Plasma equilibrium in axisymmetric poloidal magnetic field configurations in flux coordinates

A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations—a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.     

MHD stability of a finite-pressure plasma in axisymmetric configurations of the poloidal magnetic field

A set of linear integrodifferential equations is presented for the plasma displacement components that minimize the Kruskal-Oberman functional of the potential energy of an MHD perturbation. Marginal stability results when the smallest eigenvalue of this set of equations is zero.     

Peculiarities of collisionless drift instabilities in poloidal magnetic configurations

Results of the numerical analysis of collisionless drift instabilities as applied to magnetic configurations with a purely poloidal magnetic field are presented. Instabilities caused by the gradients of the ion and electron temperatures and plasma density are considered within a wide range of wavenumbers. An electromagnetic model taking into account the finite plasma pressure and magnetic field curvature is formulated for configurations with a nonuniform magnetic field.     

Model magnetic field configurations with islands

Helical perturbations of the tokamak magnetic field can give rise to magnetic islands in the vicinity of the rational magnetic surfaces at which the pitch of the magnetic field lines coincides with that of the perturbation. The widely known relationship between the magnetic island width and the perturbation amplitude is valid under the assumptions that the island width is small in comparison to the radius of the rational surface and that the perturbation amplitude is constant in the radial direction. The latter assumption indicates that the island width is small in comparison to the radial size of the region where the perturbation current is localized. The calculations carried out for four model magnetic field configurations show that the geometry of the magnetic islands depends on the extent to which the perturbation current is localize and that the width of the magnetic islands is smaller than that calculated from the familiar relationship. The larger the perturbation amplitude, the greater this difference: it may be as large as 25% for the strong perturbations arising during disruptions. The calculations are based on the solution of the geometric problem of constructing the lines of the magnetic field determined by the given distributions of the initial current and perturbation current; the equilibrium equation is not considered. The question of the direction of the perturbation current within the island relative to the direction of the initial unperturbed current is discussed. The perturbation current flowing in an island is directed opposite to the initial current with a radially decreasing density; for this reason, such an island can naturally be called a “negative” island. Together with the formation of negative islands, the formation of “positive” ones is also considered. The latter are shown to form under the following conditions: the perturbation current density should be higher than the density of the current that produces the unperturbed field and the perturbation current itself should be localized in a sufficiently narrow radial layer. The positive islands are smaller in size than negative ones.     

Role of the mean curvature in the geometry of magnetic confinement configurations

Examples are presented of how the geometric notion of the mean curvature is applied to the vector of a general magnetic field and to magnetic surfaces. It is shown that the mean curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curvature are optimum for plasma confinement in multimirror open confinement systems and rippled tori.     

Ray trajectories and electron cyclotron absorption in an axisymmetric magnetic confinement system

The propagation and cyclotron absorption of electromagnetic waves lauched into an axisymmetric magnetic confinement system in a quasi-longitudinal direction is studied in the geometrical-optics approximation. Cyclotron absorption in a dense plasma is described by solving an approximate dispersion relation that is valid for small angles between the magnetic field B and the wave vector k. This approach makes it possible to avoid the difficulty associated with a transition from large propagation angles to the case of strictly longitudinal propagation. It is shown that electron cyclotron absorption can be efficient only when the plasma density is below the critical density. The numerical results obtained for the device for producing multicharged ions in experiments at the Institute of Applied Physics of the Russian Academy of Sciences agree qualitatively with the experimental data.     

MHD stability condition of an anisotropic-pressure plasma in axially symmetric confinement systems formed by a poloidal field

A set of integrodifferential (over the longitudinal coordinate) equations for the transverse components of the plasma displacement minimizing the Kruskal-Oberman functional of the potential energy of MHD perturbations is derived. The stability condition corresponds to the absence of negative eigenvalues of this set for any magnetic surface in plasma.     

Stability of localized MHD perturbations in confinement systems with closed magnetic field lines

Conditions are determined for the stability of a finite-pressure plasma against perturbations localized near a magnetic field line in a magnetic confinement system without average minimum-B. The marginal stability (ω²=0) is achieved at the pressure profile p∝U ^?5/3 (where $U = \oint {\frac{{dl}}{B}}$ ), provided that the pressure is lower than a certain critical value above which an unstable incompressible mode in which the displacement as a function of the coordinate along the field line has zeros appears at some magnetic field line.     

Correction of the axial asymmetry of the poloidal magnetic field in the Globus-M spherical tokamak

The toroidal inhomogeneity of the poloidal magnetic field—the so-called error fields that arise due to imperfections in manufacturing and assembling of the electromagnetic system-was measured in the Globus-M spherical tokamak. A substantial inhomogeneity corresponding to the n = 1 mode, which gave rise to a locked mode and led to discharge disruption, was revealed. After compensation of this inhomogeneity with the help of special correction coils, the discharge duration increased and the global plasma parameters improved substantially. A technique for determining and compensating the n = 1 mode inhomogeneity is described, the measured dependences of the penetration threshold of the m = 2/n = 1 mode on the plasma parameters are given, and results of experiments in which record parameters for the Globus-M tokamak were achieved after correction of the poloidal magnetic field are presented.     

Improvement of collisionless particle confinement in a non-quasi-symmetric stellarator vacuum magnetic field

A non-quasi-symmetric stellarator vacuum magnetic field with an aspect ratio of about 11 is found in which collisionless particles are confined up to about 2/5 of the minor radius.     

Steady-state axisymmetric configurations of a weakly ionized plasma in the field of a rotating magnetized spherical body

Self-consistent steady-state axisymmetric configurations of a plasma envelope with a uniform anisotropic conductivity around a rotating magnetized spherical body are considered. A set of electrodynamic and magnetohydrodynamic equations is analyzed under the assumption that the mass velocity of a moving weakly ionized plasma has only the azimuthal component. The equations describing the profile of the angular frequency of the rotating plasma envelope, the magnetic field, the conduction currents, and the plasma density distribution are solved in the limit of a strong anisotropy of the conductivity of a weakly ionized gas. The applicability of the results obtained to a qualitative interpretation of the phenomena occurring in the plasmaspheres of magnetized planets is discussed.     

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.     

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.     

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.     

Convectively stable pressure profile in magnetic confinement systems with internal rings

A convectively stable pressure profile in a long multiple-mirror (corrugated) magnetic confinement system with internal current-carrying rings is calculated. The plasma energy content in the axial region can be increased by using an internal ring that reverses the on-axis magnetic field direction and gives rise to an average magnetic well near the axis. The pressure profile in the outer region—outside the magnetic well—is considered in detail. It is shown that, in the radial pressure profile, a pedestal can be formed that leads to a higher pressure drop between the center and the plasma edge. The pressure profile is calculated from the Kruskal-Oberman criterion—a necessary and sufficient condition for the convective stability of a collisionless plasma. The revealed pedestal arises near the boundary of the average magnetic well in the region of the smallest but alternating-sign curvature of the magnetic field lines due to a break in the convectively stable pressure profile. Such a shape of the stable pressure profile can be attributed to the stabilizing effect of the alternating-sign curvature of the field lines in the multiple-mirror magnetic confinement systems under consideration.     

Electromagnetic oscillations near the critical surface in a plasma: Methodological note

Agreement between different approaches to studying the propagation of electromagnetic oscillations near the critical surface is elucidated. The propagation of plane waves, electromagnetic rays, and wave beams are analyzed. The results obtained are valid when the angles between the magnetic field and the plasma density gradient are not too small.     

Relaxation of plasma rotation in toroidal magnetic confinement systems

A study is made of the relaxation of plasma rotation in nonaxisymmetric toroidal magnetic confinement systems, such as stellarators and rippled tokamaks. In this way, a solution to the drift kinetic equation is obtained that explicitly takes into account the time dependence of the distribution function, and expressions for the diffusive particle fluxes and longitudinal viscosity are derived that make it possible to write a closed set of equations describing the time evolution of the ambipolar electric field E and the longitudinal (with respect to the magnetic field) plasma velocity U₀. Solutions found to the set of evolutionary equations imply that the relaxation of these two parameters to their steady-state values occurs in the form of damped oscillations whose frequency is about 2v_T/R (where v_T is the ion thermal velocity and R is the major plasma radius) and whose damping rate depends on the ion-ion collision frequency and on the magnetic field parameters. In particular, it is shown that, for tokamaks with a slightly rippled longitudinal magnetic field, the frequency of oscillations in the range q>2 (where q is the safety factor) is, as a rule, much higher than the damping rate. For stellarators, this turns out to be true only of the central plasma region, where the helical ripple amplitude ? of the magnetic field is much smaller than the toroidal ripple amplitude δ=r/R.