Trapped Particle Instability Research Articles

Spectral line broadening of the Raman scattered waves in laser plasmas

Parametric instabilities are of a great concern especially in the context of laser fusion. Electromagnetic waves scattered by these instabilities can reach high amplitudes, thereby diverting a significant portion of the energy from the incident laser beam, which is essential for compressing and heating the fuel capsule. In addition, the emerging electrostatic wave can trap and accelerate electrons, which, upon impact, can preheat the target and thus prevent its effective compression. Here, we focus on the stimulated Raman scattering and the associated processes. Spectral broadening of the backward Raman scattering daughter wave is theoretically described process, which is caused by so-called trapped particle instability. The existence of trapped and freely moving electrons in plasma influenced by high amplitude electron plasma wave leads also to the anomalous dispersion of this wave. It is reflected in the shifts in the electrostatic spectrum which, on the other hand, must be visible also in the electromagnetic spectrum via the feed-back loop of the instability. In the present paper, we discuss scattered wave modes spectral broadening using results of numerical simulations.

The maximum-J property in quasi-isodynamic stellarators

Some stellarators tend to benefit from favourable average magnetic curvature for trapped particles when the plasma pressure is sufficiently high. This so-called maximum- $J$ property has several positive implications, such as good fast-particle confinement, magnetohydrodynamic stability and suppression of certain trapped-particle instabilities. This property cannot be attained in quasisymmetric stellarators, in which deeply trapped particles experience average bad curvature and therefore precess in the diamagnetic direction close to the magnetic axis. However, quasi-isodynamic stellarators offer greater flexibility and allow the average curvature to be favourable and the precession to be reversed. We find that it is possible to design such stellarators so that the maximum- $J$ condition is satisfied for the great majority of all particles, even when the plasma pressure vanishes. The qualitative properties of such a stellarator field can be derived analytically by examining the most deeply and the most shallowly trapped particles, although some small fraction of the latter will inevitably not behave as desired. However, through numerical optimisation, we construct a vacuum field in which 99.6 % of all trapped particles satisfy the maximum- $J$ condition.

Hybrid Zakharov-kinetic simulation of nonlinear stimulated Raman scattering

We present a novel 2D reduced numerical model for stimulated Raman scattering (SRS) in laser fusion plasmas in which envelope equations for the electromagnetic fields are coupled to a hybrid description of the electron species. Specifically, the electron distribution is split between a bulk part described by a Zakharov-like linear model and a kinetic tail discretized using a particle-in-cell-like (PIC) scheme. By avoiding to sample the bulk-electron distribution, this approach greatly reduces the numerical cost of SRS simulations compared with PIC codes, while still being able to describe the nonlinear evolution of the electron tail and trapping-related kinetic phenomena. First, our model is shown to reproduce accurately the linear Landau damping of an infinitesimal electron plasma wave (EPW) whose phase velocity falls into the tail of the electron distribution. Then, applying it to the simulation of the trapped-particle modulational instability of a large-amplitude EPW, results comparable to those of previously published 2D Vlasov simulations are obtained. Finally, we simulate the excitation of kinetic backward SRS from a single strong laser speckle (λ=0.527 μm, I=1016 W cm−2) in an underdense (ne=0.036 nc) plasma, which drives an EPW with wavenumber kλD≈0.34. The model predictions fairly agree with the results of a PIC simulation regarding the kinetic saturation mechanisms (i.e., trapped-particle instabilities), and with experimental data and Vlasov simulations related to the frequency shift of nonlinear EPWs. For this SRS simulation, we estimate that our hybrid model is over an order of magnitude less costly than an equivalent PIC simulation due to the lower particle count.

Trapped particle instability in : I homogeneous Vlasov plasmas

Kinetic instability due to trapped particles in non-linear evolution of large amplitude electrostatic waves i.e BGK (Bernstein-Greene-Kruskal) mode is investigated in a one dimensional, collisionless, homogeneous Vlasov-Poisson plasma with immobile ions and kinetic electrons. Considering a cosinusoidal density perturbation with non-linear perturbation amplitude α = 0.05 as starting point, we have demonstrated that long time BGK solutions with scale (where corresponds to the longest length scale in the system) are not stable due to Landau growth in the secondary sideband modes. This growth in the sideband modes is primarily related to trapped particles in the potential well of the perturbation which leads to trapped particle instability when the amplitude of the sidebands become comparable to that of the primary nonlinear mode. In the past, to explain trapped particle instability, two physical mechanisms were held responsible, namely, wave particle interaction phenomenon at resonance location v ϕ = ω/k and mode coupling interaction phenomenon between primary and secondary sideband modes. We have demonstrated and emphasized both the phenomena in our study. In addition, effect of primary mode perturbation amplitude on trapped particle instability is investigated and growth rates corresponding to secondary sideband modes are reported alongwith a new scaling law for detrapping time as a function of perturbation amplitude. Also, scaling of growth rates corresponding to lower and upper sideband modes with respect to perturbation amplitude is obtained which demonstrates the existence of a transition point at α = 0.075 between small and large perturbation amplitudes and growth rate scales as γ ∼ α 2 for α < 0.075 whereas γ ∼ α 1/2 for α > 0.075.

Trapped particle instability in : II inhomogeneous Vlasov plasmas

In Part I of the companion paper [], we have extensively investigated the instability due to trapped particle by large amplitude electrostatic waves i.e BGK (Bernstein-Greene-Kruskal) mode in one dimensional, collisionless, homogeneous Vlasov-Poisson plasma. In continuation of the work, in this Part II, we present the trapped particle instability in the presence of finite non-uniform background inhomogeneity of scale with equilibrium inhomogeneity amplitude A. In the process, we demonstrate and emphasize importance of the mode coupling phenomenon alongwith the wave-particle interaction phenomenon in order to understand the physical mechanisms, which are responsible for the growth of upper and lower sideband modes in inhomogeneous plasmas. It is shown that the larger the amplitude of equilibrium inhomogeneity A, smaller is the perturbation amplitude α for which destabilization sets in. Also, in addition, effect of equilibrium inhomogeneity amplitude on trapped particle instability, asymmetry in growth rates of upper and lower secondary sideband modes, alongwith novel scaling of detrapping time with respect to equilibrium inhomogeneity amplitude are reported.

Saturation of trapped particle instability induced by vortex-merging in electron plasma waves

Phase space vortex merging in electron plasma waves is researched by applying a periodically changing electric field in plasma space with Maxwellian velocity distribution. The pump electric field excites the electron plasma wave with large amplitude and electron phase space vortices are formed. The electron plasma wave can be transited to longer wave modes through vortex merging processes. Excepting the fundamental mode of electron plasma waves, two or more modes appear and grow exponentially before vortex merging. After merging, another mode replaces the initial pumping wave as the strongest wave mode. It is verified that the growth of this new strong mode is related to trapping particle instability. Vortex merging is one of the saturation mechanisms of trapped particle instability.

Collisionless microinstabilities in stellarators. Part 4. The ion-driven trapped-electron mode

Optimised stellarators and other magnetic-confinement devices having the property that the average magnetic curvature is favourable for all particle orbits are called maximum-$J$ devices. They have recently been shown to be immune to trapped-particle instabilities driven by the density gradient. Gyrokinetic simulations reveal, however, that another instability can arise, which is also associated with particle trapping but causes less transport than typical trapped-electron modes. The nature of this instability is clarified here. It is shown to be similar to the ‘ubiquitous mode’ in tokamaks and is driven by ion free energy, but requires trapped electrons to exist.

Quantitative study of the trapped particle bunching instability in Langmuir waves

The bunching instability of particles trapped in Langmuir waves is studied using Vlasov simulations. A measure of particle bunching is defined and used to extract the growth rate from numerical simulations, which are compared with theory [Dodin et al., Phys. Rev. Lett. 110, 215006 (2013)]. In addition, the general theory of trapped particle instability in 1D is revisited and a more accurate description of the dispersion relation is obtained. Excellent agreement between numerical and theoretical predictions of growth rates of the bunching instability is shown over a range of parameters.

Kinetic simulations and reduced modeling of longitudinal sideband instabilities in non-linear electron plasma waves

Kinetic Vlasov simulations of one-dimensional finite amplitude Electron Plasma Waves are performed in a multi-wavelength long system. A systematic study of the most unstable linear sideband mode, in particular its growth rate γ and quasi- wavenumber δk, is carried out by scanning the amplitude and wavenumber of the initial wave. Simulation results are successfully compared against numerical and analytical solutions to the reduced model by Kruer et al. [Phys. Rev. Lett. 23, 838 (1969)] for the Trapped Particle Instability (TPI). A model recently suggested by Dodin et al. [Phys. Rev. Lett. 110, 215006 (2013)], which in addition to the TPI accounts for the so-called Negative Mass Instability because of a more detailed representation of the trapped particle dynamics, is also studied and compared with simulations.

Collisionless microinstabilities in stellarators. I. Analytical theory of trapped-particle modes

This is the first of two papers about collisionless, electrostatic micro-instabilities in stellarators, with an emphasis on trapped-particle modes. It is found that, in so-called maximum-$J$ configurations, trapped-particle instabilities are absent in large regions of parameter space. Quasi-isodynamic stellarators have this property (approximately), and the theory predicts that trapped electrons are stabilizing to all eigenmodes with frequencies below the electron bounce frequency. The physical reason is that the bounce-averaged curvature is favorable for all orbits, and that trapped electrons precess in the direction opposite to that in which drift waves propagate, thus precluding wave-particle resonance. These considerations only depend on the electrostatic energy balance, and are independent of all geometric properties of the magnetic field other than the maximum-$J$ condition. However, if the aspect ratio is large and the instability phase velocity differs greatly from the electron and ion thermal speeds, it is possible to derive a variational form for the frequency showing that stability prevails in a yet larger part of parameter space than what follows from the energy argument. Collisionless trapped-electron modes should therefore be more stable in quasi-isodynamic stellarators than in tokamaks.

Collisionless microinstabilities in stellarators. II. Numerical simulations

Microinstabilities exhibit a rich variety of behavior in stellarators due to the many degrees of freedom in the magnetic geometry. It has recently been found that certain stellarators (quasi-isodynamic ones with maximum-J geometry) are partly resilient to trapped-particle instabilities, because fast-bouncing particles tend to extract energy from these modes near marginal stability. In reality, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here, the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with the National Compact Stellarator Experiment and the DIII-D tokamak. In gyrokinetic simulations, performed with the gyrokinetic code GENE in the electrostatic and collisionless approximation, ion-temperature-gradient modes, trapped-electron modes, and mixed-type instabilities are studied. Wendelstein 7-X and QIPC exhibit significantly reduced growth rates for all simulations that include kinetic electrons, and the latter are indeed found to be stabilizing in the energy budget. These results suggest that imperfectly optimized stellarators can retain most of the stabilizing properties predicted for perfect maximum-J configurations.

Immediate influence of heating power on turbulent plasma transport

The theory that describes immediate impact of heating power on pressure-gradient-driven turbulence and turbulent transport (without waiting for the evolution of global parameters and the mean distribution function) is reported. A new mechanism, which is an external source directly coupling with plasma fluctuations in the phase space so as to affect turbulence and transport, is investigated. Application is made to the cases of trapped particle instability and long-range fluctuations, which are driven by background microscopic fluctuations. This theory can predict an abrupt change in transport at the on/off of heating, which has been indicated by experimental observations. The derivative of heating power density by plasma pressure is a key parameter, in addition to conventional spatial gradients. The condition under which this new effect can be observed is also evaluated.

Gyrokinetic TEM stability calculations for quasi-isodynamic stellarators

Recent theoretical findings suggest that in perfectly quasi-isodynamic stellarators, the trapped-particle instability as well as the ordinary electron-density-gradient-driven trapped-electron mode are stable in the electrostatic and collisionless approximation. In these configurations contours of constant magnetic field strength B are poloidally closed, the second adiabatic invariant J is constant on flux surfaces and peaks in the centre. It follows that the diamagnetic drift frequency ω*α and the bounce-averaged magnetic drift frequency are in opposite directions, 0, everywhere on the flux surface. This is the signature of average "good curvature" for trapped particles and, thanks to this property, particles that bounce faster than the frequency of any unstable mode must draw energy from it near marginal stability. Consequently, the point of marginal stability cannot exist for the collisionless trapped-particle mode, and hence this mode will be absent. By a similar argument, the ordinary trapped-electron mode is also stable. Because perfect quasi-isodynamicity can never be reached, it is necessary to test configurations approaching quasi-isodynamicity numerically in order to probe their resilience against trapped-particle modes. Progress has been made to extend gyrokinetic simulations to stellarator geometries, enabling us to perform the first linear gyrokinetic simulations of density-gradient-driven modes in stellarator configurations approaching quasi-isodynamicity, such as Wendelstein 7-X and more recently found configurations. These simulations appear to confirm the analytical predictions.

Resilience of Quasi-Isodynamic Stellarators against Trapped-Particle Instabilities

It is shown that in perfectly quasi-isodynamic stellarators, trapped particles with a bounce frequency much higher than the frequency of the instability are stabilizing in the electrostatic and collisionless limit. The collisionless trapped-particle instability is therefore stable as well as the ordinary electron-density-gradient-driven trapped-electron mode. This result follows from the energy balance of electrostatic instabilities and is thus independent of all other details of the magnetic geometry.

Adiabatic nonlinear waves with trapped particles. III. Wave dynamics

The evolution of adiabatic waves with autoresonant trapped particles is described within the Lagrangian model developed in Paper I, under the assumption that the action distribution of these particles is conserved, and, in particular, that their number within each wavelength is a fixed independent parameter of the problem. One-dimensional nonlinear Langmuir waves with deeply trapped electrons are addressed as a paradigmatic example. For a stationary wave, tunneling into overcritical plasma is explained from the standpoint of the action conservation theorem. For a nonstationary wave, qualitatively different regimes are realized depending on the initial parameter S, which is the ratio of the energy flux carried by trapped particles to that carried by passing particles. At S &amp;lt; 1/2, a wave is stable and exhibits group velocity splitting. At S &amp;gt; 1/2, the trapped-particle modulational instability (TPMI) develops, in contrast with the existing theories of the TPMI yet in agreement with the general sideband instability theory. Remarkably, these effects are not captured by the nonlinear Schrödinger equation, which is traditionally considered as a universal model of wave self-action but misses the trapped-particle oscillation-center inertia.

Novel features of non-linear Raman instability in a laser plasma

The electron phase space evolution in a non-relativistic and homogeneous laser plasma generated by a nanosecond laser in a near infrared region in the presence of stimulated Raman scattering is investigated by a numerical simulation. The mechanism of electron acceleration in the potential wells of the plasma wave accompanying the Raman back-scattering is analyzed in a 1D Vlasov-Maxwell model. The dominant wave modes are both the backward and the forward propagating Raman waves, each accompanied by a daughter electrostatic wave. In addition to a strong interaction of plasma electrons with the primary electrostatic wave in the case of back-scattering, a cascading is observed consisting in a secondary scattering of the primary Raman back-scattered wave. This phenomenon reduces the Raman reflectivity and causes an acceleration of electrons against the direction of the heating laser beam. Moreover, the strong trapping in the primary electrostatic wave generated by the Raman back-scattering leads due to the trapped particle instability to a significant spectral broadening of the original plasma wave and a subsequent intermittent behaviour of the scattering process. The high phase velocity electrostatic daughter wave of the forward Raman scattering cannot trap the electrons directly, but there is an indication of non-resonant quasi-modes combined of this wave and of the simultaneously existing electrostatic daughter wave accompanying the Raman back-scattering. The transform method is used for a solution of the set of partial differential equations, which consists of the Vlasov equation and of the full set of Maxwell equations in a 1D approximation. A simplified Fokker-Planck collision term is added to overcome the numerical instabilities during the simulation. The model has relevance to a long scale plasma geometry, such as occurring in the indirect drive experiments near the light entrance holes of target hohlraum.

On generation of Alfvénic-like fluctuations by drift wave–zonal flow system in large plasma device experiments

According to recent experiments, magnetically confined fusion plasmas with “drift wave–zonal flow turbulence” (DW-ZF) give rise to broadband electromagnetic waves. Sharapov et al. [Europhysics Conference Abstracts, 35th EPS Conference on Plasma Physics, Hersonissos, 2008, edited by P. Lalousis and S. Moustaizis (European Physical Society, Switzerland, 2008), Vol. 32D, p. 4–071] reported an abrupt change in the magnetic turbulence during L-H transitions in Joint European Torus [P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 (1987)] plasmas. A broad spectrum of Alfvénic-like (electromagnetic) fluctuations appears from E×B flow driven turbulence in experiments on the large plasma device (LAPD) [W. Gekelman et al., Rev. Sci. Instrum. 62, 2875 (1991)] facility at UCLA. Evidence of the existence of magnetic fluctuations in the shear flow region in the experiments is shown. We present one possible theoretical explanation of the generation of electromagnetic fluctuations in DW-ZF systems for an example of LAPD experiments. The method used is based on generalizing results on shear flow phenomena from the hydrodynamics community. In the 1990s, it was realized that fluctuation modes of spectrally stable nonuniform (sheared) flows are non-normal. That is, the linear operators of the flows modal analysis are non-normal and the corresponding eigenmodes are not orthogonal. The non-normality results in linear transient growth with bursts of the perturbations and the mode coupling, which causes the generation of electromagnetic waves from the drift wave–shear flow system. We consider shear flow that mimics tokamak zonal flow. We show that the transient growth substantially exceeds the growth of the classical dissipative trapped-particle instability of the system.

Generation of fast electrons in the external corona of laser plasma by the Raman scattering

The mechanism of laser plasma generation and heating by the impinging laser beam occurs mostly by the way of oscillatory motion of electrons in the electric field of the light wave, which is damped by the electron–ion collisions, dissipating thus the energy of the coherent laser wave, the so-called inverse bremsstrahlung mechanism. However, in the external part of the corona the laser plasma is only weakly collisional and the evolution of the electron distribution function in the electron phase space is far from remaining a Maxwellian one. The predominant mechanism causing deviations from a Maxwell distribution is the particle trapping in electrostatic waves. The electrostatic waves turn up in the plasma as daughter wave in the nonlinear processes of wave transformations, the typical example of which is the Raman scattering when the heating laser wave decays in a scattered electromagnetic wave and a forward directed electrostatic electron plasma wave. The potential troughs of this electrostatic wave are trapping the electrons which travel at nearly the same speed as the wave phase velocity and later start to oscillate around the wave potential minima. The scattered Raman wave may grow strong enough to undergo a secondary scattering, which leads to a Raman cascade. The oscillating electrons have their own dynamics separate from that of the bulk thermal electrons, which gives rise to additional wave modes. One of them is the trapped particle instability, which broadens the electrostatic spectrum and leads to an intermittency. The electron phase space is hardly accessible to a direct measurement of the electron distribution function, which is changing fast on a time scale of order of picoseconds, it therefore makes sense to resort to a numerical modelling of the phase space evolution. This is effected by a solution of the Vlasov equation (including a weak collision term) simultaneously with the Maxwell equations in a 1D periodic slab model by a transform method. The parameters of the computation were chosen to be compatible with the plasma in the laser corona typically generated by the nanosecond Prague Asterix Laser System (PALS) system at the high flux end attainable by a tight focusing ∼1016 W/cm2. The results of the modelling are presented and the physically interesting points highlighted.

Stimulated Raman scattering in the presence of trapped particle instability

The behaviour of electron gas in a laser plasma corona is studied in the presence of stimulated Raman scattering (SRS). The 1D Vlasov–Maxwell model describing plasma relevant to the experiment PALS (Prague Asterix Laser System), where the nanosecond iodine laser with the wavelength of first harmonic λ vac = 1.3152 and with the power density in the focal spot I = 10 20 W/m 2 is in operation. For the solution of Vlasov equation for the electron distribution function a Fourier–Hermite transform method is used. For the numerical stabilization a small collision term is added to the Vlasov equation keeping its value realistic for the condition relevant to the PALS experiment. The dominant wave modes in our model are both the backward (SRS-B) and the forward (SRS-F) Raman scattering, each of them accompanied by the forward going electron plasma wave. Several mechanisms were identified such as the SRS-B plasma wave spectral broadening due to a trapped particle instability (TPI) or the formation of an electrostatic quasi-mode by non-resonant interaction of SRS-B and SRS-F plasma waves.

Trapped-Particle Instability Leading to Bursting in Stimulated Raman Scattering Simulations

Nonlinear, kinetic simulations of stimulated Raman scattering (SRS) under laser-fusion conditions present a bursting behavior. Different explanations for this regime have been given in previous studies: saturation of SRS by increased nonlinear Landau damping [K. Estabrook et al., Phys. Fluids B 1, 1282 (1989)]], and detuning due to the nonlinear frequency shift of the plasma wave [H. X. Vu et al., Phys. Rev. Lett. 86, 4306 (2001)]]. Another mechanism, also assigning a key role to the trapped electrons is proposed here: the breakup of the plasma wave through the trapped-particle instability.