Index of content:
Volume 2, Issue 6, June 1990

The free energy of Maxwell–Vlasov equilibria
View Description Hide DescriptionA previously derived expression [Phys. Rev. A 4 0, 3898 (1989)] for the energy of arbitrary perturbations about arbitrary Vlasov–Maxwell equilibria is transformed into a very compact form. The new form is also obtained by a canonical transformation method for solving Vlasov’s equation, which is based on Lie group theory. This method is simpler than the one used before and provides better physical insight. Finally, a procedure is presented for determining the existence of negative‐energy modes. In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed.

The magnetic pumping of plasmas with sawtooth waveforms
View Description Hide DescriptionThe pumping of plasmas by sawtooth‐waveform magnetic induction variations is studied theoretically and by means of computer simulations. A sawtooth is a cyclic waveform that is characterized by a slow increase in the magnetic induction followed by a rapid drop in the induction. Two types of sawtooth pumping are analyzed, and the types classified as to whether or not the first adiabatic invariants of the plasma particles are conserved during the rapid drops in the magnetic induction. When the invariants are conserved, the sawtooth waveforms are found to be less efficient than square waves for pumping plasmas. When the adiabatic invariants are not conserved, the pumping efficiency is found to be a slight improvement over square waves. Both types of pumping are applied to a hypothetical tokamak plasma and it is concluded that neither type of sawtooth pumping is practical for heating magnetically confined fusion plasmas.

A self‐consistent theory of radial transport of field‐aligned current by microturbulence
View Description Hide DescriptionThe radial transport of field‐aligned current resulting from collisionless microturbulence is examined self‐consistently. The self‐consistent treatment of mode coupling is shown to constrain the transport in such a way that the relaxation of current gradients is regulated solely by electrostatic fluctuations that couple to ion dissipation, even in the presence of temperature gradients and temperature anisotropy. As a consequence, the radial flux of parallel current induced by collisionless microinstabilities is insufficient to account for the dynamo in reversed field pinch plasmas.

Magnetic fluctuations due to thermally excited Alfvén waves
View Description Hide DescriptionMagnetic fluctuations resulting from the thermally excited magnetohydrodynamic waves are investigated using fluid and kinetic models to describe a stable, uniform, compressible plasma in the range above the drift wave frequency and below the ion cyclotron frequency. It is shown that the fluid model with resistivity yields spectral densities that are roughly Lorentzian and exhibit equipartition with no apparent cutoff in wavenumber space and a Bohm‐type diffusion coefficient. Under certain conditions, the ensuing transport may be comparable to classical values. For a phenomenological cutoff imposed on the spectrum, the typical fluctuating‐to‐equilibrium magnetic field ratio is found to be of the order of 10^{−10}. Physical mechanisms to obtain decay profiles of the spectra with increasing wavenumber as a result of dispersion and/or different forms of damping are investigated analytically in a cold fluid approximation and numerically, with a kinetic model. The mode dispersion resulting from the finite ion gyro‐frequency is identified as the leading effect determining the spectral profile shapes. It is found that the amplitude of fluctuations may be within a factor of the value suggested by the cold plasma model. The results from both models are presented and compared in low‐ and high‐beta regimes.

Anomalous thermal electron heating and heat transport inhibition due to parametric instabilities
View Description Hide DescriptionMeasurements are presented of strong thermal electron heating and the heat transport inhibition. An electron plasma wave heats hot electrons near the critical density. A return current is induced to keep charge neutrality. Thermal electrons are heated strongly by the resistivity of parametrically excited isotropic ion wave turbulence (anomalous joule heating). The heat transport of thermal electrons is also inhibited by the resistivity. The experimental results agree reasonably well with theory.

Intrinsic chaos in a plasma wave
View Description Hide DescriptionThe linear plasma response to an electrostatic wave is described by the Van Kampen–Case–Siewert (VCS) modal structure using Lie transform techniques. The VCS structure, which is well known to give rise to Landau damping, is also proved to be responsible for chaotic particle response and diffusion in phase space in the quasilinear limit.

Suppression of Rayleigh–Taylor and bulk convective instabilities in imploding plasma liners and pinches
View Description Hide DescriptionLinear growth of instabilities during implosion of annular and columnar plasma pinches having sharp outer boundaries is analyzed. The dominant modes of the Rayleigh–Taylor instability differ in time, radius, and growth rate from the dominant modes of the bulk convective instability. The spectra of growth rates of perturbations localized azimuthally or axially suggest explanations for filamentation and stratification of plasmas. Localized Rayleigh–Taylor modes grow fastest initially in an implosion. These modes can be suppressed completely, however, by a relatively modest axial magnetic field, which produces magnetic shear in a Z pinch. Similarly, bulk convective modes can be substantially reduced by magnetic shear. The ‘‘window of stability’’ produced by magnetic shear occurs for an axial field magnitude about equal to the azimuthal. Properly shaped current pulses in Z pinches with axial fields may therefore produce stable implosions to high compression ratios, in agreement with recent experiments. The most dangerous Rayleigh–Taylor modes for the symmetry of an annular pinch of initial aspect ratio R _{0}/δ_{0} are those with wavenumber k∼R _{0}/δ^{2} _{0}, where R _{0} is the initial pinch radius. To suppress pinch instabilities optimally, both the liner thickness and axial magnetic field should be chosen properly.

Nonlinear behavior of magnetohydrodynamic modes near marginally stable states. I. General formulation and application to the nonresonant kink modes in a reversed field pinch and to the quasi‐interchange modes in a tokamak
View Description Hide DescriptionTwo types of nonlinear equations describing the time development of modes near marginally stable states in an inhomogeneous medium are obtained through a general formulation that employs a perturbation expansion around the marginally stable state under the assumption of a single helicity. One type of nonlinear equation has a Hamiltonian form that may be interpreted as the equation of motion for a particle in the potential field of a central force; the other type leads to the Landau equation, which is well known in fluid dynamics. The former equation is obtained when the linear operator is degenerate at the marginally stable state, which corresponds to the case when the linear dispersion relation has a double root for the frequency at the marginally stable state, whereas the latter is obtained when the linear operator is nondegenerate, i.e., the linear dispersion relation has a single root. In the framework of magnetohydrodynamics, the former corresponds to the nonresonant ideal modes, and the latter to the resistive modes. The nonlinear behavior of the nonresonant kink modes in a reversed field pinch and of the quasi‐interchange modes in a tokamak are examined with the application of the general formulation. It is shown that new stable helical equilibria bifurcate near the initial axisymmetric equilibrium, so that the plasma nonlinearly oscillates around the new bifurcated equilibrium, which leads to nonlinear saturation of the nonresonant kink modes in a reversed field pinch and of the quasi‐interchange mode in a tokamak. The nonlinear stabilizing effects causing bifurcation of the equilibrium are interpreted as quasilinear effects. Compressibility reduces the nonlinear stabilizing effects through changing the quasilinear components and is important even when the modes are near marginally stable states.

Nonlinear behavior of magnetohydrodynamic modes near marginally stable states. II. Application to the resistive fast interchange mode
View Description Hide DescriptionWith the use of the general formulation developed in an earlier paper, the nonlinear evolution of the resistive fast interchange mode near the marginally stable state is obtained analytically. The nonlinear amplitude equation of the mode is shown to be of the Landau type. It is also shown that there is a stable equilibrium bifurcating from the initial equilibrium. Comparing this analytical result to numerical simulations, it is confirmed that the saturation level and the saturation time are well estimated by this Landau type of nonlinear amplitude equation.

Extending the collisional fluid equations into the long mean‐free‐path regime in toroidal plasmas. I. Plasma viscosity
View Description Hide DescriptionAn expression for the pressure anisotropy and thus for the viscous stress in the plateau regime is derived for arbitrary toroidal magnetic configurations without assuming incompressibility or the existence of flux surfaces, without neglecting the flow components perpendicular to the magnetic surface, and without restricting the flow velocity to be a constant on the flux surface. It can be employed to study low‐frequency instabilities in the long mean‐free‐path regime. A smoothly connected formula for the pressure anisotropy, valid in both the collisional fluid regime and the plateau regime, is given to facilitate the numerical computation. An alternative interpretation of the neoclassical transport theory is also obtained. It is found that if the effects of the temperature gradient are neglected, neoclassical transport fluxes can be interpreted as driven by the velocity stress.

Ion temperature effects on ion charge‐state distributions of an electron cyclotron resonant ion source
View Description Hide DescriptionA method is described for determining ion cyclotron resonance (ICR) heating effects on multiply charged‐ion energy distributions using a Monte Carlo fit to experimental time‐of‐flight spectrometer data. The method is general but is used here specifically to separate the effects of plasma ambipolar potential spread and ion temperature in an electron cyclotron resonance (ECR) heated magnetic mirror ion source (MIMI) [Phys. Fluids 2 8, 3116 (1985)]. A steady‐state equilibrium model is also developed that models the relevant atomic processes occurring in MIMI plasmas. This model and the Monte Carlo analysis are used to relate the effect of midplane ICR heating on end loss ion charge state distributions to its effect on the confined ion distributions. The model allows for collisional, moderately collisional, and collisionless confinement, specific to each charge state in the distribution. Both experiment and modeling show that increased ion temperature causes a shift to lower‐Z ion populations in both the confined and end loss charge‐state distributions.

Three‐dimensional transmission of the fast wave in ion cyclotron resonance plasma heating
View Description Hide DescriptionThe transmission of the fast Alfvén wave in second harmonic ion cyclotron resonance heating of a deuterium–hydrogen (minority) plasma is analyzed in a general three‐dimensional magnetogeometry. The approach is based on the congruent reduction of the Hermitian, anisotropic pressure fluid plasma model. The unreduced, 17th‐order, slow‐amplitude transport equation, characterizing the problem of interest, is systematically reduced to the fourth‐order irreducible transport system describing the coupling of the fast wave to the resonant components of the anisotropic pressure plasma model. This system yields a compact expression for the fast wave transmission coefficient in a general geometry and arbitrary direction of propagation of the incident wave.

Whistler‐wave electron cyclotron heating in uniform and nonuniform magnetic fields
View Description Hide DescriptionRecent laboratory experiments on single‐pass electron cyclotron heating employing whistler waves [Phys. Fluids B 1, 2010 (1989)] have shown direct evidence of nonlinear collective processes that can have a major effect on the properties of the resulting plasma. A theoretical interpretation of these experimental results is proposed herein, based on linear and nonlinear aspects of electron cyclotron heating. The observed increments in electron kinetic energy are shown to be consistent with a conventional linearized analysis of the gyroresonant interaction. The observed bursts of radiation in sidebands near the heating frequency are consistent with the nonlinear gyrophase bunching of resonant electrons in the whistler wave resulting in coherent excitation of unstable sidebands.

Growth of a spike on a laser beam in a plasma
View Description Hide DescriptionAn on‐axis spike in the intensity distribution of a Gaussian laser beam grows rapidly as the beam propagates in a laser‐produced plasma. The nonlinearity arises through the ponderomotive force and has a saturating behavior. For v _{osc} ∼v _{th,} the optimum spike radius is approximately a few c/ω_{ p 0 } and the amplification length is ∼2cω/ω^{2} _{ p 0 }, where ω_{ p 0 } is the unperturbed plasma frequency, ω is the radiation frequency, and v _{osc} and v _{th} are oscillatory and thermal velocities of electrons, respectively.

Photographic study of hot electrons generated by the two‐plasmon decay instability
View Description Hide DescriptionHot electrons generated by the two‐plasmon decay instability in a CO_{2}‐laser irradiated underdense plasma are studied by photographic methods. For laser intensities less than 5×10^{1} ^{3} W/cm^{2} the electrons are emitted in the plane of polarization. From the direction of their trajectories the wave vector distribution of the electron accelerating two‐plasmon decay waves is determined. These wave vector measurements are in good agreement with values theoretically predicted for the fastest growing mode. For laser intensities large than 6×10^{1} ^{3} W/cm^{2} space‐charge effects become important and hot electrons are detected out of the plane of polarization.

Three‐halves harmonic emission from two‐color experiments
View Description Hide DescriptionThree‐halves harmonic emission from a plasma in which the peak plasma density is n _{ c }>n>n _{ c }/4 has been temporally and spectrally resolved. This is achieved by using 0.53 μm wavelength laser light to irradiate a plasma preformed by 0.35 μm wavelength laser light. It is observed that both the backscattered red‐shifted and blue‐shifted peaks of the 3ω_{0}/2 emission from the 0.53 μm laser disappear at approximately the same time. This result constrains the possible production mechanism for backscattered blue‐shifted photons to scattering of the blue‐shifted photons or plasmons close to n _{ c }/4.

A hyperbolic model for the one‐dimensional electromagnetic pulse
View Description Hide DescriptionAn account of the current phenomenology relative to the electromagnetic pulse is obtained by having recourse to the model of a charged fluid. In particular, a differential system is deduced for the description of a one‐dimensional electric field generated by the so‐called Compton current. The system turns out to be hyperbolic, thus allowing for discontinuity waves. The evolution law for the amplitude of such waves is examined in detail. Mathematical problems resulting from vanishing of the electric field ahead of the wave front are also investigated.

Evolution of an X‐pinch plasma
View Description Hide DescriptionThe evolution of a dynamic X‐pinch plasma, formed by two or more intersecting crossed copper wires, is investigated theoretically and numerically in two dimensions to determine the behavior at the point of intersection. An X‐pinch plasma for conditions corresponding to (a) pure hydrodynamic and (b) nonlocal thermodynamic equilibrium radiation hydrodynamic simulations are considered when driven by an externally applied current.

Nonlocal fluxes at a plasma sheath
View Description Hide DescriptionThe particle and energy fluxes of electrons at the boundary of a plasma in contact with a perfectly absorbing plate are considered. In general, the fluxes are shown not to be determined by the plasma temperature and density at the plate but rather by a convolution of the plasma profiles in the vicinity of the plate. A simple empirical expression is proposed for the nonlocal fluxes, which approximately reproduces the results of a full kinetic calculation. The implications of this, to divertor plasmas near the neutralizer plate, are discussed.

Numerical simulations of perturbed Vlasov equilibria
View Description Hide DescriptionIn this work, the long‐time behavior of the solutions of the Vlasov–Poisson system when starting near a spatially homogeneous equilibrium is analyzed numerically. It is found that the asymptotic state toward which the system evolves is not a Bernstein–Green–Kruskal (BGK) equilibrium, but can rather be described by a superposition of BGK modes. Also, it is shown that the small oscillations at the plasma frequency after saturation in the one‐sided bump‐on‐tail instability are due to beating between the unstable mode and the least stable mode.