Geometric properties of magnetic field lines on toroidal magnetic surfaces in the context of plasma equilibrium

An analysis of plasma equilibrium in a magnetic confinement system includes studies of how the shape of the magnetic surfaces is distorted with varying magnitude and profile of the plasma pressure. Such studies allow one, in particular, to determine the maximum β value consistent with equilibrium, β_eq, i.e., the maximum plasma pressure above which the equilibrium in a confinement system under analysis is impossible. Since the magnetic field lines form magnetic surfaces, their global relationship with equilibrium is obvious. Here, special attention is paid to a local relationship between equilibrium and geometric properties of the magnetic field lines.     

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.     

On the boundary of an equilibrium plasma near the magnetic separatrix in a tokamak

It is shown that plasma rotation near the tokamak the wall can result in a shift of the isobaric separatrix with respect to the magnetic one. This shift is calculated analytically, and this effect is exemplified by simple plasma equilibrium states. The plasma rotation that causes the shift of the separatrices can be driven either by a nonzero radial electric field at the plasma edge or by the Hall effect, which may take place even in the absence of the electric field.     

Isodynamic axisymmetric equilibrium near the magnetic axis

Plasma equilibrium near the magnetic axis of an axisymmetric toroidal magnetic confinement system is described in orthogonal flux coordinates. For the case of a constant current density in the vicinity of the axis and magnetic surfaces with nearly circular cross sections, expressions for the poloidal and toroidal magnetic field components are obtained in these coordinates by using expansion in the reciprocal of the aspect ratio. These expressions allow one to easily derive relationships between quantities in an isodynamic equilibrium, in which the absolute value of the magnetic field is constant along the magnetic surface (Palumbo’s configuration).     

Electromagnetic oscillations near the critical surface in a magnetized plasma

It is shown that, in a plasma whose density varies across the magnetic field lines, electromagnetic oscillations that are localized near the critical surface can exist. Such oscillations can be excited spontaneously in a nonequilibrium plasma of closed magnetic confinement systems.     

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.     

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.     

Sructure of steady-state Debye layers in a low-density plasma near a dielectric surface

Exact steady-state analytic solutions describing kinetic processes in a low-density plasma layer near a dielectric surface are found in a time-dependent one-dimensional model with allowance for secondary electron emission. It is shown that, at low electron temperatures, both the electric potential and electron density monotonically decrease toward the dielectric surface (Debye layer). As the electron temperature increases, an anti-Debye layer first forms, in which the potential monotonically increases toward the wall, and regimes with a nonmonotonic potential profile then arise.     

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.     

Kinetics of a low-density plasma near a dielectric surface with account for secondary electron emission

Exact steady solutions in a one-dimensional kinetic model of the processes in a low-density plasma layer near a dielectric surface are constructed analytically with allowance for secondary electron emission. It is shown that, for low electron temperatures, the solutions describe a regime in which the electric potential and electron density decrease monotonically toward the dielectric wall (a classical Debye layer). For higher electron temperatures, there are solutions describing regimes such that the electric potential and electron density increase monotonically toward the wall (an inverse Debye layer).     

Matching a surface to a given sectional curve

We describe an algorithm to position a rigid surface so as to make its cross-section by a given plane match a given curve in that plane, a problem relevant to model-based medical imaging. After building an atlas of cross-sections of the surface and searching it for a best position to start from, each iteration of the algorithm (1) determines a vector field along the intersection curve that will improve its matching with the target curve, and (2) computes and applies a small displacement of the surface whose effect on the intersection will approximate best the required vector field. Computations use least-square techniques, an exponential formula for Lie groups of transformations, and generic properties of cross-sections. Experiments with an implementation are reported and theoretical tools for justifying and improving the algorithm, some of them based on Catastrophe Theory, are outlined.     

Numerical simulations of plasma equilibrium in a one-dimensional current sheet with a nonzero normal magnetic field component

The force balance in a thin collisionless current sheet in the Earth’s magnetotail with a given constant magnetic field component B _z across the sheet is numerically studied for the first time in a self-consistent formulation of the problem. The current sheet is produced by oppositely directed plasma flows propagating from the periphery of the sheet toward the neutral plane. A substantially improved version of a macroparticle numerical model is used that makes it possible to simulate on the order of 10⁷ macroparticles even with a personal computer and to calculate equilibrium configurations with a sufficiently low discrete noise level in the first-and second-order moments of the distribution function, which determine the stress tensor elements. Quasisteady configurations were calculated numerically for several sets of plasma parameters in some parts of the magnetotail. The force balance in the sheet was checked by calculating the longitudinal and transverse pressures as well as the elements of the full stress tensor. The stress tensor in the current sheet is found to be nondiagonal and to differ appreciably from the gyrotropic stress tensor in the Chew-Goldberger-Low model, although the Chew-Goldberger-Low theory and numerical calculations yield close results for large distances from the region of reversed magnetic field.     

Three-dimensional rotational plasma flows near solid surfaces in an axial magnetic field

A rotational flow of a conducting viscous medium near an extended dielectric disk in a uniform axial magnetic field is analyzed in the magnetohydrodynamic (MHD) approach. An analytical solution to the system of nonlinear differential MHD equations of motion in the boundary layer for the general case of different rotation velocities of the disk and medium is obtained using a modified Slezkin–Targ method. A particular case of a medium rotating near a stationary disk imitating the end surface of a laboratory device is considered. The characteristics of a hydrodynamic flow near the disk surface are calculated within the model of a finite-thickness boundary layer. The influence of the magnetic field on the intensity of the secondary flow is studied. Calculations are performed for a weakly ionized dense plasma flow without allowance for the Hall effect and plasma compressibility. An MHD flow in a rotating cylinder bounded from above by a retarding cap is considered. The results obtained can be used to estimate the influence of the end surfaces on the main azimuthal flow, as well as the intensities of circulating flows in various devices with rotating plasmas, in particular, in plasma centrifuges and laboratory devices designed to study instabilities of rotating plasmas.     

Bifurcations of axisymmetric plasma equilibrium in a tokamak

Bifurcation of solutions to the Grad–Shafranov-type equation for helically symmetric plasma near the threshold for tearing instability are analyzed. Quadratic and cubic nonlinearities were added to the linear dependence of the current density on the helical flux. Depending on the character of nonlinearity, two types of bifurcation can be observed, the “small” and the “large” ones. The small bifurcation is typical of cubic nonlinearity and reveals itself in the growth of the magnetic island from zero as the profile parameter increases above the instability threshold. The large bifurcation is typical of quadratic nonlinearity and causes jumplike formation of a large-scale magnetic island upon exceeding the instability threshold. As the profile parameter decreases below the instability threshold, the large-scale island continues to persist for some time (the hysteresis effect) and then suddenly disappears.     

Hydration dynamics near a model protein surface

The evolution of water dynamics from dilute to very high concentration solutions of a prototypical hydrophobic amino acid with its polar backbone, N-acetyl-leucine-methylamide (NALMA), is studied by quasi-elastic neutron scattering (QENS) and molecular dynamics (MD) simulation for both the completely deuterated and completely hydrogenated leucine monomer. The NALMA-water system and the QENS data together provide a unique study for characterizing the dynamics of different hydration layers near a prototypical hydrophobic side chain and the backbone of which it is attached. We observe several unexpected features in the dynamics of these biological solutions under ambient conditions. The NALMA dynamics shows evidence of de Gennes narrowing, an indication of coherent long timescale structural relaxation dynamics. The translational and rotational water dynamics at the highest solute concentrations are found to be highly suppressed as characterized by long residential time and slow diffusion coefficients. The analysis of the more dilute concentration solutions models the first hydration shell with the 2.0 M spectra. We find that for outer layer hydration dynamics that the translational diffusion dynamics is still suppressed, although the rotational relaxation time and residential time are converged to bulk-water values. Molecular dynamics analysis of the first hydration shell water dynamics shows spatially heterogeneous water dynamics, with fast water motions near the hydrophobic side chain, and much slower water motions near the hydrophilic backbone. We discuss the hydration dynamics results of this model protein system in the context of protein function and protein-protein recognition.     

Bifurcation of the equilibrium of a current-carrying plasma column

The classical (for the Grad-Shafranov equation) formulation of the equilibrium problem for a cylindrical current-carrying plasma column is shown to admit multiple solutions. The multiple solutions are bifurcational in character and appear due to the nonlinearity of the equilibrium equation. This was demonstrated analytically using a stepped current profile as an example. Bifurcational solutions found for the cylindrical case survive in toroidal geometry too.     

Ray trajectories near the electron cyclotron resonance surface in an axisymmetric magnetic trap

Characteristic features of the propagation of electromagnetic electron cyclotron waves in the vicinity of the electron cyclotron resonance surface are investigated both analytically and numerically with allowance for variation in the magnetic field strength and a corresponding variation in the magnetic field direction. It is demonstrated that variation in the magnetic field direction can qualitatively change the wave propagation pattern and can markedly affect the efficiency of electron cyclotron resonance plasma heating in an axisymmetric magnetic trap.