Volume 5, Issue 5, May 1998
Index of content:
- Basic Plasma Phenomena, Waves, Instabilities
5(1998); http://dx.doi.org/10.1063/1.872781View Description Hide Description
A new expression of and instability criterion for tearing modes is derived for arbitrary magnetic shear configuration in the low beta and large aspect ratio limit. Local solutions of an ideal external kink equation are solved analytically by means of proper expansion and transformation. An analytic expression of the criterion parameter results from the analytic solutions. The instability criterion obtained depends on the location of the resistive layer, and on a dimensionless parameter λ related to the ratio of the gradients of the equilibrium current density and of the rotational transform. Strauss’s formula and the previous instability criterion are recovered as a special case in the large- limit without a conducting wall. Considering both the boundary conditions at the plasma core and the conducting wall, the expression of is extended to include the stabilizing effect of the conducting wall. The properties of tearing instability are analyzed based on the expression of
5(1998); http://dx.doi.org/10.1063/1.872782View Description Hide Description
Helicon waves in a plasma confined by a cylinder are treated. The undamped normal modes of the helicon and Trivelpiece–Gould (TG) waves have distinctly different wave patterns at high magnetic fields but at low fields have similar patterns and therefore interact strongly. Damping of these modes, their excitation by antennas, and the rfplasma absorption efficiency are considered. Nonuniform plasmas are treated by solving a fourth-order ordinary differential equation numerically. A significant difference between this and earlier codes which divide the plasma into uniform shells is made clear. Excitation of the weakly damped wave, followed by conversion to the strongly damped TG wave which leads to high helicon discharge efficiency, is examined for realistic density profiles. A reason for the greater heating efficiency of the vs the mode for axially peaked profiles is provided.
5(1998); http://dx.doi.org/10.1063/1.872890View Description Hide Description
The theory of resistive free boundary modes localized at the plasma–vacuum interface in a plane slab equilibrium is improved and extended. If one proceeds to sufficiently small wavelengths, then the stability criterion (the current density vector must not have a component along the magnetic field vector in the interface) remains unchanged, but the unstable eigenmodes become independent of resistivity, and their growth rates diverge like the inverse square root of the wavelength.
5(1998); http://dx.doi.org/10.1063/1.872783View Description Hide Description
The spectrum of unstable perturbations of a simple magnetized atmosphere is studied. Magnetic field lines are straight, horizontal, and line tied to conducting walls at both ends. The temperature has horizontal variation across the field lines as well as vertical variation. The unstable spectrum close to marginal stability is found to be continuous from zero to a maximum growth rate. The structure of the unstable continuum modes is calculated in the dissipationless limit. The presence of singularities in the eigenfunction does not affect the growth rate significantly, nor does it lead to plasma heating.
5(1998); http://dx.doi.org/10.1063/1.872784View Description Hide Description
A simple one-dimensional model of Maxwellian injection into a cylindrical Penning–Malmberg trap is presented. This model is used to predict the radial density profile of an electron column produced by a biased cathode with an potential variation. The column density is assumed to depend upon the cathode potential voltage and the self-consistent space-charge potential as A one-parameter family of theoretical solutions describes the radial density profiles. The model’s predictions agree well with electron density profiles resulting from a spiral tungsten filament measured over a wide range in cathode voltages.
- Nonlinear Phenomena, Turbulence, Transport
5(1998); http://dx.doi.org/10.1063/1.872785View Description Hide Description
A self-consistent set of equations for the fast space–time evolution of fluctuations and the slow space–time evolution of density and flows in a toroidal plasma, relevant for simulations using field-aligned coordinates in thin flux tubes, has been derived. The methodology for the derivation of these equations is outlined for a model set of equations for the plasma edge, specific to resistive ballooning modes but readily adaptable to other instabilities. The derivation proceeds by first writing the axisymmetric and fluctuatingequations in the usual toroidal coordinate system. These are then transformed to the twisted coordinate flux-tube system. Most simulations which use twisted flux-tube computational grids transform to the field-aligned coordinate system first and then take averages to obtain the slow evolution. They however miss some terms since the two operations, namely, multiscale separation and coordinate transformation, do not necessarily commute, because of subsidiary assumptions on the box size. In the present formulation, all the relevant neoclassical effects such as the Pfirsch–Schlüter current and the Stringer spin-up as well as the toroidalReynolds stress are properly included. This set of multiscale equations is appropriate for the study of the formation and evolution of transport barriers.
5(1998); http://dx.doi.org/10.1063/1.872786View Description Hide Description
The quadratic response tensor provides a complete description of second-order wave processes in a nonlinear medium. The first exact expression is derived for the quadratic response tensor of a warm collisionless plasma, whose particles have a nonrelativistic Maxwellian velocity distribution. The exact expression is written in terms of a set of generalized plasma dispersion functions which satisfy simple symmetry properties and recursion relations and which can be expressed in terms of the standard nonrelativistic plasma dispersion function.
5(1998); http://dx.doi.org/10.1063/1.872787View Description Hide Description
The transport of magnetic field lines is investigated numerically by summing an isotropic spectrum of randomly phased magnetic fluctuations to a uniform background field. The aim of this study is to clarify the dependence of transport properties on the fluctuation correlation length which ideally should be much smaller than the periodicity scale , and to eliminate the effects due to periodicity. Two types of spectra are considered: in the first, the longest turbulence wavelength equals , and is varied in order to describe either the injection zone or the inertial range of turbulence. In this case, strong anisotropy of transport and, for moderate fluctuation levels, Lévy random walk in the direction only is found. When is decreased, both transport and anisotropy are reduced. In the second type of spectrum the longest wavelengths in the spectrum are excluded in order to have and to reduce the influence of periodicity of the magnetic fieldmodel. In this case transport is isotropic for all fluctuation levels, and the Lévy random walk is independent of periodicity, since it is found even when the transverse displacement is within the periodicity box, . On the other hand this superdiffusive regime is limited to very small fluctuation levels. Because of the efficient reduction of anisotropy and periodicity effects, the second type of spectrum appears to be more appropriate to investigate the transport properties.
5(1998); http://dx.doi.org/10.1063/1.872788View Description Hide Description
The damping of finite-amplitude magnetosonic pulses propagating perpendicular to a magnetic field in a collisionless plasma containing two ion species (majority light ions and minority heavy ions) is studied through theory and simulation. A magnetosonic pulse accelerates heavy ions in the direction parallel to the wave front; this cross-filed motion of heavy ions then generates a long-wavelength perturbation behind the original pulse. Because of these processes the original pulse is damped. The damping rate of a solitary pulse is theoretically obtained. It decreases with increasing amplitude. The damping is further investigated by means of a one-dimensional, fully electromagnetic code based on a three-fluid model. The theoretically obtained damping rate is in good agreement with the simulation result. Also, it is confirmed that small-amplitude periodic waves are not damped.
Two-dimensional bounce-averaged collisional particle transport in a single species non-neutral plasma5(1998); http://dx.doi.org/10.1063/1.872789View Description Hide Description
In this paper we describe a new theory of like particle collisional transport for a non-neutral plasma confined in a Penning trap. The theory is valid in the regime and where is the axial bounce frequency, is the rotation frequency, is the collision frequency, is the cyclotron radius, and is the Debye length. In this regime each particle can be bounce averaged into a long rod and the transport understood as arising from the drift motion of the rods due to long-range mutual interactions. This is a very different mechanism than is considered in the classical theory of transport, where a particle guiding center undergoes a step of order as a result of a velocity scatteringcollision. For the parameter range considered, the new theory predicts transport rates that are orders of magnitude larger than those predicted by classical theory and that scale with magnetic field strength like rather than The new theory differs from a previous analysis of transport due to drift interactions of charged rods, in that the finite length of the rods is taken into account. This enables transport to occur even for the case of an drift rotation frequency that is a monotonic decreasing function of radius (as was the case in recent experiments).
5(1998); http://dx.doi.org/10.1063/1.872790View Description Hide Description
Magnetosonic waves are intensively studied due to their importance in space plasmas and also in fusion plasmas where they are used in particle acceleration and heating experiments. In the present paper, the magnetosonic waves propagating in a multi-ion-species plasma perpendicular to an external magnetic field are investigated. Due to the presence of several ion species, this mode splits into two branches: high- and low-frequency modes. This opens a new channel of nonlinear interactions (between these two modes), and qualitatively changes the picture of turbulence in the long-wave region. Considering the limit of a cold collisionless plasma, a general system describing the propagation of nonlinearly coupled high- and low-frequency waves is derived. This system includes the Korteweg–de Vries (KdV), Boussinesq, and Zakharov equations as limiting cases. Solitary solutions of the system of coupled equations are obtained.
5(1998); http://dx.doi.org/10.1063/1.872791View Description Hide Description
Linear and nonlinear particle-magnetohydrodynamic (MHD) simulation codes are developed to study interactions between energetic ions and MHD modes. Energetic alpha particles with the slowing-down distribution are considered and the behavior of toroidal Alfvén eigenmodes (TAE modes) is investigated with the parameters pertinent to the present large tokamaks. The linear simulation reveals the resonance condition between alpha particles and TAE mode. In the nonlinear simulation, two TAE modes are destabilized and alpha particle losses induced thereby are observed. Counterpassing particles are lost when they cross the passing-trapped boundary. They are the major part of lost particles, but trapped particles are also lost appreciably.
- Magnetically Confined Plasmas, Heating, Confinement
5(1998); http://dx.doi.org/10.1063/1.872792View Description Hide Description
On the China Tokamak (CT-6B) [Nucl. Fusion36, 1669 (1996)], application of negative limiter bias resulted in enhanced shear in the naturally occurring shear layer near the fixed limiter radius with negligible change of plasma density, electron temperature, the parallel plasma flow, and the impurity ion radiation power. In the layer, decreased turbulent fluctuation, reduced poloidal correlation and increased nonlinear coupling of the turbulence were observed to be very possibly correlated with enhanced shear. The results suggest that there exists interaction of shear with turbulence, and an shear-induced shift in the phase angle between density and poloidal electric field fluctuations and nonlinear three-wave coupling may play an important role in suppressing edge turbulence.
5(1998); http://dx.doi.org/10.1063/1.872793View Description Hide Description
The stabilityproperties of pressure-driven ballooning and Mercier modes in general stellarator configurations are studied. A perturbation method originally introduced to study tokamakstability by Greene and Chance [Nucl. Fusion21, 453 (1981)] is generalized to three-dimensional systems. The effects of profile variation can be examined by introducing self-consistent variations in equilibrium quantities to a localized region whose amplitudes are small but whose cross-field derivative is large. In the general case, the set of equilibria are characterized by two free functions, which can be chosen to be the local variation of pressure and rotational transform profiles (or pressure gradient and the field-line-averaged parallel current). In this way, stability curves for Mercier and ballooning modes in three-dimensional equilibria are generated that are analogous to the -α curves used in studying tokamak equilibrium.
5(1998); http://dx.doi.org/10.1063/1.872794View Description Hide Description
In order to understand the individual charged particle behavior as well as plasma macroparameters (temperature, density, etc.) during the adiabatic major radius compression -compression) in a tokamak, a kinetic approach is used. The perpendicular electric field from the Ohm’s law at zero resistivity is made use of in order to describe particle motion during the -compression. Expressions for both passing and trapped particle energy and pitch angle change are derived for a plasma with high aspect ratio and circular magnetic surfaces. The particle behavior near the passing trapped boundary during the compression is studied to simulate the compression-induced collisional losses of alpha particles. Qualitative agreement is obtained with the alphas loss measurements in deuterium-tritium (D-T) experiments in the TokamakFusion Test Reactor (TFTR) [World Survey of Activities in Controlled Fusion Research [Nucl. Fusion special supplement (1991)] (International Atomic Energy Agency, Vienna, 1991)]. The plasma macroparameters evolution at the -compression is calculated by solving the gyroaveraged drift kinetic equation.
5(1998); http://dx.doi.org/10.1063/1.872795View Description Hide Description
The eigenfunctions of the equation are the solutions of a Sturm-Liouville operator and form an orthonormal basis for the expansion of magnetic fields on any closed domain. Equilibria can be formed by summing individual eigenfunctions The so-called Taylor state is a member of the general class of equilibria consisting entirely of a single mode with the lowest eigenvalue. It is interesting to note that the more general configurations formed by sums of these eigenfunctions are not necessarily zero pressure, nor are they necessarily stable. Physical insight can be gained from decomposition in this way. Stability calculations are greatly simplified in this choice of basis functions, especially for incompressible modes with . The magnetohydrodynamic(MHD)equation can be written as a bilinear form where the matrix can be composed from the integrated scalar triple products of the equilibrium basis curl-eigenfunction states and the forward and adjoint curl-eigenfunction magnetic perturbations. The problem of ideal MHD stability in simple geometries (but general equilibria) becomes tractable for PC solutions. The general procedure for stability analysis in arbitrary geometries is given, and detailed calculations for spherical equilibria are given.
Self-consistent removal of sawtooth oscillations from transient plasma data by generalized singular value decomposition5(1998); http://dx.doi.org/10.1063/1.872796View Description Hide Description
This paper addresses the problem of removing sawtooth oscillations from multichannel plasma data in a self-consistent way, thereby preserving transients that have a different physical origin. The technique which does this is called the generalized singular value decomposition (GSVD), and its properties are discussed. Using the GSVD, spatially resolved electron temperature measurements are analyzed. Special attention is paid to transient regimes in which the temperature is perturbed either by the laser blow-off injection of impurities or by pellet injection. Nonlocal transport issues are briefly discussed.
Predictive simulations of tokamak plasmas with a model for ion-temperature-gradient-driven turbulence5(1998); http://dx.doi.org/10.1063/1.872797View Description Hide Description
A drift wavetransportmodel, recently developed by Ottaviani, Horton and Erba (OHE) [Ottaviani et al., Plasma Phys. Controlled Fusion 39, 1461 (1997)], has been implemented and tested in a time-dependent predictive transport code. This OHE model assumes that anomalous transport is due to turbulence driven by ion temperature gradients and that the fully developed turbulence will extend into linearly stable regions, as described in the reference cited above. A multiplicative elongation factor is introduced in the OHE model and simulations are carried out for 12 discharges from major tokamakexperiments, including both L- and H-modes (low- and high-confinement modes) and both circular and elongated discharges. Good agreement is found between the OHE model predictions and experiment. This OHE model is also used to describe the performance of the International Thermonuclear Experimental Reactor (ITER) [Putvinski et al., in Proceedings of the 16th IAEA Fusion Energy Conference, Montréal, Canada, 1996 (International Atomic Energy Agency, Vienna, 1997), Vol. 2, p. 737.] A second version of the OHE model, in which the turbulenttransport is not allowed to penetrate into linearly stable regions, has also been implemented and tested. In simulations utilizing this version of the model, the linear stability of the plasma core eliminates the anomalous thermal transport near the magnetic axis, resulting in an increase in the core temperatures to well above the experimental values.
5(1998); http://dx.doi.org/10.1063/1.872798View Description Hide Description
Owing to the nonspherical nature of the heat deposition in the pellet ablation cloud by energy loss of incident plasmaelectrons streaming parallel to the uniform magnetic field, a nonuniform pressure distribution develops at the pellet surface. This can lead to deformation of “soft” cryogenic pellets exposed to high temperature and high density magnetized plasmas. The effect of deformation on the burning rate and stability of the condensed phase is evaluated for pellets and liquid jets.
Analytical stability condition for the ideal kink mode in a toroidal plasma with elliptic cross section5(1998); http://dx.doi.org/10.1063/1.872799View Description Hide Description
An analytical stability condition for the ideal kink mode with toroidal mode number and (dominating) poloidal mode number in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity is treated as a small parameter independent of the inverse aspect ratio (ε), and the expansion is performed up to order in the potential energy It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when is small ( is the safety factor at the magnetic axis), whereas an ellipticity of opposite sign (horizontal elongation) is stabilizing. By means of an additional expansion, in this effect is shown to be due to a -independent term in proportional to the ellipticity and to the poloidal beta value at the surface. Since the leading order term in in the absence of ellipticity is of order [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)], the ellipticity term dominates for sufficiently small values of The analytical result is in agreement with previous, numerical results [Lütjens et al., Nucl. Fusion32, 1625 (1992)], and leads, already for rather moderately vertically elongated plasmas, to stability limits in much lower than the value 0.3 valid in the limit for a plasma with circular cross section.