William Grossmann

A reformulation of lower‐hybrid wave propagation and absorption

A full wave solution of the propagation and resonant absorption of lower‐hybrid waves in an inhomogeneous plasma using the cold plasma model is carried out and compared with results obtained in an infinite homogeneous plasma medium. It is found that the density gradient coupled to finite poloidal model number allows the lower‐hybrid wave to reach the hybrid layer for the case of purely perpendicular propagation. Such a wave is evanescent in the infinite homogeneous plasma. The analysis also shows that, for the cold plasma model, roughly equal amounts of energy are absorbed by the ions and electrons in the resonant layer. The analysis may have some bearing on the experimental efforts to propagate lower‐hybrid waves into the interior of tokamak plasmas.

A stochastic model for lower hybrid wave scattering by density fluctuations

Recent tokamak experiments have shown the presence of drift‐wave‐induced random density fluctuations localized near the plasma edge. These fluctuations may be treated as a random perturbation of the refractive index that appears in the scalar wave equation obeyed by the electrostatic potential. The treatment here includes the additional effect on density gradient fluctuations through a term, kyM, appearing in the wave equation, where ky is the poloidal wavenumber of the incident wave and M≡〈ω2pe〉/ωΩce . This additional term is shown to have non‐negligible consequences in several cases of physical interest. By invoking the modern theory of stochastic differential equations, various statistics of particular functionals of the potential may be computed. An explicit calculation of the fractional power Pt transmitted through the scattering region is given as a function of the usual parameter e2LS, where e is the relative density fluctuation amplitude, L is the scale width of the scattering region, and S is the...

Particle Loss in a Three‐Dimensional Cusp

The loss of particles through an axisymmetric cusped magnetic field containment geometry is considered. An appropiate adiabatic invariant is derived for the plasma considered to be a high β, collisionless collection of particles contained within a field‐free region bounded by a mathematically sharp interface. The particles are allowed to undergo soft collisions with the interface thus giving rise to a boundary layer. A velocity space loss cone criterion analogous to the loss cone criterion in magnetic mirror machines is derived. Particle loss rates are calculated for a special form of the particle distribution function, and the results are compared with relevant experimental data. A previous treatment of the present problem considering two‐dimensional geometry is re‐examined in view of the present work concerning adiabatic invariance.

Magnetohydrodynamic stability of highly elongated axisymmetric equilibria

The magnetohydrodynamic stability of diffuse, axisymmetric toroidal plasma configurations with large elongation in the direction of the axis of symmetry and large aspect ratio is investigated. In an asymptotic expansion the elongation and aspect ratio are related and both are made large. This expansion reduces the equilibrium relations in leading order to an ordinary differential equation readily susceptible to numerical solution or, in some cases, analytical solution. The same expansion procedure is also applied to the linearized stability analysis and both partial differential equations and a Rayleigh–Ritz quotient for the eigenvalues and eigenvector result. Belt pinch and doublet shapes for various current density profiles are examined for internal and external modes for both high and low β forms of the equilibrium. Growth rates and eigenfunctions for the discrete modes are given and general features of the spectrum are discussed.

Simulations of rf-driven sheath formation in two dimensions

The results from two‐dimensional particle simulations of sheath formation around periodic metal arrays placed inside magnetized plasmas and driven by oscillating voltages are reported. The main goal is the modeling of the plasma interaction with the Faraday bars surrounding the antennas during ion cyclotron tokamak heating. The study of the time‐averaged potentials shows that the two‐dimensional sheath structure depends on both the sheath length‐to‐thickness ratio and the inclination of the magnetic lines. The equipotential surfaces form closed, nested cells between adjacent bars. When the magnetic lines are nearly perpendicular to the potential gradients, the ion motion is dominated by the E×B drift, and ion streamlines form vortices around the equipotentials. At larger inclinations of the magnetic lines, the flow decouples from the equipotentials and ion transport is mainly along the potential gradients. The critical angle for the transition from vortex circulation to field aligned flow is computed. The...

Stability of a high‐β, l=3 stellarator

The stability of an infinitely long, high β, l=3 stellarator is investiated. The calculation is carried out by using the new scyllac expansion in the sharp boundary ideal magnetohydrodynamic model. It is found that for any given size l=3 field allowed by equilibrium considerations and mode number m, an infinite but discrete set of wavenumbers k exist for which the plasma is unstable to all β; that is, the critical β equals zero. These modes can be described as long wavelength interchanges. Thus, with regard to sharp boundary stability, l=3 is less desirable than l=1 for the basic scyllac magnetic field.

Nonlinear signal processing using integration of fluid dynamics equations

Here, we describe a new and unique image sharpening method based on computational techniques developed for CFD. Our preliminary experience with this method shows its capability for nonlinear enhancement of image edges as well as deconvolution of an image with random noise. This indicates a potential application for image deconvolution from sparse and noisy data resulting from measurements of backscattered laser-speckle intensity.

Wave propagation in one‐dimensional inhomogeneous random media with an application to lower hybrid waves in fusion plasmas

Wave propagation in a one‐dimensional stratified medium, whose physical parameters are subject to random fluctuations, is considered. The complex‐valued reflection coefficient obeys, as a function of the slab width L, a stochastic Riccati differential equation, associated with an initial‐value problem. Using this fact, solving first the boundary‐value problem satisfied by the field on [0,L] is avoided. The reflection coefficient can be computed numerically by a Monte‐Carlo‐type procedure by the generation of suitable sequences of random numbers aimed at constructing realizations of the stochastic processes that enter the refractive index. An application is made to the propagation of ‘‘lower hybrid waves’’ in thermonuclear (fusion) plasmas for realistic models of the deterministic density profile using available experimental data for the statistics of the random fluctuations. Several functional forms of the density profile are considered, and the relevant physical approximations are discussed. This work ge...

Two‐Dimensional Equilibrium for High‐β Mirror Devices Including Particle Loss

Two‐dimensional equilibria for high‐β mirror devices are found by solving the Vlasov equation and Amperes law self‐consistently. Particle loss is taken into account self‐consistently by allowing particles to exist only outside of the local loss‐cone angle. It is found that for a given set of plasma parameters and applied mirror ratio (1) solutions only exist below some critical value of β, and (2) in certain cases the solutions show a nonuniqueness related to the question of stability.