Multi-dimensional Vlasov simulations on trapping-induced sidebands of Langmuir waves

In common electron-ion plasmas, some collective modes of interaction or eigenmodes can be found. And these waves are related to the charge density fluctuations. A low frequency one called ion acoustic wave corresponds to the characteristic frequency determined by ions and a high frequency one called Langmuir wavekruer ; Nicholson corresponds to the characteristic frequency determined by electrons. In inertial confinement fusion (ICF),ICF1 ; I_D ; Tik an incident pump laser can easily decay to a reflected electromagnetic wave and a Langmuir wave, which is the so-called stimulated Raman scattering (SRS).chen ; chen1 ; Brunner1 ; xiao1 ; xiao2 ; xiao3 ; xiao4 ; xiao5 Because this instability consumes pump energy and produces hot electrons, which are detrimental to fusion, it is crucial to find the saturation mechanism of SRS.Estabrook ; Vu ; yin ; Lin

At $1960s$, Kruer $et$ $al.$ revealed a trapped-particle instability that Langmuir wave can decay to sidebands,KDS and it can serve as a saturation mechanism of SRS.Brunner1 The so-called Kruer, Dawson, and Sudan (KDS) model achieves great success in explaining the longitudinal sidebands in one dimension.Brunner2 ; Krasovsky Recently, Friou et al. studied sidebands of Langmuir waves as a saturation of SRS by using Vlasov and particle in cell (PIC) code in one dimensional geometry,Friou the results from different codes agree with each other and they found that the amplitude of growth rates of longitudinal sidebands have scaling law $\gamma\sim g_{1}\phi^{g_{2}}$, where $g_{2}\sim 0.6-0.9$ for different wavenumber of Langmuir waves. Also, another theoretical model given by Dewar, Kruer and Manheimer is used to study the sideband instability.Dewar This model is based on the nonlinear Schrödinger equation and shows that the sidebands are originated from the nonlinear frequency shift by trapped electrons.Berger1 Dewar’s model was also be extended to two dimensions (so called Dewar-Rose-Yin model, $i.e.$ DRY model) to analyze the filamentation of Langmuir wave.rose1 ; rose2 ; winjum

Recently, the transverse decay channels of plasma waves get many attentions.Berger2 ; Denis1 ; Denis2 ; Chapman Berger $et$ $al.$ used both generalization of KDS model (GKDS) and DRY model to study the transverse sidebands of Langmuir wave.Berger2 The growth rates of transverse sidebands of Langmuir wave are obtained from these two models and reasonably agreed with their single-wavelength Vlasov simulations. They found that transverse sidebands can lead to the filamentation of Langmuir wave. Also, D. A. Silantyev $et$ $al.$ studied the transverse sideband of Langmuir waves.Denis1 ; Denis2 They found the scaling of the maximum growth rate is $\gamma_{max}\propto\sqrt{\phi}$ by using Vlasov-Possion simulations, where $\phi$ is the electrostatic potential of Langmuir waves. Similarly, Chapman $et$ $al.$ observed that the ion acoustic waves (IAW) have the longitudinal decay channel, two-ion decay (TID), and the transverse decay channel, off-axis instability (OAI). Although they failed to explain OAI by existing theories, they conjectured the OAI is an ion-driven trapped particle instability.Chapman We should note that the nature of TID instability is different from sideband instability. These works indicate that trapped particles can generate both longitudinal and transverse instabilities. Inspired by the transverse modulation of plasma waves, we believe that sidebands of Langmuir wave is a muti-dimensional instability, and the oblique sidebands whose wavevector has an oblique angle to the wavevector of Langmuir wave should be considered. In Berger’s Vlasov simulations, they only considered one longitudinal wavelength of Langmuir wave, thus they cannot observe the longitudinal and oblique sidebands.Berger2 Now, we fill this gap by using multi-wavelength Vlasov simulations and DRY model to study the sidebands of Langmuir waves in multi-dimensions .

In this paper, first, we use two-dimensional Vlasov-Poisson simulations to observe the nonlinear evolution of sidebands of Langmuir waves, including longitudinal sidebands, oblique sidebands and transverse bands. We should note that the oblique sidebands are just the trapped particle instability in a new direction. We find that oblique sidebands are important decay modes of Langmuir waves, which is similar to longitudinal sidebands and transverse sidebands. When the amplitudes of sidebands are comparable with that of Langmuir wave, sidebands saturate through a violent process featured as vortex merging in the phase space. This phenomenon is firstly discovered by Brunner $et$ $at$.Brunner2 Recently, Yang $et$ $al.$ also studied vortex merging by Vlasov simulations.yang The $k_{x}$, $k_{y}$ component of modes then broaden rapidly. Finally, the system is developed into a turbulent state lasting from $t\omega_{pe}=880$ to the end of simulations. Second, we extent the DRY model to two dimension and obtain the growth rates of sidebands. The Landau damping of Langmuir waves in a non-Maxwellian distribution is considered. In simulations, we obtain the scaling law for longitudinal sidebands is $\gamma\sim\phi^{0.75}$, and that for oblique sideband with $k_{y}=0.03125$ is $\gamma\sim\phi^{0.85}$. The $g_{2}$ for longitudinal sidebands in our paper agree with the results in Friou’s work,Friou $g_{2}\sim 0.6-0.9$. In our theoretical results, the scaling law for longitudinal sidebands is $\gamma\sim\phi^{0.5}$, and that for oblique sideband with $k_{y}=0.03$ is $\gamma\sim\phi^{0.66}$. Our theoretical model can describe the longitudinal sidebands, oblique sidebands, and transverse sidebands. The growth rates of sidebands qualitatively agree with our Vlasov simulation results.

This paper is structured in the following ways. Firstly, in Sec. II, we use Vlasov simulations to study the nonlinear evolution of the sidebands of Langmuir waves in two dimensions. Secondly, DRY model are performed to study the growth rate of sidebands in Sec. III. At last, the conclusion and discussion are shown in Sec. IV.

The Dewar-Rose-Yin (DRY) model was first used to study the 1D modulation instability of Langmuir wave and then extended to study 2D modulation and filamentation instabilities.Dewar ; rose1 ; rose2 ; Berger2 Here we reuse this model to predict the sidebands of Langmuir wave. In 2D situation, the modulation instability can be described by nonlinear Schrödinger equation,

$$\begin{split}i&\left(\frac{\partial}{\partial t}+\frac{\partial\omega_{k}}{\partial k_{x}}\frac{\partial}{\partial x}+\upsilon\right)\phi_{k}\\ &+\left(\frac{1}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{y}^{2}}\frac{\partial^{2}}{\partial y^{2}}+\frac{1}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{x}^{2}}\frac{\partial^{2}}{\partial x^{2}}+\Delta\omega\right)\phi_{k}=0,\end{split}$$

(3)

where $\phi_{k}(x,y,t)$ is the envelope of potential, $\nu$ is the Landau damping, and $\omega_{k}$ is the root of the dispersion relation $\epsilon(\vec{k},\omega)=0$. The fluid type of dispersion relation is used,

$$\epsilon(\vec{k},\omega)=\epsilon(k,\omega)=1-\frac{\omega_{pe}^{2}}{\omega^{2}-3k^{2}v_{te}^{2}},$$

(4)

where $k=|\vec{k}|=\sqrt{k_{x}^{2}+k_{y}^{2}}$ is the wavenumber of sidebands. The nonlinear term causing instability is the trapped-particle induced nonlinear frequency shift, $\Delta\omega$. To express it analytically, we use the adiabatic driven formula, Berger1

$$\frac{\Delta\omega}{\omega_{pe}}=-\frac{\alpha}{\sqrt{2\pi}(k_{L}\lambda_{De})^{2}}\sqrt{\frac{eE_{max}}{T_{e}k_{L}}}(v^{2}-1)exp(-\frac{v^{2}}{2})|_{v=v_{\phi}/v_{te}}.$$

(5)

where $\alpha=0.544$, $v_{\phi}$ is the phase velocity of Langmuir wave and $E_{max}=0.2m_{e}\omega_{pe}v_{te}/e$ is the electron field amplitude of the Langmuir wave, which corresponds to a potential of $|e\phi/T_{e}|=0.6$.

In order to find instability growth rate of Eq. 3, we assume two small sidebands $\omega_{k}=\omega_{L}\pm\Omega$, $k_{x}=k_{L}\pm\Delta k_{x}$ and $k_{y}=k_{Ly}\pm\Delta k_{y}$ where $\omega_{L}$, $k_{L}$ are linear frequency and wavenumber of Langmuir wave, and in our consideration, $k_{Ly}=0$. Let the Landau damping $\nu$ be absorbed by $\Omega$. Assuming a trial solution $\phi(x,y,t)=1/2(\phi_{k}(x,y,t)exp(i\vec{k}\cdot\vec{x}-i\omega_{k}t)+c.c.)$ for Eq. 3, we can obtain the dispersion relation,

$$\begin{split}&\left(\Omega-\Delta k_{x}\frac{\partial\omega_{k}}{\partial k_{x}}\right)^{2}=\left(\frac{\Delta k_{x}^{2}}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{x}^{2}}+\frac{\Delta k_{y}^{2}}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{y}^{2}}\right)\\ &\left(\frac{\Delta k_{x}^{2}}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{x}^{2}}+\frac{\Delta k_{y}^{2}}{2}\frac{\partial^{2}\omega_{k}}{\partial k_{y}^{2}}+\frac{\Delta\omega}{2}\right),\end{split}$$

(6)

where $\Delta k_{x}=k_{x}-k_{L}$, $\Delta k_{y}=k_{y}$. The growth rate of sideband is

$$\gamma(k_{x},k_{y})=Im(\Omega)+\upsilon.$$

(7)

The imaginary part of $\Omega$ exists only when $(\Delta k_{x}^{2}\frac{\partial^{2}\omega_{k}}{\partial k_{x}^{2}}+\Delta k_{y}^{2}\frac{\partial^{2}\omega_{k}}{\partial k_{y}^{2}})\Delta\omega<0$. Since $\frac{\partial^{2}\omega_{k}}{\partial k_{x}^{2}}=\frac{\partial^{2}\omega_{k}}{\partial k_{y}^{2}}\approx 3v_{te}^{2}/\omega_{L}$, thus modulation instability exists only when $\Delta\omega<0$. The maximum value of $Im(\Omega)$ and its location are easily solved from Eq. (6),

	$\displaystyle(Im(\Omega))_{m}=\frac{\Delta\omega}{4},$		(8)
	$\displaystyle(k_{x}-k_{L})^{2}+k_{y}^{2}=\frac{\Delta\omega\omega_{L}}{6v_{te}^{2}}.$		(9)

Since $\Delta\omega$ is independent of $k$, if no damping is considered $\nu=0$, a constant maximum growth rate is located along a circle of the initial Langmuir wavenumber, $(k_{L},0)$. This is, of course, not consistent with our simulation results, so we must consider the distribution of Landau damping, $\upsilon=\upsilon(k_{x},k_{y})$.

To do so, we must calculate the Landau damping in a trapping distribution function, such as the one presented at $t=200\omega_{pe}^{-1}$ in our simulation. Following the treatment of early works,Berger2 ; Divol a two-dimensional trapping distribution function is artificially constructed,

$$\begin{split}&f_{0}(v_{x},v_{y})=N_{e}f_{0,x}(v_{x})f_{0,y}(v_{y}),\\ &f_{0,y}(v_{y})=f_{M}(v_{y}),\\ &f_{0,x}(v_{x})=f_{M}(v_{x})+\delta f_{x}^{flat}(u),\\ &\delta f_{x}^{flat}(u)=P(u)\frac{1}{\sqrt{2}v_{te}}exp(-u^{2}/2),\\ &P(u)=\beta u+\gamma(u^{2}-1),\\ &u=(v_{x}-v_{\phi})/\delta v,\end{split}$$

(10)

where $f_{M}(v_{x})$ and $f_{M}(v_{y})$ are one-dimensional Maxwellian distributions. $\delta v=2\sqrt{e\phi/T_{e}}$ is the width of the plateau around the phase velocity. $\beta$ and $\gamma$ are the first and second derivatives of $f_{0,x}(v_{x})$ at phase velocity,

$$\begin{split}&\beta=(\delta\tilde{v})\tilde{v}exp(-\tilde{v}^{2}/2)|_{\tilde{v}=v_{\phi}/v_{te}},\\ &\gamma=\frac{\delta\tilde{v}}{3}(1-\tilde{v}^{2})exp(-\tilde{v}^{2}/2)|_{\tilde{v}=v_{\phi}/v_{te}},\\ \end{split}$$

(11)

where $\delta\tilde{v}=\delta v/v_{te}$. The constructed distribution function resembles Fig. 3(a), which enables the zero Landau damping near the phase velocity $v_{\phi}=\omega_{L}/k_{L}$. The kinetic dispersion function under the non-Maxwellian distribution function is

$$\epsilon(\omega,k)=1-\frac{\omega_{pe}^{2}}{k^{2}}P\int^{\infty}_{-\infty}\frac{\nabla_{v}f_{0}\cdot d\textbf{v}}{v-\omega/k}-i\pi\frac{\omega_{pe}^{2}}{k^{2}}|\nabla_{v}f_{0}|_{v=\frac{\omega}{k}},$$

(12)

where $P$ means the Cauchy principal value, $\textbf{v}=(v_{x},v_{y})$, and $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$.

Since the distribution function is almost flattened near the Langmuir wave phase velocity but decreases dramatically when $k_{y}$ increases, we assume the Landau damping varies with $k_{y}$ but is constant along $\Delta k_{x}$. Then we can numerically solve the dispersion function Eq. (12) along $k_{x}=k_{L}$ and obtain the Landau damping as a function of $k_{y}$ in Fig. 6. When $|e\phi/T_{e}|=0$, it backs to the Maxwellian distribution function, so the Landau damping is $-0.026\omega_{pe}$ as shown in Fig. 6(a). Fig. 6(b) shows the Landau damping at $k_{y}=0$ when $e\phi/T_{e}=0.6$, and the evaluated Landau damping being $0$ satisfies our designed distribution function. As $k_{y}$ increases to $0.12$ in Fig. 6(c), the phase velocity of sidebands becomes away from the trapped plateau, so the Landau damping increases to $-0.0296\omega_{pe}$. Fig. 6(d) shows the overall Landau damping increases with $k_{y}$. And the Landau damping will decrease with the potential, because when $|e\phi/T_{e}|$ increases from $0.6$ to $1.2$, the width of trapped region becomes larger in accordance with our expectation. We should note that the distribution in Eqs. 10 is artifical, thus, $v_{\phi}$ in Eqs. 10 and Eqs. 11 is adjustable to make sure the Landau damping equals to $0$, when$|e\phi/T_{e}|\neq 0$ and $k_{y}=0$.

Fig. 7 shows the final theoretical growth rates of sidebands obtained when $|e\phi/T_{e}|=0.6$. The shape of sidebands in Fig. 7(a) resembles the spectrum in Fig. 2(c). We pick out three kinds of sidebands in Fig. 7(b), (c) and (d). Fig. 7(b) represents the longitudinal sidebands, where its maximum growth rate locates at $k_{x}=0.2733,k_{y}=0$ and the maximum growth rate is $\gamma_{LS}=0.0103\omega_{pe}$. In our simulations, the corresponding quantities are $k_{x}=0.25,k_{y}=0$, and $\gamma_{LS}=0.02199\omega_{pe}$. The DRY model underestimates $\delta_{k}$ and the growth rate of sideband perhaps due to the underestimation of the nonlinear frequency shift $\Delta\omega$ in Eq. 5. Fig. 7(c) is the transverse sidebands with $k_{x}=k_{L}$. The position of transverse sideband with the maximum growth rate locates at $k_{x}=k_{L},k_{y}=0.04$, and the maximum growth rate of transverse sideband is $\gamma_{TS}=0.0052\omega_{pe}$. While in simulations they are $k_{x}=k_{L},k_{y}=0.04167$ and $\gamma_{TS}=0.00588\omega_{pe}$, respectively, which show good consistencies for transverse sidebands. This is because the transverse sidebands are mostly affected by Landau damping of sidebands,Berger2 showing the Landau damping in our calculation is accurate. Fig. 7(d) stands for the growth rates of oblique sidebands with $k_{y}=0.03$, which is between the longitudinal and transverse sidebands as indicated by our Vlasov simulations.

The theoretical shapes of sidebands with the maximum growth rates and the corresponding growth rates as a function of potential amplitude are shown in Fig. 8. To compare with the Vlasov results, a sudden driven model of nonlinear frequency shift is plotted in Fig. 8(c) and (d), in addition to the adiabatic model shown in Fig. 8(a) and (b). The difference is that in a sudden driven model, $\alpha=0.823$ in Eq. 5,Berger1 which permits a higher Langmuir wave frequency shift. Fig. 8(a) shows the amplitude dependence of the positions of sidebands. As compared with Fig. 5(a), the shape and dependence on amplitude are similar, but it shrinks inward. When turning to the sudden driven model with $\Delta\omega_{sudden}\approx 1.5\Delta\omega_{adiabatic}$, the shapes get large. In Fig. 8(b) and (d), the growth rates of longitudinal sidebands scale linearly with $|e\phi/T_{e}|^{1/2}$, which agrees with early theoretical results.Dewar ; Berger2 As the nonlinear frequency shift gets larger, Fig. 8(d) shows a better agreement with the simulation results in Fig. 5(b). In Fig. 9, we obtain the scaling law for sidebands by power law fitting under sudden approximation. $g_{2}=0.5$ for longitudinal sideband and $g_{2}=0.66$ for oblique sideband with $k_{y}=0.03$. Both in simulation and theoretical results, $g_{2}$ for oblique sideband is a little larger than that of longitudinal sidebands. This may be because the influence of Landau damping on oblique sidebands.

However, there still exist discrepancies between Vlasov simulations and DRY model, especially when $|e\phi/T_{e}|$ is large. We suspect that it is due to the underestimation of $\Delta\omega$ in DRY model. Therefore, in Fig 10(a) and (b), we plot the spectra of $E_{x}$ in the $k_{x}$-$\omega$ space and the $k_{y}$-$\omega$ from Vlasov simulations with $|e\phi/T_{e}|=0.6$, respectively. The observed nonlinear frequency shifts of Langmuir wave and sidebands is on the order of $\Delta\omega\approx 0.1\omega_{pe}$, however, theory predicts that $\Delta\omega_{adiabatic}=0.0413\omega_{pe}$ for adiabatic approximation and $\Delta\omega_{sudden}=0.0624\omega_{pe}$ for sudden approximation. Though with some flaws, the DRY model is good enough to capture the main physics of sidebands of Langmuir waves, and the growth rates of sidebands are qualitatively agreed with our 2D electrostatic Vlasov simulations.

In this paper, first, we study the evolution of Langmuir waves in a muti-wavelength system through 2D Vlasov simulations. There are three stages in our simulations. In the drive phase, an external driver is used to excited a monochromatic Langmuir waves. In the phase of sideband growth, the sidebands of Langmuir waves are excited because of the trapped electrons. Three kinds of sidebands are observed, which are the longitudinal sidebands, oblique sidebands and transverse sidebands. The oblique sidebands are arc-shaped in the k-space. The growth rates of oblique sidebands are smaller than that of the longitudinal sideband with the maximum growth rate but higher than the transverse sideband. In the turbulent state, the vortex merging happens when the amplitudes of sidebands are comparable with that of Langmuir waves. After vortex-merging, $20$ vortices merges to $15$ vortices. And The spectra of $k_{x}$ and $k_{y}$ become broader, inducing the filamentation of Langmuir waves. The amplitude dependence of sidebands are also studied by Vlasov simulations. We find that $\Delta k_{x}$ of longitudinal sidebands and $\Delta k_{x}$ and $\Delta k_{y}$ of oblique sidebands increase with the amplitude. Next, we reuse the DRY model to study the sidebands of Langmuir waves. Based on the early work,Berger2 the Landau damping with $k_{y}$ should be considered. The growth rates of three kinds of sidebands are finally obtained by DRY model, and they are qualitatively agreed with our Vlasov simulations.

The sideband instability is a saturation mechanism of SRS in ICF, and in earlier works, researchers mostly focused on the Longitudinal sidebands of Langmuir wave.Brunner1 ; Friou The Vlasov simulations and theoretical results show clearly that sideband of Langmuir wave is a multidimensional instability. So, the sidebands of Langmuir wave in oblique and tranverse direction should also be considered. In this paper, we only consider the situation in two dimensions. The sideband with $k_{z}$ component might also exist. Besides, in early works, some kinetic effects by particle trapping have been studied by using particle in cell (PIC) simulations, such as Langmuir wave bowingyin ; Lin the trapped particle side loss hxvu3 in multidimensions. These works indicate that Multidimensional effects are important for instabilities in ICF. In the future works, PIC simulations will be used to study the sidebands of Langmuir wave in multidimensions.

We are pleased to acknowledge useful discussions with T. Yang, Y. Z. Zhou and S. X. Xie. This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25050700), National Natural Science Foundation of China (Grant Nos. 11805062, 11875091 and 11975059), Science Challenge Project, No. TZ2016005 and Natural Science Foundation of Hunan Province, China (Grant No. 2020JJ5029).

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