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We present a novel method for the solution of the diffusion equation on a composite AMR mesh. This approach is suitable for including diffusion based physics modules to hydrocodes that support ALE and AMR capabilities. To illustrate, we proffer our implementations of diffusion based radiation transport and heat conduction in a hydrocode called ALE-AMR. Numerical experiments conducted with the diffusion solver and associated physics packages yield 2nd order convergence in the L2 norm.

The critical issue of divertor configuration for heat and particle flux control in a conceptual ARIES compact stellarator (CS) reactor is addressed. The goal is to determine a divertor location and...

The finite difference discretization method is used to solve the electron energy transport equation in complex 3D edge geometries using an unstructured grid. This grid is generated by field-line tracing to separate the radial and parallel fluxes and minimize the numerical diffusion connected with the strong anisotropy of the system. The influence of ergodicity on the edge plasma transport in the W7-X stellarator is investigated in this paper. Results show that the combined effect of ergodicity and the radial plasma diffusion leads to the efficient smoothing of the temperature profiles in the finite-β case.

The magnetic field between the plasma surface and wall of the National Compact Stellarator (NCSX), is mapped via field line tracing in finite beta (4%) configurations with part (20%) of the rotational transform generated by the bootstrap current. The edge plasma magnetics are more stellarator-like, with a complex 3D structure, and less like the ordered 2D symmetric structure of a tokamak. The field lines make a transition from ergodically covering a surface to ergodically covering a volume, as the distance from the last closed magnetic surface is increased. The first few toroidal transits from the starting points are highly structured, with significant flux expansion at the bean shaped cross-section. The later toroidal transits become moderately stochastic, but with Kolmogorov lengths that can be made smaller than the connection if the separation between wall and plasma is sufficiently large.

Toroidal plasmas created with negative magnetic shear in the core region offer advantages in terms of MHD stability properties. These plasmas, transiently created in several tokamaks, have exhibited high-performance as measured by normalized stored energy and neutron production rates. A critical issue with extending the duration of these plasmas is the need to maintain the off-axis-peaked current distribution required to support the minimum in the safety factor q at large radii. We present equilibrium and transport simulations that explore the use of electron cyclotron heating and current drive to maintain this negative shear configuration. Using parameters consistent with DIII-D tokamak operation (Strait E et al 1995 Phys. Rev. Lett. 75 4421, Rice B W et al 1996 Nucl. Fusion 36 1271), we find that with sufficiently high injected power, it is possible to achieve steady-state conditions employing well aligned electron cyclotron and bootstrap current drive in fully non-inductively current-driven configurations.

Pulsed reflectometry using both ordinary (O) and extraordinary (X) modes can provide time- and space-resolved measurements of the electron density, the magnitude of the magnetic field, the magnetic shear as a function of radius, and information on density and magnetic fluctuations. Such a diagnostic also yields the current profile from the curl of the magnetic field. This research addresses theoretical issues associated with the use of pulsed reflectometry with particular emphasis on applications in the Sustained Spheromak Physics Experiment (SSPX) at the Lawrence Livermore National Laboratory [E. B. Hooper et al., “Sustained Spheromak Physics Experiment,” in Proceedings of the 17th International Atomic Energy Agency (IAEA) Fusion Energy Conference, Yokohama, Japan, October 19–24, 1998, Lawrence Livermore National Laboratory Report UCRL-JC-132034 (September 29, 1998)]. Simulation results are presented for O- and X-mode mixed-polarization reflectometry and linear mode conversion in two spatial dimensions. The profile reconstruction algorithms depend on Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) formulae for group delays and linear mode conversion, which agree reasonably well with direct numerical solutions of the wave equation. Reconstructions of the electron density and modulus of the magnetic field are relatively robust in the presence of two-dimensional electron density and magnetic perturbations of the plasma.

Ultra-short-pulse reflectometry is studied by means of the numerical integration of one- and two-dimensional full-wave equations for ordinary and extraordinary modes propagating in a plasma. The numerical calculations illustrate the use of the reflection of ultra-short-pulse microwaves as an effective probe of the density or magnetic profile in the presence of density or magnetic fluctuations in the plasma. Bragg resonance effects can be identified in the reflected signals, which give information on fluctuations. It is also demonstrated that ultra-short-pulse reflectometry can be used to perform correlation reflectometry measurements in which correlation lengths for density fluctuations are deduced from the observed cross-correlation function of phase shifts as a function of frequency.

Scattering of laser light by stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS) is a concern for indirect drive inertial confinement fusion (ICF). The hohlraum designs for the National Ignition Facility (NIF) raise particular concerns due to the large scale and homogeneity of the plasmas within them. Experiments at Nova have studied laser–plasma interactions within large scale length plasmas that mimic many of the characteristics of the NIF hohlraum plasmas. Filamentation and scattering of laser light by SBS and SRS have been investigated as a function of beam smoothing and plasma conditions. Narrowly collimated SRS backscatter has been observed from low density, low‐Z, plasmas, which are representative of the plasma filling most of the NIF hohlraum. SBS backscatter is found to occur in the high‐Z plasma of gold ablated from the wall. Both SBS and SRS are observed to be at acceptable levels in experiments using smoothing by spectral dispersion (SSD).

Three-dimensional simulations are presented which calculate the time-dependent, self-consistent evolution of filamentation and stimulated Brillouin backscatter (SBBS) in nonuniform laser beams with intense hot spots. The simulations show significant reflectivity ( $g$1%) for modest average-intensity gain exponents (5--10). Many hot spots cooperate to produce a total SBBS amplification that increases with the plasma length as well as hot-spot length. Temporal beam smoothing with bandwidth comparable to the SBBS growth rate substantially reduces the nonuniform-laser-beam reflectivity.

In this paper we discuss the present divertor design for the planned TPX tokamak, which will explore the physics and technology of steady-state (1000s pulses) heat and particle removal in high confinement (2--4{times} L-mode), high beta ({beta}{sub N} {ge} 3) divertor plasmas sustained by non-inductive current drive. The TPX device will operate in the double-null divertor configuration, with actively cooled graphite targets forming a deep (0.5 m) slot at the outer strike point. The peak heat flux on, the highly tilted (74{degrees} from normal) re-entrant (to recycle ions back toward the separatrix) will be in the range of 4--6 MW/m{sup 2} with 18 MW of neutral beams and RF heating power. The combination of active pumping and gas puffing (deuterium plus impurities), along with higher heating power (45 MW maximum) will allow testing of radiative divertor concepts at ITER-like power densities.

The Landau‐fluid model is a recently introduced fluid‐moment closure scheme [G. W. Hammett and F. W. Perkins, Phys. Rev. Lett. 64, 3019 (1990)] that was designed to include kinetic dissipative effects like Landau damping in fluid calculations. The fluid‐moment hierarchy is terminated by assuming linear relationships among the retained moments in Fourier‐transform space, with coefficients determined by matching the plasma response to that obtained from a kinetic analysis. This paper generalizes the technique to the full range of ion and electron collisionality and applies it to a new fluid simulation code constructed to study laser filamentation in underdense plasmas [Berger et al., Phys. Fluids B 5, 2243 (1993)]. By matching the ion‐acoustic complex frequency derived from the fluid model with that predicted by collisional, Fokker–Planck, and kinetic analyses, the specific heat ratio, thermal conductivity coefficient, and viscosity coefficient for ions and the thermal conductivity coefficient for electrons are determined as functions of the wave number k. For frequencies much less than the pump frequency this leads to a fourth‐order polynomial dispersion relation whose spectrum includes damped ion‐acoustic waves as well as filamentation modes whose stability depends on the pump strength. An analytic instability threshold condition on the laser intensity is derived from which the relative importance of ponderomotive and thermal drives can be assessed. Expressions for the linear susceptibilities in the presence of a finite‐amplitude pump are also given, which might prove useful for understanding spectral linewidths for Thomson scattering.

The features of the fully implicit 2-D fluid code UEDGE are described. The utility of the code is demonstrated by showing bifurcations or multiple solutions of the tokamak edge plasma for both deuterium and impurity injection in the divertor.

A desire to interpret recent experiments on filamentation with and without beam‐smoothing techniques led to the development of a three‐dimensional fluid model that includes the effects of nonlocal electron transport and kinetic ion damping of the acoustic waves. The damping of the electron‐temperature perturbations that drive thermal filamentation by nonlocal electron conduction, valid in the diffusive limit, is supplemented in the present model by electron Landau damping in the collisionless limit when the wavelength of the perturbation is much less than the electron–ion scattering mean‐free path. In this collisionless limit, Landau damping of the ‘‘temperature’’ fluctuations makes ponderomotive forces universally more important than thermal forces. Simulations in plasmas of current interest illustrate the relative importance of thermal and ponderomotive forces for strongly modulated laser beams. Although thermal forces may initiate filamentation, the most intense filaments are associated with ponderomotive forces. The present simulations of filamentation model well the density perturbations observed in experiments [Young et al., Phys. Rev. Lett. 61, 2336 (1988)]. In addition, a simple criterion is obtained analytically and supported by simulations for stabilization of filamentation by laser beam‐smoothing techniques such as induced spatial incoherence and random phase plates [Eq. (1)].

The effects of plasma currents and cross-field drifts on single null tokamak divertor operation are simulated using a fully implicit 2-D fluid code. Equations solved are those for particle continuity, parallel momentum, electron energy, ion energy, electrostatic potential, and neutral gas diffusion. The core and scrape-off layer regions are separated by a magnetic separatrix, both of which are included in the simulation. The core plasma is poloidally periodic, and the inner and outer private flux regions are properly connected. The code utilizes a fully implicit method-of-lines scheme to advance the variables in time with a Krylov technique or a direct Newton iteration with a numerical Jacobian. Results are presented on the effects of currents and cross-field drifts for DIII-D single-null parameters.

Fragmentation is a fundamental material process that naturally spans spatial scales from microscopic to macroscopic. We developed a mathematical framework using an innovative combination of hierarchical material modeling (HMM) and adaptive mesh refinement (AMR) to connect the continuum to microstructural regimes. This framework has been implemented in a new multi-physics, multi-scale, 3D simulation code, NIF ALE-AMR. New multi-material volume fraction and interface reconstruction algorithms were developed for this new code, which is leading the world effort in hydrodynamic simulations that combine AMR with ALE (Arbitrary Lagrangian-Eulerian) techniques. The interface reconstruction algorithm is also used to produce fragments following material failure. In general, the material strength and failure models have history vector components that must be advected along with other properties of the mesh during remap stage of the ALE hydrodynamics. The fragmentation models are validated against an electromagnetically driven expanding ring experiment and dedicated laser-based fragmentation experiments conducted at the Jupiter Laser Facility. As part of the exit plan, the NIF ALE-AMR code was applied to a number of fragmentation problems of interest to the National Ignition Facility (NIF). One example shows the added benefit of multi-material ALE-AMR that relaxes the requirement that material boundaries must be along mesh boundaries.

Divertor heat load distributions due to thermal and alpha particles have been assessed in an NCSX-based compact stellarator reactor. A divertor plate system is envisaged, with 4 plates per field period and covering 7% of the plasma surface area. The field-line tracing technique is employed; for thermal flux, the conventional approach is used, while for alphas, their characteristic exit pattern from the plasma and subsequent gyro- orbits are approximated. For the ARIES-CS reference design point (R=7.75 m, A=4.5, B=5.7 T, beta=6.4% and Pnet=1000 MW), combined peak heat loads in the 5 -18 MW/m2 range on the plates have been obtained, assuming a 75% radiation fraction both in the core and at the edge, and a 5% alpha loss fraction. The alpha heat flux could be a dominant determining factor. Further optimization study is warranted to lower all peak heat loads to satisfy the accepted limit of les10 MW/m2.

In response to the comment on theory and simulation of light filamentation in laser‐produced plasma the authors believe that the criticism overstates the outcome of choosing one form of thermal conduction over another. (AIP)

Theoretical and experimental studies of plasma diagnostic reflectometry have been undertaken as a collaborative research project between the Lawrence Livermore National Laboratory (LLNL) and the University of California Department of Applied Science Plasma Diagnostics Group under the auspices of the Laboratory Directed Research and Development Program at LLNL. Theoretical analyses have explored the basic principles of reflectometry to understand its limitations, to address specific gaps in the understanding of reflectometry measurements in laboratory experiments, and to explore extensions of reflectometry such as ultra-short-pulse reflectometry. The theory has supported basic laboratory reflectometry experiments where reflectometry measurements can be corroborated by independent diagnostic measurements.

A divertor simulator, IDEAL, has been proposed by S. Cohen to study the difficult power-handling requirements of the tokamak program in general and the ITER program in particular. Projections of the power density in the ITER divertor reach {approximately} 1 Gw/m{sup 2} along the magnetic fieldlines and > 10 MW/m{sup 2} on a surface inclined at a shallow angle to the fieldlines. These power densities are substantially greater than can be handled reliably on the surface, so new techniques are required to reduce the power density to a reasonable level. Although the divertor physics must be demonstrated in tokamaks, a linear device could contribute to the development because of its flexibility, the easy access to the plasma and to tested components, and long pulse operation (essentially cw). However, a decision to build a simulator requires not just the recognition of its programmatic value, but also confidence that it can meet the required parameters at an affordable cost. Accordingly, as reported here, it was decided to examine the physics of the proposed device, including kinetic effects resulting from the intense heating required to reach the plasma parameters, and to conduct an independent cost estimate. The detailed role of the simulator inmore » a divertor program is not explored in this report.« less

ALE-AMR is a new hydrocode that we are developing as a predictive modeling tool for debris and shrapnel formation in high-energy laser experiments. In this paper we present our approach to implementing laser ray tracing in ALE-AMR. We present the basic concepts of laser ray tracing and our approach to eciently traverse the adaptive mesh hierarchy.

The generation of neutron/gamma radiation, electromagnetic pulses (EMP), debris and shrapnel at mega-Joule class laser facilities (NIF and LMJ) impacts experiments conducted at these facilities. The complex 3D numerical codes used to assess these impacts range from an established code that required minor modifications (MCNP - calculates neutron and gamma radiation levels in complex geometries), through a code that required significant modifications to treat new phenomena (EMSolve - calculates EMP from electrons escaping from laser targets), to a new code, ALE-AMR, that is being developed through a joint collaboration between LLNL, CEA, and UC (UCSD, UCLA, and LBL) for debris and shrapnel modelling.

The ALE-AMR code developed for NIF is a multi-material hydro-code that models target assembly fragmentation in the aftermath of a shot. The combination of ALE (Arbitrary Lagrangian Eulerian) hydro with AMR (Adaptive Mesh Refinement) allows the code to model a wide range of physical conditions and spatial scales. The large range of temperatures encountered in the NIF target chamber can lead to significant fluxes of energy due to thermal conduction and radiative transport. These physical effects can be modeled approximately with the aid of the diffusion equation. We present a novel method for the solution of the diffusion equation on a composite mesh in order to capture these physical effects.

We have developed a new 3D multi-physics multi-material code, ALE- AMR, for modeling laser/target effects including debris/shrapnel generation. The code combines Arbitrary Lagrangian Eulerian (ALE) hydrodynamics with Adaptive Mesh Refinement (AMR) to connect the continuum to microstructural regimes. The code is unique in its ability to model hot radiating plasmas and cold fragmenting solids. New numerical techniques were developed for many of the physics packages to work efficiency on a dynamically moving and adapting mesh. A flexible strength/failure framework allows for pluggable material models. Material history arrays are used to store persistent data required by the material models, for instance, the level of accumulated damage or the evolving yield stress in J2 plasticity models. We model ductile metals as well as brittle materials such as Si, Be, and B4C. We use interface reconstruction based on volume fractions of the material components within mixed zones and reconstruct interfaces as needed. This interface reconstruction model is also used for void coalescence and fragmentation. The AMR framework allows for hierarchical material modeling (HMM) with different material models at different levels of refinement. Laser rays are propagated through a virtual composite mesh consisting of the finest resolution representation of the modeled space. A new 2 nd order accurate diffusion solver has been implemented for the thermal conduction and radiation transport packages. The code is validated using laser and x-ray driven spall experiments in the US and France. We present an overview of the code and simulation results.

The authors disagree with the assertions in Ref. 1. The authors think that the expression for the electron heat flux is fairly general and the appropriate electron damping rage, eq. (56) of Ref. 2 is correct. (AIP)

Ultra-short-pulse reflectometry is studied by means of the numerical integration of one- and two-dimensional full-wave equations for ordinary and extraordinary modes propagating in a plasma. The numerical calculations illustrate the use of the reflection of ultra-short-pulse microwaves as an effective probe of the density or magnetic profile in the presence of density or magnetic fluctuations in the plasma. Bragg resonance effects can be identified in the reflected signals, which give information on fluctuations. It is also demonstrated that ultra-short-pulse reflectometry can be used to perform correlation reflectometry measurements in which correlation lengths for density fluctuations are deduced from the observed cross-correlation function of phase shifts as a function of frequency.

A symmetric tandem mirror plugging a long central cell, with plugs stabilized by a hot‐component plasma is considered. The system is taken to have a flat pressure profile with a steep edge gradient. The interaction of the precessional mode with Alfven waves generated in the central cell is then considered. This analysis is noneikonal and is valid when mΔ/r

Particle and energy confinement of ions during ion cyclotron heating (ICH) in the centre cell of a tandem mirror is simulated by a Monte Carlo code. The interaction of the ion cyclotron wave with the ions is calculated using reduced equations of motion, and Coulomb collisions are modelled using a Monte Carlo algorithm. An axial electrostatic potential profile is generated by an iteration procedure, assuming Maxwellian electrons. The axial profile of neutral gas which fuels the ions from a localized gas box is determined from a neutral particle Monte Carlo code. Steady state calculations are reported for the Tandem Mirror Experiment Upgrade (TMX-U). The calculated energy confinement time τE is about twice that measured for off-midplane fuelling. The calculated τE can be decreased to the experimental value by either introducing extra gas at the ICH antenna to model the effect of ions striking the antenna or assuming anomalous radial loss. However, the enhanced radial loss also increases the end loss current well above that measured. Experimental and calculated τE agree more closely for midplane fuelling with heating from one antenna.

Summarized in this article are the design characteristics of the magnet-system configuration constructed for use in the modified Tandem Mirror Experiment (TMX Upgrade), and a description of the resulting vacuum magnetic field. Many engineering and physics considerations and limitations governed the design. Several of the physics issues are discussed here, including single-particle drift surfaces and adiabaticity, central-cell resonant radial transport, magnetohydrodynamic stability analysis, and finite-beta equilibrium. The described design procedures can be applied to other tandem-mirror experiments or reactor studies.

An integrated study of compact stellarator power plants, ARIES-CS, has been conducted to explore attractive compact stellarator configurations and to define key research and development (R&D) areas. The large size and mass predicted by earlier stellarator power plant studies had led to cost projections much higher than those of the advanced tokamak power plant. As such, the first major goal of the ARIES-CS research was to investigate if stellarator power plants can be made to be comparable in size to advanced tokamak variants while maintaining desirable stellarator properties. As stellarator fusion core components would have complex shapes and geometry, the second major goal of the ARIES-CS study was to understand and quantify, as much as possible, the impact of the complex shape and geometry of fusion core components. This paper focuses on the directions we pursued to optimize the compact stellarator as a fusion power plant, summarizes the major findings from the study, highlights the key design aspects and constraints associated with a compact stellarator, and identifies the major issues to help guide future R&D.

The kinetic equation for particles interacting with turbulent fluctuations is derived by a new nonlinear technique which successfully corrects the difficulties associated with quasilinear theory. In this new method the effects of the fluctuations are evaluated along particle orbits which themselves include the effects of a statistically averaged subset of the possible configurations of the turbulence. The new method is illustrated by calculating the pitch angle diffusion coefficient D sub Mu Mu for particles interacting with slab model magnetic turbulence, i.e., magnetic fluctuations linearly polarized transverse to a mean magnetic field. Results are compared with those of quasilinear theory and also with those of Monte Carlo calculations. The major effect of the nonlinear treatment in this illustration is the determination of D sub Mu Mu in the vicinity of 90 deg pitch angles where quasilinear theory breaks down. The spatial diffusion coefficient parallel to a mean magnetic field is evaluated using D sub Mu Mu as calculated by this technique. It is argued that the partially averaged field method is not limited to small amplitude fluctuating fields and is hence not a perturbation theory.

The equation for the m=1 ballooning mode is derived; the dominant finite Larmor radius terms force the mode to be rigid. A dispersion relation with the wall near the plasma is obtained for arbitrary β, and it is also shown that with the wall at infinity only the vacuum curvature contributes to all orders in β.

Mirror equilibria for arbitrary mirror ratio and flux‐tube eccentricity are obtained to leading order in the plasma pressure (beta expansion) in the paraxial limit (axial scale lengths long compared with radial scale lengths). The solutions are given in terms of quadratures over known functions. The theory is applied to a tandem‐mirror configuration.

Monte Carlo simulation experiments have been performed in order to study the velocity diffusion of charged particles in a static turbulent magnetic field. By following orbits of particles moving in a large ensemble of random magnetic field realizations with suitably chosen statistical properties, a pitch‐angle diffusion coefficient is derived. Results are presented for a variety of particle rigidities and rms random field strengths and compared with the predictions of standard quasi‐linear theory and the nonlinear partially averaged field theory.

Ballooning modes in tandem mirrors with quadrupole vacuum field symmetry are investigated in the limits of zero gyroradius and large aspect ratio. The eigenmode equation derived from the guiding‐center energy principle is reduced to one dimension by the use of an eikonal representation of the cross‐field variation of the perturbation. The eigenvalue problem is solved numerically and the structure of the local dispersion relation examined in order to derive conditions for marginal stability. Applications to the TMX‐U and MFTF‐B tandem‐mirror experiments are presented.

By adjusting the electrostatic potential on the end plates of a mirror plasma, the plasma’s magnetohydrodynamic (MHD) stability can be altered. Furthermore, the plasma’s MHD response to low‐frequency modulation of the end plates can be used to pump unwanted ions that collect in a thermal barrier. Here, precise boundary conditions are derived for the MHD equations of motion to model an end plate separated from the plasma by a sheath.

Velocity sapce fokker−planc coefficients obtained in an investigation of the temperature relaxation of a magnetizaed plasma differ from those obtained by other workers in thensor structue. Although the diffusion tensor is in error, it is shown that the relaxation rates are unaffected by the mistake.

Stability of tandem mirrors to rigid‐displacement ballooning modes is studied using vacuum field geometry. Results show substantial improvement in values of beta over those predicted by ‘‘infinite‐m’’ theory. Previous work has shown that kinetic effects stabilize these latter short‐wavelength modes, suggesting that only the rigid mode survives.