Search

A plasma fluid model which takes into account ion drifts has been constructed and applied to the scrape-off layer of a tokamak with a poloidal divertor. The model equations have been solved numerically in a realistic two-dimensional geometry. The solutions are characterized by near-sonic toroidal velocities and large poloidal flows in most of the scrape-off layer, together with steep gradients in the pressure and electrostatic potential along the magnetic field near the X-point, contrary to the predictions of the standard model. The potential step at the X-point should reduce parallel heat transport and could act as an H-mode trigger.

Reconstruction techniques have been developed to obtain 2‐D images of high β plasmas by taking advantage of the large toroidal rotation rate on TFTR. The Shafranov shift, which is large in high β plasmas, is taken into account in the reconstructions. A 2‐D image of the electron temperature is reconstructed from the electron cyclotron emission with the assumption of rigid poloidal rotation. The soft x‐ray tomography is performed using the Fourier–Bessel expansion technique from two x‐ray cameras. The reconstruction techniques are tested numerically through the use of model islands. The image is reproduced from the source function with these techniques; however, the x‐ray tomography without rotation using the two cameras gives unacceptable results.

To reconstruct an image using computed tomography (CT), the axis of rotation must pivot at the same point on the reconstruction plane that the X-ray source and the CT detector assembly rotate about around the imaged object. This pivot point is used as a reference point for backprojecting pixel values to their proper coordinates. Reconstructing an image with a nonstationary axis of rotation would backproject pixel values to incorrect coordinate points. A convolution filtered backprojection algorithm has been derived for correcting images that were acquired with a nonstationary axis of rotation using the fan beam geometry with a collinear (flat) detector. The correction method accounts for the vertical displacements of the axis of rotation as the CT scanner rotates around the imaged object, as may be the case when sagging occurs. Software simulations are performed to show how the algorithm corrects for the shift in the axis of rotation. >

During the 1987 run, TFTR reached record values of QDD, neutron source strength, and Ti(0). Good confinement together with intense auxiliary heating has resulted in a plasma pressure greater than 3*105 Pascals on axis, which is at the ballooning stability boundary. At the same time improved diagnostics, especially ion temperature profile measurements, have led to increased understanding of tokamak confinement physics. Ion temperature profiles are much more peaked than previously thought, implying that ion thermal diffusivity, even in high ion temperature supershot plasmas, is greater than electron thermal diffusivity. Based on studies of the effect of beam orientation on plasma performance, one of the four neutral beamlines has been re-oriented from injecting co-parallel to counter parallel, which will increase the available balanced neutral injection power from 14 MW to 27 MW. With this increase in balanced beam power, and the addition of 7 MW of ICRF power it is planned to increase the present equivalant QDT of 0.25 to close to break-even conditions in the coming run.

A fluid model which includes ion drifts has been applied to the SOL of a tokamak with a poloidal divertor. Large parallel velocities and steep gradients in the pressure and electrostatic potential are found near the X-point, in contrast with the solutions of the standard model.

A numerical procedure is described for the evaluation of magnetic coordinates given a toroidal, scalar pressure plasma with an arbitrary magnetic field. The accurate representation of magnetic field strength in this way is invaluable for the calculation of drift orbits and transport in asymmetric plasmas.

The Compact Ignition Tokamak (CIT) is a proposed modest-size ignition experiment designed to study the physics of alpha particle heating. The basic concept is to achieve ignition in a modest-size minimum cost experiment by using a high plasma density to achieve nT{sub E} {approx} 2 x 10{sup 20}s/m{sup 3} required for ignition. The high density requires a high toroidal field (10 T). The high toroidal field allows a large plasma current (10 MA) which provides a high level of ohmic heating, improves the energy confinement, and allows a relatively high beta ({approx} 6%). The present CIT design also has a high degree of elongation (k {approx} 1.8) to aid in producing the large plasma current. A double null poloidal divertor and pellet injection are part of the design to provide impurity and particle control, improve the confinement, and provide flexibility for improving the plasma profiles. Auxiliary heating is expected to be necessary to achieve ignition, and 10-20 MW of ICRF is to be provided.

The structure of a magnetic field is determined by a one-degree-of-freedom; time-dependent Hamiltonian. This Hamiltonian is evaluated for a given field in a perturbed action-angle form. The location and the size of magnetic islands in the given field are determined from Hamiltonian perturbation theory and from an ordinary Poincare plot of the field line trajectories.

This paper presents an algorithm which solves sparse systems of linear equations of the form Ax =y, where A is non-symmetric, by the Incomplete LU Decomposition-Conjugate Gradient (ILUCG) method. The algorithm minimizes the error in the Euclidean norm ‖xi − x‖2 , where xi is the solution vector after the ith iteration and x the exact solution vector. The results of a test on one real problem indicate that the algorithm is likely to be competitive with the best existing algorithms of its type.

Two dimensional calculations of impurtiy transport in a high recycling divertor scrape-off region have been made with an updated version of the ZTRANS Monte Carlo computer code. The calculations use plasma parameters for the Doublet 3 divertor, as computed by the Planet Fluid Transport Code. The effects of electric field, particle drift velocities, and thermal forces are included in the calculations. For all impurity species studied, it is found that impurity transport is dominated by frictional forces, over most of the scrape-off region. Light impurities, however, impinge substantially closer to the divertor plate center than do heavy impurities, which tend to impinge at the outer plate boundary. 8 refs., 4 figs., 2 tabs.

The symbolic style of programming referred to as Symbolic Algol I [1] appears to have a number of advantages when applied to the solution of sets of nonlinear partial-differential equations. Programs written in that style are clear, elegant, and concise and their modular structure enables large parts of the programs to be used over and over again for many different problems. Such programs, however, tend to be slow because they involve a large number of nested procedure calls at execution time. Finite-difference methods in several dimensions require in general that a relatively small number of equations be solved a large number of times and much is gained if these nested procedure calls are executed only once. This is achieved by a generator or translator program, written in Algol, which processes input written in a related style named Symbolic Algol II. Usually only finite-difference equations in very compact symbolic form are input, while output is completely explicit and can be in a number of computer languages. Of greatest interest are Assembler code modules automatically produced in this way. They are competitive in speed with fully hand-optimized Fortran versions and are produced effortlessly and error free, so that complex sets of equations can readily be programmed or alterations made. For production runs these modules can be incorporated, into a Fortran control program.

The poloidal magnetic divertor operating in the cool-high density plasma mode represents a good solution to the tokamak first wall erosion and impurity transport problems. It also, in general, improves the energy confinement time in the beam-heated discharges. In the existing experiments, the neutralizer plates are oriented close to normal to the flux surfaces in the poloidal plane, but the planned high-power experiments require that the plate be sharply angled to reduce the power flux density. We have studied the effect of changing the plate angle on the radial plasma temperature and density profiles at the plate. We find that some plate orientations shift the electron flux profile relative to the power flux profile, resulting in unfavorable temperature profiles. The question of existence of a steady-state solution was also addressed. For a steady state to exist, the radial neutral fluxes have to exactly cancel the radial diffusive ion fluxes. Depending on the diffusion coefficients, the plasma density may have to increase with radius to satisfy this condition. The implications of these conclusions for the INTOR, and CIT experiments is also discussed.

The INTOR divertor operation has so far been modeled only in a simplified rectangular geometry, which omitted some of the important features of the actual INTOR divertor. Recently, the PLANET plasma transport code upgrades have allowed us to model the open magnetic geometry of the INTOR divertor realistically, in particular, with regard to the trapped neutral gas region between the two separatrices and the open geometry. The effect of this neutral gas build-up on the plasma density, temperature, and the energy loss has been examined. The neutral hydrogen interaction with the plasma, and the interaction of the resulting charge-exchange neutrals with the walls is discussed. The power flux profiles at the plates were computed under the assumption of constant radial transport coefficients. It was found that the change in these profiles from the x- point to the plates is due mostly to the compression in the poloidal magnetic field. The heat loads on the neutralizer plate and first wall have been characterized, and the erosion of the divertor walls and the neutralizer plates is discussed.

A Monte Carlo method of evaluating typical particle and energy transport coefficients is given for the case in which the particle drift orbits are a significant fraction of the plasma radius. The method is applied to a preliminary design for a helical axis (heliac) stellarator experiment.

A simple empirical particle transport model has been constructed with the purpose of gaining insight into the L- to H-mode transition in tokamaks. The aim was to construct the simplest possible model which would reproduce the measured density profiles in the L-regime, and also produce a qualitatively correct transition to the H-regime without having to assume a completely different transport mode for the bulk of the plasma. Rather than using completely ad hoc constructions for the particle diffusion coefficient, we assume D = 1/5 chi/sub total/, where chi/sub total/ approx. = chi/sub e/ is the thermal diffusivity, and then use the kappa/sub e/ = n/sub e/chi/sub e/ values derived from experiments. The observed temperature profiles are then automatically reproduced, but nontrivially, the correct density profiles are also obtained, for realistic fueling rates and profiles. Our conclusion is that it is sufficient to reduce the transport coefficients within a few centimeters of the surface to produce the H-mode behavior. An additional simple assumption, concerning the particle mean-free path, leads to a convective transport term which reverses sign a few centimeters inside the surface, as required by the H-mode density profiles.

Electron cyclotron emission measurements taken at 20 locations in the horizontal midplane during a sawtooth crash have been analysed based on the assumption of fast rigid rotation of the plasma. Due to this fast rotation (approx.100..mu..sec), which remains fairly constant throughout the sawtooth crash, we have been able to make time-to-space reconstructions of half the poloidal plane using points which are separated in time by not more than 40..mu..sec. The existence of a temperature flattening in the precursor phase, which we interpret as an m = 1 temperature island, is clearly demonstrated, and its location and width agree well with local emissivity measurements from soft x-ray tomography viewing the same poloidal plane. The rotating temperature island in the precursor phase, the outward movement of the region of high T/sub c/ during the crash phase, and the shape of T/sub e/ during the crash phase, and the shape of T/sub e/ distribution after the crash during the successor phase have all been documented in a time sequence of color contours. 4 refs., 10 figs.

In the realistic simulation of physical systems whether in 2-D or 3-D there arise often situations where the fluid flow has a preferred direction. An example of this is a magnetically confined plasma where the flow is predominently along the magnetic field lines. To avoid a large numerical diffusion and hence inaccuracy, it is often necessary to adopt a flux coordinate system with coordinates following closely the contours given naturally by the physics of the problem, an example for this is shown.

The structure of a magnetic field is determined by a one-degree of freedom, time-dependent Hamiltonian. This Hamiltonian is evaluated for a given field in a perturbed action-angle form. The location and the size of magnetic islands in the given field are determined from Hamiltonian perturbation theory and from an ordinary Poincare plot of the field line trajectories.

The tomographic method used for deriving soft x-ray local emissivities on TFTR, using one horizontal array of 60 soft x-ray detectors, is described. This method, which is based on inversion of Fourier components and subsequent reconstruction, has been applied to the study of a sawtooth crash. A flattening in the soft x-ray profile, which we interpret as an m = 1 island, is clearly visible during the precursor phase and its location and width correlate well with those from electron temperature profiles reconstructed from electron cyclotron emission measurement. The limitations of the Fourier method, due notably to the aperiodic nature of the signals in the fast crash phase and the difficulty of obtaining accurately the higher Fourier harmonics, are discussed. 9 refs., 13 figs.

Expansion techniques have led to new tools for designing stellarators and interpreting experimental results. The stellarator expansion is coupled with Hamiltonian techniques to investigate the effect of plasma pressure on magnetic surface structure. An expansion for nonplanar-axis systems is described. A reduced-equation initial-value formalism is used to create three-dimensional heliac equilibria. A low-..beta.. expansion with an auxiliary expansion about the magnetic axis makes possible the determination of island widths in heliacs and allows analytic determination of the effect of equilibrium modifications associated with plasma pressure on radial transport.

This paper is an extension of the previous paper, A Program Generator for the Incomplete LU-Decomposition-Conjugate Gradient (ILUCG) Method which appeared in Computer Physics Communications. In that paper a generator program was presented which produced a code package to solve the system of equations Ax/sub approx./ = b/sub approx./, where A is an arbitrary nonsingular matrix, by the ILUCG method. In the present paper an alternative generator program is offered which produces a code package applicable to the case where A is symmetric and positive definite. The numerical algorithm used is the Incomplete Cholesky Conjugate Gradient (ICCG) method of Meijerink and Van der Vorst, which executes approximately twice as fast per iteration as the ILUCG method. In addition, an optional preprocessor is provided to treat the case of a not diagonally dominant nonsymmetric and nonsingular matrix A by solving the equation A/sup T/Ax/sub approx./ = A/sup T/b/sub approx./.

Astronomical objects for which gravitational forces are dominant are usually spherical in form, and many computer calculations have been successfully carried out in this 1-dimensional regime. Examples include stellar structure and evolution [1], stellar pulsation [2] and collapse [3], and the dynamics of star clusters [4]. When rotation is important the geometry typically becomes axially symmetric (or 2-dimensional) as in a disc-shaped galaxy, and if a magnetic field is added one reaches the full 3-dimensional case. There are many situations in cosmical physics and astrophysics where it would be useful to be able to obtain detailed computer solutions of sets of coupled time-dependent 3-dimensional magnetofluid equations. Depending on the problem these equations should include a variety of physical effects, e.g. gravitation, special or general relativity, ionization, radiation or pair production processes and so on.

A numerical model based on the assumption that pulsars are spinning magnetized neutron stars is studied. A two-fluid model which includes inertial terms coupled self-consistently with Maxwell's equations is used. In this first paper of the series the case with aligned rotation and magnetic axes is examined with the assumption that the only source of charged particles is the star itself. It is found that under these conditions the magnetic field lines are closed everywhere and the particles in the equatorial region close to the light cylinder R/sub L/ reach sufficiently high energies to escape from the magnetosphere. With the Crab pulsar parameters, 1x10$sup 38$ ergs s$sup -1$ are carried away by relativistic particles and an equatorial region of high energy density is found near R/sub L/. A tentative conclusion is made that the described behavior is due to the assumption regarding the source of particles which results in 100 percent charge separation and a space-charge-limited emission from the star. In this space-charge-limited regime not enough current is available to open the field lines, but the electrostatic forces are large enough to force motion across the field lines. An artificially large particle rest mass has been used in this papermore » to ensure a smoother transition between the nonrelativistic and highly relativistic regions of space. Solutions which correspond to an essentially negligible rest mass in the nonrelativistic region will be discussed in subsequent papers. (AIP)« less

This paper is an extension of our previous paper “A program generator for the Incomplete LU-decomposition-Conjugate Gradient (ILUCG) method” which appeared in Computer Physics Communications. In that paper we presented a generator program which produced a code package to solve the system of equations Ax = bb, where A is an arbitrary nonsingular matrix, by the ILUCG method. In the present paper we offer an alternative generator program which produces a code package applicable to the case where A is symmetric and positive definite. The numerical algorithm used is the Incomplete Cholesky Conjugate Gradient (ICCG) method of Meijerink and Van der Vorst which executes approximately twice as fast per iteration as the ILUCG method. In addition, we provide an optional preprocessor to treat the case of a not diagonally dominant nonsymmetric and nonsingular matrix A by solving the equation ATAx = ATb.

A method is developed for evaluating transport coefficients in asymmetric geometries using the Monte Carlo method. The method is applied to the stellarator.