Measurement of the Saturation Length of the Self-Modulation Instability
Max Planck Institute for Physics（马克斯·普朗克物理研究所）；CERN（欧洲核子研究组织）；UCL（伦敦大学学院）；PARTREC, UMCG, University of Groningen（帕尔特雷克，乌得勒支大学医学中心，格罗宁根大学）；Uppsala University（乌普萨拉大学）；University of Manchester（曼彻斯特大学）；Cockcroft Institute（科克克罗夫特研究所）；Lancaster University（兰开斯特大学）；GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa（等离子体与核聚变研究所，技术高等学院，里斯本大学）
A. Clairembaud, M. Turner, M. Bergamaschi, L. Ranc, F. Pannell, J. Mezger, H. Jaworska, N. van Gils, J. Farmer, P. Muggli, the AWAKE Collaboration
The self-modulation (SM) instability transforms a long charged particle bunch traveling in plasma into a train of microbunches that resonantly drives large-amplitude wakefields. We present the first determination of the saturation length of SM using experimental and numerical results. The saturation length is the distance over which wakefields reach their maximum amplitude along the plasma. By varying the plasma length and measuring the radius of the transverse distribution of the bunch, we find that the saturation length of SM decreases with plasma density and initial field amplitude, e.g., when seeding. The saturation length is a fundamental parameter of the instability, and these results are key for understanding SM and designing plasma wakefield accelerators driven by long bunches, such as AWAKE, or by long laser pulses for radiation production.
The beam hosing instability is analyzed theoretically for a witness beam in the quasi-linear regime of plasma accelerators. In this regime, the hosing instability saturates, even for a monoenergetic bunch, at a level much less than standard scalings predict. Analytic expressions are derived for the saturation distance and amplitude and are in agreement with numerical results. Saturation is due to the natural head-to-tail variations in the focusing force, including the self-consistent transverse beam loading.
We show that the hosing instability can be suppressed after the saturation of the self-modulation instability of a long particle bunch if the plasma density perturbation is linear. We derive scalings for maximum bunch tilts and seeds for the self-modulation instability to ensure stable propagation beyond saturation of self-modulation. Numerical solutions of the reduced hosing equations and three-dimensional particle-in-cell simulations confirm our analytical findings. Our results may also apply when a train of particle bunches or laser pulses excites a linear wake.
The saturation mechanism of the Weibel instability is investigated theoretically by considering the evolution of currents in numerous cylindrical beams that are generated in the initial stage of the instability. Based on a physical model of the beams, it is shown that the magnetic field strength attains a maximum value when the currents in the beams evolve into the Alfven current and that there exist two saturation regimes. The theoretical prediction of the magnetic field strength at saturation is in good agreement with the results of two-dimensional particle-in-cell simulations for a wide ran…
The growth and saturation of magnetic fields due to the Weibel instability (WI) have important implications for laboratory and astrophysical plasmas, and this has drawn significant interest recently. Since the WI can generate a large magnetic field from no initial field, the maximum magnitudes achieved can have significant consequences for a number of applications. Hence, an understanding of the detailed dynamics driving the nonlinear saturation of the WI is important. This work considers the nonlinear saturation of the WI when counter-streaming populations of initially unmagnetized electrons …
A novel regime of the saturation of the Pierce-type ion-sound instability in bounded ion-beam-plasma system is revealed in 1D PIC simulations. It is found that the saturation of the instability is mediated by the oscillating virtual anode potential structure. The periodically oscillating potential barrier separates the incoming beam ions into two groups. One component forms a supersonic beam which is accelerated to an energy exceeding the energy of the initial cold ion beam. The other component is organized as a self-consistent phase space structure of trapped ions with a wide energy spread - …
The temporal evolution of the magnetic field associated with electron thermal Weibel instability in optical-field ionized plasmas is measured using ultrashort (1.8 ps), relativistic (45 MeV) electron bunches from a linear accelerator. The self-generated magnetic fields are found to self-organize into a quasi-static structure consistent with a helicoid topology within a few ps and such a structure lasts for tens of ps in underdense plasmas. The measured growth rate agrees well with that predicted by the kinetic theory of plasmas taking into account collisions. Magnetic trapping is identified as…
Spatio-temporal evolution of Buneman instability has been followed numerically till its quasilinear quenching and beyond, using an in-house developed electrostatic 1D particle-in-cell simulation code. For different initial drift velocities $k_{L}v_{0}/ω_{pe} \approx 0.1 \, - \, 1$ and for a wide range of electron to ion mass ratios (m/M), growth rate obtained from simulation agrees well with the numerical solution of the fourth order dispersion relation. Quasi-linear saturation of Buneman instability occurs when ratio of electrostatic field energy density ($\sum\limits_{k} |E_{k}|^{2}/8π$) to …
We present an investigation for the generation of intense magnetic fields in dense plasmas with an anisotropic electron Fermi-Dirac distribution. For this purpose, we use a new linear dispersion relation for transverse waves in the Wigner-Maxwell dense quantum plasma system. Numerical analysis of the dispersion relation reveals the scaling of the growth rate as a function of the Fermi energy and the temperature anisotropy. The nonlinear saturation level of the magnetic fields is found through fully kinetic simulations, which indicates that the final amplitudes of the magnetic fields are propor…
A large set of numerical simulations of magnetohydrodynamic (MHD) turbulence induced by the magnetorotational instability (MRI) is presented. Revisiting the previous survey conducted by Sano et al. (2004), we investigate the gas pressure dependence of the saturation level. In ideal MHD simulations, the gas pressure dependence is found to be very sensitive to the choice of a numerical scheme. This is because the numerical magnetic Prandtl number varies according to the scheme as well as the pressure, which considerably affects the results. The saturation level is more sensitive to the numerical…
The nonresonant cosmic ray instability, predicted by Bell (2004), is thought to play an important role in the acceleration and confinement of cosmic rays (CRs) close to supernova remnants. Despite its importance, the exact mechanism responsible for the saturation of the instability has not been determined, and there is no first-principle prediction for the amplitude of the saturated magnetic field. Using a survey of self-consistent kinetic hybrid simulations (with kinetic ions and fluid electrons), we study the saturation of the non-resonant streaming instability as a function of the parameter…
The time evolution and saturation of the Weibel instability at the ion Alfvén current are presented by ab initio particle-in-cell simulations. We found that the ion Weibel current in 3D could evolve into the Alfvén current where the magnetic field energy is sustained at 1.5\% of the initial beam kinetic energy. The current filaments are no longer isolated at saturation, but rather connected to each other to form a network structure. Electrons are continuously heated during the coalescence of the filaments, which is crucial for obtaining sustained magnetic fields with much stronger levels than …
An analytical model for the self-modulation instability of a long relativistic proton bunch propagating in uniform plasmas is developed. The self-modulated proton bunch resonantly excites a large amplitude plasma wave (wake field), which can be used for acceleration of plasma electrons. Analytical expressions for the linear growth rate and the number of exponentiations are given. We use the full three-dimensional particle-in-cell (PIC) simulations to study the beam self-modulation and the transition to the nonlinear stage. It is shown that the self-modulation of the proton bunch competes with …
The Weibel instability driven by two symmetric counter-streaming relativistic electron plasmas, also referred to as current-filamentation instability, is studied in a constant and uniform external magnetic field aligned with the plasma flows. Both the linear and non linear stages of the instability are investigated using analytical modeling and Particle-In-Cell (PIC) simulations. While previous studies have already described the stabilizing effect of the magnetic field, we show here that the saturation stage is only weakly affected. The different mechanisms responsible for the saturation are d…
We study the evolution of the self-modulation instability using bunches with finite rise times. Using particle-in-cell simulations we show that unlike long bunches with sharp rise times, there are large variations of the wake amplitudes and wake phase velocity when bunches with finite rise times are used. These results show that use of bunches with sharp rise times is important to seed the self-modulation instability and to ensure stable acceleration regimes.
The current filamentation instability, which generically arises in the counterstreaming of supersonic plasma flows, is known for its ability to convert the free energy associated with anisotropic momentum distributions into kinetic-scale magnetic fields. The saturation of this instability has been extensively studied in symmetric configurations where the interpenetrating plasmas share the same properties (velocity, density, temperature). In many physical settings, however, the most common configuration is that of asymmetric plasma flows. For instance, the precursor of relativistic collisionles…
A new non-dissipative mechanism is proposed for the saturation of the axisymmetric magnetorotational (MRI) instability in thin Keplerian disks that are subject to an axial magnetic field. That mechanism relies on the energy transfer from the MRI to stable magnetosonic (MS) waves. Such mode interaction is enabled due to the vertical stratification of the disk that results in the discretization of its MRI spectrum, as well as by applying the appropriate boundary conditions. A second order Duffing-like amplitude equation for the initially unstable MRI modes is derived. The solutions of that equat…
Two counter-propagating cool and equally dense electron beams are modelled with particle-in-cell (PIC) simulations. The electron beam filamentation instability is examined in one spatial dimension. The box length resolves one pair of current filaments. A small, a medium-sized and a large filament are considered and compared. The magnetic field amplitude at the saturation time of the filamentation instability is proportional to the filament size. It is demonstrated, that the force on the electrons imposed by the electrostatic field, which develops during the nonlinear stage of the instability, …
The success of direct laser-driven inertial confinement fusion (ICF) relies critically on the efficient coupling of laser light to plasma. At ignition scale, the absolute stimulated Raman scattering (SRS) instability can severely inhibit this coupling by redirecting and strongly depleting laser light. This Letter describes a new dynamic saturation regime of the absolute SRS instability. The saturation occurs when spatiotemporal fluctuations in the ion-acoustic density detune the instability resonance. The dynamic saturation mitigates the strong depletion of laser light and enhances its transmi…
We have studied the modulation instability of obliquely propagating ion acoustic waves in a collisionless magnetized warm plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. We have derived a nonlinear Schr{ö}dinger equation using the standard reductive perturbation method to describe the nonlinear amplitude modulation of ion acoustic wave satisfying the dispersion relation of ion acoustic wave propagating at an arbitrary angle to the direction of the external uniform static magnetic field. We have investigated the correspondence between t…
Amplitude modulation of quantum ion-acoustic waves (QIAWs) in a quantum electron-pair-ion plasma is studied. It is shown that the quantum coupling parameter $H$ (being the ratio of the plasmonic energy density to the Fermi energy) is ultimate responsible for the modulational stability of QIAW packets, without which the wave becomes modulational unstable. New regimes for the modulational stability (MS) and instability (MI) are obtained in terms of $H$ and the positive to negative ion density ratio $β$. The growth rate of MI is obtained, the maximum value of which increases with $β$ and decrease…
The impact of superthermal electrons on dispersion properties of isotropic plasmas and on the modulational instability of a monochromatic Langmuir wave is studied for the case when the power-law tail of the electron distribution function extends to relativistic velocities and contains most of the plasma kinetic energy. Such an energetic tail of electrons is shown to increase the thermal correction to the Langmuir wave frequency, which is equivalent to the increase of the effective electron temperature in the fluid approach, and has almost no impact on the dispersion of ion-acoustic waves, in w…
We investigate numerically the Princeton magneto-rotational instability (MRI) experiment and the effect of conducting axial boundaries or endcaps. MRI is identified and found to reach a much higher saturation than for insulating endcaps. This is probably due to stronger driving of the base flow by the magnetically rather than viscously coupled boundaries. Although the computations are necessarily limited to lower Reynolds numbers ($\Re$) than their experimental counterparts, it appears that the saturation level becomes independent of $\Re$ when $\Re$ is sufficiently large, whereas it has been …
