---
abstract: 'It is shown that partial incoherence, in the form of stochastic phase noise, of a Langmuir wave in an unmagnetized plasma gives rise to a Landau-type damping. Starting from the Zakharov equations, which describe the nonlinear interaction between Langmuir and ion-acoustic waves, a kinetic equation is derived for the plasmons by introducing the Wigner-Moyal transform of the complex Langmuir wave field. This equation is then used to analyze the stability properties of small perturbations on a stationary solution consisting of a constant amplitude wave with stochastic phase noise. The concomitant dispersion relation exhibits the phenomenon of Landau-like damping. However, this damping differs from the classical Landau damping in which a Langmuir wave, interacting with the plasma electrons, loses energy. In the present process, the damping is non-dissipative and is caused by the resonant interaction between an instantaneously-produced disturbance, due to the parametric interactions, and a partially incoherent Langmuir wave, which can be considered as a quasi-particle composed of an ensemble of partially incoherent plasmons.'
address:
- |
    $^1$ *Dipartimento di Scienze Fisiche, Università di Napoli ”Federico II” and INFN Sezione di Napoli, Complesso Universitario di M.S. Angelo,\
    Via Cintia, I-80126 Napoli, Italy*
- |
    $^2$ *Institut für Theoretische Physik IV, Ruhr-Universität Bochum\
    D-4470 Bochum, Germany*
- |
    $^3$ *Dipartimento di Fisica Generale, Università di Torino,\
    Via Pietro Giuria 1, 10125 Torino, Italy*
- '$^4$ *Department of Electromagnetics, Chalmers University of Technology, SE-412 96, Göteborg, Sweden*'
author:
- 'R. Fedele$^1$, P.K. Shukla$^2$, M. Onorato$^3$, D. Anderson$^4$, and M. Lisak$^4$'
title: |
     [*Submitted*]{}\
    Landau damping of partially incoherent Langmuir waves
---

PACS number(s): 52.35.Mw, 52.35.Fp, 52.25.Dg, 03.40.Kf

Keywords: quasi-particles, Langmuir envelopes, Landau damping, modulational instability, nonlinear Schroedinger equation, Zakharov equations.

More than fifty years ago, Landau [@9] discovered a damping of electron plasma waves whose linear dynamics is governed by the Vlasov-Poisson system of equations. This damping was caused by resonant wave-electron interactions. The Landau theory clearly shows that the decay rate of the wave energy due to the interaction with the resonant electrons of the system (satisfying $p=\omega/k$, where $p$ denotes the electron velocity and $\omega$ and $k$ are the frequency and wave number of the electron plasma wave ) is proportional to the first derivative of the equilibrium distribution function $\rho_0 (p)$ of the electrons. Typically the shape of $\rho_0 (p)$ is such that $d\rho_0 (p=\omega/k)/dp<0$, which implies that there are more particles with $p < \omega/k$ (which gain energy from the wave) than with $p
>\omega/k$ (which give energy to the wave). Consequently, the net result is that the wave is damped. However, finite amplitude Langmuir waves in plasmas can be created when some free energy sources, such as electron and laser beams, are available in the system. In the past, several authors [@nishikawa]-[@Zakmushrub] have considered the nonlinear coupling between high-frequency Langmuir and low-frequency ion-acoustic waves. Physically, this coupling occurs because a large amplitude plasma wave, interacting with small density ripples, produces a current which becomes the source for an envelope of Langmuir waves. The low-frequency ponderomotive force of the latter reinforces the density modulation in such a way as to produce a space-charge electric field, which varies on a time scale longer than the electron plasma period. Thus, within the slowly-varying envelope approximation, the dynamics of the Langmuir wavepacket is governed by a nonlinear Schrödinger equation (NLSE), while the non-resonant density ripples follow an ion-acoustic wave equation driven by a ponderomotive force created by the Langmuir wave). In his classic paper, Zakharov [@Zakharov] demonstrated that these nonlinearly coupled equations admit a class of modulationally unstable stationary solutions and also exhibit the phenomenon of wave collapse.\
In this paper, we present a Landau damping of partially coherent Langmuir waves whose dynamics is governed by the Zakharov equations. In fact, we show that the latter can be reduced to a kinetic-like equation for a Wigner-Moyal-like transform for the amplitude of the electric field associated with the Langmuir wave. Based on this kinetic equation we consider the dynamics of small perturbations on a partially coherent background solution consisting of a constant amplitude and a stochastically varying phase. The concomitant dispersion relation depends strongly on the spectral energy density of the Langmuir wave and a Landau-type damping of the perturbations is found due to the resonant interaction between the Langmuir quasi-particle and the instantaneously created ripples.

As it is well known, the nonlinear propagation of a one–dimensional Langmuir wave–packet in an unmagnetized plasma with unperturbed density $n_0$, is governed by the Zakharov equations [@Zakharov] $$i{\partial E\over\partial t}~+~{3 v_{te}^{2}\over
2\omega_{pe}}{\partial^2 E\over\partial
x^2}~-~\omega_{pe}{\delta n\over 2n_0}E~=~0~~~,
\label{zakharov-1}$$ $$\left({\partial^2\over\partial t^2}~-~c_{s}^{2}{\partial^2\over\partial x^2}
\right){\delta n\over 2n_0}~=~-{c_{s}^{2}\over 2}{\partial^2\over\partial x^2}
\left({|E|^2 -|E_0|^2\over 4\pi n_0 T_e}\right)~~~,
\label{zakharov-2}$$ where $E$ is the slowly varying complex electric field amplitude, $\delta n$ is the density fluctuation, $v_{te}$ is the electron thermal velocity, $\omega_{pe}$ is the electron plasma frequency, $T_e$ is the electron temperature, $c_{s}$ is the sound speed, and $E_0$ is the unperturbed electric field amplitude. Moreover, $x$ and $t$ play the role of configurational space coordinate (longitudinal coordinate with respect to the wavepacket centre) and time, respectively. We will show that the nonlinear propagation of a Langmuir wavepacket, governed by the above Zakharov system, can be suitably described in phase space by a kinetic–like equation within a framework similar to the one of the [*Vlasov-Poisson*]{} system. A similar approach has recently been used to study the modulational instability of a large-amplitude electromagnetic wavepacket as well as the coherent instability analysis of intense charged–particle beam dynamics [@5; @6]. Within this framework, we show that a phenomenon similar to the classical Landau damping occurs for partially coherent plasma waves. In particular, this phenomenon tends to suppress the modulational instability. To this end, the above system of equations (\[zakharov-1\]) and (\[zakharov-2\]) can be cast in the following form $$i\alpha{\partial \Psi\over\partial s}~+~{\alpha^{2}\over
2}{\partial^2
\Psi\over\partial
x^2}~-~U~\Psi~=~0~~~,
\label{zakharov-1-bis}$$ $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U~=~-{\partial^2\over\partial x^2}
\left(\langle|\Psi|^2 -|\Psi_0|^2\rangle\right)~~~,
\label{zakharov-2-bis}$$ where the angle brackets account for the statistical ensemble average [*ála*]{} Klimomtovich [@a] due to the partial incoherence of the waves, $s=\sqrt{3}\lambda_{De}\omega_{pe}t$ ($\lambda_{De}\equiv
v_{te}/\omega_{pe}$ is the electron Debye length), $\alpha
=\sqrt{3}\lambda_{De}$, $U=\delta n / 2n_0$, $\mu=\sqrt{m_{e}/3m_{i}}$ ($m_{e}$ and $m_{i}$ are the electron and the ion masses, respectively), and $$\Psi~=~{\mu E\over \sqrt{8\pi n_0 T_e}}~~~,
\label{Psi}$$ $$\Psi_0~=~{\mu E_0\over \sqrt{8\pi n_0 T_e}}~~~.
\label{Psi-zero}$$ Note that (\[Psi-zero\]) can be written as $$\rho_0
\equiv |\Psi_0|^2=\mu^2{\epsilon\over\epsilon_{T}}~~~,
\label{Psi-zero-bis}$$ where $\epsilon\equiv E_{0}^{2}/ 8\pi$ and $\epsilon_T\equiv n_0 T_e$ are the electric energy density and the electron thermal energy density, respectively. According to the theory modelled by the Zakharov system (\[zakharov-1-bis\]) and (\[zakharov-2-bis\]), $\epsilon /\epsilon_T >>1$ (i.e, the ponderomotive effect is larger than the thermal pressure effect). Note also that Eq. (\[zakharov-2-bis\]) implies that $U$ is a functional of $|\Psi|^2$, namely $U=U\left[|\Psi|^2\right]$. Consequently, once (\[zakharov-2-bis\]) is combined with (\[zakharov-1-bis\]), the latter becomes a nonlinear Schrödinger equation (NLSE) with the nonlinear term $U\left[|\Psi|^2\right]$. This implies that, as in Quantum Mechanics, one can introduce the density matrix [@a] as a sort of “two-points correlation function” and then the Wigner-Moyal transform [@b] which is solution of the von Neumann-Weyl equation [@c]. In case of surface gravity waves, whose dynamics and instability is described in deep water by a nonlinear Schrödinger equation, such a kind of approach has been given in the pioneering works by Alber [@e], Crawford et al. [@f] and Janssen [@g]. More recently, a similar approach has been developed for the propagation of electromagnetic wavepackets in nonlinear media where an appropriate kinetic equation is able to show a random version of the modulational instability [@5; @6; @j]. In particular, this approach can be also seen as an application of the Klimontovich statistical average method [@j]. The transition to the phase-space $x-p$ ($p\equiv dx/ds$ is the conjugate variable of $x$) allows us to write a von Neumann equation [@a] for $w(x,p,s)$. In fact, although, we are dealing with a classical system, by following Alber [@e], one can introduce the correlation function (whose corresponding meaning in Quantum Mechanics is nothing but the density matrix for mixed states) $$\rho(x,x',s)= \langle\Psi^*(x,s) \Psi (x',s)\rangle~~~,
\label{wigner}$$ where the statistical ensemble average takes into account the incoherency of the waves. The technique was successfully introduced in the field of statistical quantum mechanics to describe the dynamics of a system in the classical space language [@e]-[@g]. Thus one can define the following Wigner-Weyl transform [@b] $$w(x,p,s) = {1 \over 2 \pi \alpha}
\int_{-\infty}^{\infty}
\rho\left(x + { y \over 2},~ x - { y \over
2}, ~s\right)~\exp\left(i{p y\over \alpha} \right)~dy~~~,
\label{12-0}$$ which allows us to transit from the configuration space to the phase space. In fact, $w(x,p,s)$ is governed by the following kinetic-like equation (von Neumann-like equation [@a] ) $${\partial w\over \partial s} + p {\partial w\over
\partial x}
- \sum_{n=0}^{\infty}{(-1)^{n}\over \left(2n+1\right)!}
\left( { \alpha \over 2} \right)^{2 n}
{\partial^{2n+1}U\over\partial x^{2n+1}}
{\partial^{2n+1}w\over\partial p^{2n+1}}~=~0~~~.
\label{a4}$$ On the other hand, since (\[12-0\]) implies that $$\langle |\Psi|^2\rangle~=~\int_{-\infty}^{\infty}w(x,p,s)~dp~~~,
\label{x-projection}$$ the functional $U\left[|\Psi|^2\right]$ can be expressed as a functional of $w$, namely $U=U\left[w\right]$. Consequently, (\[zakharov-2-bis\]) can be written as $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U~=~-{\partial^2\over\partial x^2}
\left(\int_{-\infty}^{\infty}w~dp -\int_{-\infty}^{\infty}w_0~dp\right)~~~,
\label{zakharov-2-ter}$$ where $w_0$ is the Wigner-Moyal transform of $\Psi_0$ ($\rho_0
=|\Psi_0|^2 = \int_{-\infty}^{\infty}~w_0~dp$). The above scheme allows us to carry out a stability analysis of the stationary Langmuir wave by considering the linear dispersion relation of small amplitude perturbations. We start from the equilibrium state: $w = w_0 (p)$, $U=U_0=0$, and perturb the system according to $$w(x,p,s)=w_0 (p) + w_1 (x,p,s)~~~,
\label{rho-perturbation}$$ $$U(x,s)~=~U_0~+~U_1(x,s)~=~U_1(x,s)~~~,
\label{U-perturbation}$$ where $w_1 (x,p,s)$ and $U_1 (x,s)$ are first-order quantities. Consequently, (\[a4\]) and (\[zakharov-2-ter\]) can be linearized as follows $${\partial w_1\over \partial s} + p {\partial w_1\over
\partial x}
=~\sum_{n=0}^{\infty}{(-1)^{n}\over \left(2n+1\right)!}
\left( { \alpha \over 2} \right)^{2 n}
{\partial^{2n+1}U_1\over\partial x^{2n+1}}
w_{0}^{(2n+1)}~~~,
\label{a4-2}$$ $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U_1~=~-{\partial^2\over\partial x^2}
\left(\int_{-\infty}^{\infty}w_1~dp\right)~~~,
\label{zakharov-2-linearized}$$ where $w_{0}^{(2n+1)}\equiv d^{2n+1}\rho_0/dp^{2n+1}$. By introducing the Fourier transform of $U_1(x,s)$ and $w_1
(x,p,s)$, i.e. $$U_1(x,s)=\int_{-\infty}^{\infty}~dk~\int_{-\infty}^{\infty}~d\omega~
\widetilde{U_1}(k ,\omega)~\exp\left(ik x-i\omega
s\right)~~~,
\label{U-1-bis}$$ $$w_1(x,p,s)=\int_{-\infty}^{\infty}~dk~\int_{-\infty}^{\infty}~d\omega~
\widetilde{w_1}(k,p,\omega)~\exp\left(ik x-
i\omega s\right)~~~,
\label{w-1}$$ and substituting (\[U-1-bis\]) and (\[w-1\]) into Eq.s (\[a4-2\]) and (\[zakharov-2-linearized\]), we readily obtain the following dispersion relation $$\omega^2~-~\mu^2 k^2~=~k^2\int_{-\infty}^{\infty}~{w_0
\left(p+\alpha k /2\right)~-~w_0
\left(p-\alpha k /2\right)\over \alpha k}~
{dp\over p-\omega /k}~~~.
\label{dispersion-relation}$$

We consider a Lorenzian spectrum of the form: $$w_0(p)=\frac{\rho_0} {\pi} \frac{\Delta} {\Delta^2+p^2},$$ where $\Delta$ is the width of the spectrum. Dispersion relation (\[dispersion-relation\]) becomes: $$\left(\omega^2-\mu^2 k^2\right)\left(\omega^2-\alpha^2 k^4/4 +
2 i k \Delta  \omega-k^2 \Delta^2 \right)
~=~\rho_0 k^4~~~,
\label{lorenzian-dispersion-relation}$$ As the simplest case, we take the limit of $\Delta \rightarrow 0$. This case corresponds to a coherent monochromatic wave-train and the dispersion relation reduces to $$\left(\omega^2-\mu^2 k^2\right)\left(\omega^2-\alpha^2 k^4 /4\right)
~=~\rho_0 k^4~~~,
\label{monochromatic-dispersion-relation}$$ which predicts the well known result of a purely growing instability [@Zakharov].

In the following, we concentrate on the limit of small $\alpha k << 1$ and $\omega /k >>1$. We will show that during its nonlinear evolution the Langmuir wavepacket can exhibit a phenomenon, similar to the Landau damping [@Dawson], between parametrically-driven non-resonant density ripples and Langmuir quasi-particles. It is easy to verify that the condition $\alpha k << 1$ (i.e., $\lambda_{De} k<<1$) implies that $${w_0\left(p+\alpha k /2\right)~-~w_0
\left(p-\alpha k /2\right)\over \alpha k}~\approx
dw_{0}/dp~\equiv w_{0}^{'}~~~.
\label{k-small-approximation}$$ Consequently, (\[dispersion-relation\]) becomes $$\omega^2~=~k^2~\int~{w_{0}^{'}
\over p-\omega /k}~dp~~~,
\label{k-small-dispersion-relation}$$ where, using the assumption $\omega /k >>1$, we can neglect the term involving $\mu^2$ because $\mu <<1$. Note that (\[k-small-dispersion-relation\]) is formally identical to the linear dispersion relation that holds for warm plasma waves [@9] or for non-monochromatic charged-particle bunches in circular accelerating machines for an arbitrary complex coupling impedance [@7]. Consequently, the dispersion relation (\[k-small-dispersion-relation\]) allows us to predict a sort of Landau damping [@9], as described in plasma physics for linear plasma waves as well as in charged-particle beam physics. A similar effect, which has been called [*quantum-like Landau damping*]{} (QLLD), has recently been predicted for the nonlinear propagation of a large-amplitude electromagnetic waves in nonlinear media, such as optical fibers and plasmas as well as for the quantum-like description of charged-particle beams in an accelerating machine [@5; @6; @j]. In order to investigate this phenomenon also for large-amplitude Langmuir waves, we apply the standard procedure [@14] used for the Landau damping of a linear plasma wave, which involves integrating the above dispersion relation in the complex plane, evaluating both the Cauchy principle value (PV) and the semi-residue in $p= \omega /k$. The result is $$\omega^2~=~k^2 \int_{PV}~{w_{0}^{'}
\over p-\omega /k}~dp~+
~i\pi k^2 w_{0}^{'}(\beta)~~~,
\label{landau-dispersion-relation}$$ For a symmetric sufficiently smooth stationary background distribution, $w_0 (p)$, we have $$\omega^2~=~\left(1+{3\Delta^{2}
\over \rho_{0}}\right)k^2~+
~i\pi k^2 w_{0}^{'}(\beta)~~~,
\label{reactive-dispersion-relation-2}$$ where $\Delta\equiv \left(\langle p^2 \rangle\right)^{1/2}$ denotes the r.m.s. width of $w_0 (p)$, and $\beta \equiv
\rho_{0}^{1/4}\left(1+3\Delta^{2} / 4\rho_{0}^{2}\right)$. Note that, according to the above hypothesis, we have $\beta >>1$ and $\Delta^2 /\rho_0 <<1$. This implies that $\rho_0 =|\Psi_0|^2
>>1$, and from (\[Psi-zero-bis\]) it follows that $\epsilon
/\epsilon_T >>1/\mu^2$. Eq. (\[reactive-dispersion-relation-2\]) clearly shows that the damping rate is proportional to the derivative of the Wigner distribution $w_0$. This is formally similar to the expression for the Landau damping of a linear plasma wave in a warm unmagnetized plasma. In fact, writing $\omega$ in the complex form $\omega =\omega_R + i\omega_I$, substituting it in (\[landau-dispersion-relation\]) and then separating the real and imaginary parts, we obtain $$\omega_{R}^{2}~-~\omega_{I}^{2}~=~k^2\int_{PV}~{w_{0}^{'}
\over p-\omega /k}~dp~~~,
\label{real-part}$$ and $$\omega_{I}~=~{\pi \over 2} {k^2\over \omega_R}
w_{0}^{'}(\beta)~~~.
\label{imaginary-part}$$ The Landau damping rate of the Langmuir wave in the appropriate unities is $\gamma =\sqrt{3}\omega_{pe}\lambda_{De}\omega_I$. For example, for a Gaussian wavepacket spectrum, i.e. $w_0 (p)= \left(\rho_0
/\sqrt{2\pi \Delta^{2}}\right)\exp\left(-p^2 /2\Delta^{2}
\right)$, one obtains $$\gamma = -\left({3\pi\over 8}\right)^{
1/2}{\mu^{5/2}
\over\Delta^3}
\left({\epsilon\over \epsilon_T}\right)^{5/4}
~\omega_{pe}\lambda_{De}k~\exp\left[-{\mu
\over 2\Delta^{2}}\left({\epsilon\over\epsilon_T}
\right)^{1/2}\right]~~~,
\label{imaginary-part-1}$$ where higher-order terms have been neglected. The present damping mechanism differs from the standard Landau damping in that here $w_0 (p)$ does not represent the equilibrium velocity distribution of the plasma electrons, but it can be considered as the “kinetic” distribution of all Fourier components of the partially incoherent large-amplitude Langmuir wave in the warm plasma. In terms of plasmons, we realize that $w_0 (p)$ represents the distribution of the partially incoherent plasmons that are distributed in $p$-space (i.e. in $k$-space) with a finite “temperature”. Consequently, the QLLD described here is due to the partial incoherence of the wave which corresponds to a finite-width Wigner distribution of the plasmons.

In this paper, we have carried out an analysis for a Landau-type damping in plasmas containing partially incoherent Langmuir waves. Starting from the Zakharov system of equations and employing the Wigner-Moyal quasi-distribution function for the complex Langmuir wave electric field amplitude, it has been shown that the Zakharov system of equations can be converted into a pair of coupled evolution equations in phase space. For a monochromatic and coherent wave ($w_0(p) =\rho_0\delta
(p)$), we have obtained a dispersion relation which exhibits a purely growing instability [@Zakharov]. Furthermore, for a broad-band Langmuir wave spectrum, corresponding to a partially incoherent constant amplitude wave, we have investigated the properties of small perturbations for $\alpha k<<1$ (i.e., $\lambda_{De} k<<1$), and very large dimensionless phase velocity (i.e., $\omega /k
>>1$). Our dispersion relation for this case exhibits a sort of weak Landau damping, very similar to the one predicted for the Vlasov-Poisson system for linear plasma waves. The physical origin of this phenomenon is attributed to the “non-monochromatic” behaviour of the Wigner spectrum of the Langmuir wave, which can be considered as an “ensemble of incoherent plasmons”. Actually, a broad-band spectrum of incoherent plasmons, forming the “wave packet”, interact individually with the self-excited density ripples. Similar to the standard Landau damping, where the electrons interact individually with a linear plasma wave and statistically produce a net transfer of energy from the wave to the particles, the plasmons interact individually with the ripples and produce a change of energy which is more significant around the resonance, which is determined by the condition $p=\omega /k$. Finally, we observe that since the system can be modulationally unstable [@Zakharov], this phenomenon acts in competition with the modulational instability.

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