Thin current sheets in collisionless plasma: Equilibrium structure,plasma instabilities,and particle acceleration

The review is devoted to plasma structures with an extremely small transverse size, namely, thin current sheets that have been discovered and investigated by spacecraft observations in the Earth’s magnetotail in the last few decades. The formation of current sheets is attributed to complicated dynamic processes occurring in a collisionless space plasma during geomagnetic perturbations and near the magnetic reconnection regions. The models that describe thin current structures in the Earth’s magnetotail are reviewed. They are based on the assumption of the quasi-adiabatic ion dynamics in a relatively weak magnetic field of the magnetotail neutral sheet, where the ions can become unmagnetized. It is shown that the ion distribution can be represented as a function of the integrals of particle motion—the total energy and quasi-adiabatic invariant. Various modifications of the initial equilibrium are considered that are obtained with allowance for the currents of magnetized electrons, the contribution of oxygen ions, the asymmetry of plasma sources, and the effects related to the non-Maxwellian particle distributions. The theoretical results are compared with the observational data from the Cluster spacecraft mission. Various plasma instabilities developing in thin current sheets are investigated. The evolution of the tearing mode is analyzed, and the parameter range in which the mode can grow are determined. The paradox of complete stabilization of the tearing mode in current sheets with a nonzero normal magnetic field component is thereby resolved based on the quasi-adiabatic model. It is shown that, over a wide range of current sheet parameters and the propagation directions of large-scale unstable waves, various modified drift instabilities—kink and sausage modes—can develop in the system. Based on the concept of a turbulent electromagnetic field excited as a result of the development and saturation of unstable waves, a mechanism for charged particle acceleration in turbulent current sheets is proposed and the energy spectra of the accelerated particles are obtained.     

Study of Current Sheets in Three-Dimensional Magnetic Fields with an X-Line by Holographic Interferometry

Results are presented from experimental studies of the spatial electron density distribution in current sheets formed in three-dimensional magnetic configurations with X-lines. The electron density is measured by using two-exposure holographic interferometry. It is shown that plasma sheets can form in a magnetic configuration with an X-line in the presence of a sufficiently strong longitudinal magnetic-field component B_∥ when the electric current is excited along the X-line. As the longitudinal magnetic-field component increases, the electron density decreases and the plasma sheet thickness increases; i.e., the plasma is compressed into a sheet less efficiently.     

Kinetic models of current sheets with a sheared magnetic field

Thin current sheets, whose existence in the Earth’s magnetotail is confirmed by numerous spacecraft measurements, are studied analytically and numerically. The thickness of such sheets is on the order of the ion Larmor radius, and the normal component of the magnetic field (B _z ) in the sheet is almost constant, while the tangential (B _x ) and shear (B _y ) components depend on the transverse coordinate z. The current density in the sheet also has two self-consistent components (j _x and j _y , respectively), and the magnetic field lines are deformed and do not lie in a single plane. To study such quasi-one-dimensional current configurations, two kinetic models are used, in particular, a numerical model based on the particle-in-cell method and an analytical model. The calculated results show that two different modes of the self-consistent shear magnetic field B _y and, accordingly, two thin current sheet configurations can exist for the same input parameters. For the mode with an antisymmetric z profile of the B _y component, the magnetic field lines within the sheet are twisted, whereas the profiles of the plasma density, current density component j _y , and magnetic field component B _x differ slightly from those in the case of a shearless magnetic field (B _y = 0). For the symmetric B _y mode, the magnetic field lines lie in a curved surface. In this case, the plasma density in the sheet varies slightly and the current sheet is two times thicker. Analysis of the dependence of the current sheet structure on the flow anisotropy shows that the sheet thickness decreases significantly with decreasing ratio between the thermal and drift plasma velocities, which is caused by the dynamics of quasi-adiabatic ions. It is shown that the results of the analytical and numerical models are in good agreement. The problems of application of these models to describe current sheets at the magnetopause and near magnetic reconnection regions are discussed.     

Time Evolution of the Macroscopic Characteristics of a Thin Current Sheet in the Course of Its Formation in the Earth’s Magnetotail

A numerical model is developed that allows tracing the time evolution of a current sheet from a relatively thick current configuration with isotropic distributions of the pressure and temperature in an extremely thin current sheet, which plays a key role in geomagnetic processes. Such a configuration is observed in the Earth’s magnetotail in the stage preceding a large-scale geomagnetic disturbance (substorm). Thin current sheets are reservoirs of the free energy released during geomagnetic disturbances. The time evolution of the components of the pressure tensor caused by changes in the structure of the current sheet is investigated. It is shown that the pressure tensor in the current sheet evolves in two stages. In the first stage, a current sheet with a thickness of eight to ten proton Larmor radii forms. This stage is characterized by the plasma drift toward the current sheet and the Earth and can be described in terms of the Chu–Goldberger–Low approximation. In the second stage, an extremely thin current sheet with an anisotropic plasma pressure tensor forms, due to which the system is maintained in an equilibrium state. Estimates of the characteristic time of the system evolution agree with available experimental data.     

Magnetic structure of current sheets in magnetic fields with a singular X line

Direct measurements of magnetic fields in a plasma show that current sheets can form in magnetic configurations with an X line in the presence of a longitudinal magnetic field. It is found that, in a plane perpendicular to the X line and to the direction of the main current, the current sheet has two very different dimensions. The tangential and normal components of the magnetic field and current density in the sheet are determined. The influence of the initial conditions (such as the strength of the longitudinal magnetic field, the gradient of the transverse field, and the plasma ion mass) on the current sheet parameters is investigated.     

Classical and quantum description of electron trapping during the Cherenkov beam instability in plasma

A nonlinear quantum theory of the Cherenkov instability of a nonrelativistic monoenergetic electron beam in a cold plasma is constructed. It is shown that the instability of a low-density beam is almost purely quantum in nature and results from the emission of one quantum of a plasma wave—a plasmon—by the beam electrons. The number of emitted (and absorbed) plasmons increases with beam density, so, in the limit of high-density beams, the instability becomes a classical Cherenkov beam instability in plasma. Some analytic solutions and estimates are found, detailed numerical results are obtained, and the evolution of the quantum distribution function of the beam electrons in different regimes of the beam instability is investigated.     

Theory of the polarization bremsstrahlung from thermal electrons scattered by the Debye sphere of an ion in a plasma

The polarization bremsstrahlung from thermal electrons scattered by the Debye sphere of an ion in a plasma is studied in the quasiclassical approximation. The model of the local plasma frequency is used to check the validity of the asymptotic expression for the polarizability of the electron cloud of an ion in the high-frequency range. This asymptotic expression is then used to derive a formula for the intensity of the total effective polarization bremsstrahlung. The R factor (the ratio of the contribution from the polarization bremsstrahlung to the contribution from conventional static bremsstrahlung) is obtained as a function of the plasma coupling parameter and electron density in order to analyze the role of the polarization bremsstrahlung in the total bremsstrahlung of the thermal plasma electrons. The spectral intensity of the effective polarization bremsstrahlung is calculated in the rotational approximation, which was previously employed in the theory of conventional static bremsstrahlung. It is shown that the spectral intensity of the polarization bremsstrahlung from thermal electrons scattered by the Debye sphere around an ion, as compared with the polarization bremsstrahlung by fast superthermal electrons, decreases more gradually with increasing frequency.     

Plasma acceleration in current sheets formed in helium in two- and three-dimensional magnetic configurations

The processes of heating and acceleration of plasma in current sheets formed in 2D and 3D magnetic configurations with an X-line in helium plasma have been investigated using spectroscopic methods. It is found that, in 2D magnetic configurations, plasma flows with energies of 400?C1000 eV, which are substantially higher than the ion thermal energy, are generated and propagate along the width (the larger transverse dimension) of the sheet. In 3D configurations, the influence of the longitudinal (directed along the X-line) component of the magnetic field on the plasma parameters in the current sheet has been studied. It is shown that plasma acceleration caused by the Amp??re force can be spatially inhomogeneous in the direction perpendicular to the sheet surface, which should lead to sheared plasma flows in the sheet.     

Drift modes of a quasi-two-dimensional current sheet

Stability of a plasma configuration consisting of a thin one-dimensional current sheet embedded into a two-dimensional background current sheet is studied. Drift modes developing in plasma as unstable waves along the current direction are considered. Dispersion relations for kink and sausage perturbation modes are obtained depending on the ratio of parameters of thin and background current sheets. It is shown that the existence of the background sheet results in a decrease in the instability growth rates and a significant increase in the perturbation wavelengths. The role of drift modes in the excitation of oscillations observed in the current sheet of the Earth’s magnetotail is discussed.     

Two-dimensional plasma density distributions in low-pressure gas discharges

The plasma density distribution in a two-dimensional nonuniform positive column of a low-pressure gas discharge is studied in the hydrodynamic approximation with allowance for ion inertia. Exact solutions are derived for discharges in a rectangular and a cylindrical chamber. Asymptotic solutions near the coordinate origin and near the critical surface are considered. It is shown that, for potential plasma flows, the flow velocity component normal to the plasma boundary is equal to the ion acoustic velocity. The results obtained can be used to analyze the processes occurring in low-pressure plasmochemical reactors.     

Suprathermal plasma flows in current sheets formed in two- and three-dimensional magnetic configurations

Dynamics of the thermal and directed motions of argon plasma ions in current sheets formed in various magnetic configurations was investigated experimentally Measurements in three-dimensional magnetic configurations with an X line were carried out for the first time. The results of these measurements were compared with the data obtained in experiments with two-dimensional magnetic configurations. The ion temperature and the energies and velocities of directed plasma flows within the current sheet were determined by analyzing the shapes of argon ion spectral lines broadened due to the Doppler effect. It is found that, under the given experimental conditions, the axial magnetic field does not affect the ion temperature and plasma acceleration in the sheet.     

Determination of plasma density from data on the ion current to cylindrical and planar probes

To improve probe methods of plasma diagnostics, special probe measurements were performed and numerical models describing ion transport to a probe with allowance for collisions were developed. The current–voltage characteristics of cylindrical and planar probes were measured in an RF capacitive discharge in argon at a frequency of 81 MHz and plasma densities of 10¹⁰–10¹¹ cm^–3, typical of modern RF reactors. 1D and 2D numerical models based on the particle-in-cell method with Monte Carlo collisions for simulating ion motion and the Boltzmann equilibrium for electrons are developed to describe current collection by a probe. The models were used to find the plasma density from the ion part of the current–voltage characteristic, study the effect of ion collisions, and verify simplified approaches to determining the plasma density. A 1D hydrodynamic model of the ion current to a cylindrical probe with allowance for ion collisions is proposed. For a planar probe, a method to determine the plasma density from the averaged numerical results is developed. A comparative analysis of different approaches to calculating the plasma density from the ion current to a probe is performed.     

System of Kinetic Equations for Collisionless Space Plasma in the Approximation of Field-Aligned Force Equilibrium for Electrons

A system of kinetic equations describing relatively slow large-scale processes in collisionless magnetoplasma structures with a spatial resolution on the order of the proton thermal gyroradius is derived. The system correctly takes into account the electrostatic effects in the approximation of field-aligned force equilibrium for electrons. The plasma is considered quasineutral, and the magnetic field is described by the Ampère equation. The longitudinal component of the electric field is found explicitly from the equality of the field-aligned component of the electric force acting on plasma electrons and the divergence of the electron pressure tensor. The electric field component orthogonal to the magnetic field is determined by the distributions of the number densities, current densities, and stress tensors of all plasma species in the instantaneous long-range approximation described by a system of time-independent elliptic equations. Versions of the system of equations adapted to the case of magnetized electrons described by the Vlasov equation in the drift approximation, as well as to the case in which all plasma species are magnetized, are derived. The resulting systems of equations allow creating numerical models capable of describing large-scale processes in nonuniform collisionless space plasma.     

Study of the structure and dynamics of current sheets in three-dimensional magnetic configurations with an X line by holographic interferometry

Results are presented from studies of the structure and dynamics of current sheets in three-dimensional magnetic configurations with an X line by means of holographic interferometry. It is found that the efficiency of plasma compression into the sheet is reduced as the longitudinal magnetic field B _z , directed along the X line, increases. This effect is attributed to the enhancement of the longitudinal component of the magnetic field within the sheet and to the corresponding increase in the magnetic pressure. It is shown that the formation of a plasma sheet lags behind the beginning of the plasma current pulse, the delay time being close to the characteristic Alfvén time.     

Oscillation spectrum of an electron gas with a small density fraction of ions

The problem is solved of the stability of a nonneutral plasma that completely fills a waveguide and consists of magnetized cold electrons and a small density fraction of ions produced by ionization of the atoms of the background gas. The ions are described by an anisotropic distribution function that takes into account the characteristic features of their production in crossed electric and magnetic fields. By solving a set of Vlasov-Poisson equations analytically, a dispersion equation is obtained that is valid over the entire range of allowable electric and magnetic field strengths. The solutions to the dispersion equation for the m = +1 main azimuthal mode are found numerically. The plasma oscillation spectrum consists of the families of Trivelpiece-Gould modes at frequencies equal to the frequencies of oblique Langmuir oscillations Doppler shifted by the electron rotation and also of the families of “modified” ion cyclotron (MIC) modes at frequencies close to the harmonics of the MIC frequency (the frequencies of radial ion oscillations in crossed fields). It is shown that, over a wide range of electric and magnetic field strengths, Trivelpiece-Gould modes have low frequencies and interact with MIC modes. Trivelpiece-Gould modes at frequencies close to the harmonics of the MIC frequency with nonnegative numbers are unstable. The lowest radial Trivelpiece-Gould mode at a frequency close to the zeroth harmonic of the MIC frequency has the fastest growth rate. MIC modes are unstable over a wide range of electric and magnetic field strengths and grow at far slower rates. For a low ion density, a simplified dispersion equation is derived perturbatively that accounts for the nonlocal ion contribution, but, at the same time, has the form of a local dispersion equation for a plasma with a transverse current and anisotropic ions. The solutions to the simplified dispersion equation are obtained analytically. The growth rates of the Trivelpiece-Gould modes and the behavior of the MIC modes agree with those obtained by numerical simulation.     

Distributions of the ion temperature,ion pressure,and electron density over the current sheet surface

The distributions of the ion temperature, ion pressure, and electron density over the width (the major transverse dimension) of the current sheet have been studied for the first time. The current sheets were formed in discharges in argon and helium in 2D and 3D magnetic configurations. It is found that the temperature of argon ions in both 2D and 3D magnetic configurations is almost uniform over the sheet width and that argon ions are accelerated by the Ampère force. In contrast, the distributions of the electron density and the temperature of helium ions are found to be substantially nonuniform. As a result, in the 2D magnetic configuration, the ion pressure gradient across the sheet width makes a significant contribution (comparable with the Ampère force) to the acceleration of helium ions, whereas in the 3D magnetic configuration, the Ampère force is counterbalanced by the pressure gradient.     

An efficient method for extracting plasma ions in laser isotope separation systems

The possibility of using a Hall-current accelerator to extract ions from a partially ionized plasma produced by selective laser isotope photoionization of atomic vapor is examined. A mechanism for ion acceleration is investigated using one-dimensional time-dependent equations of two-fluid magnetohydrodynamics. The current cutoff due to the ion space charge is prevented by electron emission. It is shown that, at an accelerating voltage of 25–50 V and emission current density of several mA/cm², the ion component is accelerated throughout the entire plasma volume up to a velocity of ～10⁵ cm/s in a few microseconds. The influence of resonant charge exchange and secondary ionization by electrons on both the acceleration dynamics and selectivity degradation is taken into account. It is shown that the Hall-current extractor allows one to avoid selectivity degradation even when the plasma size exceeds the charge-exchange mean free path by one order of magnitude.     

Heliospheric current sheet and effects of its interaction with solar cosmic rays

The effects of interaction of solar cosmic rays (SCRs) with the heliospheric current sheet (HCS) in the solar wind are analyzed. A self-consistent kinetic model of the HCS is developed in which ions with quasiadiabatic dynamics can present. The HCS is considered an equilibrium embedded current structure in which two main plasma species with different temperatures (the low-energy background plasma of the solar wind and the higher energy SCR component) contribute to the current. The obtained results are verified by comparing with the results of numerical simulations based on solving equations of motion by the particle tracing method in the given HCS magnetic field with allowance for SCR particles. It is shown that the HCS is a relatively thin multiscale current configuration embedded in a thicker plasma layer. In this case, as a rule, the shear (tangential to the sheet current) component of the magnetic field is present in the HCS. Taking into account high-energy SCR particles in the HCS can lead to a change of its configuration and the formation of a multiscale embedded structure. Parametric family of solutions is considered in which the current balance in the HCS is provided at different SCR temperatures and different densities of the high-energy plasma. The SCR densities are determined at which an appreciable (detectable by satellites) HCS thickening can occur. Possible applications of this modeling to explain experimental observations are discussed.