Volume 5, Issue 6, June 1998
 REVIEW ARTICLE


Thermal equilibria and thermodynamics of trapped plasmas with a single sign of charge
View Description Hide DescriptionPlasmas consisting exclusively of particles with a single sign of charge (e.g., pure electron plasmas and pure ion plasmas) can be confined by static electric and magnetic fields (e.g., in a Penning trap) and also be in a state of global thermal equilibrium. This important property distinguishes these totally unneutralized plasmas from neutral and quasineutral plasmas. This paper reviews the conditions for and structure of the thermal equilibrium states and then develops a thermodynamic theory of the trapped plasmas. Thermodynamics provides hundreds of general relations (Maxwell relations) between partial derivatives of thermodynamic variables with respect to one another. Thermodynamic inequalities place general and useful bounds on various quantities. General and relatively simple expressions are provided for fluctuations of the thermodynamic variables. In practice, trapped plasmas are often made to evolve through a sequence of thermal equilibrium states through the slow addition (or subtraction) of energy and angular momentum (say, by laser cooling and torque beams). A thermodynamic approach to this late time transport describes the evolution through coupled ordinary differential equations for the thermodynamic variables, which is a huge reduction in complexity compared to the partial differential equations typically required to describe plasmatransport. These evolution equations provide a theoretical basis for the dynamical control of the plasmas.
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 LETTERS


Sheath over a rough surface in a tilted magnetic field
View Description Hide DescriptionIn this paper a sheath near the wall whose surface is “rough,” containing random bumps with a height comparable to the distance between them is considered. The magnetic field is assumed to form a shallow angle with the wall (as in a tokamakdivertor) and the scale of nonuniformities is assumed greater than the ion gyroradius. The following new effects caused by surface roughness are identified: (1) The fraction of the surface “wetted” by the plasma becomes much smaller than for a flat wall. (2) The ion subsheath is broadened and the diamagnetic current in it acquires a complex threedimensional structure. (3) Crossfield nonuniformities of the plasma potential appear in the bulk plasma.
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 ARTICLES


Rotating toroidal equilibria of quasineutral and nonneutral plasmas
View Description Hide DescriptionStarting from the fundamental ion and electron fluid equations, a “master” equilibrium equation set is constructed for axisymmetric toroidal plasmas. These equations retain the effects of different ion and electron densities, temperatures, sheared rotation, and electric forces. The master equation is then examined in various limits including the nonneutral plasma limit. An ordering is assumed which makes the effects of the Reynolds stress,pressure, magnetic stress, and electric field stress all comparable emphasizing the natural transition from the nonneutral case to the quasineutral one. For subrelativistic flows, charge separation effects are only significant for nearly forcefree plasmas. A set of equilibrium equations are derived for three different cases: (A) (B) , and (C) and . In the pure electron plasma case, the resulting equilibrium equation includes the effects of toroidal rotation, poloidal magnetic field, and electron pressure extending the equation of Daugherty and Levy [Phys. Fluids 10, 155 (1967)].

Effect of rotation on the Rayleigh–Taylor instability of two superposed magnetized conducting plasma
View Description Hide DescriptionThe study of the Rayleigh–Taylor (RT) instability of two superposed conducting fluids acted upon by a uniform horizontal magnetic field is discussed. The usual magnetohydrodynamic(MHD)equations, with modifications to include the effect of finite Larmor radius corrections (FLR) and rotation, are taken for the description of the problem. The fluid is assumed to be dusty, and dust in the form of suspended particles is considered. The equations of the problem are linearized, and using necessary boundary conditions, a general dispersion relation for the RT instability of two superposed plasma is obtained. The dispersion relation shows no contribution of the magnetic field on the condition of the RT instability. It is also found that rotation, FLR, and suspended particles do not affect the condition of the RT instability. The different special cases are discussed to find the effect of different parameters on the growth and condition of the RT instability.

On the resonant particle dynamics in the field of a finiteamplitude circularly polarized wave propagating along the axis of a magnetic trap
View Description Hide DescriptionThe motion of electrons in gyroresonance with a quasimonochromatic transverse electromagnetic wave propagating along a weakly nonuniform axially symmetric external magnetic field is analyzed. With allowance made for spatial dependencies of the wave amplitude and wave number, approximate equations of motion are derived via an averaging over the fast Larmor gyration. Wave amplitude finiteness effects and interrelationships between exact and approximate invariants of the motion are discussed.

Interchange instabilities in a partially ionized plasma
View Description Hide DescriptionIn a partially ionized plasma, the charged particle population is coupled to the neutrals through charge exchange,ionization and recombination. An interchange instability is identified in which the driving factor is the neutral pressure gradient and the dominant collisional process is charge exchange. Regimes are considered in which the mean free path for neutral collisions with ions is small in comparison to the macroscopic length scales. The instability is analogous to a gravitational interchange mode with an effective gravity where is the collision frequency of ions with neutrals. The neutral flow results from a balance between the neutral pressure gradient and collisional friction with the ions. An arbitrary dispersion relation is derived using fluid equations to describe the ions and the Boltzmann equation to describe the neutrals. This dispersion relation contains viscous and inertial effects and is substantially altered in the presence of realistic parallel wavelength due to a coupling between the unstable interchange mode and a stabilizing shear Alfvén mode. Reasonable conditions under which the modes may exist are examined and the possible relevance to divertor plasmas is considered.

Relativistic guidingcenter dynamics in the presence of strong flows
View Description Hide DescriptionA Lagrangian approach to relativistic particle guidingcenter dynamics in strongly magnetized plasmas, based on a covariant formulation, is presented, proving that a fourdimensional transformation is required in order to define the correct relativistic gyrophaseaveraged Lagrangian. Comparisons with previous approaches show that these become potentially inconsistent, particularly for magnetoplasmas exhibiting relativistic massflow velocities and/or deeply relativistic particles, causing the paradox of hyperrelativistic guidingcenter velocities. To avoid this inconvenience, the relativistic velocity addition law is adopted to describe the particle motion with respect to the reference frame locally at rest with the fluid. Based on the adoption of the relativistic Hamilton principle, new expressions are determined for the relativistic guidingcenter velocity transformation, the relativistic magnetic moment and the relativistic gyrokinetic equations of motion.

Numerical simulation and theory of generation of electromagnetic waves in the presence of whistler turbulence
View Description Hide DescriptionThe generation of mode electromagnetic waves from whistlerturbulence by plasmamasereffect is studied both numerically and theoretically. Simulations using a twodimensional electromagnetic and relativistic particle code show that high frequency electromagnetic waves with righthanded polarization (mode) can be generated by a plasmamaser mechanism from low frequency whistler waves excited by electron temperature anisotropy. Results obtained theoretically are in good agreement with numerical findings.

Phase conjugation in the low reflectivity regime by nearly degenerate fourwave mixing in a homogeneous plasma
View Description Hide DescriptionAccounting for both the long and shortwavelength density gratings by considering the polarizations of all four electromagnetic waves parallel to each other and a finite tilt angle between the signal and pump waves, the steadystate phase conjugation in the low reflectivity regime by nearly degenerate fourwave mixing in a homogeneous plasma is investigated. The response of the density gratings caused by the beating of the signal and pump waves is considered from the theory of stimulated Brillouin scattering in a twocomponent plasma. The expression for the power reflectivity of the conjugate wave for the arbitrary difference frequency between the signal and pump waves and that for resonant fourwave mixing are obtained. Numerical results have been discussed in light of the experiment reported in the literature. It is noted that the measured values of the difference frequency corresponding to either density grating resonance, and the corresponding maximum reflectivity of the conjugate wave, provide a diagnostic tool for the electron temperature and the electron–ion temperature ratio, respectively.

Selfconsistent meanfield forces in turbulent plasmas: Current and momentum relaxation
View Description Hide DescriptionThe properties of turbulentplasmas are described using the twofluid equations. Global constraints for the fluctuationinducedmeanfield forces that act on the ion and electron fluids are derived. These constraints imply functional forms for the parallel meanfield forces in the Ohm’s law and the momentum balance equation. These forms suggest that the fluctuations attempt to relax the plasma to a state where both the current and the bulk plasma momentum are aligned along the mean magnetic field with proportionality constants that are global constants. Observations of flow profile evolution during discrete dynamo activity in reversed field pinch experiments are interpreted.

Interaction of sound waves with slow dissipative layers in anisotropic plasmas in the approximation of weak nonlinearity
View Description Hide DescriptionNonlinear theory of driven magnetohydrodynamics waves in the slow dissipative layer in anisotropic plasmas developed by Ballai, Ruderman, and Erdélyi [Phys. Plasmas 5, 252 (1998)] is used to study the nonlinear interaction of sound waves with static onedimensional anisotropic plasmas. The magnetic configuration consists of a nonhomogeneous magnetic slab sandwiched by a homogeneous magneticfree plasma and by a homogeneous magnetic plasma. Sound waves launched from the magneticfree plasma propagate into the inhomogeneous region interacting with the localized slow dissipative layer and are partially reflected, dissipated or transmitted by this region. The nonlinearity parameter, introduced by Ballai, Ruderman, and Erdélyi, is assumed to be small and a regular perturbation method is used to obtain analytical wave solutions. The main effects caused by nonlinearity in the dissipative layer are the decrease in the absorption coefficient and the generation of higher harmonics in the outgoing wave in addition to the fundamental one. We also found that the nonlinear coefficient of wave energy absorption does depend on the particular type of dissipation operating in the slow dissipative layer.

External inductance of large to ultralow aspectratio tokamak plasmas
View Description Hide DescriptionThis paper presents a method for calculating the external inductance and mutual inductance coefficients of tokamak plasma configurations in a consistent way. The method actually solves the external equilibrium problem, linking the poloidal equilibrium fields with the value of the total plasma current and the geometric parameters that describe the plasma cross section. This link imposes constraints upon the values of the inductance for superconducting tori obtained by a previous method described by S. P. Hirshman and G. H. Neilson [Phys. Fluids 29, 790 (1986)]. Only if these constraints are properly taken into account do their results correspond to real tokamak equilibrium configurations. The present method is illustrated by calculating the external equilibrium parameters for a wide range of values of the tokamak aspect ratio. Of particular interest are the results for the external inductance, the elongation, and the vertical equilibrium field in the low aspectratio range

A full wave theory of highharmonic fast wave absorption in highbeta plasmas
View Description Hide DescriptionA theory of fast wave absorption in a highbeta plasma is given. A reduced, secondorder, ordinary differential equation has been used which includes all collisionless electron dissipation mechanisms and ion cyclotron damping over many harmonics. This is relevant to the highharmonic fast wave heating scheme proposed by Ono [Phys. Plasmas 2, 4075 (1995)]. The parameters appropriate to the National Spherical Tokamak Experiment NSTX [J. Spitzer et al., Fusion Technol. 30, 1337 (1996)] have been used to investigate the absorption characteristics of electrons and ions under highbeta, highharmonic conditions.

Structure and dynamics of current sheets at Alfvén resonances in a differentially rotating plasma
View Description Hide DescriptionAlfvén resonances, where the local flow speed relative to the boundary is equal to the local Alfvén speed, introduce novel dynamical features in a differentially rotating plasma. The spatial structure and dynamics of current sheets in such plasmas is investigated analytically as well as numerically. The current sheets at Alfvén resonances tend to powerlaw singularities. The growth of current sheets is algebraic in time in the linear regime and saturates in the presence of dissipation without the intervention of nonlinear effects. These results have significant implications for forced reconnection and Alfvén wave dissipation in laboratory and space plasmas.

Radiation fronts in tokamak divertor plasmas
View Description Hide DescriptionThe impurity radiation from a tokamakdivertor can be significantly enhanced due to the formation of a Vshaped impurity radiation front and the effects of perpendicular plasmaheat conduction. For some conditions, the transition to a Vshaped radiation front can have a bifurcation character, such that the impurity radiation region jumps from the divertor targets to the entrance into the divertor. The divertor geometry can play a significant role in the formation of a Vshaped impurity radiation front. The analysis of the energy transport and impurity radiation suggest that divertor geometries with “vertical” target, neutral gas recirculation where sidewalls, or their combination are the most attractive from the point of view of the formation of a Vshaped radiation front and the plasmaenergy loss.

Rayleigh–Taylor instability: Comparison of hybrid and nonideal magnetohydrodynamic simulations
View Description Hide DescriptionThe evolution of the Rayleigh–Taylor instability in a low β, twodimensional plasma is studied using a hybrid code and a nonideal magnetohydrodynamic(MHD) code. The Rayleigh–Taylor instability was chosen as a test case because it is an important mixing process at boundary layers, and because it exhibits different behaviors in the conventional and nonideal limits. The nonideal MHD effects considered are the Hall term and finite Larmor radius (FLR) corrections. Three cases are considered in detail: conventional MHD, weak nonideal effects, and strong nonideal effects. In the conventional MHD regime the usual “bubble and spike” behavior of the Rayleigh–Taylor instability is observed. In the weak nonideal MHD regime long wavelength modes, reminiscent of the Kelvin–Helmholtz instability, dominate nonlinearly but very short wavelength filaments develop at the boundary interface. In the strong nonideal MHD regime, smallscale structures dominate and the boundary layer relaxes via a diffusionlike process rather than a largescale nonlinear mixing process. In general, the hybrid and fluid simulations are in good agreement. The differences, both physical and numerical, are discussed.

Improved Ohmic confinement induced by multipulse turbulent heating in the Hefei Tokamak6M
View Description Hide DescriptionThe improved Ohmic confinement phase (Hmode) has been observed during the turbulent heating (TH) pulse on the Hefei Tokamak6M (HT6M) [World Survey of Activities in Controlled Fusion Research, Nuclear Fusion Special Supplement (International Atomic Energy Agency, Vienna, 1991), p. 190]. The electron temperature and density profiles become steeper, and a more negative radial electric field well is developed across the normal Ohmic phase (Lmode) to Hmode transition, at the plasma edge. The edge toroidal and poloidal velocities and and the main ion pressure gradient are substantially enhanced after the TH pulse. It is found that across the transition, and all play significant roles for regulating the profile, and the negative well shape of in the Hmode is dominantly maintained by the poloidal rotation. The time evolution indicates that prior to the transition plays a key role in inducing the rapid variations of the and its shear. The density fluctuation suppression is independent of the sign of the shear and curvature and consistent with the theoretical models of Shaing et al. and Zhang and Mahajan, while the sign has an appreciable effect on the suppression of the plasma potential fluctuations. This fact reveals that the dependence of these two fluctuation suppressions on the shear are different, suggesting that the existing L→H transition theories which consider only a single fluctuating field should be improved.

On the theory of internal kink oscillations
View Description Hide DescriptionIn this paper a time evolution equation for internal kink oscillations is derived. It is valid for both stable and unstable plasma regimes, and incorporates the response of an energetic particle population. A linear analysis reveals a parallel between (i) the time evolution of the spatial derivative of the internal kink radial displacement and (ii) the time evolution of the perturbed particle distribution function in the field of an electrostatic wave (Landau problem). It is shown that diamagnetic drift effects make the asymptotic decay of internal kink perturbations in a stable plasma algebraic rather than exponential. However, under certain conditions the stable root of the dispersion relation can dominate the response of the onaxis displacement for a significant period of time. The form of the evolution equation naturally allows one to include a nonlinear, fully toroidal treatment of energetic particles into the theory of internal kink oscillations.

Global consistency for thin flux tube treatments of toroidal geometry
View Description Hide DescriptionIn order to avoid a selfmapping problem due to a very long correlation length along a field line, present treatments of magnetized plasmaturbulence in flux tube geometry often extend the computational domain to several times the field line connection length. This is shown to result in the admission of nonphysical parallel wavelengths, possibly corrupting the solution even if it is converged. Also shown is that if a flux tube domain is constructed by thinning the full spectrum of allowed wavelengths, the same boundary condition results as if the domain were the entire flux surface. The only consistent remedy is therefore to extend the domain in the perpendicular coordinate within the flux surface, possibly all the way back to the entire flux surface, remaining consistent with the global boundary conditions.

Optimal design of feedback coils for the control of external modes in tokamaks
View Description Hide DescriptionA formalism is developed for optimizing the design of feedback coils placed around a tokamak plasma in order to control the resistive shell mode. It is found that feedback schemes for controlling the resistive shell mode fail whenever the distortion of the mode structure by the currents circulating in the feedback coils becomes too strong, in which case the mode escapes through the gaps between the coils, or through the centers of the coils. The main aim of the optimization process is to reduce this distortion by minimizing the coupling of different Fourier harmonics due to the feedback currents. It is possible to define a quantity which parametrizes the strength of the coupling. Feedback fails for The optimization procedure consists of minimizing subject to practical constraints. If there are very many evenly spaced feedback coils surrounding the plasma in the poloidal direction then the optimization can be performed analytically. Otherwise, the optimization must be performed numerically. The optimal configuration is to have many, large, overlapping coils in the poloidal direction.
