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PHYSICS OF PLASMAS

VOLUME 7, NUMBER 9

SEPTEMBER 2000

Detailed comparison of simulated and measured plasma profiles in the scrape-off layer and edge plasma of DIII-D G. D. Porter, R. Isler, J. Boedo, and T. D. Rognlien Lawrence Livermore National Laboratory, L-637, P.O. Box 808, Livermore, California 94551-9900

共Received 24 January 2000; accepted 5 May 2000兲 The results of detailed comparisons between experimental measurements of the scrape-off layer and divertor plasmas and simulations using the UEDGE code for a DIII-D discharge 关J. Luxon et al., Proceedings of the 11th International Conference on Plasma Physics and Controlled Nuclear Fusion 共International Atomic Energy Agency, Vienna, 1986兲, Vol. I, p. 159兴 are reported. The simulations focus on understanding the flow of both fuel and impurity particles throughout the edge and scrape-off layer 共SOL兲 plasma. The core impurity content and the core hydrogen ionization rate can be explained by sputtering and recycling in the divertor region alone. The model reproduces most of the detailed experimental measurements. The simulations include the effect of intrinsic impurities, assumed to be carbon originating from sputtering of the plasma facing surfaces. The simulations accurately reproduce the total radiated power, although the spatial profile of radiation is somewhat narrower in the simulation. The measured carbon density on closed field lines is reproduced well with the simulation. Comparison of carbon emission lines indicates the total carbon sputtering yield is a factor of 2 to 4 less than expected, although the total radiated power and core carbon content are insensitive to the sputtering yield. The agreement between simulation and experiment permits more meaningful interpretation of the experimental measurements. © 2000 American Institute of Physics. 关S1070-664X共00兲02708-7兴

共e兲共f兲共g兲共h兲

I. INTRODUCTION

This paper describes comprehensive UEDGE modeling of the well diagnosed scrape-off layer 共SOL兲 plasma in DIII-D1 discharge 94002. In addition to the usual large array of plasma diagnostics which DIII-D has brought to bear upon the study of the physics of the plasma edge and SOL2—radial profiles of density and temperatures at two poloidal locations; spectroscopic measurements of line radiation; toroidally viewing measurement of specific line emission; detailed profiles of divertor heating; multi-camera arrays of bolometers to determine the profile of radiated power—the DIII-D team has recently added measurement of the parallel flows of both impurities and the primary ion species.3–6 This impressive diagnostic capability can now be used to test virtually all aspects of the plasma and neutral models in the UEDGE code. The model validation results reported here permit a detailed description of the flow of both the deuterium and carbon ionization states through the edge and SOL plasma. In an effort to prove the validity of the model, this report describes many detailed results which can be compared with measurements. We start with a summary of the conclusions to provide a broader picture before delving into the details. The diagnostics used to validate the physics in the UEDGE model includes 共a兲共b兲共c兲共d兲

The simulations use the most current theoretical models for all phenomena with the exception of the cross field transport, which is determined empirically. The turbulence driven cross field diffusion is modeled by spatially constant diffusion coefficients for particles and thermal energies, and the magnitude of the coefficients is obtained by fitting the density and temperature profiles measured on the experiment. Carbon impurities are assumed to arise from sputtering on the plasma facing walls. The sputtering yield obtained by the University Toronto group7–9 is used. The simple cross-field diffusivity used in the simulation is sufficient to obtain radial profiles of the electron density, electron temperature, and ion temperature which are consistent with the profiles measured just before the onset of an Edge Localized Mode 共ELM兲, validating the model for ‘‘between ELM’’ conditions. The effects of the ELMs are not explored in this paper. The simulations indicate the upstream density profile is strongly affected by the divertor recycling process, and is therefore sensitive to the model for particle removal. The model used here assumes the private flux wall, and the outer wall in the divertor region, are saturated, hence there is no pumping of fuel neutrals on these surfaces. A 5% sticking coefficient for neutrals in the main chamber is assumed. This is sufficient to remove particles at the same rate as the neutral beam input, providing steady state solutions of the prepuff phase of the simulated discharge. The electron temperature profile in the

Thomson scattering for upstream plasma profiles; Charge Exchange Recombination spectroscopy 共CER兲 for upstream plasma profiles, and core carbon content; Divertor Thomson for downstream plasma profiles; Infrared television 共IRTV兲 for divertor heating profile;

1070-664X/2000/7(9)/3663/18/$17.00

Filter scope array for D ␣ emission profiles; Divertor SPRED for impurity emission spectra; Bolometer arrays for radiated power profile; Spectroscopy for carbon ion flow velocities.

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© 2000 American Institute of Physics

Downloaded 17 Aug 2001 to 132.239.1.230. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

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Phys. Plasmas, Vol. 7, No. 9, September 2000

divertor region is determined in part by radiation from intrinsic impurities, thus this radiation plays an important role in determining the fraction of recycling neutrals which penetrate to the closed field lines. In turn, ionization of recycling neutrals control the details of the upstream density profile. Consistency of the simulated and measured density profile suggests the fluid neutral model in UEDGE accurately reproduces this recycling process. The recycling process leads to a significant convective power loss in the High confinement mode 共H-mode兲 pedestal region 共1.5 MW for the discharge analyzed here, 20% of the input power兲, affecting the temperature at the top of the pedestal, and hence affecting core energy transport. The model used here is consistent with measurement of the total radiated power; the flow velocity of low ionization states of carbon; the line emission of high ionization states of carbon; and the density of carbon on the closed field lines in the plasma edge. This suggests that the transport model used for the carbon flow is reasonably accurate. Inconsistency with the line emission of low ionization states of carbon suggests the sputtering yields are overestimated. Most of the carbon sputtered from the plasma facing walls is quickly ionized and flows to the plates. A small amount is transported up the ion temperature gradient parallel to the magnetic field where it can diffuse radially to the closed field lines. The amount of carbon which accumulates around the separatrix is determined by an effective potential well which is established by a balance between the ion thermal gradient force and drag off the primary fuel ion flow. The depth of this well determines the density of carbon, and hence the flux of carbon into the closed field lines. Thus the fueling of carbon to the core is insensitive to the magnitude of the wall carbon sources. Although we have not attempted to compare simulation and experiment in detail during the detached phase of this discharge, we find many of the detachment phenomena are reproduced in the simulation when additional deuterium gas is introduced as is done experimentally. The variation in the total radiation power and the divertor heating power is reproduced in the simulation. In addition, the approximate doubling of the carbon density on closed field lines at detachment is reproduced. A general description of the discharge which is simulated in this paper is described in Sec. II; the UEDGE model and simulation results of the attached phase are described in Sec. III. The simulated effect of gas injection is described in Sec. IV, showing that it induces detachment as seen in experiment. The paper closes with conclusions in Sec. V. II. DISCHARGE DESCRIPTION

Discharge 94002 was obtained 28 October, 1997 as part of an experimental campaign to measure the plasma flow characteristics in the divertor region of DIII-D under both attached and detached plasma conditions. The intention was to determine the flows under ‘‘standard’’ H-mode operation. To this end, the plasma configuration was held fixed throughout the discharge, and several similar discharges were taken under each operating condition. This permitted

FIG. 1. Magnetic equilibrium for shot 94002, 1750 ms. The view of the impurity flow velocity diagnostics shown in detail in 共b兲.

the maximization of the total data obtained, and hence provides an extensive set of data for validation of the plasma modeling codes. The discharge was started in the standard H-mode conditions: high neutral beam heating power relative to the H-mode power threshold, no additional injected gas, lower single null plasma equilibrium, and no explicit neutral pumping 共the clean walls are assumed to pump since the density equilibrates with particle input via neutral beams兲. After reaching temporal equilibrium under these conditions, additional deuterium gas was introduced to enhance the total radiated power and thereby to transition to operation with detached divertor plasmas. A. Plasma configuration

The magnetic reconstruction of the plasma configuration during the prepuff, attached phase is shown in Fig. 1, together with the geometry of several relevant diagnostics. This reconstruction is obtained using EFIT,10 with a constraint on the position of the separatrix at 共R,Z兲⫽共1.94,0.809兲, a position determined using a hyperbolic tangent fit to the measured electron temperature profile, and defining the separatrix to lie one-half of a temperature scale length outside the position of the maximum temperature gradient.11 This has moved the separatrix 0.6 cm 共along the Thomson measurement line兲 inward from the position obtained with an unconstrained EFIT solution, corresponding to a 0.3 cm shift at the outer midplane. The field of view of each of the instrumented fibers used for determining the flow velocities spectroscopically is shown in detail in Fig. 1共b兲. The field of view of each fiber is a projection of the straight line from the turning mirror located above the pumping baffle to the floor on the far side of the device onto a poloidal plane 共constant toroidal angle兲. Toroidal symmetry is assumed throughout this paper. B. Temporal behavior

The temporal behavior of this discharge is shown in Fig. 2. The plasma current is constant at about 1.4 MA throughout the period of interest. The neutral beam power increases from 2.2 to 9.2 MW at 1200 ms. There is a brief excursion



Detailed comparison of simulated and measured plasma . . .

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FIG. 3. Behavior of the core electron content in shot 94002.

FIG. 2. Temporal behavior of discharge 94002. The injected neutral gas amplitude is shown in 共a兲关amp兴; the injected beam power in 共b兲关kW兴; the D ␣ emission at the outer strike point in 共c兲; and the line averaged density 共solid line兲 and the density near the axis 共dotted line兲 in 共d兲关m⫺3 兴.

from Low confinement mode 共L-mode兲 to H-mode just before the power increase, but the plasma returns to L-mode for about 100 ms after the power increase, then drops into H-mode for the remainder of the discharge. This suggests the H-mode power threshold is slightly more than the initial 2.2 MW. As shown in Fig. 2共a兲, additional gas is introduced at 2000 ms to enhance radiation and detach the divertor plasma. This is the origin of the slow rise in both the D ␣ monitor signal from near the outer strike point 共fs04兲 and the electron density signals 关the central density, n e 共0兲, and the line averaged density along R⫽1.94, 具 n e (V2) 典 ]. C. Behavior of the core particle content and particle flow

Adequate simulation of the plasma flows in the SOL depends in part on the determination of the particle input rate and how the particles flow across the separatrix between the closed flux surfaces and the SOL. A large particle flux from the core to the SOL leads to large fuel ion flows along the

field to the divertor in the SOL. On the other hand, if there is little flux across the separatrix, fuel ion flows in the SOL will be small. In turn, the parallel flow of impurity ions is determined by a balance between various parallel forces, primarily a balance between the drag force on the fuel ions, and a force arising from the parallel ion temperature gradient. Particles are input to the system via neutral beams and injected cold gas. At equilibrium, these particles are either removed by a pump 共either a cold cryopump in DIII-D or wall pumping on clean carbon walls兲, or the particle content in the core will rise. In addition to the gas input shown in Fig. 2共a兲, there is an input of energetic particles which is approximately 20 amp/MW of the neutral beam heating power. The disposition of these particles is considered in this section. The temporal behavior of the total core electron content, i.e., 2V 兰 n e ␳ d ␳ , is shown in Fig. 3. This signal is obtained by integrating the Thomson data from the axis to the separatrix, ␳ ⫽1, assuming the density is constant on flux surfaces. There is a brief increase at 1200 ms when the plasma slips momentarily into H-mode, then a sustained increase at 1300 ms when it again goes into H-mode. The rate of rise at 1300 ms is consistent with a particle source rate of about 1.6 kA. The particle content equilibrates at about 135 Coul for 0.5 s after the H-mode transition, then rises when additional gas is injected. The rate of rise after the gas puff at 2000 ms is about 25 amp. The ‘‘noise’’ on the particle content is attributable to particle losses associated with ELM activity. One can estimate the particle fluxes, and by inference the core source rates, by examination of the density and temperature profiles at the edge. The temporal behavior of several


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FIG. 4. Behavior of the particle flux across the separatrix. The electron density at the separatrix is in panel 共a兲; the electron temperature on the separatrix in 共b兲; several components of the input power in 共c兲; the inferred perpendicular thermal diffusivity at the separatrix in 共d兲; the total radial particle flux at the top of the pedestal in 共e兲; and the total radial particle flux at the separatrix in 共f兲.

plasma parameters relevant to the determination of the particle flux across the separatrix, together with the estimated particle flux, is shown in Fig. 4. The density and temperature at the separatrix are shown in panels 共a兲 and 共b兲. The various power flow channels are required to estimate the cross field thermal and particle diffusivities, in panel 共c兲, and the estimated thermal diffusivity in 共d兲. The radial particle flux across the top of the H-mode pedestal and the separatrix are shown in panels 共e兲 and 共f兲, respectively. The particle flux is calculated assuming the parameter ␣ ⫽2.5D/ ␹ ⫽0.6. This value has been determined by examining a small number of discharges, and should be viewed as somewhat uncertain.12 This parameter is determined by comparing the rate of rise of the core particle content at the L- to H-mode transition with the particle flux across the separatrix just before the transition. In shot 94002, the particle flux just before 1300 ms is about 2 kA, while the rate of rise is about 1.6 kA. This suggests that the actual value of ␣ may be 20% larger than that assumed, a difference well within the uncertainty in determining the parameter. The separatrix particle flux increases from about 1 to 2 kA after the gas puff at 2000 ms. This means the gas puff increases the particle flux across the separatrix by 1000 A, but only increases the core particle content at a rate of 25 A

共the rate of rise following the gas puff at 2000 ms in Fig. 3兲. This result suggests that ion recycling at the divertor increases the core ionization rate by 1000 A 共greater than the neutral input rate兲 but that only about 2.5% of the gas which is ionized inside the separatrix actually goes toward increasing the core density; the remainder flows rapidly back to the SOL. D. Power flow

The temporal behavior of the input and exhaust powers for discharge 94002 are shown in Fig. 5. The distribution of power at 1750 ms is shown in the table. The total input power ( P beam⫹ P ⍀ ) is 9.44 MW, and the exhaust power ( P rad⫹ P div) is 10.22 MW. The extra 0.78 MW may be either an indication of the experimental uncertainty, or it could reflect that part of the radiated power is absorbed in the divertor tiles, and is measured by the IRTV. It will be shown later that there is indeed a significant reabsorption of radiated power in the divertor plates, thus portions of the exhaust power are double counted. There is an increase in the radiated power, and a decrease in the divertor power at about 2500 ms, indicating the time at which the divertor plasma detaches from the plates. The UEDGE analysis is focused on




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FIG. 5. Temporal behavior of the power flow channels in shot 94002. The input power channels are in 共a兲; the radiated power in 共b兲; and the divertor heating power in 共c兲. Details of the power flow at 1750 ms are specified in the table.

the attached phase at a time well before this process, 1750 ms. The effect of gas puffing in the UEDGE simulations is discussed briefly in Sec. IV. III.

UEDGE

ANALYSIS

A. Description of code and parameters

The UEDGE code is a two-dimensional 共2D兲 multi-fluid plasma code used to simulate the plasma in the edge and scrape-off layers of toroidal devices.13,14 The code solves the fluid equations for particle, energy, and momentum balance on a 2D poloidal flux based grid structure. The electrostatic potential is determined self-consistently in the SOL, but the effect of cross-field drifts is not included. Thus the main effect of the potential is parallel currents in the SOL. The plasma is simulated on normalized poloidal flux surfaces in the range 0.96⭐⌿ N ⭐1.08, covering the region between the top of the H-mode pedestal to near the flux surface which strikes a limiter 共the center post兲 in DIII-D. 共The limiter flux surface is actually at ⌿ N ⫽1.1, the simulation is restricted to inside the 1.08 surface to permit adequate grid resolution around the separatrix.兲 The grid structure is not orthogonal, permitting accurate simulation of the plate geometry of DIII-D. This geometry has a significant effect on the behavior of recycling neutrals. A nine-point stencil is used to perform numerical differences on the nonorthogonal grid. The parallel transport of the fuel ions 共deuterium兲 is assumed to be classical Spitzer–Harm15 with flux limits applied to partially account for kinetic effects. Parallel transport of impurity ions is determined from a force balance approach in which the impurity pressure gradient is deter-

mined by balancing the total force from parallel thermal gradients and drag on the primary ion species.16,17 The crossfield transport is assumed to be determined by turbulence18 and is modeled with thermal and particle diffusivities which are input to the code. Determination of these diffusivities is one of the major uncertainties in the simulation of the SOL plasma. Many of the features of the experimentally measured profiles of electron density and temperature, and ion temperature, are reproduced using a simple spatially constant model for perpendicular transport. The deuterium ion density on the innermost core flux surface is used as a boundary condition, and is determined by experimental measurement. The electron and ion temperature on the innermost core flux surface is determined by the choice of thermal diffusivity and the radial power flow in the electron and ion channels, which is specified as a boundary condition. Neutral deuterium transport is modeled with a reduced Navier–Stokes model in which the parallel momentum of the neutrals is included while the cross-field neutral transport is due to charge exchange diffusion.19,20 Neutrals originate from two sources in the simulation: a specified input rate at the outer and/or private flux walls, and recycling from the ion flux to the divertor plates. The ion recycling coefficient at the plate is not well understood in existing experiments. We assume the divertor plates are saturated early in the discharge and use a recycling coefficient of 1.0 in all results reported here. We do not consider neutral transport in the region between the outermost flux surface and the plasma facing walls of the device. Rather, we specify a neutral albedo at the outermost flux surface 共referred to as the outer wall in the remainder of this paper兲, and at the innermost flux surface in


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the private flux region 共referred to as the inner wall兲 as boundary conditions for the neutral density. The effective albedoes are not well understood experimentally. We assume the private flux wall is saturated because of the high neutral pressure in that region and use a neutral albedo of 1.0 there. The outer wall is more complicated in that the neutral pressure in the divertor region is quite high 共order 10⫺3 Torr兲 while that in the main chamber is low 共order 10⫺5 Torr兲. We assume a neutral albedo of 0.95 in the main chamber and 1.0 near the divertor in the results reported here. This means that some particle removal is done at the outer wall. This pumping rate is balanced at steady state by a finite input of deuterium ions across the innermost core flux surface. Ideally, this particle input rate should correspond to the particle input rate from the neutral beams used to heat the plasma. The radial particle flux across the separatrix is the sum of the particle flux across the innermost core flux surface, and the total ionization source on the closed flux surfaces arising from input and recycling neutrals. The final parameters which are needed for accurate simulation of the SOL plasma are the production rate of impurities in the calculational domain. Carbon impurities are assumed to arise from sputtering of the carbon walls which completely surround the plasma in DIII-D. Carbon is obtained from a combination of physical sputtering, for ions or neutrals which impinge upon the carbon at high energies (E I ⬎27.6 eV兲, and chemical sputtering from ions and neutrals of all energies. The yield of carbon atoms from physical sputtering is fairly well known, but the yield from chemical sputtering is somewhat controversial.7,8,21–24 We use a modified version of the chemical sputtering yields of Davis8 in the simulations discussed here. As suggested in private conversations with Davis, the sputtering yield for ion energies of 5 eV and less is assumed to be 20% of the yield measured at 10 eV, and the yield increases linearly between 5 and 10 eV. All six ionization states of carbon are included in the simulation.

TABLE I. Comparison of

UEDGE

P sep 共MW兲 SOL P rad 共MW兲 P in div 共MW兲 P out div 共MW兲 I sep共kA兲

simulation and experiment. 94002

UEDGE

8.00 6.09 1.04 2.97 1.2

8.0 6.45 1.29 2.22 1.23

rity radiation in the simulation. The simulated total particle flux across the separatrix is within 5% of that inferred from the experiment. This agreement is expected since the experimental value was inferred from measurement of the radial profile of the density and temperature, and the perpendicular diffusivities were determined by matching the UEDGE simulation to the same profiles. The simulated particle flux across the innermost core flux surface is only 123 A, compared to 1200 A across the separatrix. The flux across the innermost flux surface compares reasonably with the 180 A expected from 9 MW of neutral beam injection. These results indicate that there is an additional 1100 A of recycling neutrals ionized on the closed flux surfaces between the 96% surface and the separatrix. 2. Upstream profiles

The simulated radial profiles of the electron density and the electron and ion temperature at the outer midplane are compared with those measured at 1750 ms in Fig. 6. The experimental data from the Thomson scattering system are obtained approximately every 6 ms when one of seven lasers is fired. The timing of the lasers, and hence the timing of the measurement, is not synchronized relative to any plasma

B. Simulation of attached phase „1750 ms…

1. Global parameters

As described above, the radial diffusivity of both particles and thermal energy is determined by matching the upstream measurement of the radial profile of the density and temperature as measured by the Thomson and CER systems. A reasonable fit is obtained with D⬜ ⫽0.1 m2 /s,

␹ i ⫽ ␹ e ⫽0.4 m2 /s. The power and particle flow obtained using these diffusivities are compared to those inferred from the experiment in Table I. The power across the separatrix is a boundary condition for the code, assuming equal power flow in the electron and ion channels. As can be seen in Table I, the total radiated power agrees well with that measured with the bolometer arrays. 共Note that the bolometer power quoted in this table refers to the power radiated in the edge and SOL, i.e., the power radiated in the core has been subtracted.兲 Approximately 86% of the radiated power arises from carbon impu-

FIG. 6. Comparison of simulated and experimental profiles of the electron density 共a兲 and temperatures 共b兲. The data are projected to the outer midplane.



phenomena. In particular, the lasers fire at arbitrary times relative to the occurrence of Edge Localized Modes 共ELMs兲. It is well known that the ELMs modify the profile of both the edge density and temperature.25 In an effort to postpone the discussion of the effect of ELMs, the Thomson data are selected so that the data represent the profile measured 2 ms (⫾1 ms兲 prior to the onset of an ELM 共data points denoted by open circles兲. The multiplicity of data shown in Fig. 6 arises because all data taken 2 ms before an ELM in a 200 ms window around the analysis time 共1750⫾100 ms兲 are shown. The ELM frequency during this period is about 200 Hz. Time dependent simulations indicate the plasma relaxes with a time constant on the order of 1 ms, a fraction of the ELM period. This makes it feasible to examine the between ELM behavior without resorting to a full time dependent treatment of the ELMs. The consistency between the simulated and measured radial profiles indicates the perpendicular transport coefficients used for the simulation represent a reasonable estimate of the transport between ELMs. For comparison, the experimental profile of electron density and temperature obtained by averaging over all laser pulses, essentially averaging over ELMs, is also shown. The ELMs produce a clear effect on the density profile—a high density plateau in the SOL—but less effect on the temperature profile. The consistency of the simulated and measured density profile gradient at the separatrix leads to a consistency in the total particle flux 共see Table I and Fig. 4兲. The simulated electron and ion temperature profiles are a good match to experiment. The fact that such good agreement is obtained with a thermal diffusivity twice that inferred from the experiment 共see Fig. 4兲 requires some explanation. The analysis done to obtain the particle flux shown in Fig. 4 is somewhat simplified. The radial power flow is obtained by calculating the conductive (⫺n ␹ ⵜT) and convective (⫺2.5TDⵜn) radial power flux using the radial gradient of the density and temperature at the outer midplane. The total power is determined by multiplying by the surface area of the plasma. In effect this assumes the poloidally averaged radial gradients are those measured at the outer midplane. However, the gradients are largest at the outer midplane since the poloidal flux surfaces are maximally compressed there. Hence we are overestimating the average radial gradients, and thus underestimating the particle and thermal diffusivity, in the analysis used for Fig. 4. The UEDGE simulation, on the other hand, is matched to the radial profiles at the outer midplane, but the radial profiles elsewhere are calculated consistently with the poloidal flux expansion. These results indicate that the average radial gradient is approximately one-half that at the outer midplane. The agreement between the simulated and measured upstream midplane profiles is attributable largely to the fact that the radial diffusivities are determined by matching the experiment. However, examination of the detailed effects is illuminating. The detailed radial density profile is determined by a combination of the ionization source profile and particle transport. The small radial diffusivities used here are consistent with a transport barrier in the edge, as expected for H-mode plasmas. The good fit shown in Fig. 6共a兲 requires


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FIG. 7. Comparison of the measured and simulated divertor heat flux for the attached phase of discharge 94002.

inclusion of the effect of impurity radiation to properly model the ionization source profile. Without the power loss due to radiation, the plasma temperature near the plates is much higher, creating a short mean free path for recycling neutrals. This means that fewer recycling neutrals penetrate to the closed surfaces, reducing the total ionization current. The radial gradient of the density is then essentially constant, i.e., the radial profile is linear, so that the particle flux at the separatrix is similar to that at the innermost flux surface. On the other hand, the existence of significant neutral penetration to the closed lines when the impurity radiation is included, together with the assumption of spatially constant particle diffusivity, means the radial density gradient must increase between the innermost flux surface and the separatrix so that the radial particle flux can accommodate particles introduced by ionization of recycled neutrals. Clearly the latter more accurately characterizes the measured profile. Note that the total ionization current on closed surfaces is only 1.7% of the total neutral current off the plates for the simulation shown in Fig. 6. This small core regionization efficiency must be accurately modeled to simulate the data. 共Note that this core reionization current also leads to a 1.5 MW convective radial power loss. This large power affects the details of the density and temperature profiles in the pedestal region, probably affecting the core energy confinement.12兲 Thus accurate simulation of the upstream radial profiles provides not only a means of determining simple anomalous transport coefficients, but provides a test for other physics in the UEDGE model. 3. Divertor profiles

A more thorough test of the UEDGE physics models is obtained by comparing the simulated and measured profiles of parameters in the divertor region. In this case, the plasma parameters are determined not only by the choice of the cross-field transport coefficients, but also by models for neutral recycling, particle pumping, and impurity production and transport. There are many relevant divertor diagnostics on DIII-D. We start with a comparison of the plate heating profile in Fig. 7. As can be seen, the agreement between the simulation and measurement is remarkably good near the strike points. This agreement is also reflected in the total power to the inner and outer divertor, as seen in Table I, confirming many of the physics models in UEDGE, not just the parallel thermal conductivity. For example, only about


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10% of the simulated power to the inner divertor arises from parallel heat transport or as a result of recombination of the ion current to the plate. The remaining divertor heating power arises from reabsorption of radiation losses from the plasma above the plate. The line radiation from both carbon impurities and hydrogenic excitation and recombination is assumed to be emitted isotropically. That portion which is directed to the carbon divertor plates is assumed to be absorbed heating the plate. The spatial profile of the total heating power is determined by the spatial profile of the radiation. Note that the inner strike point lies on the 45° tile 共see Fig. 1兲, while the divertor heating power profile is plotted versus major radius in Fig. 7. The heating profile along the floor is broader by 40% than is shown in the figure because of this geometry effect. Hence, the heating profile is rather broad, as expected from radiation which is centered a few centimeters off the floor. The agreement between the measured and simulated heating profiles therefore suggests that the model for carbon production and transport is in reasonable agreement with experiment. Details of the radiation profiles are described later. Approximately 50% of the divertor heating on the outer plate arises from reabsorbed radiation. Note that the recognition that a significant portion of the divertor heating power arises from reabsorbed radiation helps to understand the experimental power balance described in Fig. 5. The total measured ‘‘exhaust’’ power described there exceeds the input power. In fact the radiated power is being double counted since it appears in both the bolometer analysis and in the divertor heating power. The simulated plate heating power at the outer strike point is significantly less than measured for radii well outside the strike point. This difference is postulated to be due to the effect of ELMs. The IRTV diagnostic does not have sufficient temporal resolution to determine the heating profile between ELMs, but is an integral measurement. There is significant radial particle flux associated with the ELM activity, increasing the plate ion current with a wider footprint at the plate than between ELMs. The broadening effect of the ELMs is evidenced in the density profile shown in Fig. 6. Hence the temporally integrated heating power will be broader than expected between ELMs as simulated here. Another diagnostic of the divertor region which can be compared with the UEDGE simulation is the total D ␣ emission profile. This is a measure of the plasma density and temperature, and the neutral density in the divertor region. This diagnostic is comprised of six channels which view the lower divertor, and seven channels which view the upper, as shown in Fig. 1. The emission measured and simulated on the upper divertor is approximately a factor of 100 less than that measured on the lower, and is ignored here. The emission across the floor is also measured with a 1D spectroscopic diagnostic. The D ␣ emissivity determined by the spectrometer is consistent with that measured by the fiber scopes. The measured profile across the divertor floor is compared with the simulation in Fig. 8. The simulation of the emission seen in the SOL is in reasonable agreement with that measured on the low field side, but agrees less well on the high field side. 共See the data on the innermost two channels and the data on the outermost three channels in Fig.

Porter et al.

FIG. 8. Comparison of simulated and measured D ␣ emission profile across the divertor floor.

8.兲 Although it is felt that the agreement on the high field side is satisfactory, the difference between the high and low field sides warrants some discussion. The 2D profile of the deuterium neutral density in the divertor region is shown in Fig. 9. The neutral density near the divertor is a few times 1020 m⫺3 on the high field side, and a few times 1019 m⫺3 on the low field side. The scale length of the neutral density profile is a few centimeters on the high field side, and a few millimeters on the low field side. The difference in the neutral density profile arises because the plasma on the high field side is detached, while that on the low field side remains attached. The electron temperature in the volume with high neutral density on the high field side is around 1 eV, and volume recombination of the deuterium ions flowing down the field lines is an important process. 共The conclusion of a detached, recombining plasma on the high field side is consistent with the deuterium spectroscopy.兲 Hence the D ␣ emission measured near the inner strike point arises from a combination of ionization and recombination. The contribu-

FIG. 9. 共Color兲 Profile of the neutral deuterium density in the divertor region. The inset shows details of the density near the outer strike point.




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consistent between simulation and experiment. If the simulation is overestimating the electron density by a factor of 3, but correctly simulating the emission, it follows that the neutral hydrogen density is underestimated by a factor of 3. Such a large error in the neutral density should be reflected in the core ionization rate, and hence in the upstream density profiles. Since the upstream density profile is well simulated, we conclude the experimental density is underestimated because of the large reflected light signal. 4. ImpurityÕradiation comparison

FIG. 10. Comparison of simulated and measured electron density and temperature near the outer strike point on discharge 94002.

tion of these two processes is about equal in the simulations. The emissivity arising from recombination is sensitive to details of the electron temperature profile, and may lead to a greater difference between simulation and measurement on the high field side than the low field side. The emission seen in the private flux region, viewed through the X-point region, is rather seriously underestimated by the simulation. This discrepancy is frequently seen in simulations of DIII-D, and is not understood. The UEDGE simulation results can also be compared with data from the divertor Thomson system. This diagnostic measures the profile of the electron density and temperature over a vertical line extending approximately 20 cm from the divertor floor along the line R⫽1.48 m 共see Fig. 1兲. These data are obtained in the SOL just outside the outer strike point on discharge 94002. The data are shown, together with the simulated results, in Fig. 10. The simulated density is approximately a factor of three higher than that measured, although the temperature is in reasonable agreement with measurement. The discrepancy in the density is believed to arise because of abnormally large reflected light signals obtained during the experiment. The density is obtained from an absolute intensity measurement by subtracting the background light. When the reflected light is large, this subtraction leads to large errors in the density. The divertor Thomson diagnostic measures the electron density and temperature near the viewpoint of the fifth channel of the D ␣ monitors 共see Fig. 8兲. The simulation shows good agreement with the measured emission on this channel, but the simulation has a factor of three higher density than measured with the Thomson system. The total emission scales as n e n 0 具 ␴ v 典 excitation . The excitation cross section is determined largely by the electron temperature, which seems relatively

The final diagnostic set with which the UEDGE simulation is compared are the impurity diagnostics. These include two bolometer cameras which determine the total radiated power profile, flow diagnostics which determine the parallel velocity of specific impurity ionization states, and spectroscopic measurements of impurity line intensities. Starting with the bolometer measurements, the 2D profile of the simulated total radiation in the divertor region is shown in Fig. 11, together with the simulated and measured signals on each of the 48 channels of the bolometer arrays. The geometry of the bolometer array is shown in Fig. 12. As noted in Table I, the total radiated power is consistent with that inferred from the bolometer array. We examine here the details of the emission profiles. It is important to note, however, that 75% of the total radiated power in the simulation arises from cells with less than 10% of the peak power density. Hence large discrepancies in the peak emission powers will not seriously affect the total radiated power. As shown in the 2D profile, there is intense radiation off the inner plate, and at the outer plate. This radiation is predominately from carbon, with the impurity radiation 86% of the total radiated power. Basically, the intense radiation bands seen in Fig. 11共a兲 lie near the 6 to 8 eV electron temperature where carbon line radiation is strong. 共The radiated power is roughly evenly split between CII, CIII, and CIV in the simulation.兲 The radiation seen very near the inner plate arises from collisionalradiative recombination of deuterium ions. Unfortunately, a high resolution 2D image of the experimental radiation pattern can not be meaningfully compared with Fig. 11共a兲. A tomographic inversion of the data obtained from the two bolometer cameras is routinely done.26 However, differences in the 2D profile obtained from the inversion and that obtained in the UEDGE simulation can arise from either limitations of the spatial resolution in the bolometer cameras, or from limitations in the physics model in UEDGE. To determine the consistency of the simulated radiation profile with the experiment, the signal on each of the 48 bolometer channels from the radiation profile obtained in the simulation is calculated, including the finite width of the bolometer views. The result is compared with the experimental signals in Figs. 11共b兲–11共d兲. Two simulated signals are shown, one arising from deuterium radiation only 共both from excitation and recombination兲, and the second from the total radiation. Of course, the bolometers are sensitive only to total radiation, so this is to be compared with the simulation. The relationship between channel number and view is shown in Fig. 12. Channels 0 through 23 comprise the upper


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FIG. 11. 共Color兲 Simulated 2D profile of the total radiated power 共a兲, and comparison between simulated and measured bolometer signals on discharge 94002 共b兲–共d兲. The data from all bolometer channels are shown in 共d兲, the data near the inner strike point are expanded in 共b兲, and those near the outer strike point in 共c兲.

camera view, and channels 24 through 47 the lower. The experimental signals in channels 11 through 23 and channels 32 through 47 are believed to arise largely from radiation in the core plasma which is not simulated. Hence the difference between experiment and simulation on these channels is not relevant. Channels 6 and 24 view the outer strike point from the upper and lower camera, respectively. The simulated signal in each of these channels is a factor of 2.4 and 2.6 higher than the experiment, respectively. The simulated signal on channel 5 is 52% of the experiment, and that on channel 7 is about 75% of the experiment. These results indicate that the radiation seen in the experiment at the outer strike point is distributed over a somewhat broader area than obtained in the simulation. The inner strike point is viewed by channel 9 of the upper bolometer camera, and channel 27 of the lower. Note that channel 27 views along the separatrix through the X-point to the inner strike point. This channel integrates the intense emission seen along the separatrix in Fig. 11共a兲. The simulated emission on channel 27 is 2.4 times that seen experimentally. The simulated emission on channel 9 is about 3.4 times that seen experimentally. Note, however, that the signal simulated for channel 10, viewing just outside the inner strike point, agrees quite well with experiment, as do channels 28, 29 and 30. Thus it appears that the simulated radiation pattern on both the inner and outer divertor legs is somewhat more localized than seen experimentally. The broader radiation pattern seen in the experiment may arise from the effect of the ELMs. The bolometer analysis integrates over a time period long relative to the ELMing fre-

FIG. 12. Geometry of the two bolometer cameras.



FIG. 13. Comparison of simulated and measured emission spectra along the line R⫽1.49 m. The pale vertical lines at the simulated lines are used only to guide the eye between the simulated and measured spectra.

quency. Since the ELMs broaden the density profile in the SOL, they would also be expected to broaden the radiation profile in the divertor region. Another diagnostic used to assess the model of impurity behavior is the divertor SPRED. This diagnostic determines the emission spectra in the range of 100 to 1700 Å with a vertical view along the line R⫽1.49 m, viewing the divertor region just outside the outer strike point as seen in Fig. 1共a兲. The UEDGE code can only simulate emission from deuterium and carbon in this spectral range. The calculated emission spectra are compared with those measured in Fig. 13. The simulated and measured emission for CIV and D 共Lyman alpha兲 is within a factor of 2 to 3. The simulated emission for CIII is about a factor of 5 high, and that for CII is at least an order of magnitude high. Agreement for the deuterium emission is expected since the simulations agreed with the fiberscope data which determine the emission amplitude in the D ␣ line. The low measured emission for the Lyman alpha line may indicate the importance of reabsorption.27,28 The neutral density near the outer strike point is high enough 共see Fig. 9兲 that reabsorption of the L ␣ line may be important. It is clearly important near the inner strike point. Detailed calculations similar to those done by Scott29 are required to properly assess this hypothesis. The trend of reasonable agreement between simulation and experiment for high ionization states of carbon, and poor agreement for low ionization states, may indicate some inaccuracy in the carbon sputtering model. Neutral carbon which emerges from the outer plate is rapidly ionized. The lowest ionization states are recycled back to the plate. However, there is emission as the carbon is ionized, and it appears this emission is significantly higher in the simulation than the experiment. On the other hand, the higher ionization states emit at higher electron temperatures 共ionization potential of 24.4 eV for CII, 47.9 eV for CIII, and 64.5 eV for CIV兲, and hence radiate further off the plate. The density of these higher ionization states is determined not only by the sputtering yield from the plate, but also by parallel transport to regions of higher temperature. The results shown in Fig. 13 suggest we are accurately simulating the density of the high


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ionization states, but overestimating the density of the low ionization states. Perhaps these results are an indication that the sputtering yield is reduced from that expected for a pure carbon surface by, for example, boronization in the experiment. A simulation with the sputtering yield reduced a factor of 4 was done as a means of assessing the viability of this hypothesis. The total carbon input decreased by a factor of only 2 because the plasma and neutral densities readjust, increasing the ion and neutral fluxes to the walls. The total radiated power decreased by only 30% 共from 6.4 to 4.4 MW兲, indicating that the radiated power depends only weakly on the sputtering yields. The simulated line emissivity seen on the divertor SPRED was reduced by about a factor of 3 for the CIV lines, but a factor of 40 for the CII lines. The simulated and measured emissivity of all carbon lines were in much better agreement. These results are consistent with the hypothesis that the sputtering yields obtained on DIII-D are lower than expected, by a factor of between 2 and 4. Experimental evidence has recently been found that the sputtering yield has indeed been significantly reduced over the last several years of DIII-D operation.30 5. Simulation of flow patterns

Next, consider the comparison of the simulation and measurement of the ion flow velocities in the SOL. The simulated emissivity of the 6581 Å doublet of singly ionized carbon, CII, together with the 2D profile of the C ⫹1 parallel velocity in Fig. 14. This doublet is used in the experiment to determine the parallel flow velocity of this ionization state. As expected, the emissivity is intense near the T e ⫽8 eV contours in the private flux region and in the SOL. This temperature contour lies near the plate at the outer strike point since the plasma is attached there, but off the inner plate where the plasma is detached. The sign convention used for the parallel velocity is that flow from the inner to outer plate is positive. The flows are seen to be predominately toward the plate on the inner divertor. The flow is also toward the plate on the outer divertor, near the plate, but is away from the plate over a large region of the SOL on the outside. The diagnostic measures the flow in the SOL on the outside, as seen in Fig. 1共b兲. The diagnostic which measures the carbon flow velocity has been described by Isler.4–6 Briefly, the parallel velocity of specific carbon ionization states is determined by measuring the visible spectrum of emission lines seen in essentially toroidal views. The geometry of the views, projected to a poloidal plane, are shown in Fig. 1共b兲 for each of the five instrumented view fibers. Although the diagnostic measures the total emission along the view of each fiber, the emission for low ionization states is quite localized, as shown in Fig. 14共a兲. The details of the comparison of simulated and measured C ⫹1 flow velocity as seen in fiber 4 are shown in Fig. 15. The spectrum determined experimentally is fit with a model which assumes the existence of blue- and redshifted carbon emission. Each of these lines is assumed to be split by the Zeeman effect. The fit provides a parallel velocity for both components 共the magnitude of the shift in the wave length兲 and an estimate of the radial location of the emission


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FIG. 14. 共Color兲 Emission profile of the 6581 Å doublet of CII 共a兲, and profile of the parallel velocity for CII 共b兲.

for each line 共the magnitude of the Zeeman splitting兲. This analysis provides the two data points seen in Fig. 15共b兲. The projection of the view of fiber 4 is shown in Fig. 15共c兲. The diagnostic views the plasma through a view port just above

the pumping baffle, and the view is divided into a portion from above the baffle to the tangency point 共labeled as upper, and plotted as a dashed line兲 and a portion from the tangency point to the exit point on the opposite side of the torus 共la-

FIG. 15. Simulated total emission of the 6581 Å doublet of CII along the upper and lower halves of the fiber 4 view 共a兲, the simulated and experimental parallel velocity of the CII ion 共b兲, and the detailed view of fiber 4 共c兲.




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FIG. 16. Simulated emission profiles and simulated and experimental profiles of the parallel velocity of the CII ions for fibers 2, 3, 5, and 6.

beled as lower, and plotted as a solid line兲. We plot the intensity of the emission of the CII line on both the upper and lower portions of the view in Fig. 15共a兲, and the parallel velocity along the view in Fig. 15共b兲. The emission seen by fiber 4 is brightest at all radii on the lower portion of the view, inferring that the measurement of the velocity is obtained in the lower portion of the fiber view. The difference in the simulated intensity of emission at the location of the blue- and redshifted measurement position exceeds that seen experimentally. Experimentally, the difference in the emission at the two locations can not exceed a factor of 10 to 20 to enable adequate resolution of the two lines. We use the simulated emissivity to indicate regions of brightest emission, and thus to guide interpretation of the flow measure-

ment. The measured and simulated parallel velocities on the lower portion of the view are in excellent agreement for both the blue- and redshifted velocities. Negative velocities 共away from the plate兲 are seen in the SOL near the X-point, and positive velocities 共toward the plate兲 are seen near the plate in the outer SOL. The comparison between measured and simulated parallel velocities seen on the other four instrumented fibers is shown in Fig. 16 关the views of all fibers are shown in Fig. 1共b兲兴. Fiber 2, which views into the private flux region, requires a little study to properly interpret. The emission near the inner measurement radius 共R⫽1.4 m兲 is highest on the upper portion of the view, indicating intense emission in the SOL just below the X-point. However, the emission seen at


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R⫽1.48 m is dominated by the lower portion of the view, i.e., in the SOL near the outer strike point. The simulated emission profile therefore indicates the parallel velocity measured at R⫽1.4 m should be compared with the simulated velocity on the upper portion of the curve, and that measured at 1.48 m should be compared with the simulation on the lower portion. Viewed this way, there is reasonable agreement between simulation and measurement. Although the simulation does not adequately reproduce the measurement on fiber 6 and the velocity at R⫽1.45 m of fiber 5, the agreement between experiment and simulation is fairly good. It is clear, however, that the large shear of the velocity pattern requires better spatial resolution on the experiment to adequately test the models used in UEDGE. The flow of the primary fuel ions is measured using a Mach probe which is inserted in the divertor region at R⫽1.48 m 关along the series of ‘‘⫹’s’’ labeled Divertor Thomson in Fig. 1共a兲兴.3,31 Unfortunately, this probe was not operated on shot 94002, and was inserted at 2500 ms on similar shots taken the same day. This time is after the additional gas puff, and is very near the time the outer leg detaches. Hence, it is not possible to make a detailed comparison of simulation and experiment for the primary ion flow in the attached phase on this discharge. We have obtained excellent agreement with this diagnostic with simulations of other discharges. In general we find the flow velocity is near Mach 1 at the plate when the plasma is attached. The spatial extent of the high flow velocity region off the plate depends on detail on gas puffing, i.e., on how near the plasma is to detachment. As the plasma approaches detachment, the high flow region extends further off the plate. At detachment the flow typically maximizes at Mach 1 共or higher兲 near the ionization front, and decreases toward the plate. These general characteristics are seen in both simulation and experiment. 6. Global impurity flow behavior

The discussion in the previous section focused on analysis of the parallel flow of singly ionized carbon (C ⫹1 ). This is of interest only because it is measured experimentally, and hence provides a basis of assessing the accuracy of the impurity flow model in the code. In reality, we are interested in the global behavior of the impurity flow, i.e., how the carbon flows from the source of molecular and atomic neutrals, into the SOL plasma, and thence to the core plasma. We would like to control the flow to enhance the total radiation in the divertor region, but impede the penetration of carbon to the core. Knowledge of the carbon ionization state is not important in this more global picture, only the overall behavior of the total carbon is. Understanding the details of the carbon flow from the walls to the closed flux surfaces is a difficult task. The carbon originates as neutral hydrocarbon molecules or neutral atoms from sputtering on the walls and plates of the device. Transport of molecular hydrocarbons is not considered here. Rather, all carbon is assumed to originate at the walls as neutral atoms, and a diffusive neutral model is used to determine the density of neutral carbon. The neutral carbon is ionized in the SOL plasma, and is transported both parallel

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FIG. 17. Vector plot of impurity flow pattern of carbon for the simulation.

UEDGE

and perpendicular to the magnetic flux surfaces. Perpendicular transport is assumed anomalous, with the same particle diffusivity for all carbon ionization states and for the deuterium ions. Parallel transport is via a force balance model, as described previously. The picture is complicated because the carbon ionization state changes as the carbon is transported through the plasma, depending on the local electron temperature. Hence, we are faced with the challenge of describing the composite flow of six carbon ionization states. We start by showing a vector plot of the poloidal flow pattern of the total carbon in Fig. 17. This figure shows the magnitude and direction of the total carbon flow 共all ionization states, conductive and convective兲 in each cell. Understanding this complex flow pattern is the first step in finding ways to control it. Recall that the only source of carbon is sputtering from all walls surrounding the plasma. This means that the carbon originates at the outer boundaries of the vector plot. The question to address is how does the carbon migrate from the walls to the flux surfaces in the SOL which surround the closed flux surfaces? Carbon can get to the core only by radial diffusion from these poloidal flux surfaces in the UEDGE model. The largest carbon source is found on the plates since the ion and neutral deuterium flux is largest there. Much of this carbon, however, is simply ionized in the SOL near the plate, and flows back to the plate where it is absorbed. Hence many of the vectors in Fig. 17 indicate carbon circulating near the plates. This carbon plays little role in the fueling of carbon on the closed flux surfaces, although a small fraction of carbon originating on the plates will penetrate to the SOL which surrounds the closed surfaces, and hence can contribute to core fueling. There is a general flow of carbon from the inner to the outer plate in the private flux region. This flow is fueled in part by carbon originating off the private flux wall



through chemical sputtering. Note there is significant radial flow from the private flux into the SOL near the X-point because of the high density gradient formed across the separatrix. The low ionization states of carbon have a high density in the private flux region, and a low density in the higher temperature SOL. This creates large density gradients which produce radial flow from the private flux region to the SOL. Some of the carbon flowing into the SOL through this channel again flows back to the plate, and is of little interest. However, some of it flows above the X-point, and fuels the SOL in the region which communicates to the closed surfaces. Similarly, there are circulation patterns for the carbon flowing off the outer wall, some of which reach the poloidal flux surfaces near the separatrix surrounding the closed field lines, particularly on the high field side of the SOL. These results indicate the carbon in the SOL surrounding the closed poloidal flux surfaces is obtained from three sources: a small fraction of carbon from the plates is transported along the field line up the ion temperature gradient; carbon originating from chemical sputtering on the private flux wall is transported radially down steep impurity density gradients across the separatrix in the divertor legs, and thence along the field lines; and carbon originating on the outer wall is transported by radial diffusion down the impurity density gradient. Consider next how the carbon accumulates in the SOL and flows into the core. The vector plot shown in Fig. 17 does not permit sufficient resolution to identify the radial flow from the SOL into the closed lines. The details of the radial flux of each of the carbon ionization states into the closed flux surfaces are shown in Fig. 18共a兲. The sign convention used here is that a positive radial flux is from the closed surfaces to the SOL, out of the core. The core carbon content is fueled predominately by inward flowing C ⫹4 . The carbon is quickly ionized to fully stripped C ⫹6 . The high ionization states of carbon build up until there is an inverted density profile 共higher density on the closed surfaces兲, and then the inward flow of C ⫹4 is balanced by an outward flow of C ⫹6 . Note that the carbon flows into the core predominately from just above the X-point, and out of the core at the outer midplane. The large radial flux of C ⫹4 seen 1m off the inner plate 共just above the X-point on the high field side兲 arises because the C ⫹4 accumulates in the SOL at this position, builds up a large density, and diffuses down the density gradient into the core. Carbon accumulates at this poloidal position because of the poloidal structure of the net force on the C⫹4 ion shown in Fig. 18共b兲. Define an effective potential, ␾, as the integral of the net force on the C ⫹4 ion, i.e., F net⫽⫺ⵜ ␾ . The force balance model used to determine the parallel impurity transport creates a pressure gradient in the impurity species which balances this net force. Extrema of the effective potential, i.e., points with zero parallel gradient, correspond to nulls in the net force on the impurity ion. There is a local potential minimum near the force stagnation point 共F⫽0兲 about 1m off the inner plate, and a second minimum near the force stagnation point just above the X-point on the low field side 共at 4.8 m from the inner plate兲. Examination of the detailed forces indicates these are points at which the drag force from the deuterium ions is balanced by the upward force from the


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FIG. 18. Poloidal distribution of the radial carbon flux across the separatrix 共a兲, and the poloidal distribution of the net force, and the density of the C ⫹4 species at the separatrix 共b兲.

parallel ion thermal gradient. The C ⫹4 ions accumulate in the local potential minima until the forces arising from their pressure gradients balance the externally imposed forces. There is a maximum in the C ⫹4 density at the potential minima, and hence a maximum in the radial flux into the core. There is a potential maximum about 2.6 m above the inner plate, corresponding to another null in the net force. Since there is very little parallel thermal gradient in this region, this force null corresponds to a deuterium flow stagnation point. The forces on either side of this potential maximum move impurities away, creating a local minimum in the C ⫹4 density. The ultimate measure of success in modeling the source and transport of carbon in the tokamak is the determination of the carbon density on the closed flux surfaces. This density is determined experimentally from the CER diagnostic, measuring the absolute density of C ⫹6 . The result is shown in Fig. 19, which shows that the density during the attached phase is around 1.4⫻1018 m⫺3 . The UEDGE simulation described in this paper obtains an average density of 1.3 ⫻1018 m⫺3 between the 96% flux surface and the separatrix at the outer midplane where the measurement is made, in excellent agreement with the experiment. IV. EFFECT OF GAS PUFFING

As described previously, simulation indicates the inner leg of discharge 94002 is detached throughout the shot. Spectroscopy of the deuterium lines in the inner leg indicates significant recombination, consistent with electron temperatures of about 1 eV, experimental evidence of detachment.


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FIG. 20. Variation of the simulated total radiated power 共a兲 and total divertor heating power 共b兲 with external neutral gas input rate.

FIG. 19. Temporal behavior of the C ⫹6 density at the edge of discharge 94002. This is essentially the total carbon density since the electron temperature in this region exceeds 200 eV.

The outer leg, however, is attached. Detachment of the outer leg is achieved experimentally by puffing deuterium gas into the main chamber. The experimental signature for detachment is the increase in total radiated power, and the decrease in the divertor heating power 共especially the outer divertor兲 seen in Fig. 5. Many phenomena happen when the outer leg detaches. The perpendicular thermal diffusivity increases approximately 50% 关see Fig. 4共d兲兴. The density at the separatrix increases 关see Fig. 4共a兲兴. The ion current to the plate changes, and hence the carbon source distribution changes. The parallel thermal gradients change, leading to changes in the impurity force wells, and concomitant changes in the core impurity content 共see Fig. 19兲. Detailed simulation of all these phenomena will require effort similar to that expended in describing the simulation of attached plasmas to this point of the report. Such detailed simulation is beyond the scope of the present paper. However, it is reasonable to explore the changes associated with detachment in a qualitative way by simply asking what happens to the divertor plasma when additional gas is introduced without making any other changes. The results of these simulations are described here. There are many qualitative similarities between the simulation of detachment obtained in this simple fashion and the experiment. The simulated response of the two primary exhaust power channels 共radiation and divertor heating兲 to a puff of deuterium gas is shown in Fig. 20. There is a 30% increase in the total radiated power, and a 45% decrease in the outer divertor heating power at detachment in the simulation, similar to those seen experimentally. Experimentally the plasma

never reaches steady state after the introduction of additional deuterium. Rather, the total core electron content and the edge density rise roughly linearly with time. Although it is possible to simulate time dependent plasmas with UEDGE, it is much more straightforward to simulate steady state conditions. The simulations shown in Fig. 20 are fully converged, steady state solutions at each input gas current. If, however, we start with the 400 A case and increase the thermal and particle diffusivities by 50%, as suggested by the experimental results, we are not able to obtain a steady state solution. Rather, the total radiated power slightly exceeds the input power, and the plasma cools throughout the calculational domain. If the code is run in the time dependent mode, 5 s after the diffusivities are changed the plasma is evolving with a time constant of several hundred milliseconds, and the core plasma is below 1 eV over much of the closed flux surfaces. The physics models in UEDGE are no longer applicable. Cooling of the edge plasma will affect the core plasma, which is held fixed in the simulation by boundary conditions. This suggests the plasma would collapse in the experiment if the discharge were run for a long period of time. The simulated response of the carbon density at the outer midplane to the injection of deuterium gas in the main chamber is shown in Fig. 21. The core carbon density rises by a factor of two when the outer divertor plasma detaches. This is similar to what is seen in the experiment, as shown in Fig. 19. V. DISCUSSION

A detailed comparison between simulation of the edge/ SOL plasma and experimental results on a discharge in DIII-D has been presented. As might be expected for both simulation and measurement of a region with extremely complex physics, there is both good news and bad news. The



FIG. 21. Simulated variation of the core carbon density with input neutral gas current.

simulation is consistent with many of the diagnostics when a very simple, spatially constant, model for cross field transport of both particle and thermal energy is used. Specifically, we find consistency with the following: 共1兲 The upstream profile of the electron density, and the electron and ion temperature. 共2兲 The electron temperature profile near the outer strike point. 共3兲 The heating profile on the divertor plates. 共4兲 The profile of D ␣ emission in the SOL at the divertor plates. 共5兲 The total radiated power, with carbon radiation being dominant. 共6兲 The line intensity of high ionization states of carbon. 共7兲 The parallel flow velocity of C ⫹1 ions in the low field side of the divertor region. 共8兲 The total carbon density on the closed flux surfaces near the separatrix. Unfortunately, we also find some inconsistencies. It is these inconsistencies which will enable improvements in the model, hence it is worthwhile to enumerate them in some detail. 共1兲 The simulated total radiated power in the divertor region is concentrated in a narrower region near the strike points than indicated by the bolometer arrays. 共2兲 The simulated emission of low ionization states of carbon is higher than measured. 共3兲 The simulated emission of D ␣ radiation in the private flux region 共near the X-point兲 is lower than measured. Examination of these lists of successes and failures of the model indicates the model can reproduce the experiment in a global sense, but fails in details. For example, we find consistency in the total radiated power and the carbon density on the closed field lines. However, the simulations obtain line radiation emission from the low ionization states of carbon which are more than an order of magnitude higher than measured experimentally. This suggests the sputtering yield used in the simulation is higher than that seen in the experiment. We find that the emissivity of low ionization states of carbon is sensitive to the sputtering yield, but that the total radiated power is not. The sputtering yield of the carbon walls is


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apparently a factor of 2 to 4 lower than expected from the sputtering models of the University of Toronto group.8 The calculated total carbon density in the edge of the closed field lines is insensitive to the sputtering yield, and is in excellent agreement with that measured. This result permits some confidence in the impurity transport model. This model indicates that the parallel transport of carbon impurities is determined by a balance between the ion temperature gradient and the drag forces. The temperature gradients are insensitive to the sputtering yields since the divertor plasma is somewhat self-regulatory in terms of the total radiated power. Hence the transport of impurities from the walls to the core plasma is weakly dependent on the source strength of the carbon. We have proposed a model which explains this insensitivity. We have not attempted a complete simulation of the detached phase of this discharge, but have restricted the analysis to exploring the effect of adding gas from the outer wall, keeping all other parameters constant. The effect on total radiated power, divertor heating power, and core carbon content is shown to be consistent with experiment. ACKNOWLEDGMENTS

We gratefully acknowledge the help of the entire DIII-D experimental crew without whom we would have been unable to compare any simulation results with experiment. In addition we thank W. P. West, M. E. Fenstermacher, M. E. Rensink, and N. Wolf for many helpful discussions. This work was supported at LLNL by the U.S. Department of Energy under Contract Nos. W-7405-ENG-48 and DEAC03-89ER5114. 1

J. Luxon, P. Anderson, F. Batty et al., in Proceedings of the 11th International Conference on Plasma Physics and Controlled Nuclear Fusion 共International Atomic Energy Agency, Vienna, 1986兲, Vol. I, p. 159. 2 M. E. Fenstermacher, S. L. Allen, D. N. Hill et al., Contrib. Plasma Phys. 36, 127 共1996兲. 3 J. A. Boedo, R. Lehmer, R. A. Moyer et al., J. Nucl. Mater. 266–269, 783 共1999兲. 4 R. C. Isler, N. H. Brooks, A. W. Leonard, and W. P. West, J. Nucl. Mater. 266–269, 376 共1999兲. 5 R. C. Isler, N. H. Brooks, W. P. West et al., Phys. Plasmas 6, 541 共1999兲. 6 R. C. Isler, N. H. Brooks, W. P. West et al., Phys. Plasmas 6, 1837 共1999兲. 7 J. W. Davis, A. A. Haasz, and P. C. Stangeby, J. Nucl. Mater. 155–157, 234 共1988兲. 8 J. W. Davis and A. A. Haasz, J. Nucl. Mater. 241–243, 37 共1997兲. 9 G. Haas, P. Bachmann, D. Du¨chs et al., J. Nucl. Mater. 196–198, 481 共1992兲. 10 L. L. Lao, H. S. John, R. D. Stambaugh et al., Nucl. Fusion 25, 1611 共1985兲. 11 G. D. Porter, J. Miller, M. Brown et al., Phys. Plasmas 5, 1410 共1998兲. 12 G. D. Porter and the DIII-D team, Phys. Plasmas 5, 4311 共1998兲. 13 T. D. Rognlien, P. N. Brown, T. B. Campbell et al., Contrib. Plasma Phys. 34, 362 共1994兲. 14 T. Rognlien, J. Milovich, M. Rensink, and G. Porter, J. Nucl. Mater. 196–198, 347 共1992兲. 15 L. Spitzer and R. Harm, Phys. Rev. 89, 977 共1953兲. 16 M. Keilhacker, R. Simonini, A. Taroni, and M. L. Watkins, Nucl. Fusion 31, 535 共1991兲. 17 G. R. Smith, P. N. Brown, R. B. Campbell et al., J. Nucl. Mater. 220– 222, 1024 共1995兲. 18 X. Xu and R. H. Cohen, Phys. Plasmas 36, 202 共1996兲. 19 D. A. Knoll, P. R. McHugh, S. I. Krasheninnikov, and D. J. Sigmar, Phys. Plasmas 3, 293 共1996兲.


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