Investigation of the Langdon effect on the nonlinear evolution of SRS from the early-stage inflation to the late-stage development of secondary instabilities

As shown in figures 2 and 3, SRS grows convectively with a low saturation level at the larger k_l λ_De, but grows quickly to a high early-stage saturation level at the smaller k_l λ_De. This issue can be predicted by the linear criterion ${\nu }_{l0}\leqslant 2{\gamma }_{0}\sqrt{{v}_{l}/{v}_{s}}$ for absolute SRS, where ν_l0 is the initial Landau damping. Since ν_l0 decreases rapidly with decreasing k_l λ_De, there exists one critical ${[{k}_{l}{\lambda }_{\text{De}}]}_{c}$ at which the equality of the criterion can be satisfied. For ${k}_{l}{\lambda }_{\text{De}} > {[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, the SRS growth is convective. The reflectivity can saturate at a lower level and nonlinear effects are quite weak, as presented in figures 4(c) and (d). In such cases, the reflectivity calculated by the Tang's formula (22) agrees well with the simulation results of the TWC model, as shown in figures 4(a) and (b). For ${k}_{l}{\lambda }_{\text{De}}< {[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, the SRS growth is absolute, implying that SRS can keep growing until saturated by nonlinear effects, as shown by the time evolution of reflectivity in figures 4(c) and (d), which is calculated by the TWC model and hence the only nonlinear effect is the pump depletion. Since nonlinear effects only become important at large reflectivity, the saturated reflectivity for absolute SRS growth can not be too small. Consequently, when ${[{G}_{\text{R}}]}_{\text{c}}$ at ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ is small and hence the convectively saturated level is low, there would be a sharp increase of reflectivity (especially, much sharper than the prediction by the Tang's model) at the transition from convective to absolute SRS, as exampled in figure 4(b). In comparison, the change of R near ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ is much more gradual in figure 4(a), since for ${[{G}_{\text{R}}]}_{\text{c}} > 15$ (cf table 1), the reflectivity due to convective amplification is already sufficiently large to incur nonlinear saturation effects.

Zoom In Zoom Out Reset image size

Table 1. Summary of critical parameters for the transition from convective to absolute SRS growth from the linear TWC model and from the Vlasov–Maxwell simulation. The homogeneous He plasma with n_e = 0.1 n_c , T_i = T_e /5 and length of  200λ₀  (λ₀ = 351 nm) is taken.

I15	m	Linear TWC			Vlasov–Maxwell
		${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$	$\frac{{[{\nu }_{l0}]}_{\text{c}}}{0.01{\omega }_{0}}$	${[{G}_{\text{R}}]}_{\text{c}}$	${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$	$\frac{{[{\nu }_{l0}]}_{\text{c}}}{0.01{\omega }_{0}}$	${[{G}_{\text{R}}]}_{\text{c}}$
2.82	2	0.278	0.208	20.5	0.42	2.58	1.68
	2.9	0.338	0.241	16.4	0.465	2.15	1.84
0.11	2	0.238	0.0358	4.99	0.269	0.154	1.12
	2.9	0.295	0.0429	3.92	0.316	0.112	1.45

In addition to decreasing k_l λ_De, increasing m also leads to a decrease of ν_l0, and thus can result in the transition from convective to absolute SRS in certain regions of k_l λ_De. Comparing the lines with asterisks in figures 4(c) and (d), for the same laser and plasma parameters, SRS is in the convective regime when m = 2 but can grow absolutely and eventually saturated by nonlinear effects when m = 2.9. Consequently, ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ becomes larger for a greater m, as listed in table 1, indicating that the Langdon effect can broaden the parameter range for absolute SRS instability as shown in figure 4. In table 1, it is found that the difference in ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ for different m results in close (slightly higher for increasing m) values of ${[{\nu }_{l0}]}_{\text{c}}$ in the linear prediction. This is because in equation (23), γ₀ and v_s are almost independent of k_l λ_De, while according to the matching condition of SRS, it can be proved that ${v}_{l}\approx 3{v}_{\text{the}}^{2}{k}_{l}/{\omega }_{l}$ increases moderately with larger ${[{k}_{l}{\lambda }_{\text{De}}]}_{c}$ at greater m, in contrast to the strong rise of ν_l0 with increasing k_l λ_De.

Considering that nonlinear kinetic effects can not be included by the TWC model, results from Vlasov–Maxwell simulations are shown in figure 5 for some cases. Due to the nonlinear kinetic effects, the convective SRS in the linear prediction can become absolute. One example is presented in figure 5(c), where the initial growth of SRS agrees well with the TWC model before t < 500T₀, after which the TWC model predicts convective saturation while the Vlasov–Maxwell simulation exhibits the continual growth of SRS toward much higher reflectivity. A detailed investigation shows that with the growth of the EPW field, resonant electrons with v_x ≈ v_phl are trapped, resulting in flattening of the EEDF around v_phl, as shown in figure 5(d). Correspondingly, the Landau damping is reduced [34, 35] while the EPW frequency is downshifted causing a frequency mismatch [36–38]. This in turn induces a frequency upshift of the scattered wave that tends to restore the frequency matching resonance, as shown in the bottom panel of figure 5(c) for the adjustment period  500T₀ < t < 2700T₀. The frequency mismatch induces a phase mismatch δ_mis and thus a reduction of growth rate by  cos δ_mis, impairing the SRS growth, while the nonlinear reduction of ν_l favors the SRS growth. In the case shown in figure 5(c), the competition between these two factors results in oscillation of the reflectivity during  1500T₀ < t < 2500T₀, wherein significant frequency mismatch exists in the plasma region further away from the left boundary (e.g. x/λ₀ = 140 in figure 5(c)). After a period of adjustment, ultimately at t > 2700T₀ over a large plasma region the frequency upshift of the scattered wave becomes sufficiently large to compensate the frequency downshift of the EPW, reducing the frequency mismatch to a low level. Consequently, the nonlinear absolute growth rate ${\gamma }_{\text{abs,NL}}\approx 2{\gamma }_{0}\sqrt{{v}_{l}/{v}_{s}}\,\cos \,{\delta }_{\text{mis}}-{\nu }_{\text{NL}}$ exceeds zero over a large plasma region, and SRS enters into the absolute growth period. Until when t > 3100T₀ the EPW amplitude in the plasma region near the left boundary (e.g. x/λ₀ = 60 in figure 5(c)) is so large that the frequency downshift of the EPW can not be completely compensated to maintain the SRS resonance, resulting in saturation of the absolute SRS growth.

Zoom In Zoom Out Reset image size

To illustrate the Langdon effect on the early-stage saturation of SRS in Vlasov–Maxwell simulation, we need to measure the early-stage saturated reflectivity R_sat,e. However, as shown in figures 2(b)–(f) and 3(b)–(f), for the absolute SRS, the reflectivity versus time after the saturation is irregular and nonstationary, making it necessary to clearly define R_sat,e in a reasonable way. Here we define R_sat,e as the averaged reflectivity over the time period from the onset of early-stage saturation  (t_sat) until the onset of significant secondary instabilities  (t_early). In practice, t_sat can be identified as the time when the growth of the reflectivity becomes flattened, and t_early can be defined as the time until when  90% of the EPW field energy is contained within  [0.98k_lc , 1.02k_lc ], as indicated in figures 5(a) and (b). For the convective SRS growth, on the other hand, the reflectivity would eventually become constant with time or in some cases oscillate in a regular way and hence have a constant mean value; correspondingly, it is natural to identify the saturated reflectivity R_sat as the steady (mean) value of the reflectivity.

In Vlasov–Maxwell simulation, R_sat,e versus k_l λ_De is shown in figures 6(a) and (b) for different m, whereas t_sat versus k_l λ_De is displayed in figures 6(c) and (d) for cases with absolute SRS growth. As a comparison, R_sat,e with an alternative definition of t_early, i.e. the time until when 90% of the EPW energy is contained within  [0.95k_lc , 1.05k_lc ], as well as the time-averaged reflectivity over the entire simulation time with t ⩾ t_sat, is also shown. As seen, R_sat,e is insensitive to the individual choice in the definition of t_early. In fact, for relatively large k_l λ_De, kinetic electron trapping plays a key role. In such cases, the nonlinear evolution in the early stage is predominately determined by (i) nonlinear reduction of Landau damping, (ii) nonlinear frequency downshift of EPW and the accompanied frequency upshift of the scattered wave, and (iii) pump depletion that becomes important when the reflectivity rises to a high level; while secondary instabilities that significantly broaden the k_l -spectrum of EPW and denote the end of the early stage, mainly consist of trapped particle instability and generation of beam acoustic modes (cf section 3.2). As shown in figures 2(a)–(f) and 3(a)–(e), broadening of the k_l -spectrum of EPW is sudden and abrupt. Consequently, t_early can be delineated with a great accuracy, yielding a robust R_sat,e. Things are different for the three low- k_l λ_De cases indicated by open squares in figures 6(b) and (d), where the early-stage saturation is predominately caused by pump depletion, after which broadly featured LDI begins to become important, resulting in broadening of the k_l -spectrum. As shown in figure 3(f), in these cases broadening of the k_l spectrum is much more gradual, making t_early prone to individual choices. However, R_sat,e remains insensitive to individual choices of t_early since R_sat,e is mainly contributed by the first reflectivity peak with large width and high amplitude, which is mainly saturated by the pump depletion and hence always contained within t < t_early.

Zoom In Zoom Out Reset image size

In figures 6(a) and (b), with decreasing k_l λ_De, an abrupt rise in R_sat,e appears at one critical ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, where the transition from convective to absolute SRS growth occurs. ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ as obtained from the Vlasov–Maxwell simulations, as well as the initial Landau damping ν_l0 and the convective gain G_R corresponding to ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, is listed in table 1 for both m = 2 and m = 2.9 at I₁₅ = 2.82 and I₁₅ = 0.11. It can be seen that due to nonlinear kinetic effects, ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ can far exceed that from the TWC model, leading to much smaller ${[{G}_{\text{R}}]}_{\text{c}}$ and hence a sharp rise in the reflectivity at the transition. Nevertheless, even in the presence of kinetic effects, ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ is larger for a greater m. Also the values of ${[{\nu }_{l0}]}_{\text{c}}$ corresponding to ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$ are quite close for different m, as in the prediction of the TWC model. However, contrary to the TWC model, in the Vlasov–Maxwell simulation ${[{\nu }_{l0}]}_{\text{c}}$ is lower for larger m. To comprehend this point, notice that after a proper account of the nonlinear modification to ν_l and δω_l by kinetic effects, equations (16)–(18) can still be used to describe the early-stage evolution of SRS. Various approximations for the nonlinear evolution of ν_l and δω_l have been proposed in the literature, of which one relatively simple bounce-averaged model is given in references [35, 39] as

$$\begin{equation}{\nu }_{l}=\frac{{\nu }_{l0}}{1+\frac{3{\pi }^{2}}{128}{\int }_{0}^{t}{\omega }_{B}(t)\mathrm{d}t}\end{equation} \tag{ 34 }$$

$$\begin{equation}\delta {\omega }_{l}={\left.1.09{\omega }_{B}\frac{{\omega }_{\text{pe}}^{2}}{{k}_{l}^{3}(\partial {\mathcal{D}}_{r}/\partial {\omega }_{l})}\frac{{\partial }^{2}({f}_{e0}/{n}_{e0})}{\partial {v}_{x}^{2}}\right\vert }_{{v}_{\text{phl}}}\end{equation} \tag{ 35 }$$

where the bouncing frequency ${\omega }_{B}=\sqrt{e{E}_{l}{k}_{l}/{m}_{e}}$. In this modified TWC model, when other parameters such as γ₀, v_l , v_s and v₀ are kept nearly the same, the initial Landau damping ν_l0 and δω_l /ω_B that depicts the strength of nonlinear frequency shift, determine the nonlinear evolution of SRS, and hence whether the absolute SRS growth can occur or not. Since the increase of either the Landau damping or the frequency shift would impair the SRS growth, it can be expected that with increasing δω_l /ω_B , a lower ν_l0 is required for the onset of absolute SRS growth. This is indeed the effect of increasing m, which leads to greater δω_l /ω_B at the same ν_l0, as elucidated in figure 7. Consequently, ${[{\nu }_{l0}]}_{c}$ must be smaller for increasing m to overcome the effect of stronger nonlinear frequency shift at greater m. It can be further understood that the different variation of δω_l and ν_l0 with m ultimately results from δω_l ∝ ∂² f_e0/∂² v_x , in contrast to ν_l0 ∝ ∂f_e0/∂v_x [7]. This general understanding should hold even though the simplified model specified by equations (34) and (35) is not precise, indicating that apart from the Landau damping, the influence of super-Gaussian EEDFs on the nonlinear frequency shift is also an important factor to affect the early-stage development of SRS.

Zoom In Zoom Out Reset image size

Below ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, R_sat,e is quite insensitive to k_l λ_De, except for an intermediate range of k_l λ_De ∼ 0.33–0.4 for m = 2.9 and k_l λ_De ∼ 0.28–0.35 for m = 2, where R_sat,e increases with decreasing k_l λ_De as shown in figure 6(a). For the high k_l λ_De range ( k_l λ_De > 0.4 for m = 2.9 and k_l λ_De > 0.35 for m = 2), the phase velocity of the EPW is located in the bulk region of the EEDF, leading to strong Landau damping and also strong electron-trapping induced nonlinearity (e.g., ν_l0/γ₀ ≫ 1 and δω_l /γ₀ ≫ 1). Despite the difference in the initial Landau damping, after the adjustment period, the absolute SRS growth become similar for different k_l λ_De and m, until when the reflectivity reaches the level $\sim 0.02$, where secondary instabilities become important and the early stage ends, as shown in figure 5(a). As a result, the dependence of R_sat,e on both k_l λ_De and m is much weaken. In the intermediate range of k_l λ_De with weaker Landau damping and kinetic nonlinearity, the Landau damping and the kinetic nonlinear shift are comparable to γ₀ (e.g., ν_l0 ∼ 0.5–2.9 and δω_l /γ₀ ∼ 0.3–1.3 for k_l λ_De ∼ 0.28–0.35 at m = 2, I₁₅ = 2.82 and δn_e /n_e0 = 0.02). Thus, the nonlinear adjustment is insufficient to smear the effects of decreasing initial Landau damping and nonlinear frequency shift when k_l λ_De decreases or m increases. Consequently, R_sat,e increases with decreasing k_l λ_De or increasing m. For the low k_l λ_De range (k_l λ_De < 0.33 for m = 2.9 and k_l λ_De < 0.28 for m = 2) with even smaller initial Landau damping, after the adjustment period, the Landau damping is nearly negligible. The absolute growth rate ${\gamma }_{\text{abs,NL}}\lesssim \mathrm{max}[{\gamma }_{\text{abs}}]\equiv 2{\gamma }_{0}\sqrt{{v}_{l}/{v}_{s}}$, and also the evolution of SRS towards the early-stage saturation, becomes quite similar for different k_l λ_De and m, as shown in figure 5(b). As a result, R_sat,e again becomes nearly independent of k_l λ_De and m.

In figures 6(c) and (d), it can be seen that the saturation time t_sat is quite large near ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$. As shown in figure 5(c), here an oscillating plateau of the reflectivity is formed before the absolute growth period, significantly lengthening the adjustment period. This is because the counteracting effects of nonlinear Landau damping reduction and the frequency shift induced phase mismatch, nearly balance during the adjustment period. A slight decrease of k_l λ_De or increase of m, weakens both the Landau damping and the frequency shift, thus breaks the balance and significantly reduces the adjustment time, leading to a sharp drop of t_sat with decreasing k_l λ_De or increasing m near ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, as shown in figures 6(c) and (d). This decreasing trend of t_sat holds until for k_l λ_De far below ${[{k}_{l}{\lambda }_{\text{De}}]}_{\text{c}}$, where t_sat tends to a value nearly independent of k_l λ_De and m. Here, the SRS growth is absolute even in the absence of nonlinear kinetic effects, the adjustment period is negligible, and t_sat is primarily contributed by the absolute growth period. The absolute growth rate ${\gamma }_{\text{abs}}\approx 2{\gamma }_{0}\sqrt{{v}_{l}/{v}_{s}}$ is nearly independent of k_l λ_De and m, and hence so is t_sat.

Now we examine the impacts of the Langdon effect on the nonlinear saturation of SRS in the late stage when secondary instabilities are important. Since the nonlinear behavior and dominant saturation mechanism in the late stage are quite different between the high k_l λ_De regime  (0.25 ≲ k_l λ_De ≲ 0.45) which is investigated under a high intensity drive  (I₁₅ = 2.82), and the low k_l λ_De regime  (0.18 ≲ k_l λ_De ≲ 0.3) that is studied under a low intensity drive  (I₁₅ = 0.11), in the following we discuss them separately.

For I₁₅ = 2.82 and T_e = 3.2 keV as shown in figures 2(c) and (d) for m = 2 and m = 2.9 respectively, a burst behavior of the reflectivity is exhibited. In the active phase, the trapped particle induced nonlinearity ultimately results in a chaotic state, wherein a nearly continuous k_l -spectrum of the EPW field spreading from k_lc ≈ 1.5ω₀/c to about ω₀/c is generated. As can be seen, the downward broadening of the wavenumber is quite abrupt. It is caused by vortex-merging processes in the phase space, which begins with pairwise merging of phase-space 'holes' or vortices near the phase velocity of EPW [40, 41], as shown in figure 8 near t ≈ 1500T₀ for m = 2 and t ≈ 920T₀ for m = 2.9, respectively. This in turn gives rise to broadband incoherent EPW field consisting of beam acoustic modes [16–18]. The decay of the resonant (and usually downshifted) EPW into BAMs breaks the three wave resonance condition of SRS, and thus serves as an efficient saturation mechanism for SRS. The corresponding typical k_l –ω_l spectrum is shown in figure 9(b), where the BAM feature is obvious. Note that the adjusted resonant point at  (ω_l , k_l ) ≈ (0.37ω₀/c, 1.50ω₀/c), which corresponds to the intersection of BAMs with the Stokes curve, is downshifted relative to the linear resonance mode (ω_lc , k_lc ) = (0.39ω₀, 1.47ω₀/c). Correspondingly, the backscattered feature in the transverse electric field at  (ω_s , k_s ) ≈ (0.63ω₀, − 0.55ω₀/c), as shown in figure 9(a), is upshifted relative to the linear resonance mode  (ω_sc , k_sc ) ≈ (0.61ω₀, −0.52ω₀/c). For m = 2 where the bursts are well separated, the temporal-spatial evolution of the EPW, the scattered wave and the pump wave is shown in figures 10(a)–(c). It can be seen that BAMs are developed between  1400T₀ and  1800T₀, while their convection and damping lead to oscillation of the reflectivity during this period. Then at 2000T₀, a strong peak occurs. Accordingly, the laser pump is depleted, while the decay to BAMs is significantly enhanced by the strong EPW field. As a result, the reflectivity quickly drops to near zero at t ∼ 2200T₀. Then, the pump intensity is restored. However, the BAM packet needs a much longer time  (∼L/v_l ) to damp or convect through the plasma [42]. Inside the packet, the rapid decay to BAMs caused by the large EPW field keeps the (upshifted) backscattered light at a low level. As this scattered light moves outside the packet into the unperturbed plasma region behind the packet, it is off-resonance with  Δω ≈ δω_l . This limits its further amplification, and also produces a beat pattern separated at τ = 2π/Δω [43]; correspondingly, many high frequency minor bursts appears during the quiescent period between 2500T₀ and  3000T₀. When the packet has almost convected out of the plasma  (t ∼ 3200T₀), a new major burst coming from new SRS growth in the plasma, now almost clear of the incoherent BAMs, occurs. Again, the pump depletion that takes effect instantaneously when the reflectivity is large, and the generation of BAMs whose effects last a long time, begin to suppress the reflectivity. The continual suppression and recovery of SRS lead to a sequence of major bursts with period about 1300T₀, intervened by many minor bursts. For m = 2.9, the decay to BAMs and the pump depletion play similar roles and act as the predominant saturation mechanism for t_early < t < 4000T₀. Nevertheless, as shown in figures 10(d) and (f), due to the smaller Landau damping and hence the greater growth rate and gain of SRS at larger m, significant SRS re-growth can occur behind the packet even when less than half the plasma is clear of the incoherent BAM field, permitting several packets to coexist and interact. For example, at t ∼ 2000T₀, new packet II is formed near the left boundary when the previous packet I has just convected half through the plasma. The bursts of the reflectivity at t ∼ 2200T₀ are generated deep inside packet I at x ≈ 150λ₀, and further amplified across packet II on its path to the left boundary, similar to the high gain case in [43]. So, the bursts become overlapped, while the significant interaction between bursts leads to a less regular burst behavior. Consequently, compared to m = 2, the quiescent period is significantly reduced, and the average reflectivity is enhanced. Besides, over a long timescale t > 4000T₀, in addition to decay to BAMs, rescattering also become important, featured by the remarkable k_l -features near −0.54ω₀/c [53]. The typical ω_s –k_s spectrum of the scattered wave, and the corresponding ω_l –k_l spectrum of the EPW, are shown in figures 9(c) and (d), respectively, where the feature with  (ω_l , k_l ) ≈ (0.32ω₀, −0.54ω₀/c) and  (ω_s , k_s ) ≈ (0.3ω₀, 0) is due to rescattering of the primary upshifted backward scattered wave with  (ω_s , k_s ) ≈ (0.64ω₀, −0.54ω₀/c). Consequently, the SRS becomes more chaotic, leading to more irregular variation of the reflectivity.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For T_e = 2.8 keV and I₁₅ = 2.82 as shown in figures 2(e) and (f), the evolution of SRS after t > t_early can be further divided into three periods: I. In the initial period with t < 4000T₀, the generation of BAMs plus the pump depletion remains the dominant saturation mechanism. The SRS reflectivity consists of overlapped bursts for both m = 2 and m = 2.9, and though less apparent, still the quiescent period is shorter for m = 2.9. II. In the intermediate period with  4000T₀ < t < 6000T₀ for m = 2 and  4000T₀ < t < 7000T₀ for m = 2.9, rescattering aids in limiting the SRS level. III. In the final period with t > 6500T₀ for m = 2 and t > 8500T₀ for m = 2.9, the decay to low- k Langmuir branch (including modes due to rescattering and forward SRS) is significantly enhanced, leading to the k_l -spectrum continuously ranging from k_lc down to zero. As shown in figure 9(f) for the typical ω_l –k_l spectrum of the EPW, both BAMs and the low- k Langmuir branch with features due to rescattering and forward SRS contained are nearly fully occupied. Correspondingly, the ω_s –k_s spectrum of the scattered wave shown in figure 9(e) exhibits features of rescattering with (ω_s , k_s ) around (0.3ω₀, 0) and forward SRS with (ω_s , k_s ) around (0.68ω₀/c, 0.61ω₀/c). Such fully developed incoherence results in the drop of SRS reflectivity in the final period, as demonstrated in figures 2(e) and (f).

The variation of the late-stage saturation mechanism with T_e , together with the corresponding average reflectivity, is summarized in figure 11(a) for m = 2 and m = 2.9 at I₁₅ = 2.82. For all cases, initially the dominant saturation mechanism is decay to BAMs plus the pump depletion. In this period, the lower Landau damping and hence greater gain for increasing m results in shorter quiescent period and hence greater average reflectivity. With decreasing T_e or increasing m, rescattering and decay to low-k_l Langmuir branch can become important in the later period, usually leading to a more chaotic state with reduced average reflectivity. Consequently, the dependence of the averaged reflectivity on m becomes more uncertain in the later period, and the average reflectivity can be smaller for increasing m in some cases.

Zoom In Zoom Out Reset image size

For T_e = 1.5 keV and I₁₅ = 0.11 as shown in figures 3(c) and (d), in the initial period after t > t_early (t < 12000T₀ for m = 2 and t < 7000T₀ for m = 2.9), the dominant secondary process is trapped particle instability (TPI) [44–46], featured by the appearance of two sidelobes with k_l = k_lc ± Δk_TPI around the primary component k_lc . The sidelobe frequency is resonant with the electron bouncing frequency in the wave frame, thus satisfying Δω_TPI − Δk_TPI v_phl = ±ω_B [45], where Δω_TPI ≡ ω_TPI − ω_lc ,  Δk_TPI = k_TPI − k_lc , and v_phl = ω_lc /k_lc is the EPW phase velocity. As shown in figure 12(a) for the typical ω_l –k_l spectrum of the EPW when TPI dominates, the most significant mode occurs along the dispersion relation of EPW, thus Δω_TPI/Δk_TPI = v_l , giving Δk_TPI = ±ω_B /(v_phl − v_l ). For T_e = 1.5 keV, using  2πeE_l /m_e ω₀ c ≈ 0.01 estimated from the simulation data, it can be obtained  Δk_TPI ≈ ±0.26ω₀/c, consistent with the sidelobe locations on the k_l -spectrum as shown in figures 3(c) and (d).

Zoom In Zoom Out Reset image size

As time evolves, Langmuir decay instability [21, 23], where a primary Langmuir wave (LW) decays into a secondary Langmuir wave and an ion acoustic wave (IAW), begins to become important. LDI satisfies the matching condition

$$\begin{equation}\begin{aligned}\hfill {\omega }_{l,i}& ={\omega }_{l,i+1}+{\omega }_{a}\hfill \\ \hfill {k}_{l,i}& ={k}_{l,i+1}+{k}_{a}\hfill \end{aligned}\end{equation} \tag{ 36 }$$

where i denotes the stage number of the Langmuir cascade with i = 0 corresponding to the primary LW, and ω_a and k_a are the frequency and wave number of the IAW, respectively. The dispersion relation for the LW can be approximated by ${\omega }_{l,i}^{2}={\omega }_{\text{pe}}^{2}+3{k}_{l,i}^{2}{v}_{\text{the}}^{2}$, while the dispersion relation for the IAW is approximately ω_a = |k_a |c_s , where ${c}_{s}\approx \sqrt{Z{T}_{e}/M}$ is the acoustic velocity with Z and M being the charge and mass of the ion species. Substituting these dispersion relations into equation (36) yields k_l,i+1 ≈ −k_l,i + Δk_LDI and k_a ≈ 2k_l,i , where the wavenumber difference between two successive cascade step is

$$\begin{equation}{\Delta}{k}_{\text{LDI}}\approx \frac{2{c}_{s}{\omega }_{l}}{3{v}_{\text{the}}^{2}}.\end{equation} \tag{ 37 }$$

The LDI threshold can be estimated by [47]

$$\begin{equation}\frac{{{\epsilon}}_{0}{E}_{\text{l,LDI}}^{2}}{{n}_{0}{T}_{e}}=16\frac{{\nu }_{e}}{{\omega }_{e}}\frac{{\nu }_{a}}{{\omega }_{a}}\end{equation} \tag{ 38 }$$

where ν_a /ω_a ≈ 0.099 for T_e /T_i = 5 in He plasma considered here. For m = 2 and T_e = 1.5 keV, when t ≈ 10000T₀, E_l /E_l,LDI ≈ 1.14, LDI with one cascade step is excited and results in the reduction of reflectivity in the later period, as seen from figure 3(c). The corresponding ω_l –k_l spectrum of the electrostatic field is shown in figure 12(b), where the features at k_l ≈ −1.51ω₀/c + 0.07ω₀/c due to the secondary LW and at k_l ≈ 3ω₀/c due to IAW are obvious. For m = 2.9, due to decreasing ν_l with increasing m, E_l,LDI is reduced while E_l is generally greater, leading to E_l /E_l,LDI ≈ 15 at t ≈ 7000T₀. Consequently, at least five LDI cascade steps are apparent from figures 3(d) and 12(c) for t > 10000T₀, while the sidelobes of TPI become insignificant. This indicates that LDI cascade becomes the dominant saturation mechanism, limiting the reflectivity to a very low level about  5% for t > 15000T₀.

For T_e = 1.2 keV and m = 2 as shown in figure 3(e), in the initial period t < 8000T₀, TPI plus the pump depletion is still the dominant saturation mechanism. However, when t ∼ 8000T₀, E_l/E_l,LDI ≈ 4, so LDI cascade can be excited and the reflectivity in the later period is significantly reduced, as shown in figures 3(e) and 12(d). For T_e = 1.2 keV and m = 2.9 as shown in figure 3(f), ν_l /ω_l ∼ 10⁻⁶ is quite small. As a result, SRS is strongly driven and grows rapidly at the early time, leading to E_l/E_l,thr ≈ 100 and δn_e /n_e ≈ 0.06 at t = 5000T₀. Broad-featured LDI is developed, as shown in figure 12(e) for  6000T₀ < t < 8000T₀. As more cascade steps are excited, the k_l -spectrum is gradually broadened (in contrast to the sudden broadening of the k_l -spectrum in other cases). When t ∼ 10000T₀, multiple cascade steps with broad spectral features have been excited, as shown in figure 12(f). In this strongly-driven regime, SRS is quite turbulent, and the instantaneous reflectivity varies over a wide range. This leads to a greater average reflectivity in the later period compared to m = 2.

The variation of the late-stage saturation mechanism with T_e , together with the corresponding average reflectivity, is summarized in figure 11(b) for m = 2 and m = 2.9 at I₁₅ = 0.11. Except for the three strongly driven cases with low T_e (T_e = 0.8 keV and m = 2, T_e = 0.8 keV and m = 2.9, T_e = 1.2 keV and m = 2.9), initially the dominant mechanism is TPI plus the pump depletion, while LDI and LDI cascade can develop over time for some T_e and m, significantly reducing the reflectivity. Increasing m is favorable for excitation of LDI or LDI cascade since the LDI threshold is reduced while the EPW amplitude is typically greater before the onset of LDI due to the lower Landau damping. This can in turn result in a much stronger drop of the reflectivity in the later period than m = 2, thus the reflectivity at m = 2.9 in the later period can be smaller than m = 2.