LANGMUIR WAVE DECAY IN INHOMOGENEOUS SOLAR WIND PLASMAS: SIMULATION RESULTS

Let us first consider simulations performed for a plasma with a small average level of density fluctuations, i.e., Δn ≃ 0.001. Figure 1 shows the growth with time of the normalized spectral energy density ${W}_{L}={\displaystyle \sum }_{k}{| {E}_{k}| }^{2}$ of the Langmuir waves excited by the beam and the corresponding time evolution of the beam velocity distribution $f(v)$; W_L grows until saturation near ${\omega }_{p}t\simeq 60000,$ whereas $f(v)$ widens toward lower velocities, asymptotically forming a plateau with a vanishing slope $\partial f(v)/\partial v\simeq 0$. Indeed, as Δn is very below the threshold $\sim 3{({v}_{T}/{v}_{b})}^{2}$ determined in our previous works (Krafft et al. 2013; Volokitin et al. 2013), the dynamics of the system roughly presents the same features as those described by the quasilinear theory of the weak turbulence in homogeneous plasmas (Vedenov & Ryutov 1975, p. 3; see also Volokitin et al. 2013) or other close models (e.g., O'Neil et al. 1971; Volokitin & Krafft 2004; Krafft et al. 2005, 2010; Krafft & Volokitin 2006, 2010; Zaslavsky et al. 2006, 2007; Krafft & Volokitin 2013). In this case it was shown (e.g., Krafft et al. 2013) that the density inhomogeneities weakly influence the development of the beam instability and that the main features observed are (i) the formation of a plateau in the velocity distribution function $f(v)$ at asymptotic times, (ii) the dependence of the wave spectral energy density, scaling as ${| {E}_{k}| }^{2}\propto ({n}_{b}/{n}_{0}){({\omega }_{k}/k)}^{4}$ in the velocity domain above the thermal region where the beam can excite Langmuir waves, and (iii) a very small amount of accelerated beam electrons if Δn is not vanishing (e.g., Krafft et al. 2013).

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Up to the time ${\omega }_{p}t\simeq 30000,$ most features of the system's evolution are in agreement with the predictions of the quasilinear theory of Langmuir waves. However, for ${\omega }_{p}t\gtrsim 30000$, short-wavelength density oscillations, which have been identified as IS waves, appear and grow with time along the full length L of the system, as shown by Figure 2  which presents the Langmuir wave energy density ${| E| }^{2}$ and the density fluctuations $\delta n/{n}_{0}$ as a function of the space coordinate $z/{\lambda }_{{\rm{D}}}$ at three different times ${\omega }_{p}t=22000,$ 28000, and 35000. The short-wavelength density perturbations $\delta {n}_{{is}}/{n}_{0}$ (see Equation (12)) present rather large amplitudes whereas the Langmuir wave packets reveal energy densities ${| E| }^{2}$ peaking up to around 0.01. To study these fluctuations in more detail and separate them from the background long-wavelength fluctuations $\delta n/{n}_{0}$, we filter $\delta n$ and the normalized plasma velocity u, which consists of removing all the Fourier harmonics $\delta {n}_{k}$ and u_k with $k\lt {k}_{*}\sim 2$–$3{k}_{b}$ (${k}_{b}={\omega }_{p}/{v}_{b}$); we obtain the short-wavelength density and plasma velocity in the form

$$\begin{eqnarray}&&\delta {n}_{{is}}=\displaystyle \sum _{k\gt {k}_{*}}\delta {n}_{k}(t){e}^{{ikz}},\quad {u}_{{is}}=\displaystyle \sum _{k\gt {k}_{*}}{u}_{k}(t){e}^{{ikz}}.\end{eqnarray} \tag{ 12 }$$

Note that it is not essential to determine the exact value of ${k}_{*}$ and that we remove the parasitic oscillations that remain after this operation at the edges of the chosen space portion where the filtering is applied. Despite its arbitrariness, this procedure constitutes an effective method for analyzing the properties of the excited IS oscillations.

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Using Equation (12), we present in Figure 3 the time and space variations of the turbulence parameter ${| E| }^{2}$ and the short-scale oscillations $\delta {n}_{{is}}/{n}_{0}\ \mathrm{over}$ the full time range of the simulation. The dark lines traveling downward (Figure 3, left) correspond to Langmuir waves propagating along the beam direction with the group velocity ${v}_{g}/{v}_{T}\simeq 3{k}_{b}{\lambda }_{{\rm{D}}}\simeq 0.2$ (${k}_{b}{\lambda }_{{\rm{D}}}\simeq 0.07$). At ${\omega }_{p}t\gtrsim 30000,$ IS waves appear (Figure 3, right) that propagate in the beam direction with a velocity around the normalized IS velocity ${c}_{s}/{v}_{T}$. Simultaneously (at ${\omega }_{p}t\gtrsim 30000$), Langmuir waves propagating in the direction opposite to the beam with the group velocity $-{v}_{g}/{v}_{T}\simeq -0.2$ can be observed. This picture is in full agreement with the development of a nonlinear decay process during which a Langmuir wave ${\mathcal{L}}$ (${\omega }_{L},{k}_{L}$) transfers energy to a counterpropagating Langmuir wave ${\mathcal{L}}^{\prime}$ of frequency ${\omega }_{{L}^{\prime }}$ and wavenumber ${k}_{{L}^{\prime }}\lt 0$ and an IS wave ${\mathcal{S}}^{\prime}$ (${\omega }_{{S}^{\prime }},{k}_{{S}^{\prime }}\gt 0$). During this interaction the conditions of parametric resonance ${\omega }_{L}=$ ${\omega }_{{L}^{\prime }}+{\omega }_{{S}^{\prime }}$ and ${k}_{L}=$ ${k}_{{L}^{\prime }}+{k}_{{S}^{\prime }}$ should be verified. As is well known from the theory developed in homogeneous plasmas (Kadomtsev 1965; Nicholson & Goldman 1978), those are satisfied if ${k}_{{L}^{\prime }}\simeq {k}_{0}-{k}_{L}$ and ${k}_{{S}^{\prime }}\simeq 2{k}_{L}-{k}_{0}$, with ${k}_{L}\simeq {k}_{b}$ and ${k}_{0}{\lambda }_{{\rm{D}}}=2{c}_{s}/3{v}_{T}\simeq 0.03;$ corresponding wave dispersions are ${\omega }_{L}\simeq {\omega }_{p}(1+3{k}_{L}^{2}{\lambda }_{{\rm{D}}}^{2}/2)$ and ${\omega }_{S}\simeq {c}_{s}{k}_{{S}^{\prime }}.$ We show below that in the present case these resonance conditions are fulfilled.

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Let us study the time evolution of the Langmuir and IS waves' spectra. To distinguish between the IS waves propagating in the positive and the negative directions (i.e., in the direction of the beam propagation and opposite to it, respectively), we calculate the spectra of the Riemann invariants ${R}_{+}=\rho +u$ and ${R}_{-}=\rho -u,$ where $\rho =\delta n/{n}_{0}$. In the absence of Langmuir waves, ${R}_{+}$ is conserved along the line ${dz}/{dt}={c}_{s}$ for IS waves propagating in the positive direction (increasing z in Figure 3) and ${R}_{-}$ is conserved along the line ${dz}/{dt}=-{c}_{s}$ for IS waves propagating in the negative direction (decreasing z in Figure 3). Thus, the spectrum of the energy density of the IS waves can be represented by ${S}_{k},$ with ${S}_{k}={[{(\rho +u)}^{2}]}_{k}$ for $k\geqslant 0$ and ${S}_{k}={[{(\rho -u)}^{2}]}_{k}$ for $k\lt 0$ : S_k is the spectrum of the IS waves with $k\geqslant 0$ propagating in the positive direction and of the IS waves with $k\lt 0$ propagating in the negative direction.

The spectra of the Langmuir and the IS waves' energy densities ${| {E}_{k}| }^{2}$ and S_k are calculated in Figure 4 for the same times as the profiles of ${| E| }^{2}$ and $\delta n/{n}_{0}$ in Figure 2. The main peak centered near $k{\lambda }_{{\rm{D}}}\simeq {k}_{b}{\lambda }_{{\rm{D}}}\simeq 0.07$ in the ${| {E}_{k}| }^{2}$ spectrum corresponds to waves ${\mathcal{L}}$ excited by the beam instability near the most unstable mode around ${\omega }_{k}/k\simeq {v}_{b}-{\rm{\Delta }}{v}_{b}\simeq 12.7{v}_{T}$ (Figure 4, upper left); note that ${| {E}_{k}| }^{2}$ broadens with time toward higher $k\gtrsim {k}_{b}$ (i.e., lower phase velocities ${\omega }_{k}/k\lt {v}_{b}$); no waves with small $k\gt 0$ are visible as wave scattering on the density fluctuations $\delta n$ is very weak. A second peak (waves ${\mathcal{L}}^{\prime}$) appears at ${\omega }_{p}t\simeq 22000$ near $k{\lambda }_{{\rm{D}}}\simeq -0.04$; it grows with time, eventually reaching the same amplitude as the other one at $k{\lambda }_{{\rm{D}}}\simeq 0.07.$ Meanwhile, IS waves ${\mathcal{S}}^{\prime}$ are excited around $k{\lambda }_{{\rm{D}}}\simeq 0.11$, as shown by the S_k spectrum at ${\omega }_{p}t\simeq 22000$ (Figure 4, lower left); this peak grows in correspondence with that at $k{\lambda }_{{\rm{D}}}\simeq -0.04$ (note that the peak near k = 0 in the S_k spectrum corresponds to the initial density oscillations and should not be considered here). One can identify the first cascade of the decay process ${\mathcal{L}}\to {\mathcal{L}}^{\prime} +{\mathcal{S}}^{\prime}$. Indeed, the theory of three-waves' resonant decay in a homogeneous plasma predicts that for a parent Langmuir wave at ${k}_{L}{\lambda }_{{\rm{D}}}\simeq 0.07$ we should get ${k}_{{L}^{\prime }}{\lambda }_{{\rm{D}}}\simeq ({k}_{0}-{k}_{L}){\lambda }_{{\rm{D}}}\simeq -0.04$ and ${k}_{{S}^{\prime }}{\lambda }_{{\rm{D}}}\simeq (2{k}_{L}-{k}_{0}){\lambda }_{{\rm{D}}}\;\simeq 0.11$, which fits very well with the observations (Figure 4). Further (at ${\omega }_{p}t\gtrsim 65000$, not shown here), a second cascade of decay occurs, i.e., ${\mathcal{L}}^{\prime} \to {\mathcal{L}}^{\prime\prime} +{\mathcal{S}}^{\prime\prime} ,$ providing Langmuir waves with ${k}_{{L}^{\prime\prime }}{\lambda }_{{\rm{D}}}\simeq ({k}_{L}-2{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.02$ and IS waves $\mathrm{near}\ {k}_{{S}^{\prime\prime }}{\lambda }_{{\rm{D}}}=(-2{k}_{L}+3{k}_{0}){\lambda }_{{\rm{D}}}\simeq -0.05,$ that propagate in the direction opposite to the beam drift, i.e., with a group velocity $-{c}_{s}/{v}_{T}$. The secondary decay process is weaker than the first one and the amplitudes of the involved IS waves are smaller. Nevertheless, the cascading process occurs two times, as expected by the theory (no further cascades as ${k}_{L}/{k}_{0}\lt 3$); unless damping of waves is included (what is not the case here), there is no threshold for such decay processes.

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The energy of the IS waves starts to grow when the Langmuir turbulence is almost saturated; thus, one can expect to observe a clear linear stage of the decay instability. Indeed, as shown in Figure 5, the energy ${\int }_{L}{| \delta {n}_{{is}}/{n}_{0}| }^{2}{dz}/L$ of the short-scale IS oscillations (integrated over the full simulation box) grows exponentially within the time range $21000\lesssim {\omega }_{p}t\lesssim \mathrm{37\; 000}$, whereas the corresponding Langmuir energy ${\displaystyle \int }_{L}{| E| }^{2}{dz}/L$ changes only slightly. The growth rate Γ of the IS energy can be estimated as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Gamma }}}{{\omega }_{p}}\simeq \displaystyle \frac{1}{2}{\rm{\Delta }}\mathrm{ln}({\displaystyle \int }_{L}{| \displaystyle \frac{\delta {n}_{{is}}}{{n}_{0}}| }^{2}\displaystyle \frac{{dz}}{L})/{\rm{\Delta }}({\omega }_{p}t)\simeq 2-3{\rm{}}\;{\rm{}}{10}^{-4},\end{eqnarray} \tag{ 13 }$$

which is indeed smaller than but close to the maximum growth rate of the decay instability given by Galeev et al. (1975) for monochromatic waves in homogeneous plasmas

$$\begin{eqnarray}&&\displaystyle \frac{{\gamma }_{{\rm{D}}}}{{\omega }_{p}}\simeq \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{{\omega }_{S}}{{\omega }_{p}}\displaystyle \frac{{| E| }^{2}}{16\pi {n}_{0}{T}_{e}}}.\end{eqnarray} \tag{ 14 }$$

Here ${\omega }_{S}$ is the frequency of the IS waves; then, as ${\omega }_{S}/{\omega }_{p}\simeq 2{c}_{s}{k}_{b}/{\omega }_{p}\simeq 0.006$ and ${| E| }^{2}/16\pi {n}_{0}{T}_{e}\simeq {10}^{-4}$ (average value) we get ${\gamma }_{{\rm{D}}}\simeq 4$ ${10}^{-4}{\omega }_{p}.$ The uncertainty in the determination of the average value of ${| E| }^{2}/16\pi {n}_{0}{T}_{e}$ in Equation (14) and the fact that ${\gamma }_{{\rm{D}}}$ is calculated for monochromatic waves in homogeneous plasmas can explain the difference between Γ and the theoretical value ${\gamma }_{{\rm{D}}}.$ Note also that, as expected, the nonlinear growth rates Γ and ${\gamma }_{{\rm{D}}}$ are significantly smaller than the typical minimum frequency ${\omega }_{S}=0.006$ of the waves involved in the decay, ensuring that the collective response of the IS waves can take place as well as the exchanges of energy during the wave–wave coupling.

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Finally, we can conclude that our simulations, which were carried out for a small average level Δn ≃ 0.001 of density fluctuations, are consistent with the theory of weak turbulence for Langmuir waves resonantly interacting with IS waves through the channel ${\mathcal{L}}\to {\mathcal{L}}^{\prime} +{\mathcal{S}}^{\prime}$. Note that, obviously, the beam instability does not excite a monochromatic wave but a broad wave spectrum. Then, as shown by the above figures, many waves of this wide packet can decay, producing a broad spectrum of daughter waves. The fastest Langmuir decay occurs for waves with the largest field amplitudes, as shown by Equation (14), and the waves with larger k can undergo more decay cascades than those with smaller k. After the occurrence of several cascades, the Langmuir energy can be accumulated within the region $-{k}_{0}/2\lt k\lt {k}_{0}/2$, where waves cannot continue to decay but where the process of scattering off ions can become effective (see e.g., Cairns 2000; Kontar & Pecseli 2002). However, this problem is not considered here.

The influence of the background density inhomogeneities on the system's dynamics becomes significant only when Δn approaches or exceeds the threshold $3{({v}_{T}/{v}_{b})}^{2}$, as shown by our previous works (Krafft et al. 2013; Volokitin et al. 2013) and the study we present herein concerning wave decay. We will consider in details successively two cases, when Δn = 0.01 and Δn = 0.02.

When the average level of density fluctuations Δn becomes of the order or larger than the threshold $3{({v}_{T}/{v}_{b})}^{2}$, the dynamics of the system is strongly influenced by the scattering of the waves on the density inhomogeneities. Indeed, the wave–particle resonance conditions are violated due to the random variations of the waves' phase velocities ${\omega }_{k}/k\simeq {\omega }_{p}(z)/k$; as a consequence, and compared to the homogeneous plasma case, the rate of growth of the Langmuir wave energy emitted during the bump-on-tail instability is decreased, while the relaxation time of the beam is increased. Moreover, due to their interactions with the density inhomogeneities, Langmuir waves with larger k can transfer part of their energy to Langmuir waves with smaller k, which in turn can damp and accelerate beam electrons of velocities $v\gt {v}_{b}$ up to $2{v}_{b}$ and more. Note also that in plasmas with ${\rm{\Delta }}n\gtrsim 3{({v}_{T}/{v}_{b})}^{2}$ Langmuir waveforms tend to form spatially localized and clumpy wave packets.

The presence of fluctuating density gradients generating various processes of wave reflection, refraction, and scattering as well as wave energy focusing alter the development of parametric instabilities occurring in plasmas with long-wavelength density inhomogeneities; some of the first reasons are, for example, the random variations of the wave frequencies and thus of the resonance conditions between waves, ${\omega }_{k}={\omega }_{{k}^{\prime }}+{\omega }_{{k}^{\prime\prime }}$, and the modification of the distribution of Langmuir wave energy in the k-space. Thus, the influence of ${\rm{\Delta }}n\gtrsim 3{({v}_{T}/{v}_{b})}^{2}$ on the nonlinear dynamics of waves during Langmuir turbulence is important, as revealed by the following simulation results.

Figure 6 shows the time evolution of the Langmuir wave spectral energy W_L and of the beam electron velocity distribution $f(v)$ in a plasma with ${\rm{\Delta }}n\simeq 0.01,$ near the threshold $3{({v}_{T}/{v}_{b})}^{2}$. One observes the presence of accelerated electrons (right panel) and the linked saturation of the wave energy growth (left panel, to compare with Figure 1).

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Profiles of the Langmuir wave energy ${| E| }^{2}$ and the density fluctuations $\delta n/{n}_{0}$ along the simulation box of length L are presented in Figure 7 for three different times ${\omega }_{p}t=30000,$ 36000, and 52000. One can see that Langmuir wave packets are focused and localized within a wide space region where the density fluctuations are forming a well (upper panels). Note that such observations were also done for density perturbations forming humps and not only depletions, as in the case presented here. After some time (middle panels) the local structure of the Langmuir waves, which form a set of more or less separated packets, is conserved despite the modification of the density profile. However, after the appearance of IS waves of noticeable intensity, the structure of the Langmuir packets becomes much more irregular and chaotic in the space region where short-scale density oscillations are rising (lower panels). This is due to the decay of Langmuir waves and to the consequent redistribution of energy between them, as shown below.

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The spectra of the Langmuir and IS waves, shown in Figure 8 for the same times as in Figure 7, present features similar to those observed in Figure 4 for the case of small ${\rm{\Delta }}n\simeq 0.001$; indeed, one can observe peaks corresponding to the development of a decay instability. However, due to the presence of randomly varying density inhomogeneities, some differences exist between the theory of decay in quasi-homogeneous plasmas and the observations, as will be discussed below.

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First, the wavenumber of the Langmuir mode observed near $k{\lambda }_{{\rm{D}}}\simeq -0.05$ at ${\omega }_{p}t\simeq 30000$ and 36000 is consistent with the value expected for a first decay cascade, i.e., ${k}_{{L}^{\prime }}{\lambda }_{{\rm{D}}}\simeq ({k}_{0}-{k}_{L}){\lambda }_{{\rm{D}}}\simeq -0.06$; the value of k_L is determined by the location of the highest peak with $k\gt 0$ in the Langmuir spectrum at ${\omega }_{p}t\simeq 30000$ (upper left panel, Figure 8), i.e., ${k}_{L}{\lambda }_{{\rm{D}}}\simeq 0.09$ (here ${k}_{0}{\lambda }_{{\rm{D}}}\simeq 0.03$ and ${k}_{b}{\lambda }_{{\rm{D}}}\simeq 0.055$; note that the most excited Langmuir waves have phase velocities ${\omega }_{k}/k\leqslant {v}_{b},$ so that ${k}_{L}\gt {k}_{b}$). At the same time, the largest peak in the IS spectrum is centered near $k{\lambda }_{{\rm{D}}}\simeq 0.15$, which is very close to the expected wavenumber ${k}_{{S}^{\prime }}{\lambda }_{{\rm{D}}}\;\simeq (2{k}_{L}-{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.15$ of IS waves produced during a first decay cascade. Moreover, the peak at $k{\lambda }_{{\rm{D}}}=0.025$ in the Langmuir spectrum (upper left and middle panels) represents the mode ${k}_{{L}^{\prime\prime }}{\lambda }_{{\rm{D}}}=({k}_{{L}^{\prime }}-{k}_{{S}^{\prime\prime }}){\lambda }_{{\rm{D}}}=({k}_{L}-2{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.03$ generated during a second decay cascade, for which the corresponding IS wavenumber is expected at ${k}_{{S}^{\prime\prime }}{\lambda }_{{\rm{D}}}\;=(-2{k}_{L}+3{k}_{0}){\lambda }_{{\rm{D}}}\simeq -0.09$, which is close to the mode at $k{\lambda }_{{\rm{D}}}\simeq -0.1$ visible in the IS spectrum (lower middle panel). Let us stress that the complex peak structure in the low-frequency spectra can be attributed to the fact that several decay instability regions exist and interfere one with another. Moreover, effects due to Langmuir waves' reflections on the density humps are essential and lead to the widening of the peaks in the corresponding wave spectra.

Second, one important difference compared to the quasi-homogeneous plasma case is that wave decay occurs in localized space-time regions. Indeed, for a given moment in the time range when decay occurs, there are not many occurrences of decay spread accross the full range of z (as in Figure 3), but the development of such a process can be observed only in two or three localized space regions. In this case, the processes of scattering, reflection, and/or 1D refraction modify the waves' propagation locally, depending, for example, on the presence along the waves' paths of more or less sharp gradients, deep wells, or wide humps, so that these waves can lose some part of their energy during their propagation, reducing by this fact the possibility of being submitted to further decay processes along z. To understand how a locally arising nonlinear interaction of Langmuir waves with density fluctuations develops, let us focus our attention on finite portions $[{z}_{1},{z}_{2}]$ of the simulation box. Figure 9 shows the spatio-temporal evolution of the Langmuir wave energy ${| E| }^{2}$ and the short-scale IS density fluctuations $\delta {n}_{{is}}/{n}_{0}$ in the subbox $[{z}_{1},{z}_{2}]=[13000,24000]{\lambda }_{{\rm{D}}}$, during the time interval $30000\lesssim {\omega }_{p}t\lesssim 54000$. One can see the formation of three main regions of IS wave activity (right panel), which broaden and eventually intersect. Note the emergence of IS waves traveling in the direction opposite to the beam propagation with a group velocity around $-{c}_{s}/{v}_{T}$, starting, for example, at $z\simeq 21000{\lambda }_{{\rm{D}}}$ and ${\omega }_{p}t\simeq 45000$: they correspond to the development of a second cascade of decay instability (see also above).

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The left panel shows that the interactions of Langmuir packets with density fluctuations occur during limited and rather short time durations. Then, as their group velocity ${v}_{g}/{v}_{T}$ significantly exceeds the IS velocity ${c}_{s}/{v}_{T}$, the Langmuir waves can propagate away from these interaction regions; further, part of their energy can be transferred to Langmuir packets arising from a first decay cascade and propagating in the opposite direction, with a group velocity $-{v}_{g}/{v}_{T}$ (see, for example, the waves crossing near $z\simeq 21000{\lambda }_{{\rm{D}}}$ and ${\omega }_{p}t\simeq 47000$ in the left panel). The beatings between Langmuir waves propagating in opposite directions lead to the generation of IS waves. Processes involving such beatings form a part of the cascade of energy transfer in the weak turbulence theory; however, some differences exist compared to the homogeneous plasma case: due to the presence of background density fluctuations, the decay processes are occurring locally, i.e., only in specific time and space regions.

Figure 10 shows the "local" wave spectra corresponding to Figure 9, i.e., computed within a specific subbox [13000, 24000]${\lambda }_{{\rm{D}}}$ during a limited time $30000\lesssim {\omega }_{p}t\lesssim 54000$. Those reveal more clearly than the global ones—corresponding to the full space-time domain (Figure 8)—the development of a cascade of energy transfer during the interaction of Langmuir and IS waves. This is particularly visible in the low-frequency spectra, with peaks appearing clearly in the vicinity of $k{\lambda }_{{\rm{D}}}=0.15\ \mathrm{and}$ $-0.1$. In the high-frequency spectra, peaks are excited near $k{\lambda }_{{\rm{D}}}=-0.065,$ 0.025, 0.065 and 0.09 (upper left panel); the largest one, which corresponds to beam-driven waves, is located at $k{\lambda }_{{\rm{D}}}\simeq 0.09.$ As discussed above, a first decay cascade of the Langmuir wave ${k}_{L}{\lambda }_{{\rm{D}}}\simeq 0.09$ starts, giving rise to a counterpropagating Langmuir wave at ${k}_{{L}^{\prime }}{\lambda }_{{\rm{D}}}\simeq ({k}_{0}-{k}_{L}){\lambda }_{{\rm{D}}}$ $\simeq -0.06$ and to an IS wave with ${k}_{{S}^{\prime }}{\lambda }_{{\rm{D}}}=(2{k}_{L}-{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.15.$ Further (at ${\omega }_{p}t\gtrsim 30000$), a second decay cascade occurs, providing the peak at ${k}_{{L}^{\prime\prime }}{\lambda }_{{\rm{D}}}=({k}_{L}-2{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.03$ (upper right panel) and an IS wave $\mathrm{at}\ {k}_{{S}^{\prime\prime }}{\lambda }_{{\rm{D}}}=(-2{k}_{L}+3{k}_{0}){\lambda }_{{\rm{D}}}\simeq -0.09$, propagating opposite to the beam. We obviously recover the same results as above, but with significantly less ambiguity in the interpretation of the spectral peaks.

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Figure 11 shows the profiles of the Langmuir wave energy ${| E| }^{2}$ and the density fluctuations $\delta n/{n}_{0}$ within the same spatio-temporal area as in Figures 9–10. IS waves' short-scale oscillations appear first at ${\omega }_{p}t=35000$ within the region $17000{\lambda }_{{\rm{D}}}\lt z\lt 20000{\lambda }_{{\rm{D}}}$; further they extend over wider space domains, eventually occupying for ${\omega }_{p}t\gtrsim 50000$ the subarea $[13000,24000]{\lambda }_{{\rm{D}}}$. However, they do not extend over the whole space profile (see also Figure 7). One can observe that the various space regions of IS waves' generation are separated one from the other. Moreover, in each area where these waves appear, the growth of their energy stops when the Langmuir packets have traveled away and left the interaction region due to the large difference between the IS and the Langmuir waves' group velocities, ${v}_{g}\gg {c}_{s}$ (compare, for example, the panels at ${\omega }_{p}t=35000$ and ${\omega }_{p}t=40000$). The escaping Langmuir waves can then interact with IS waves in other space-time regions where such oscillations appear.

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Figure 12 shows the time variation of the IS and Langmuir waves' energies, ${\int }_{{L}_{s}}{| \delta {n}_{{is}}/{n}_{0}| }^{2}{dz}/{L}_{s}$ and ${\int }_{{L}_{s}}{| E| }^{2}{dz}/{L}_{s}$, respectively, obtained by simulations involving a plasma with density fluctuations of larger average level, i.e., ${\rm{\Delta }}n\simeq 0.02$ (other parameters being the same as in Figure 1) and computed in the limited domain ${L}_{s}=[13000{\lambda }_{{\rm{D}}},24000{\lambda }_{{\rm{D}}}]$. One can see that, while the plasma waves' energy varies only slightly, the energy of the IS short-scale oscillations $\delta {n}_{{is}}/{n}_{0}$ exhibits two periods of exponential growth; each of them corresponds to the growth of a localized IS wave packet rising in a specific region of the subbox L_s. The first period, between ${\omega }_{p}(t)\simeq 24000$ and ${\omega }_{p}t\simeq 28000,$ corresponds to the crossing of two Langmuir packets of positive but different group velocities; indeed, when a Langmuir packet passes through a density hump, its group velocity can be significantly reduced due to 1D refraction effects. The second period, i.e., $42000\lesssim {\omega }_{p}t\lesssim 46000$ (see also Figure 9), corresponds to the occurrence of a first cascade of Langmuir waves' decay, whose growth rate can be estimated as ${\rm{\Gamma }}=2.7$ ${10}^{-4}{\omega }_{p}$ (with ${\gamma }_{{\rm{D}}}\simeq 4$ ${10}^{-4}{\omega }_{p},$; see Equations (13)–(14)).

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Let us present in this case a typical example of Langmuir decay involving several cascades. Figure 13 shows the spatio-temporal variation of the Langmuir wave energy density ${| E| }^{2}$ and of the short-scale density oscillations $\delta {n}_{{is}}/{n}_{0}$, in the subbox $[5000,8000]{\lambda }_{{\rm{D}}}$ and the time interval $4500\leqslant {\omega }_{p}t\leqslant 9000,$ when the beam instability is saturated. In the left panel, beam-driven Langmuir waves are propagating with ${v}_{g}\gt 0$ through the region $[5000,6700]{\lambda }_{{\rm{D}}}$ for ${\omega }_{p}t\lesssim 58000$; near ${\omega }_{p}t\simeq 58300$ and $z\simeq 6800{\lambda }_{{\rm{D}}},$ their amplitudes are strongly enhanced when they cross another Langmuir packet traveling with ${v}_{g}\lt 0$. At the same time, IS waves propagating with the group velocity ${c}_{s}/{v}_{T}\gt 0$ are excited (right panel).

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In Figure 14, which shows the corresponding profiles of ${| E| }^{2}$ and $\delta n/{n}_{0}$ (including $\delta {n}_{{is}}/{n}_{0}$) at six moments of time, one can see that ${| E| }^{2}$ reaches its maximum when the waves approach the local density hump near $z\simeq 6800{\lambda }_{{\rm{D}}}$ (left middle panel at ${\omega }_{p}t\simeq 58300$); at this point, both packets propagating in opposite directions interact and, as a result, some large part of the Langmuir energy is reflected and propagates away with ${v}_{g}\lt 0$ (see Figure 13, left, and Figure 14, top right). A similar event can be observed later, near ${\omega }_{p}t\simeq 60000$ and $z\simeq 7300{\lambda }_{{\rm{D}}}$ (Figure 13, left): the Langmuir wave decays. Figure 14 at ${\omega }_{p}(t)\simeq 60600$ and 63000 shows how the counterpropagating packet separates itself from the packet with ${v}_{g}\gt 0$, eventually taking with it almost all the Langmuir energy. The two peaks at ${\omega }_{p}t\simeq 63000$ (Figure 14 , bottom right) can be clearly identified in Figure 13 as the two packets propagating with ${v}_{g}\lt 0$, in the time range $60000\lesssim {\omega }_{p}t\lesssim 65000$. Moreover, near ${\omega }_{p}t\simeq 66000$ and $z\simeq 6000{\lambda }_{{\rm{D}}}$, a second decay cascade occurs, giving rise to Langmuir packets propagating with ${v}_{g}\gt 0$ and IS waves' oscillations propagating with $-{c}_{s}/{v}_{T}\lt 0$ (Figure 13).

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It is well known from theory (Musher et al. 1995) that decay processes involve several cascades of energy transfer to waves with longer wavelengths. The successive occurrence of two decay cascades was discussed and presented above for the cases of very small or finite average levels of external density fluctuations, i.e., for Δn ≃ 0.001 and Δn ≃ 0.01 (see, e.g., Figure 10). However, one must stress that decay cascading in a plasma with background density fluctuations of finite Δn presents specific features, which we examine now in more detail on the basis of simulation results presented in Figures 12–14. Note first that according to the decay resonance conditions, only three cascades are expected to occur for the beam and plasma parameters considered here. During these successive processes, Langmuir wavenumbers decrease more and more, starting from the value ${k}_{L}{\lambda }_{{\rm{D}}}\simeq 0.11$ corresponding to the most excited beam-driven Langmuir waves (not shown here); then, as a consequence of the resonance conditions, the Langmuir wave products of the first, second, and third decay cascades are characterized by the following wavenumbers: ${k}_{{L}^{\prime }}{\lambda }_{{\rm{D}}}\;\simeq ({k}_{0}-{k}_{L}){\lambda }_{{\rm{D}}}\simeq -0.08,$ ${k}_{{L}^{\prime\prime }}{\lambda }_{{\rm{D}}}\simeq ({k}_{L}-2{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.05$, and ${k}_{{L}^{\prime \prime \prime }}{\lambda }_{{\rm{D}}}\simeq (3{k}_{0}-{k}_{L}){\lambda }_{{\rm{D}}}\simeq -0.02$, whereas for the IS daughter waves we have ${k}_{{S}^{\prime }}{\lambda }_{{\rm{D}}}\simeq (2{k}_{L}-{k}_{0}){\lambda }_{{\rm{D}}}\simeq 0.19,$ ${k}_{{S}^{\prime\prime }}{\lambda }_{{\rm{D}}}\;\simeq (-2{k}_{L}+3{k}_{0}){\lambda }_{{\rm{D}}}\simeq -0.13$,  and ${k}_{{S}^{\prime \prime \prime }}{\lambda }_{{\rm{D}}}\simeq (2{k}_{L}-5{k}_{0}){\lambda }_{{\rm{D}}}\;\simeq 0.07$, respectively. A fourth cascade is not possible as the resonance conditions are not fulfilled. An essential point lies in the fact that the Langmuir waves ${\mathcal{L}}^{\prime\prime}$ coming from the second cascade propagate in the direction of the beam drift and have larger phase velocities than those excited first by the beam instability. So the second decay cascade can play an important role in the acceleration of beam electrons above the initial beam velocity v_b.

The first two decay cascades are very clearly observed in Figures 13–14. For ${\omega }_{p}t\gtrsim 56000$ and $z\lesssim 6800{\lambda }_{{\rm{D}}}$, two wave packets propagate along the beam direction (${v}_{g}\gt 0$) and a third one in the opposite direction (${v}_{g}\lt 0$) (Figure 13, left). In the vicinity of $z\simeq 6900{\lambda }_{{\rm{D}}},$ the packet with ${v}_{g}\gt 0$ and with the largest amplitude collides near ${\omega }_{p}t\simeq 58000$ with the packet propagating with ${v}_{g}\lt 0$ (see also Figure 14 at ${\omega }_{p}t\simeq 58300$). IS waves are generated as a result of the beating between these two colliding packets (Figure 13, right); during this process, the weaker amplitude packet with ${v}_{g}\lt 0$ gets energy from the more intense one with ${v}_{g}\gt 0$. This mechanism could be considered as the first cascade of wave energy transformation, but the actual situation is more complicated. Indeed, almost simultaneously, the colliding packet with ${v}_{g}\gt 0$ reflects on the density hump presenting a maximum at $z\simeq 7000{\lambda }_{{\rm{D}}}$ (Figure 14 at ${\omega }_{p}t\simeq 58300$). The propagating second packet with ${v}_{g}\gt 0$ consequently collides with the amplified packet with ${v}_{g}\lt 0$ and with the reflected one (Figure 13, left; Figure 14 at ${\omega }_{p}t\simeq 59300$). Their beatings also generate IS waves. At ${\omega }_{p}t\gtrsim 60000$ two wave packets are traveling in the direction opposite to the beam propagation (Figure 14 at ${\omega }_{p}t\simeq 63000$). Later, near ${\omega }_{p}t\simeq 67000$ and $z\simeq 6000{\lambda }_{{\rm{D}}},$ a process starts which can be called the second decay cascade, involving one of these packets and generating IS waves propagating in the negative direction (at group velocity $-{c}_{s}/{v}_{T}$) with no signatures of any Langmuir wave reflection (Figure 13, left). At last, near (${\omega }_{p}t\simeq 73500$, $z\simeq 6800{\lambda }_{{\rm{D}}}$) and (${\omega }_{p}t\simeq 76000$, $z\simeq 7800{\lambda }_{{\rm{D}}}$), structures similar to a third decay can be observed on the space-time evolution pattern (Figure 13, left); one can indeed observe a small but noticeable enhancement of the IS wave emission along the direction of the beam propagation, near ${\omega }_{p}t\simeq 73500$ and $z\simeq 6800{\lambda }_{{\rm{D}}}$ (Figure 13, right), which is expected from a third cascade; however, as the wave packets have amplitudes that are too weak, we are not able to state that this process is actually and certainly a third cascade. Indeed, Figure 15 presents the corresponding local spectra of Langmuir and IS waves at the three time moments ${\omega }_{p}t=62000,$ 68000, and 74000; they show that if peaks at ${k}_{{L}^{\prime \prime \prime }}{\lambda }_{{\rm{D}}}\simeq -0.02$ (upper right panel) and ${k}_{{S}^{\prime \prime \prime }}{\lambda }_{{\rm{D}}}\simeq 0.07$ (lower right panel) are visible, they are either too weak (IS spectrum) or mixed with other effects (Langmuir spectrum). However, at ${\omega }_{p}t=62000$ and ${\omega }_{p}t=68000$, the position of some peaks in the IS and Langmuir spectra are very close to those predicted by the resonant wave decay theory in homogeneous plasmas for the first and the second cascades (see the discussion in the previous paragraph). As mentioned above, the spectrum of Langmuir waves for ${\omega }_{p}t=74000$ does not show clearly distinguishable peaks, which can be explained by two effects, i.e., the propagation of the waves in the inhomogeneous plasma and their exchanges of energy with the beam. At the same time, three main peaks can be observed in the IS spectrum: two correspond to the wavenumbers of the IS wave products of the two first decay cascades, i.e., ${k}_{{S}^{\prime }}{\lambda }_{{\rm{D}}}\simeq 0.19$ and ${k}_{{S}^{\prime\prime }}{\lambda }_{{\rm{D}}}\simeq -0.13$, whereas the third one near $k{\lambda }_{{\rm{D}}}\simeq -0.2$ is likely due to a nonresonant wave–wave interaction.

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Finally, it is important to stress at this stage that it is generally very difficult, if not impossible, to separate the processes of wave reflection on density fluctuations from that of parametric wave–wave interactions; both effects are usually working together in a inhomogeneous plasma, and the origin of the counterpropagating (i.e., with ${v}_{g}\lt 0$) Langmuir waves produced cannot be determined conclusively in most cases. Note also that reflections of Langmuir waves on long-wavelength density inhomogeneities can, in some conditions, favor the emergence of nonlinear decay processes, owing to the appearance of parametric interaction processes involving waves traveling in the direction opposite to the beam propagation. One must also take into account the significant role of the variations of the long-wavelength density fluctuations due to their own dynamics; for example, the local positive gradient of the density hump in Figure 14 disappears for ${\omega }_{p}t\gtrsim 63000$ and conditions for IS wave generation in the vicinity of this region become less favorable than when Langmuir energy can focus on the gradients of the humps near some reflection points.