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One-Dimensional Transport Code Modeling of the Divertor-Limiter Region in Tokamaks
Abstract
A model of the divertor-limiter scrapeoff region has been incorporated into the BALDUR one-dimensional tokamak transport code. Simulations of the proposed Toroidal Fusion Test Reactor (TFTR), and Poloidal Divertor (PDX) experiments and existing Alcator-A tokamak experiments have been carried out for ohmic and neutral beam heated cases. In particular, we have studied how the edge conditions and energy-loss mechanisms in PDX depend upon plasma density, and compared our results with analytic estimates. The sensitivity of the results to changes in the transport coefficients and scrapeoff model is discussed with particular reference to the power loading on the TFTR limiter.
The power deposition and wall material erosion rates due to charge-exchange neutral atoms resulting from a recycling source at limiters in the TEXTOR and Tokamak Fusion Test Reactor tokamaks are reported. The analysis is carried out using a recently developed finite element, two-dimensional toroidal geometry diffusion theory, neutral atom transport theory, and the computer code FENAT. The power deposition and material erosion are highest at the limiter. The first wall suffers very little erosion except for the portion near the limiter.
Neutral pumping rates are calculated for pump-limiter and divertor options of a next step tokamak ignition device using a method that accounts for the coupled effects of neutral transport and plasma transport. For both pump limiters and divertors the plasma flow into the channel surrounding the neutralizer plate is greatly reduced by the neutral recycling. The fraction of this flow that is pumped can be large (>50%) but in general is dependent on the particular geometry and plasma conditions. It is estimated that pumping speeds ≳10⁵ L/s are adequate for the exhaust requirements in the pump-limiter and the divertor cases.
Outline: Section 2 simply describes, without attempt at explanation, what happens electrically when a plasma is in contact with a solid surface. The practical implications of the interaction are briefly described. Section 3 provides a physical explanation of why these effects occur and deduces initial estimates of the plasma-solid voltage difference and the spatial extent of this voltage drop. Section 4 deduces the Bohm Criterion (ion drift velocity out of the plasma must equal the ion acoustic speed) using ion fluid models. The Criterion is obtained from both the plasma and the sheath equations separately. Section 5 deduces simple formulae for the particle and energy flux which is transmitted by a sheath, both for electrically floating and biased objects. Section 6 gives a brief indication of how the sheath particle and energy transmission characteristics influence the modeling of the edge plasma in magnetically confined plasma devices, while Section 7 gives a similar brief introduction to their use in the interpretation of plasma probe data.
There have been significant advances in impurity control modeling in the last five years. The models for impurity control has grown in sophistication from simple, almost heuristic models to two and three-dimensional codes embodying realistic geometries and many of the important plasma wall and atomic physics processes. Perhaps more importantly, the growth in the models has helped to increase our understanding of how divertors and limiters work as impurity control devices and has aided in the design of improved divertors and limiters. The greatest success of the models has been in helping to identify the key role that local recycling can play in producing a “high recycling” divertor in which a cool, dense plasma is produced next to neutralizer plate. This cool plasma allows the possibility of removing the heating power (neutral beams, ICRF, alpha particles, etc.) from the plasma without the generation of large amounts of impurities. The low temperature reduces the sheath potential and thus reduces the energy of the ions and neutrals that strike the neutralizer plate to below the sputtering threshold of a number of candidate wall materials, eliminating the divertor plate as a source of impurities.
When large probes are used to measure plasma properties in tokamak scrape-off layers, the presence of the probe influences the plasma being measured. A theory is presented for probe interpretation which makes it possible to infer undisturbed values of T/sub e/, T/sub i/, n/sub e/, etc. The first requirement in such an analysis is for analytic models of the scrape-off layer, which can then be modified to allow for the presence of the probe. The present work gives such a model for the collisionless scrape-off layer, and criteria are stated governing its applicability. Such models may be of use for other applications, e.g., predictions of power loading to limiter surfaces, the magnitude of wall sputtering, etc.
The density limit observed in tokamak experiments is thought to be due to a radiative collapse of the current channel. A transport code coupled with a magnetohydrodynamic (MHD) equilibrium routine is used to determine the detailed, self-consistent evolution of the plasma profiles in tokamak discharges with radiated power close to or equaling the input power. The present work is confined to Ohmic discharges in steady state. It is found that the shape of the density profile can have a significant impact on the variation of the maximum electron density with plasma current. Analytic calculations confirm this result.
Plasma drift or rotation is caused at the edge of tokamaks by various mechanisms including flow to surfaces and neutral injection. These drift velocities can be measured using electrostatic probes and a theory is presented providing a method of probe analysis to deduce drift velocity, plasma density, and temperature. Initial probe experiments using this probe analysis yield credible values of drift velocity, but comparison with an independent technique, e.g., spectroscopic Doppler shifts, is sought.
The control of tritium in TFTR has both important safety and environmental implications. Current modelling techniques allow realistic predictions of the tritium inventory in and permeation through in-torus components. The source of the tritium was computed using a three-dimensional description of the neutral particle flux on the TFTR vacuum vessel wall from the neutral transport code DEGAS, under plasma conditions modelled with the one-dimensional transport code BALDUR. The movable limiter (graphite) was modelled using an extension of the Local Mixing Model for hydrogen retention and isotope exchange. In particular, the calculations consider the inventory and recyling for the movable limiters (which are expected to experience transient temperatures up to 2000°C) as well as the bumper limiters and protective plates which will remain somewhat cooler. Transient effects are included. The tritium transport in the stainless steel wall was modelled using the DIFFUSE code with materials parameters taken from the most recent literature. It appears that for the expected material properties, specifically the hydrogen recombination constant measured in a clean TFTR environment, and operating scenario, there will be essentially no tritium permeation through the stainless steel walls or bellows and less than 0.1 kCi of tritium inventory in the stainless steel first wall. The limiters contain as much as 5 kCi of tritium.
Ionisation occuring in the scrape-off region of magnetically-confined plasmas can result in significant changes in the density profile. An analytic model is presented and results are compared with a computer model result for INTOR. Sub-layers are identified in the scrape-off layer, including a density plateau, and criteria for the existence of such layers are presented.
This paper describes two experiments on edge plasma transport in the Caltech Research Tokamak (BT = 4 kG, R = 45 cm, a = 15 cm, T(edge) ≅25 eV, n(edge)~2 × 1012 cm-3). The first aims to understand the mechanism of edge plasma cross-field transport by attempting to measure with Langmuir probes the local E ̃ and n ̃ and their correlation in order to compute the fluctuation-induced particle flux Λrad = c · 〈 E ̃pol· n ̃〉/BT. The second experiment aims at demonstrating control of the plasma-limiter interaction through use of a local divertor coil mounted inside the limiter itself.
The two-dimensional flow of a collision-dominated hydrogen scrape-off plasma in an axisymmetric tokamak is examined. This flow is described by a set of equations which contain the dominant terms in a maximal ordering appropriate to high-density experimental divertors and reactor scrape-off plasmas. Comparison of the theory to estimates of scrape-off parameters in the Doublet III expanded boundary plasmas suggests that analysis of classical and neoclassical processes alone may be sufficient to predict plasma transport in high-density scrape-off plasmas of practical importance.
Ionization near the target plate is shown to play an important role in biasing experiments. Our previous SOL model, which calculates the induced radial electric field, is found to be inadequate to treat the new divertor geometry of TdeV. When recycling is included via the measured emission near the plate, the upgraded model correctly reproduces all the observed electric currents and fields during biasing in the new divertor configuration. A simple divertor model using this calculated field has been developed to simulate the evolution of the divertor ion and neutral parameters under the action of neutralization plate biasing. Using a 1D adiabatic fluid model for the divertor ions, a 1D convective representation for the SOL neutrals and a 0D calculation for the plenum pressure, this divertor model satisfactorily simulates most of the TdeV biasing experiments at all biasing voltages and all toroidal field directions at low line-averaged densities. The weaker agreement at high densities is largely a consequence of the crudeness of the general divertor physics rather than of the deficiency of the biasing physics implemented in the model. The model is finally used to explain the polarity asymmetries observed in divertor efficiencies during biasing, and to demonstrate that no mechanism other than plate current saturation is required to interpret the saturation of toroidal rotation observed in the SOL at large biasing voltages of either polarity.
Effects of the plasma boundary can have a substantial influence on the behaviour of the entire plasma in tokamaks. Progress in the field, particularly that over the last decade, is reviewed, with emphasis on experimental observation. Simple modelling for interpretation is also included.
Computer simulations with special versions of the one dimensional BALDUR predictive transport code are carried out to investigate the particle confinement of helium and hydrogen, the energy confinement and the burn control in the high density scenario of the ITER (CDA) physics phase. The code uses empirical transport coefficients for ELMy H mode plasmas, an improved model of the scrape-off layer (SOL), an impurity radiation model for helium and iron, and fast burn control by neutral beam injection feedback. A self-sustained thermonuclear burn is achieved for hundreds of seconds. The necessary radiation corrected energy confinement time τE is found to be 4.2 s, which is attainable according to the ITER H mode scaling. In the ignited ITER, a significant dilution of the DT fuel by helium takes place. Steady state helium fractions of up to 8% are obtained, which are found to be compatible with self-sustained burn. The SOL model yields self-consistent electron densities and temperatures at the separatrix (ne = 5.8 × 1019 m-3, Te = 80 eV)
Probe measurements of the edge plasmas of the Macrotor and Microtor tokamaks are described. Limiter scrape-off layer thicknesses as measured with Langmuir probes are for Macrotor λ7–10 cm and for Microtor λ ~ 1 cm, both values being consistent with the Bohm diffusion rate. Heat deposition measured with thermocouples attached to small probes shows an anomalously high heat flux, particularly from the electron drift direction. It is argued that the anomalous particle diffusion is most likely associated with the large edge density fluctuations, while the anomalous heat flux is most likely due to a directed high-energy electron population.
The requirements for ignition in a tokamak reactor with INTOR-like parameters were studied using a one-dimensional transport code. With empirical electron energy diffusivity
e
, ignition was obtained with 60–75 MW of neutral beam injection at a volume average pressure ratio =4–5% under a variety of conditions. Changing
e
gave ignition at the same if the plasma minor radius varied asa
e
1/2
. The maximum impurity concentration which allows ignition was found to be comparable to that for the much simpler case of a homogeneous plasma with radiative losses only. In long pulse simulations with efficient helium pumping, the maximum toroidal field ripple which allowed ignition was 2.0% (peak-to-peak) at the plasma edge. Ignition was maintained with over 99% recycling of helium ash using 5% less than maximum ripple.
The design of limiters for fusion devices depends critically on the transport properties of the plasma edge. Recent data from Langmuir probe experiments and calorimeter probe experiments on the poloidal divertor experiment (PDX) have provided information about the transport of particles and energy in the edge region of a plasma. These data have been used as input to a simple one dimensional model in which the particle and thermal diffusion coefficients are permitted to be functions of the density and temperature. The transport coefficients deduced from the data are then applied to the cases of the TFTR device and a device like FED/INTOR to predict the optimum design for limiters or pump limiters. The results indicate the optimum limiter should have a concave shape when viewed from the plasma. The effect of the uncertainties in the deduced coefficients on the optimum design performance are also discussed.
A new one-dimensional transport code named TASK/TX, which is able to describe dynamic behavior of tokamak plasmas, has been developed. It solves simultaneously a set of flux-surface averaged equations composed of Maxwell’s equations, continuity equations, equations of motion, heat transport equations, fast-particle slowing-down equations and two-group neutral diffusion equations. The set of equations describes plasma rotations in both toroidal and poloidal directions through momentum transfer and evaluates the radial electric field self-consistently. The finite element method with a piecewise linear interpolation function is employed with a fine radial mesh near the plasma surface. The Streamline Upwind Petrov–Galerkin method is also used for robust calculation. We have confirmed that the neoclassical properties are well described by the poloidal neoclassical viscous force. The modification of density profile during neutral beam injection is presented. In the presence of ion orbit loss, the generation of the inward radial electric field and torque due to radial current is self-consistently calculated.
A version of the BALDUR plasma transport code which calculates the evolution of plasma parameters is documented. This version uses an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modelling tokamak plasmas.
A graphite-shielded probe was recently installed in the divertor region of PDX to continuously monitor local electron temperature, electron density (from the ion saturation current), and plasma floating potential throughout divertor discharges. In ohmically heated deuterium plasmas, the electron temperature near the separatrix was 6 to 12 eV; these values confirm the low Te inferred from the density dependence of Balmer line emission from the divertor plasmas. During neutral beam heating, PDX divertor discharges were characterized by a sharp transition at which time the main chamber plasma density increased rapidly, the divertor Hα emission dropped, and the global energy confinement increased abruptly. At later times, edge relaxation oscillations, characterized by spikes in the Hα emission, occurred and were accompanied by a clamp in the density rise and lower confinement time. Limited scans of the temperature and density measured by the divertor probe indicated that these parameters changed with discharge conditions primarily near the separatrix. With the onset of neutral beam injection the temperature and density rose by a factor of 1.5 and 2–4 respectively. Transient drops in Te to values as low as 2 eV and concomitant rises in ne were sometimes observed near the time of the transition into the high confinement mode. Later in the discharge, the values returned to their pre-H-mode level. TV camera observations of the divertor probe revealed a “shadow” along the field lines indicating a well-defined flow in the vicinity of the separatrix.
Recent progress in models for poloidal divertors has both helped to explain current divertor experiments and contributed significantly to design efforts for future large tokamak (INTOR, etc.) divertor systems. These models range in sophistication from zero-dimensional treatments and dimensional analysis to two-dimensional models for plasma and neutral particle transport which include a wide variety of atomic and molecular processes as well as detailed treatments of the plasma-wall interaction. This paper presents a brief review of some of these models, describing the physics and approximations involved in each model. We discuss the wide variety of physics necessary for a comprehensive description of poloidal divertors. To illustrate the progress in models for poloidal divertors, we discuss some of our recent work as typical examples of the kinds of calculations being done.
The most promising concepts for power and particle control in tokamaks and other fusion experiments rely upon atomic processes to transfer the power and momentum from the edge plasma to the plasma chamber walls. This places a new emphasis on processes at low temperatures (1-200 eV) and high densities (10^20-10^22 m^-3). The most important atomic processes are impurity and hydrogen radiation, ionization, excitation, recombination, charge exchange, radiation transport, molecular collisions, and elastic scattering of atoms, molecules and ions. Important new developments have occurred in each of these areas. The best available data for these processes and an assessment of their role in plasma wall interactions are summarized, and the major areas where improved data are needed are reviewed. Comment: Preprint for the 11th PSI meeting, postscript with 22 figures, 40 pages
An axisymmetric divertor is investigated over a wide range of plasma parameters in the DIVA tokamak. Empirical scaling for scrape-off-layer plasma parameters is obtained. In an optimum discharge, 75% of the energy loss by conduction and convection and 33% of the total particle loss are guided into the divertor. By reducing the plasma-wall interactions and shielding impurity influx from the plasma, the divertor decreases both low- and high-Z impurities. Consequently, the radiation loss is reduced by a factor of 2–4, and the energy confinement time increases by a factor of 2.5.
Volumetric radiative loss measurements, correlated with temperature in the range of 10 000 to 26 000°K, have been made on an argon plasma. Pressures of 0.5, 1.0, and 2.0 atm have been used. The 1.0-atm measurements have been corrected for both absorption and ultraviolet emission and the results agree with those of Emmons in the common temperature range. The 6965 Ar I line has also been studied yielding lineshifts, halfwidths, absorption and emission coefficients. The line shift and halfwidth results are below theoretical predictions. Transition probabilities determined from both emission and absorption studies are found to be in reasonable agreement.
The electrical conductivity, translational and reactive thermal conductivity and viscosity have been computed for ionized argon in thermodynamic equilibrium at pressues from 0.001-1000 atm and temperatures to 35 000°K. Comparison of the values with experiments shows reasonable agreement.
Calculations are presented for the transport properties of a two-temperature argon plasma based on the Chapman-Enskog and perturbed-Lorentzian solutions of the Boltzmann equations. The convergence problems known to exist in the Chapman-Enskog solution for the properties of a one-temperature argon plasma also occur here; this difficulty is removed by using the perturbed Lorentzian solution at low and moderate degrees of ionization. Also, it is found that the lowest order perturbed Lorentzian and Chapman-Enskog approximate solutions provide upper and lower bounds to the actual solution at low and moderate degrees of ionization. Finally, owing to the Ramsauer minimum in the electron-atom cross section, the electrical conductivity of the two-temperature argon plasma is found to decrease with increasing electron temperature (at fixed heavy particle temperature and electron density) at low degrees of ionization, and increase with increasing electron temperature at higher degrees of ionization.
Measurements of the plasma density and electron temperature profiles in the shadow of the limiter in the Alcator tokamak are presented. These results indicate that plasma is transported across the field in this edge region at rates considerably higher than classical diffusion would predict. It also appears that the particle confinement time in this region is much less than in the central core of the plasma.
A new approach to prevent the influx of high-Z impurities into the core of a tokamak discharge by using RF power to modify the edge plasma temperature profile is discussed. This concept is based on spectroscopic measurements on PLT (Princeton Large Torus) during ohmic heating and ATC (Adiabatic Toroidal Compressor) during RF heating.
A simplification of the Chapman-Enskog method for the calculation of transport properties of two-temperature, partially ionized gas mixtures in a magnetic field is presented. The simplication is achieved by exploiting the fact that the electron mass is much smaller than that of any other constituent of the gas. A systematic study is presented of the tensor coefficients for the electron electrical conductivity, thermal conductivity, and thermal diffusion. The interplay between the degrees of ionization and magnetic field in determining the sign of the transverse electron thermal diffusion coefficient has been examined. Simple formulas for the thermal diffusion coefficients in the presence of magnetic field have been obtained. The unified theory of Kihara and Aono for charged particle interactions was applied to obtain improved values of the transport coefficients when the number of particles in a Debye sphere is not large. These have been compared with theories based on a cut-off Coulomb potential, and on a shielded-Coulomb potential. It was found that the unified theory and the shielded-Coulomb theory are in close agreement. Mixture rules were given for all three transport coefficients, including the case when there is a magnetic field present. Simplifications introduced into the Chapman-Enskog method were also demonstrated in obtaining the transport coefficients of the heavy particles of a three component, two temperature plasma. (Author)
Requirements for the material to be used as the first surface of limiters in TFTR are to (1) withstand a heat flux of 1 kW/cm<sup>2</sup> for a pulse length of 1.5 s and a duty cycle of 1/200 for 10<sup>5</sup> cycles, (2) withstand the thermal and electromagnetic loads from 10<sup>4</sup> plasma current disruptions lasting about 200 μs, (3) generate impurities at a rate low enough to meet impurity control requirements (which depend on the atomic number of the material) for TFTR, and (4) have tritium retention characteristics consistent with tritium inventory requirements for TFTR. An extensive set of material tests using electron beams, neutral beams, and plasma bombardment have been carried out to identify materials which can meet the thermal requirements of (1) and (2) above. In most cases these tests have been combined with calculations of the expected thermomechanical response of the material to determine the reasons for the result observed. These studies have identified two major classes of materials which are acceptable: (1) graphite which is coated to control hydrocarbon formation and (2) copper cladded with a lower sputtering yield material.
Monte Carlo techniques have been used to calculate neutral gas distributions in tokamaks. The algorithm uses track length estimators, suppression of absorption, and splitting with Russian roulette to reduce the variance, so that the algorithm is economic. The resultant package is small in memory requirements and relatively fast ( ∼15 seconds of PDP-10 KI time per neutral density profile). The tokamak is modeled as an infinite cylinder, and the plasma parameters are specified as a function of the radial coordinate of the cylinder. The effects of wall reflection and sputtering yields can be easily computed within the model. The algorithm has been incorporated in a one-dimensional tokamak transport code. The code has also been used to predict the energy spectra of charge-exchange neutrals which form the basis for measurements of ion temperatures in tokamaks.
Numerical computations have been performed using rigorous kinetic theory to determine the transport properties such as viscosity, thermal conductivity, electrical conductivity and electron diffusion coefficient, for an argon plasma at elevated electron temperatures (5000 K to 20,000 K) and pressures (0.001 atm to 1000 atm) with the ratio of electron temperature to heavy particle temperature ranging from 1.0 to 3.0. The values for the transport properties for a plasma in thermal equilibrium could be obtained as a special case from the present scheme of calculations. Comparison with previous results for the special case of a plasma in thermal equilibrium shows good agreement.
A calculation procedure for heat, mass and momentum transfer in three-dimensional para-bolic flowsNumerical analysis of the anode region of high in-tensity arcs
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