3D Magnetic Holes in Collisionless Plasmas

Recent multispacecraft observations in the Earth’s magnetosphere have revealed an abundance of magnetic holes—localized magnetic field depressions. These magnetic holes are characterized by the plasma pressure enhancement and strongly localized currents flowing around the hole boundaries. There are several numerical and analytical models describing 2D configurations of magnetic holes, but the 3D distribution of magnetic fields and electric currents is studied poorly. Such a 3D magnetic field configuration is important for accurate investigation of charged particle dynamics within magnetic holes. Moreover, the 3D distribution of currents can be used for distant probing of magnetic holes in the magnetosphere. In this study, a 3D magnetic hole model using the single-fluid approximation and a spatial scale hierarchy with the distinct separation of gradients is developed. It is shown that such 3D holes can be obtained as a generalization of 1D models with the plasma pressure distribution adopted from the kinetic approach. The proposed model contains two magnetic field components and field-aligned currents. The magnetic field line configuration resembles the magnetic trap where hot charged particles bounce between mirror points. However, the approximation of isotropic pressure results in a constant plasma pressure along magnetic field lines, and the proposed magnetic hole model does not confine plasma along the field direction.     

Pseudosymmetry near a magnetic surface in a plasma confinement system

The possibility is demonstrated of finding vacuum equilibrium magnetic configurations with an exactly pseudosymmetric nonparaxial boundary magnetic surface in the vicinity of which the pseudosymmetry condition is satisfied approximately. Equations are derived for calculating the boundary surface from a prescribed particular dependence of the magnetic field strength in special magnetic flux coordinates. In calculations, magnetic coordinates serve as ordinary angular coordinates, while their “magnetic” character is specified by additional integral conditions. As an example, a “tubular” orthogonal magnetic surface is calculated analytically.     

Analytic examples of force-free toroidal MHD equilibria

Analytically described toroidal (axisymmetric and three-dimensional) equilibrium magnetic field configurations with a “flat” current density, j=λB (λ = const), are proposed. Such configurations are superpositions of several force-free two-dimensional configurations with plane, axial, or helical coordinate symmetry. Each of them is generated by an exact partial solution to the corresponding Grad-Shafranov equation. A variety of toroidal configurations thus obtained allows one to model topological changes of magnetic surfaces, such as magnetic axis splitting (doublets) in axisymmetric equilibrium configurations and the appearance and interaction of magnetic islands and ergodic lines in three-dimensional configurations.     

Boundary of the adiabatic motion of a charged particle in a dipole magnetic field

The motion of a charged particle in a dipole magnetic field is considered using a quasi-adiabatic model in which the particle guiding center trajectory is approximated by the central trajectory, i.e., a trajectory that passes through the center of the dipole. A study is made of the breakdown of adiabaticity in the particle motion as the adiabaticity parameter χ (the ratio of the Larmor radius to the radius of the magnetic field line curvature in the equatorial plane) increases. Initially, for χ?0.01, the magnetic moment μ of a charged particle undergoes reversible fluctuations, which can be eliminated by subtracting the particle drift velocity. For χ?0.1, the magnetic moment μ undergoes irreversible fluctuations, which grow exponentially with χ. Numerical integration of the equations of motion shows that, during the motion of a particle from the equatorial plane to the mirror point and back to the equator in a coordinate system related to the central trajectory, the analogue of the magnetic moment μ is conserved. In the equatorial plane, this analogue undergoes a jump. The long-term particle dynamics is described in a discrete manner, by approximating the Poincaré mapping. The existence of the regions of steady and stochastic particle motion is established, and the boundary between these regions is determined. The position of this boundary depends not only on the adiabaticity parameter χ but also on the pitch angle. The calculated boundary is found to agree well with that obtained previously by using the model of a resonant interaction between particle oscillations associated with different degrees of freedom.     

Trapping and Scattering of a Relativistic Charged Particle by Resonance in a Magnetic Field and an Electromagnetic Wave

A study is made of the dynamics of a relativistic charged particle in an electromagnetic wave (with an electrostatic component) in a constant uniform magnetic field. A system with a high-frequency wave is a Hamiltonian system with two degrees of freedom and with fast and slow variables. The trapping of a particle into resonance and its scattering on resonance in such a system is described.     

Islands and ergodicity of the plasma current in toroidal magnetic confinement systems

Magnetic and current structures arising due to resonant perturbations of an equilibrium current-carrying magnetic configuration are analyzed using the Hamiltonian formalism. Special attention is paid to axisymmetric tokamak and pinch configurations. It is shown that, due to the very different dependences of the magnetic and current rotational transforms on the plasma pressure, the resonances (islands) of the magnetic field may not coincide with those of the current. The perturbed force-free equilibrium of a cylindrical pinch in which the field and current islands overlap is analyzed. The long-lived ribbon structures observed in the JET tokamak are explained as a manifestation of a force-free magneto-current island.     

Plasma equilibrium in a double-dipole magnetic confinement system with a separatrix

Results are presented from theoretical studies of plasma equilibrium consistent with the convective stability of ideal interchange modes in axisymmetric configurations with an outward-decreasing field that may have a separatrix limiting the plasma volume. A two-dimensional numerical code is developed to solve the Grad-Shafranov equation with a convectively stable pressure distribution at an arbitrary value of β. The problem is solved for an actual geometry of the magnetic field produced by thin current rings. Configurations of a double-dipole confinement system are calculated for the parameters measured in experiments carried out in the Magnetor device, as well as for higher β values. A configuration of a model mirror system with a divertor is also calculated. The code allows one to optimize confinement systems operating at high β values at which equilibrium still can exist.     

Theory of heat transport in a magnetized high-temperature plasma

The transport of charged particles across a strong magnetic field with a small random component is studied in the double diffusion approximation. It is shown that the density of the particles whose initial distribution is stretched along the field satisfies a subdiffusion equation with fractional derivatives. A more general initial particle distribution is also considered, and the applicability of the solutions obtained is discussed.     

Development of beam-plasma instability during the injection a low-energy electron beam into the ionospheric plasma

Results are presented from an active experiment on the injection of charged particle beams into the ionospheric plasma. The experiment was carried out in 1992 onboard the Intercosmos-25 satellite and the Magion-3 daughter satellite (APEX). A specific feature of this experiment was that both the ion and electron beams were injected upward, in the same direction along the magnetic field. The most interesting results are the excitation of HF and VLF-LF waves and the generation of fast charged particle flows, which were recorded on both satellites.     

Asymmetric configurations of a thin current sheet with a constant normal magnetic field component

A possible mechanism for the formation of a quasi-equilibrium asymmetric current sheet in the magnetospheric tail due to the asymmetry of peripheral plasma sources is analyzed using a self-consistent particle- in-cell model of a thin collisionless current sheet with a constant normal magnetic field component. For the case in which the current sheet is produced by only one source, quasi-equilibrium sheet configurations with maximum possible asymmetry are obtained for different input parameters of the model. In such configurations, the equilibrium force balance is satisfied with high accuracy and the shape of the current density profile remains nearly symmetric, but the current sheet itself is slightly shifted from the source as compared to the symmetric case. The configurations obtained using numerical simulations are compared with those calculated using the previous analytical model of a thin current sheet. It is found that the results provided by these models agree well both qualitatively and quantitatively.     

Effect of the magnetic field on the resonant particle acceleration

The acceleration of charged particles trapped by a potential wave in a magnetic field is investigated as applied to the problem of the generation of fast particles in a laser plasma. The conditions for unlimited particle acceleration are determined, and the spectra of fast particles are found.     

Two-dimensional electrodynamic structure of the normal glow discharge in an axial magnetic field

Results are presented from numerical simulations of an axisymmetric normal glow discharge in molecular hydrogen and molecular nitrogen in an axial magnetic field. The charged particle densities and averaged azimuthal rotation velocities of electrons and ions are studied as functions of the gas pressure in the range of 1–5 Torr, electric field strength in the range of 100–600 V/cm, and magnetic field in the range of 0.01–0.3 T. It is found that the axial magnetic field does not disturb the normal current density law.     

Local surface equilibrium equations for currentless magnetic configurations

Invariant local surface equilibrium equations are derived that interrelate the absolute value of the magnetic field B, the absolute value of the gradient of the magnetic flux |?Φ|, the local shear s, and the plasma pressure on nested equilibrium magnetic surfaces in currentless configurations. Examples of applying these equations to analysis of symmetric and isodynamic equilibria are considered.     

Particle confinement in axisymmetric poloidal magnetic field configurations with zeros of B: Methodological note

Collisionless particle confinement in axisymmetric configurations with magnetic field nulls is analyzed. The existence of an invariant of motion—the generalized azimuthal momentum—makes it possible to determine in which of the spatial regions separated by magnetic separatrices passing through the magnetic null lines the particle occurs after it leaves the vicinity of a magnetic null line. In particular, it is possible to formulate a sufficient condition for the particle not to escape through the separatrix from the confinement region to the external region. In the configuration under analysis, the particles can be lost from a separatrix layer with a thickness on the order of the Larmor radius because of the nonconservation of the magnetic moment μ. In this case, the variations in μ are easier to describe in a coordinate system associated with the magnetic surfaces. An analysis is made of the applicability of expressions for the single-pass change Δμ in the magnetic moment that were obtained in different magnetic field models for a confinement system with a divertor (such that there is a circular null line).     

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.     

Loal three-dimensional MHD stability in equilibrium optimized coordinates

The linear equation for ideal magnetohydrodynamic ballooning modes in three-dimensional configurations is derived in the coordinate system that is optimal for the representation of the equilibrium state. The magnetic field lines in this coordinate system, however, are not straight. The form of the Mercier criterion that is currently in use is recovered from the asymptotic analysis of the ballooning equation. To determine the parallel-current density, a magnetic differential equation expressed in the optimal coordinates must be inverted.     

Trapping of high-energy electrons into regime of surfatron acceleration by electromagnetic waves in space plasma

The phenomenon of trapping of weakly relativistic charged particles (with kinetic energies on the order of mc ²) into a regime of surfatron acceleration by an electromagnetic wave that propagates in plasma across a weak external magnetic field has been studied using nonlinear numerical calculations based on a solution of the relativistic equations of motion. Analysis showed that, for the wave amplitude above a certain threshold value and the initial wave phase outside the interval favorable for the surfing regime, the trajectory of a charged particle initially corresponds to its cyclotron rotation in the external magnetic field. For the initial particle energies studied, the period of this rotation is relatively short. After a certain number (from several dozen to several thousand and above) of periods of rotation, the wave phase takes a value that is favorable for trapping of the charged particle on its trajectory by the electromagnetic wave, provided the Cherenkov resonance conditions are satisfied. As a result, the wave traps the charged particle and imparts it an ultrarelativistic acceleration. In momentum space, the region of trapping into the regime of surfing on an electromagnetic wave turns out to be rather large.     

Local geometrical properties of magnetic configurations with nested equilibrium magnetic surfaces

The complete set of universal local relationships between geometrical (the curvature and torsion of the force lines of the magnetic field and the field complementary to it) and magnetic (|B|, |?Φ|, b · (? × b), and the local shear s) quantities in currentless magnetic configurations comprising a system of equilibrium nested magnetic surfaces, including those with several magnetic axes, is derived. Possible applications of these relationships are discussed.     

Distinctive features of collisionless gradient drift instabilities in a high-β plasma in a highly nonuniform magnetic field

A set of Vlasov-Maxwell equations for collisionless electromagnetic drift instabilities of high-β plasma configurations with a nonuniform magnetic fields is solved. The effect of the transverse static magnetic field variation and magnetic field line curvature, as well as the plasma temperature and density gradients, is considered. It is shown that, in a nonuniform magnetic field, the behavior of the instabilities differs substantially from that in a uniform field. Electromagnetic modes propagating strictly transverse to the lines of the static magnetic field are analyzed in detail, and unstable solutions are obtained for both extraordinary and ordinary waves. Numerical results show that, in the latter case, instability occurs when the magnetic field decreases toward the periphery and the plasma temperature and density gradients are oppositely directed.