Kinetic Electrostatic Electron Nonlinear Research Articles

Vlasov–Maxwell simulations of backward Raman amplification of seed pulses in plasmas

AbstractWe study the problem of the amplification of an ultra-short seed pulse via stimulated Raman backscattering (SRB) from a long pump pulse (assumed to have an envelope with a constant amplitude), in an underdense plasma. The SRB interaction couples the pump light wave to a daughter light seed wave propagating in the opposite direction, scattered off an electron plasma wave. In recent numerical simulations, it has been observed that besides stimulated Raman backward scattering (SRBS) and stimulated Raman forward scattering, other high-frequency kinetic instabilities can occur when modified distribution functions exist during the evolution of the system. In particular, we showed the prominent role played by kinetic electrostatic electron nonlinear (KEEN) waves (Afeyan et al., 2004). We continue this work by applying a relativistic Vlasov–Maxwell code to study stimulated KEEN wave scattering (SKEENS) and its role in the SRBS short pulse amplification processes. An analysis of the full spectrum of waves participating in the amplification processes is presented. The absence of spurious noise in grid-based Vlasov codes allows us to follow the evolution of the system with a kinetic (collisionless) description. This affords us a glimpse at the intricate phase-space structures such as trapped particle orbits, which coexist and interact nonlinearly in the electron distribution function.

Nonlinear regime of electrostatic waves propagation in presence of electron-electron collisions

The effects are presented of including electron-electron collisions in self-consistent Eulerian simulations of electrostatic wave propagation in nonlinear regime. The electron-electron collisions are approximately modeled through the full three-dimensional Dougherty collisional operator [J. P. Dougherty, Phys. Fluids 7, 1788 (1964)]; this allows the elimination of unphysical byproducts due to reduced dimensionality in velocity space. The effects of non-zero collisionality are discussed in the nonlinear regime of the symmetric bump-on-tail instability and in the propagation of the so-called kinetic electrostatic electron nonlinear (KEEN) waves [T. W. Johnston et al., Phys. Plasmas 16, 042105 (2009)]. For both cases, it is shown how collisions work to destroy the phase-space structures created by particle trapping effects and to damp the wave amplitude, as the system returns to the thermal equilibrium. In particular, for the case of the KEEN waves, once collisions have smoothed out the trapped particle population which sustains the KEEN fluctuations, additional oscillations at the Langmuir frequency are observed on the fundamental electric field spectral component, whose amplitude decays in time at the usual collisionless linear Landau damping rate.

Nonlinear nature of kinetic undamped waves induced by electrostatic turbulence in stimulated Raman backscattering

The influence of low-frequency waves of kinetic nature induced by electron trapping in backward Stimulated Raman Scattering (SRS) is investigated. Semi-lagrangian Vlasov-Maxwell simulations are carried out not only for periodic boundary conditions but also in the case of an open plasma with parabolic shape, in optical mixing. We provide a numerical example of generation of KEEN (kinetic electrostatic electron nonlinear) waves nonlinearly induced from the SRS through a mechanism we first here elucidate. In particular we identify a process of backward scattering of the SRS probe light from the so generated KEEN waves, which may provide a mechanism for the possible experimental observation and measurement of such nonlinear structures.

Simulations of kinetic electrostatic electron nonlinear (KEEN) waves with variable velocity resolution grids and high-order time-splitting

KEEN waves are non-stationary, nonlinear, self-organized asymptotic states in Vlasov plasmas. They lie outside the precepts of linear theory or perturbative analysis, unlike electron plasma waves or ion acoustic waves. Steady state, nonlinear constructs such as BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force generated by two crossing laser beams, for instance, used to drive them. Smaller amplitude drives manage to devolve into multiple highly-localized vorticlets, after the drive is turned off, and may eventually succeed to coalesce into KEEN waves. Fragmentation once the drive stops, and potential eventual remerger, is a hallmark of the weakly driven cases. A fully formed (more strongly driven) KEEN wave has one dominant vortical core. But it also involves fine scale complex dynamics due to shedding and merging of smaller vortical structures with the main one. Shedding and merging of vorticlets are involved in either case, but at different rates and with different relative importance. The narrow velocity range in which one must maintain sufficient resolution in the weakly driven cases, challenges fixed velocity grid numerical schemes. What is needed is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational effort. This is needed for low amplitude runs. There, global coherent structures take a long time to set up, such as KEEN waves, if they do so at all. The new code’s performance is compared to uniform grid simulations and the advantages are quantified. The birth pains associated with weakly driven KEEN waves are captured in these simulations. Canonical KEEN waves with ample drive are also treated using these advanced techniques. They will allow the efficient simulation of KEEN waves in multiple dimensions, which will be tackled next, as well as generalizations to Vlasov-Maxwell codes. These are essential for pursuing the impact of KEEN waves in high energy density plasmas and in inertial confinement fusion applications. More generally, one needs a fully-adaptive grid-in-phase-space method which could handle all small vorticlet dynamics whether pealing off or remerging. Such fully adaptive grids would have to be computed sparsely in order to be viable. This two-velocity grid method is a concrete and fruitful step in that direction.

On the nature of kinetic electrostatic electron nonlinear (KEEN) waves

An analytical theory is proposed for the kinetic electrostatic electron nonlinear (KEEN) waves originally found in simulations by Afeyan et al. [arXiv:1210.8105]. We suggest that KEEN waves represent saturated states of the negative mass instability (NMI) reported recently by Dodin et al. [Phys. Rev. Lett. 110, 215006 (2013)]. Due to the NMI, trapped electrons form macroparticles that produce field oscillations at harmonics of the bounce frequency. At large enough amplitudes, these harmonics can phase-lock to the main wave and form stable nonlinear dissipationless structures that are nonstationary but otherwise similar to Bernstein-Greene-Kruskal modes. The theory explains why the formation of KEEN modes is sensitive to the excitation scenario and yields estimates that agree with the numerical results of Afeyan et al. A new type of KEEN wave may be possible at even larger amplitudes of the driving field than those used in simulations so far.

Vlasov on GPU (VOG project)

This work concerns the numerical simulation of the Vlasov-Poisson equation using semi-Lagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.

Persistent subplasma-frequency kinetic electrostatic electron nonlinear waves

Driving a one-dimensional collisionless Maxwellian (Vlasov) plasma with a sufficiently strong longitudinal ponderomotive driver for a sufficiently long time results in a self-sustaining nonsinusoidal wave train with well-trapped electrons even for frequencies well below the plasma frequency, i.e., in the plasma wave spectral gap. Typical phase velocities of these waves are somewhat above the electron thermal velocity. This new nonlinear wave is being termed a kinetic electrostatic electron nonlinear (KEEN) wave. The drive duration must exceed the bounce period τB of the trapped electrons subject to the drive, as calculated from the drive force and the linear plasma response to the drive. For a given wavenumber a wide range of KEEN wave frequencies can be readily excited. The basic KEEN structure is essentially kinetic, with the trapped electron density variation being almost completely shielded by the free electrons, leaving just enough net charge to support the wave.

Stimulated-Raman-scatter behavior in a relativistically hot plasma slab and an electromagnetic low-order pseudocavity

Particle simulations on a flat-topped somewhat underdense (typically n0/nc = 0.6) plasma slab by Nikolic [Phys. Rev. E 66, 036404 (2002)] were seen to give transient stimulated scattering behavior with frequency shift [omega0 - omegas(approximately omegap)] considerably less than the plasma frequency omegap. This has been linked to the electron acoustic wave (EAW) and the scattering was thus seen as another example of stimulated electron acoustic scattering inferred by Montgomery [Phys. Rev. Lett. 87, 155001 (2001)] from experiments on low-density plasmas. Montgomery had noted the difficulty of how one could have a very narrow observed scattering from a wave whose damping was at least initially very high. Our Vlasov-Maxwell simulations for such somewhat underdense (n0/nc > or = 0.25) plasmas show that the simulation resonance was in fact determined by the beating of the pump with a new "radiating pseudocavity" electromagnetic mode for the slab at a frequency close to omegap with relatively low loss. This allows the initial narrow-band excitation of the kinetic electrostatic electron nonlinear (KEEN) waves (the nonlinear "cousins" of EAWs) at a well-defined frequency (omegaK approximately omega0 - omegap < omegap) which is not necessarily the value given by the EAW dispersion relation. (The KEEN wave characteristics have been discussed by Afeyan [33rd AAAC (2003), #238, IFSA 2003].) The consideration of such a mechanism is relevant to moderately underdense hot plasmas.