Particle dynamic in an axi-symmetric mirror machine with an extremely high plasma pressure equal to pressure of vacuum magnetic field (so-called regime of diamagnetic confinement) is investigated. Extrusion of magnetic field from central region due to plasma diamagnetism leads to non-conservation of the magnetic moment and can result in chaotic movement and fast losses of particles. The following mechanisms can provide particle confinement for unlimited time: absolute confinement of particles with high azimuthal velocity and conservation of adiabatic invariant for particle moving in smooth magnetic field. The criteria of particle confinement and estimations of lifetime of unconfined particles are obtained and verified in direct numerical simulation. Particle confinement time in the diamagnetic trap in regime of gas-dynamic outflow is discussed.
The effects of the ion Larmor radius on magnetic reconnection are investigated by means of numerical simulations, with a Hamiltonian gyrofluid model. In the linear regime, it is found that ion diamagnetic effects decrease the growth rate of the dominant mode. Increasing ion temperature tends to make the magnetic islands propagate in the ion diamagnetic drift direction. In the nonlinear regime, diamagnetic effects reduce the final width of the island. Unlike the electron density, the guiding center density does not tend to distribute along separatrices and at high ion temperature, the electrost…
The plasma equilibrium in a linear trap at $β\approx 1$ (or above the mirror-instability threshold) under the topology-conservation constraint evolves into a kind of diamagnetic "bubble". This can take two forms: either the plasma body greatly expands in radius while containing the same magnetic flux, or, if the plasma radius is limited, the plasma distribution across flux-tubes changes, so that the same cross-section contains a greatly reduced flux. If the magnetic field of the trap is quasi-uniform around its minimum, the bubble can be made roughly cylindrical, with radius much larger than t…
The article provides a kinetic description of the plasma equilibrium in the Beklemishev diamagnetic trap, where the traditional approach based on the theory of magnetic drifts is not applicable, since the ions move in a substantially non-circular orbit, the diameter of which is approximately equal to the diameter of the diamagnetic bubble. The ion distribution function was found in the collisionless approximation, neglecting the diamagnetic electron current. The radial profile of the magnetic field, the plasma density, the current density, and the components of the pressure tensor are calculat…
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, and 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, an…
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…
Using fully kinetic simulations, we study the suppression of asymmetric reconnection in the limit where the diamagnetic drift speed >> Alfven speed and the magnetic shear angle is moderate. We demonstrate that the slippage between electrons and the magnetic flux facilitates reconnection, and can even result in fast reconnection that lacks one of the outflow jets. Through comparing a case where the diamagnetic drift is supported by the temperature gradient with a companion case that has a density gradient instead, we identify a robust suppression mechanism. The drift of the x-line is slowed dow…
A nonlinear dispersion of a general stationary wave in collisionless plasma is obtained in a non-differential form from a single-particle oscillation-center Hamiltonian. For electrostatic oscillations in nonmagnetized plasma, considered as a paradigmatic example, the linear dielectric function is generalized, and the trapped particle contribution to the wave frequency shift $Δω$ is found analytically as a function of the wave amplitude $a$. Smooth distributions yield $Δω\sim a^{1/2}$, as usual. However, beam-like distributions of trapped electrons result in different power laws, or even a loga…
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.
The collisionless interaction of an expanding high-energy plasma cloud with a magnetized background plasma in the presence of a dipole magnetic field is examined in the framework of a 2D3V hybrid (kinetic ions and massless fluid electrons) model. The retardation of the plasma cloud and the dynamics of the perturbed electromagnetic fields and the background plasma are studied for high Alfven-Mach numbers using the particle-in-cell method. It is shown that the plasma cloud expands excluding the ambient magnetic field and the background plasma to form a diamagnetic cavity which is accompanied by …
Electrostatic drift instabilities driven by ion and electron temperature gradients (ITG and ETG instabilities) are considered for the ranges of perpendicular wave number values from the inverse ion thermal gyroradius up to the electron thermal gyroradius. The analysis is carried out in the framework of the local kinetic approach. Scaling laws for instability parameters are presented. These scaling laws can be used as a basis of collisionless plasma transport rates in magnetic traps with purely poloidal magnetic field topology.
The nonlinear frequency shift is derived in a transparent asymptotic form for intense Langmuir waves in general collisionless plasma. The formula describes both fluid and kinetic effects simultaneously. The fluid nonlinearity is expressed, for the first time, through the plasma dielectric function, and the kinetic nonlinearity accounts for both smooth distributions and trapped-particle beams. Various known limiting scalings are reproduced as special cases. The calculation avoids differential equations and can be extended straightforwardly to other nonlinear plasma waves.
A collisionless plasma possesses a certain amount of "available energy", which is that part of the thermal energy that can be converted into field energy. Here, a calculation is presented of the available energy carried by trapped electrons in a slender non-omnigenous flux tube of plasma. This quantity is compared with gyrokinetic simulations of the nonlinear saturated radial energy flux resulting from turbulence driven by collisionless trapped-electron modes in various stellarator and a tokamak. The numerical calculation of available energy is extremely fast and shows a strong correlation wit…
A high energy density plasma embedded in a neutral gas is able to launch an outward-propagating nonlinear electrostatic ionization wave that traps energetic electrons. The trapping maintains a strong sheath electric field, enabling rapid and long-lasting wave propagation aided by field ionization. Using 1D3V kinetic simulations, we examine the propagation of the ionization wave in the presence of a transverse MG-level magnetic field with the objective to identify qualitative changes in a regime where the initial thermal pressure of the plasma exceeds the pressure of the magnetic field ($β>1$).…
Effects induced by the finite number $N$ of particles on the evolution of a monochromatic electrostatic perturbation in a collisionless plasma are investigated. For growth as well as damping of a single wave, discrete particle numerical simulations show a $N$-dependent long time behavior which differs from the numerical errors incurred by vlasovian approaches and follows from the pulsating separatrix crossing dynamics of individual particles.
Collisionless Larmor Coupling and Blob Formation in a Laser-Plasma Expanding into a Magnetized Ambient Plasma
University of California–Los Angeles（加州大学洛杉矶分校）；Laboratoire d’Optique Appliquée (LOA), CNRS, École polytechnique, ENSTA, Institut Polytechnique de Paris（应用光学实验室，法国国家科学研究中心，巴黎综合理工学院，ENSTA，巴黎理工学院）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
A largely unsolved theoretical issue in controlled fusion research is the consistent \textit{kinetic} treatment of slowly-time varying plasma states occurring in collisionless and magnetized axisymmetric plasmas. The phenomenology may include finite pressure anisotropies as well as strong toroidal and poloidal differential rotation, characteristic of Tokamak plasmas. Despite the fact that physical phenomena occurring in fusion plasmas depend fundamentally on the microscopic particle phase-space dynamics, their consistent kinetic treatment remains still essentially unchalleged to date. The goal…
The magnetic energy of nonlinear Alfven waves in compressible plasmas may be ponderomotively coupled only to ion-acoustic quasi-modes which modulate the wave phase velocity and cause wave-front steepening. In the collisionless plasma with $β\not=0$, the dynamics of nonlinear Alfven wave is also affected by the resonant particle-wave interactions. Upon relatively rapid evolution (compared to the particle bounce time), the quasi-stationary wave structures, identical to the so called (Alfvenic) Rotational Discontinuities, form, the emergence and dynamics of which has not been previously understoo…
Nonlinear dynamics of collisionless non-neutral plasma in an external electrical trapping field is considered. Time-dependent solution of the nonlinear Vlasov-Poisson equations are obtained. The influence of initial conditions on the dynamics of charged layer is discussed.
A lot of plasma physics problems are not amenable to exact solutions due to many reasons. It is worth mentioning among them, for example, nonlinearity of the motion equations, variable coefficients or non lineal conditions on known or unknown borders. To solve these problems, different types of approximations which are combinations of analytical and numerical simulation methods are put into practice. The problem of plasma behavior in numerous varieties of a minimum-B magnetic trap where the plasma is heated under electron cyclotron resonance (ECR) conditions is the subject of numerical simulat…
Numerical simulations recover ultra slow electron holes (EH) of electron-acoustic genre propagating stably well below the ion acoustic speed where the ion response disallows any known pure electron perturbation. The reason of stability of EH at high ion temperature ($T_{i}> T_{e}$) is traced to the loss of neutralizing cold ion response. In a background of cold ions, $θ=T_e/T_i\gg 1$, they have an ion compression that accelerates to jump over a forbidden velocity gap and settle on the high velocity tail of the electron distribution $f_e$, confirming to a recently identified limit of the nonlin…
Collisionless shocks occur in various fields of physics. In the context of space and astrophysics they have been investigated for many decades. However, a thorough understanding of shock formation and particle acceleration is still missing. Collisionless shocks can be distinguished into electromagnetic and electrostatic shocks. Electromagnetic shocks are of importance mainly in astrophysical environments and they are mediated by the Weibel or filamentation instability. In such shocks, charged particles gain energy by diffusive shock acceleration. Electrostatic shocks are characterized by a str…
In the vinicity of phase-space resonances for a given species of plasma, the particle distribution function is flattened as free energy is exchanged between the plasma and resonant electromagnetic waves. Here, we present action-angle variables which explicitly separate the adiabatic invariant from the contribution that arises from perturbation of the Hamiltonian over the course of a single orbit. Then, we perform similar analysis for the orbit period, allowing one to identify a generating function $ψ$ which modifies the zeroth order contribution to the orbit period (unperturbed orbit) in the c…
The possible occurrence of equilibrium off-equatorial tori in the gravitational and electromagnetic fields of astrophysical compact objects has been recently proved based on non-ideal MHD theory. These stationary structures can represent plausible candidates for the modelling of coronal plasmas expected to arise in association with accretion discs. However, accretion disc coronae are formed by a highly diluted environment, and so the fluid description may be inappropriate. The question is posed of whether similar off-equatorial solutions can be determined also in the case of collisionless plas…
Mirror modes in collisionless high-temperature plasmas represent macroscopic high-temperature quasi-superconductors. We explicitly calculate the bouncing electron contribution to the ion-mode growth rate, diamagnetic surface current responsible for the Meissner effect, and the weak attracting electric field. The mean electric field turns out to be negligible. Pairing is a second-order effect of minor importance. The physically important effect is the resonant interaction between bouncing electrons and the thermal ion-sound background. It is responsible for the mirror mode to evolve as a phase …
Quasisymmetry and omnigeneity of an equilibrium magnetic field are two distinct properties proposed to ensure radial localization of collisionless trapped particles in any stellarator. These constraints are incompletely explored, but have stringent restrictions on a magnetic geometry. This work employs an analytic approach to understand the implications of the constraints. The particles move in an intrinsically three dimensional equilibrium whose representation is given by earlier work of Weitzner and its extension here. For deeply trapped particles a local equilibrium expansion around a minim…
When a steady-state cylindrical plasma discharge is centrally fuelled, the collisionless radial electron flux is canonically coupled to an axial current. The identification and analysis of this transport driven current, previously reported in collisionless simulations [W. J. Nunan and J. M. Dawson, Phys. Rev. Lett. $\mathbf{73}$, 1628 (1994)], is addressed analytically and extended to the collisional regime by means of first-principles kinetic models. Collisionless radial transport is described with the standard quasilinear model and collisional velocity anisotropy relaxation with the Landau k…
Electron motion in an oblique shock wave is studied by means of a one-dimensional, relativistic, electromagnetic, particle simulation code with full ion and electron dynamics. It is found that an oblique shock can produce electrons with ultra-relativistic energies; Lorentz factors with $γ\gto 100$ have been observed in our simulations. The physical mechanisms for the reflection and acceleration are discussed, and the maximum energy is estimated. If the electron reflection occurs near the end of a large-amplitude pulse, those particles will then be trapped in the pulse and gain a great deal of …
A cold electron beam propagating through a background plasma is subject to filamentation process due to theWeibel instability. If the initial beam radius is large compared with the electron skin depth and the beam density is much smaller than the background plasma density, multiple filaments merge many times. Because of this non-adiabatic process, the beam perpendicular energy of initially cold beam grows until all filaments coalesce into one pinched beam with the beam radius much smaller than initial radius and smaller than the electron skin depth. It was shown through particle-in-cell simula…
A simplified model of particle transport at a quasiparallel one-dimensional collisionless shock is suggested. In this model the MHD-turbulence behind the shock is dominated by a circularly polarized, large amplitude Alfvén wave originated upstream from the turbulence excited by particles leaking from the downstream medium. It is argued that such a wave having significantly increased its magnetic field during the transmission through the shock interface can effectively trap thermal ions, regulating their leakage upstream. Together with a background turbulence this wave also plays a fundamental …
Magnetic reconnection, especially in the relativistic regime, provides an efficient mechanism for accelerating relativistic particles and thus offers an attractive physical explanation for nonthermal high-energy emission from various astrophysical sources. I present a simple analytical model that elucidates key physical processes responsible for reconnection-driven relativistic nonthermal particle acceleration (NTPA) in the large-system, plasmoid-dominated regime in two dimensions. The model aims to explain the numerically-observed dependencies of the power-law index $p$ and high-energy cutoff…
We report the properties of a novel type of sub-proton scale magnetic hole found in two dimensional PIC simulations of decaying turbulence with a guide field. The simulations were performed with a realistic value for ion to electron mass ratio. These structures, electron vortex magnetic holes (EVMHs), have circular cross-section. The magnetic field depression is associated with a diamagnetic azimuthal current provided by a population of trapped electrons in petal-like orbits. The trapped electron population provides a mean azimuthal velocity and since trapping preferentially selects high pitch…
We present results from the first 3D kinetic numerical simulation of magnetorotational turbulence and dynamo, using the local shearing-box model of a collisionless accretion disc. The kinetic magnetorotational instability grows from a subthermal magnetic field having zero net flux over the computational domain to generate self-sustained turbulence and outward angular-momentum transport. Significant Maxwell and Reynolds stresses are accompanied by comparable viscous stresses produced by field-aligned ion pressure anisotropy, which is regulated primarily by the mirror and ion-cyclotron instabili…
Particle motion in a cylindrical multiple-cusp magnetic field configuration is shown to be highly (though not completely) chaotic, as expected by analogy with the Sinai billiard. This provides a collisionless, linear mechanism for phase randomization during monochromatic wave heating. A general quasilinear theory of collisionless energy diffusion is developed for particles with a Hamiltonian of the form $H_0+H_1$, motion in the \emph{unperturbed} Hamiltonian $H_0$ being assumed chaotic, while the perturbation $H_1$ can be coherent (i.e. not stochastic). For the multicusp geometry, two heating …
A study of collisionless external shocks in gamma-ray bursts is presented. The shock structure, electromagnetic fields, and process of electron acceleration are assessed by performing a self-consistent 3D particle-in-cell (PIC) simulation. In accordance with hydrodynamic shock systems, the shock consists of a reverse shock (RS) and forward shock (FS) separated by a contact discontinuity (CD). The development and structure are controlled by the ion Weibel instability. The ion filaments are sources of strong transverse electromagnetic fields at both sides of the double shock structure over a len…
Hybrid-kinetic numerical simulations of firehose and mirror instabilities in a collisionless plasma are performed in which pressure anisotropy is driven as the magnetic field is changed by a persistent linear shear $S$. For a decreasing field, it is found that mostly oblique firehose fluctuations grow at ion Larmor scales and saturate with energies $\sim$$S^{1/2}$; the pressure anisotropy is pinned at the stability threshold by particle scattering off microscale fluctuations. In contrast, nonlinear mirror fluctuations are large compared to the ion Larmor scale and grow secularly in time; margi…
A kinetic treatment is developed for collisionless magnetized plasmas occurring in high-temperature, low-density astrophysical accretion disks, such as are thought to be present in some radiatively-inefficient accretion flows onto black holes. Quasi-stationary configurations are investigated, within the framework of a Vlasov-Maxwell description. The plasma is taken to be axisymmetric and subject to the action of slowly time-varying gravitational and electromagnetic fields. The magnetic field is assumed to be characterized by a family of locally nested but open magnetic surfaces. The slow colli…
A new magnetic field generation mechanism in electrostatic shocks is found, which can produce fields with magnetic energy density as high as 0.01 of the kinetic energy density of the flows on time scales $ \tilde \, 10^4 \, ω_{pe}^{-1}$. Electron trapping during the shock formation process creates a strong temperature anisotropy in the distribution function, giving rise to the pure Weibel instability. The generated magnetic field is well-confined to the downstream region of the electrostatic shock. The shock formation process is not modified and the features of the shock front responsible for …
Kinetic simulations of 3D collisionless magnetic reconnection with a guide field show a dramatic enhancement of energetic electron production when compared with 2D systems. In the 2D systems, electrons are trapped in magnetic islands that limit their energy gain, whereas in the 3D systems the filamentation of the current layer leads to a stochastic magnetic field that enables the electrons to access volume-filling acceleration regions. The dominant accelerator of the most energetic electrons is a Fermi-like mechanism associated with reflection of charged particles from contracting field lines.
Dissipative processes cause collisionless plasmas in many systems to develop nonthermal particle distributions with broad power-law tails. The prevalence of power-law energy distributions in space/astrophysical observations and kinetic simulations of systems with a variety of acceleration and trapping (or escape) mechanisms poses a deep mystery. We consider the possibility that such distributions can be modeled from maximum-entropy principles, when accounting for generalizations beyond the Boltzmann-Gibbs entropy. Using a dimensional representation of entropy (related to the Renyi and Tsallis …
Using analytical theory and hybrid-kinetic numerical simulations, we demonstrate that, in a collisionless plasma, long-wavelength ion-acoustic waves (IAWs) with amplitudes $δn/n_0 \gtrsim 2/β$ (where $β\gg{1}$ is the ratio of thermal to magnetic pressure) generate sufficient pressure anisotropy to destabilize the plasma to firehose and mirror instabilities. These kinetic instabilities grow rapidly to reduce the pressure anisotropy by pitch-angle scattering and trapping particles, respectively, thereby impeding the maintenance of Landau resonances that enable such waves' otherwise potent collis…
Ions in Hall effect thrusters are often characterized by a low collisionality. In the presence of acceleration fields and azimuthal electric field waves, this results in strong deviations from thermodynamic equilibrium, introducing kinetic effects. This work investigates the application of the 14-moment maximum-entropy model to this problem. This method consists in a set of 14 PDEs for the density, momentum, pressure tensor components, heat flux vector and fourth-order moment associated to the particle velocity distribution function. The model is applied to the study of collisionless ion dynam…
We herein investigate shock formation and particle acceleration processes for both protons and electrons in a quasi-parallel high-Mach-number collisionless shock through a long-term, large-scale particle-in-cell simulation. We show that both protons and electrons are accelerated in the shock and that these accelerated particles generate large-amplitude Alfvénic waves in the upstream region of the shock. After the upstream waves have grown sufficiently, the local structure of the collisionless shock becomes substantially similar to that of a quasi-perpendicular shock due to the large transverse…
Diffusive shock acceleration at collisionless shocks is thought to be the source of many of the energetic particles observed in space. Large-scale spatial variations of the magnetic field has been shown to be important in understanding observations. The effects are complex, so here we consider a simple, illustrative model. Here, we solve numerically the Parker transport equation for a shock in the presence of large-scale sinusoidal magnetic-field variations. We demonstrate that the familiar planar-shock results can be significantly altered as a consequence of large-scale, meandering magnetic l…
Collisionless Phase Mixing Mimics Diffusive Transport in Radiation Belt Observations
University of Helsinki（赫尔辛基大学）；University of California, Los Angeles（加州大学洛杉矶分校）；University of Texas, Arlington（德克萨斯大学阿灵顿分校）；University of Birmingham（伯明翰大学）；University of Exeter（埃克塞特大学）；Air Force Research Laboratory（空军研究实验室）；Institute of Atmospheric Physics of the Czech Academy of Sciences（捷克科学院大气物理研究所）
Slow dynamical changes in magnetic-field strength and invariance of the particles' magnetic moments generate ubiquitous pressure anisotropies in weakly collisional, magnetized astrophysical plasmas. This renders them unstable to fast, small-scale mirror and firehose instabilities, which are capable of exerting feedback on the macroscale dynamics of the system. By way of a new asymptotic theory of the early nonlinear evolution of the mirror instability in a plasma subject to slow shearing or compression, we show that the instability does not saturate quasilinearly at a steady, low-amplitude lev…
