic Alfvén waves were also studied using either kinetic theory⁵⁹ or a two-fluid formalism.⁶⁰ However, the description of decay processes in a beam-plasma system using the usual electrostatic normal modes needs the interaction of different waves in order to match the three-wave decay condition. On the other hand, the present single-mode description employs a modified-beam mode that has a frequency range large enough to carry out the decay process.

The paper is organized as follows. In Sec. II, numerical solutions of the electrostatic dispersion relation of a beam-plasma system are analyzed, providing support for the single-mode modeling. In Sec. III, the equations which establish the temporal evolution of particles and waves are presented, and the simulation results obtained both from the usual two-mode approach and single-mode approach are compared. Finally, in Sec. IV, we present our final remarks.

For a system described by non-equilibrium (shifted Maxwellians) distribution functions, as it occurs in the beam-plasma system, the dispersion relation for Langmuir waves (Bohm–Gross) $ωL2=ωpe2+3vte2k2$ and the dispersion relation for ion-sound waves $ωS2=(cS2k2)/(1+λDe2k2)$ are often assumed to remain valid. Here, $ωpe=4πn0e2/me$ is the electron plasma frequency, $vte=Te/me$ is the thermal velocity of electrons, and $cS=Te/mi$ is the ion-sound speed. In the above, $n0$ is the total plasma density, $me$ and e are the mass and charge of the electron, respectively, $Te$ is the electron temperature in energy units, $mi$ is the ion mass, and $λDe=Te/4πn0e2$ is the Debye length. In this case, it is hypothesized that deviations from the distribution in relation to the (non-shifted) Maxwellian will not significantly alter the dispersion of normal plasma oscillation modes, and their contributions are considered only in the absorption/emission coefficient.

However, more careful studies on the effect of the electron beam, not only on the absorption/emission coefficient but also on the dispersion relation of the modes, have shown that in the case of a beam-plasma system, the Langmuir and ion-sound dispersion relations are only valid under very restrictive conditions.^10,45,61,62

A more careful analysis developed by Cairns⁴⁵ and Cairns and Fung,⁶³ resulted in even more restrictive conditions for the validity of this approach. The results obtained by the authors indicate that the usual weak turbulence theory must be modified to describe the behavior of a beam-plasma system when the beam is of higher intensity, meaning higher density or velocity. The necessary modifications involve the consideration of the oscillation modes relevant to the case of intense beams, as well as the use of the respective dispersion relations and coupling terms of waves in order to circumvent the inconsistency arising when considering beams of higher intensity.^13,64

Nevertheless, we hasten to emphasize that the developments presented in this paper aim solely to indicate a transition of regimes found when the beam energy is increased. This is noteworthy since the parameters involved in such a transition are on the edge of validity of the weak turbulence theory, used as the theoretical framework.

When it is assumed that the beam is very tenuous, which is tantamount to a low energy beam, and consequently the system is weakly unstable, an analytic solution is possible for the dispersion equation, and results in the well-known Bohm–Gross dispersion relation for the high frequency range (the Langmuir mode), and the ion-sound mode for the low frequency range. However, for the case of high energy beams, a numerical solution is called for.

With this criteria in mind, the solutions of Eq. (2) are sought in the complex frequency plane and are written as $ω(k)=ωr(k)+iγ(k)$⁠, where $ωr(k)$ is the dispersion relation and $γ(k)$ is the damping (⁠ $γ<0$⁠) or growth (⁠ $γ>0$⁠) rate. Figure 1 shows complex roots of Eq. (2) for a beam with fixed density $nb/n0=10−3$ and temperatures $Te/Ti=1$ and $Tb/Te=1$⁠, and different velocities, $vb/vte=7,8,9,10$⁠. Only the high frequency solutions (determined by the electronic populations) are shown.

There are always three solutions of the dispersion equation, corresponding to three normal modes of propagation, two of which are always stable [ $γ(k)<0$ for all k] and one is unstable $(γ>0)$ in a given spectral range of wavenumber values. One of the solutions, called the modified-Langmuir mode, always starts with $ωr≈ωpe$ at $k=0$⁠. This mode reduces to the usual Langmuir mode when $vb=ve=0$ and $nb=0$⁠. The other two solutions of Eq. (2) correspond to the new beam modes that are identified as $ω1B(k)$ and $ω2B(k)$⁠, with $ωr1B>ωr2B$ for any k. The color codes of the normal modes are as follows. The unstable mode is always depicted by the black curve, the related stable beam or modified-Langmuir mode by red curve, and the other beam mode by blue curve. It can be seen that by increasing the beam intensity by increasing its drift velocity, the dispersion relations change and the growing mode no longer follows the modified-Langmuir branch, jumping instead to the beam mode $ω1B$⁠. This transition is visible between the solutions for $P=0.70$ and $P=0.80$⁠.

The dispersion relations $ω±(k)$ provide good approximations for the beam modes in the small wavenumber regime when $P≲1$⁠, but are in principle valid for any temperature ratio and in the results shown by Fig. 1, $Tb=Te$⁠. However, when one of the electronic populations is much hotter than the other, the full expression of $ω+(k)$ in Eq. (3) can be identified with the electron-acoustic mode and electron/electron acoustic instability.^66,67 For instance, when the core population is hot and the beam electrons are cold $(Tb≪Te)$⁠, the beam modes are definitely fast waves for the beam electrons and $ω+(k)$ reproduces Eq. (1) of Ref. 67 or the known electron-acoustic relation when $vb=0$⁠.⁶⁶ Therefore, the beam modes $ω1,2B(k)$ obtained as numerical solutions of the dispersion Eq. (2), asymptotically reduce to electron-acoustic modes when one population is much hotter than the other.

Figure 2 shows the dispersion curves for a beam with fixed drift velocity $vb/vte=7$ and different densities $nb/n0=10−3,10−2,10−1$⁠, and $2×10−1$⁠, again with temperatures $Te/Ti=1$ and $Tb/Te=1$⁠. Likewise as shown in Fig. 1, it can be seen that by increasing the beam intensity, now by increasing its density, the dispersion relations also change, between $P=0.70$ and $P=1.51$⁠, in such a way that the growing mode no longer follows the Langmuir mode branch.

For a typical case, with velocity $vb/vte=10$ and density $nb/n0=10−3$⁠, we have $P=1.00$ and $s=1.26$⁠. When the beam velocity is increased, with $vb/vte=7,8,9,10$⁠, we obtain the following values for the parameters $P=0.70;0.80;0.90;1.00$ and $s=1.80;1.57;1.40;1.26$⁠. These values obtained for the s and P parameters, which correspond to the transition of the numerical solution, are consistent with the classification made by Cairns.⁴⁵ 

In essence, for increasing beam intensities, either increasing density or drift velocity, the dispersion relation undergoes a topological change, and the growing branch changes accordingly, corresponding to the modified-Langmuir mode when $P≲0.8$ or the $ω1B(k)$ beam mode otherwise. Therefore, a proper analytical approach would call for an algebraic dispersion relation describing the actual growing mode.

Figure 3 illustrates the pertinence of using Eq. (4) to describe the growing mode for high energy beams. Along with the numerical solution and the fitting expression for the growing mode, it is displayed the dispersion relations for Langmuir, ion-sound and beam modes, for comparison.

At this point, it must be emphasized that the model dispersion relation $ωB(k)$ is only valid for regions in the parameter space where the $ω1B(k)$ mode is both topologically detached from the modified-Langmuir branch and the unstable mode. This corresponds to the panels of Figs. 1 and 2 with $P≳0.8$⁠.

Several studies show results involving beam-plasma systems where the beam is of low intensity and the system is described by the weak turbulence theory. In particular, it is shown that in some cases the quasilinear theory alone satisfactorily describes the evolution of the system even for parameters that are beyond the original limits of the theory.^{29,31,33,46,68}

Nevertheless, the weak turbulence theory allows one to establish the kinetic equations of waves and particles for Langmuir (L) and ion-sound (S) modes, accounting for quasilinear and higher order nonlinear effects. A full account of the theoretical formulation is presented in, e.g., Refs. 25, 29, 33, and 46. Therefore, only the basic equations are shown here.

With the equations involving the Langmuir and ion-sound modes, we obtain results for the evolution of particles and waves, with the parameters $nb/ne=10−2$⁠, $vb/vte=4$⁠, $Tb/Te=1$⁠, and $Te/Ti=7$⁠. The results are shown in terms of normalized wave intensity $Iqβ=Ikβ/23/2λDe3nemevte2$⁠, where $q=kvte/ωpe$⁠, $β$ stands for the wave modes L, S, and B, and the normalized distribution function $fe=nFe$⁠. For consistency with the treatment that consider the single B mode, discussed below in Sec. III C, only the decay terms in Eqs. (6) and (7) were included.

Figure 4 shows the temporal evolution of the beam-plasma system according to the traditional approach. In Fig. 4(a), it is possible to observe the formation of a superthermal tail in the electron distribution due to absorption of the backscattered mode around $vb/vte=−4$ due to nonlinear effects. Figures 4(b) and 4(c) show the evolution of Langmuir and ion-sound waves. In Fig. 4(b), it is noticeable an increase in the intensity of the L mode at $kvte/ωpe≈vte/vb$ until the formation of a plateau in the electron distribution, after which ensues the formation of a secondary wave peak, of the backscattered Langmuir wave, in the region $kvte/ωpe≈−vte/vb$⁠. In Fig. 4(c), it is possible to observe the formation of a peak of $IqS$ at $kvte/ωpe≈2vte/vb$⁠, and after some time, the formation of the secondary peak at $kvte/ωpe≈−2vte/vb$⁠.

These results are typical for a beam-plasma system and represent a benchmark for comparison with the B mode approach.

In order to perform the intended simulations with the B mode, the initial level of wave intensities is required. A consistent theory of the evolution of the beam-plasma system, treated as an initial-value problem, demands the starting intensity levels of the plasma eigenmodes.

These expressions have been adopted in previous works as initial conditions for the beam-plasma problem, when it was assumed that the wave eigenmodes remained the same as in an equilibrium plasma. Nonetheless, not only a beam-plasma system is not in thermal equilibrium, but also the resulting dispersion relations are topologically different from the traditional wave modes, as can be seen in Figs. 1 and 2. Consequently, the beam-modified modes are not able to tap on the initial level of fluctuations of Langmuir and ion-sound modes, because they occur along different curves in the $ω×k$ diagram.

The top panel of Fig. 5 shows the electrostatic thermal noise as contour curves. The contours peak along the dispersion relation of the Langmuir mode, determined by the numerical solution of $ε(k,ωL)=0$⁠, and shown as the dashed line marked with the label L. The contour lines at the bottom correspond to emissions in the ion-sound $(S)$ range. Also superimposed to the contour lines are the Bohm–Gross dispersion relation $(BG)$ and the beam mode model $(B)$⁠.

Mode B is incorporated into the modified weak turbulence theory proposed in this work as follows (more details in the appendix). First of all, the B mode is assumed to occur along the dispersion relation (4) throughout the whole dynamical evolution of the beam-plasma system. This is admittedly a shortcoming, since a fully self-consistent treatment would take into account the fact that as the particle distributions and wave intensities evolve, the linear dispersion relations of the normal modes of oscillations will change as well. However, since we are employing a model dispersion relation for the B mode anyway, we will for now assume that its time evolution can be disregarded as a first approximation.

Figure 6 shows the evolution of the particles and the spectral intensity of mode B waves with velocity $vb/vte=8$ and density $nb/n0=10−3$⁠, including the three-wave decay terms. It can be seen in the top panel that particles lose energy to waves until the formation of the plateau and the formation of a superthermal tail due to the absorption of the backscattered mode due to the nonlinear effects. The middle panel shows the evolution of mode B waves, where we see the increase in intensity until the formation of a plateau in the particle distribution, and after that the formation of a secondary peak of backscattered mode B waves. The bottom panel shows that the ratio of B mode wave energy to particle energy is on the edge of validity of the theoretical approach, which requires this quantity to be small in comparison to unity.

The results of the temporal evolution of waves and particles obtained for high intensity beams show that the behavior of this system is qualitatively similar to the behavior of the system with low intensity beams, i.e., the traditional weak-turbulence formalism. This result can be explained by the fact that, although the usual theory considers the coupling of waves involving two distinct modes of oscillation, ion-sound and Langmuir, while the present work considers the existence of a single mode, the dispersion relation of the mode considered, the B mode, has a region of low frequency as well as a region of high frequency, which end up playing the role of the ion-sound and Langmuir waves, respectively.

The usual approach, which takes into account nonlinear effects present in the system, is done under the formulation of the weak turbulence theory. In systems where electrostatic oscillations are dominant, this approach considers the interaction between two electrostatic oscillation modes, consisting of the ion-sound mode and the Langmuir (or Bohm–Gross) mode. The dispersion relations that determine these modes are strictly valid for low intensity beams, which for the present study translates into low beam density or low beam drift velocity.

This question was addressed by searching for the numerical solution of the dispersion relation of a beam-plasma system, where the beam is of high intensity. Based on the results obtained from the numerical solution, the dispersion relation was modeled in closed mathematical form in order to allow for algebraic manipulation. The model dispersion relation was used to calculate the coefficients that determine the time evolution of the particle velocity distribution function and the spectral intensity of the waves.

Different densities and drift velocities were considered, representing high intensity beams. The general behavior of waves and particles was compared with those obtained under the weak turbulence theory approach, but calculated considering the interaction between the Langmuir and the ion-sound modes.

Both cases, that is, the usual approach and the approach presently introduced, are different not only by the use of different dispersion relations, which determine the wave–particle and wave–wave interactions, but also by the fact that the number of modes is distinct: two modes in the usual approach and one mode in the presently introduced approach.

The results show similarities between these two approaches, regarding the behavior of the distribution function and the spectral intensity of the waves. This similarity can be explained by the fact that, although the usual theory considers the coupling of waves involving two different oscillation modes, ion-sound and Langmuir, while the present work considers the existence of a single mode, the dispersion relation of the single mode shows both low and high frequency regions, for small and higher wave number, respectively. These two distinct regions eventually play the role of the ion-sound and Langmuir waves, respectively, and this characteristic is responsible for producing the similarity between the two cases.

The findings contribute to the consistency of weak turbulence theory, suggesting an extension of its validity in the case of beam-plasma systems where the beams are of high intensity. Nevertheless, further developments on the subject would additionally account for the proper temporal growth of wave intensity³⁷ and the temporal changes in dispersion relations.³⁸ In particular, the latter aspect is ensued by the transient character of the topological changes in the dispersion relations as the beam undergoes a plateau formation. In contrast, the mode B decay approach assumes that the mode will remain as a separate branch throughout the course of dynamical evolution. In this regard, it would be adequate to check the validity of this approach by comparing the theory against a one-dimensional electrostatic particle-in-cell simulation. This might be the subject of a future work.

G.T.I. acknowledges Ph.D. fellowship from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) – Brazil. R.G. acknowledges support provided by CNPq (Brazil), Grant No. 313330/2021-2.

The authors have no conflicts to disclose.

Guilherme T. Irumé: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Joel Pavan: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Writing – original draft (equal). Rudi Gaelzer: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.