Self-consistent plasma-neutral modeling in tokamak plasmas with a large-area toroidal belt limiter

Transcription

1 PHYSICS OF PLASMAS VOLUME 6, NUMBER 7 JULY 1999 Self-consistent plasma-neutral modeling in tokamak plasmas with a large-area toroidal belt limiter D. S. Gray, M. Baelmans, a), b) J. A. Boedo, D. Reiter, a) and R. W. Conn Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, California Received 31 March 1998; accepted 12 April 1999 Plasma-neutral phenomena in the edge plasma and scrape-off layer of the Torus Experiment for Technology Oriented Research G.H. Wolf and the TEXTOR Team, J. Nucl. Mater. 122&123, with the toroidal belt Advanced Limiter Test ALT-II D.M. Goebel et al., J. Nucl. Mater , are simulated using the code package B2-EIRENE D. Reiter et al., Plasma Phys. Controlled Fusion 33, Spatially-constant, anomalous radial transport coefficients (D,V, ) are used for fitting measured electron temperature and density profiles. Primary neutral fluxes are determined by plasma fluxes to material surfaces, and D emissions are predicted from them. Comparison of the predicted D emission with measurements indicates a critical need, in predictive modeling, for a self-consistent model of fluxes to material surfaces that are parallel to the magnetic field. Appropriate factors are calculated for deducing D source rates from D emissions measured in various locations, taking into account molecular processes and spatially varying plasma parameters; values range from 17 to 28 ions/photon. Ion fluxes lost to pumps or the wall must be explicitly re-introduced as neutral fluxes at the outer boundary American Institute of Physics. S X I. INTRODUCTION Couplings of two-dimensional plasma fluid codes with Monte Carlo neutral transport codes have become valuable tools for the modeling of tokamak boundary phenomena, employed both for existing devices and for the design of planned machines such as the International Thermonuclear Experimental Reactor ITER. 1 Such modeling has been performed mostly for axisymmetric divertor machines. In this work, the code package B2-EIRENE 2,3 has been applied to Torus Experiment for Technology Oriented Research TEXTOR 4 with the toroidal belt Advanced Limiter Test ALT-II. 5 Some results of this modeling see Figs. 5, 6, 8, and 9 also appear in a separate publication 6 giving an overview of experimental work at TEXTOR; the present paper discusses the limiter modeling in detail. The B2-EIRENE package provides a coupling of a twodimensional 2-D plasma fluid code B2 7 and a 3-D Monte Carlo neutral transport code EIRENE 8. In the implementation described here, B2-EIRENE is used to model the boundary plasma in TEXTOR over the minor radii from approximately 5 cm inside to 5 cm outside the last closed flux surface, the region in which the interaction of neutral hydrogen with the plasma is of greatest importance. There are two main motivations for this work. One is to develop a method for determining appropriate ions-perphoton factors F ip for deducing particle sources from measured D emissions. Since D measurements are made over a Also at Forschungszentrum Jülich GmbH, Institut für Plasmaphysik, Jülich, Germany. b Present address: Katholieke Universiteit Leuven, Afdeling T.M.E., Heverlee, Belgium. finite volumes, in which there are spatial variations in plasma parameters and in the relative contributions of atomic and molecular processes, it is desired to calculate factors that are averaged over the observed volumes, and that depend on only a small number of global parameters, such as the lineaveraged electron density. Values of F ip as high as 28 are found, which is about a factor of 2 higher than values typically used in the experimental literature. Another motivation is to use the TEXTOR/ALT-II case for model validation, since B2-EIRENE has not previously been applied to a large-area toroidal limiter geometry. It is found that the absence of a self-consistent model for fluxes to material surfaces that are parallel to the magnetic field seriously limits the accuracy of predictions in a limiter geometry. This may have implications also for divertor designs, such as the baffle structures in ITER. II. IMPLEMENTATION OF B2-EIRENE FOR ALT-II A. Experimental description TEXTOR is a circular-cross-section tokamak with major radius R 1.75 cm and plasma minor radius a cm. The standard toroidal magnetic field and plasma current are 2.25 T and 350 ka. Plasmas with Ohmic heating alone and with neutral beam injection NBI have been simulated in this work. In the NBI cases, the beam power was 1.3 MW co-injected. ALT-II is a toroidal belt pump limiter, located poloidally at 45 below the midplane on the outboard side. Its poloidal width is 28 cm, and the thickness of its graphite tiles was 1.7 cm in these experiments. Behind these tiles, plasma flows through channels scoops to target plates for X/99/6(7)/2816/10/$ American Institute of Physics

2 Phys. Plasmas, Vol. 6, No. 7, July 1999 Self-consistent plasma-neutral modeling in tokamak TABLE I. Coefficients used to describe radial transport. n D V Case Heating cm 3 m 2 /s m/s m 2 /s 1 OH OH OH conbi conbi pumping. The limiter is radially movable; in these experiments, the last closed flux surface was at a 45.0 cm. A total of five tokamak plasma discharges have been simulated in this work. The cases, numbered 1 through 5, are identified in Table I. The last two simulate discharges with neutral beam heating. The simulated discharges were in deuterium. There is only one ion species present in the B2 calculations in this implementation: D. No impurities are included in this work. In order to choose certain important inputs to the plasma code boundary conditions transport coefficients, radiation loss, profile measurements of electron temperature, density, and radiation have been employed. A reciprocating double Langmuir probe, located at the outboard midplane, provides measurements of electron temperature and density. 9 The probe inserts approximately halfway into the modeled region. For smaller minor radii, measurements of T e and n e are taken from electron cyclotron emission and laser interferometer diagnostics, respectively. The radiation profile has been measured using bolometer arrays. 10 This measurement has been used to determine both the radiation emitted within the B2 mesh and the transport heat flux through the inner boundary of the mesh. The D emission predicted by the codes is compared to quantitative measurements made using a video camera. 6 B. Modeling description In a B2-EIRENE implementation, the two codes share a common computational mesh, representing nested magnetic flux surfaces, divided up in the poloidal dimension. B2 calculates plasma parameters and fluxes from fluid equations, consistent with given boundary conditions BC s, classical parallel transport and anomalous radial transport coefficients, and source rates arising from neutral processes. The source rates for the fluid equations are provided by EIRENE. EIRENE, in turn, uses the plasma solution calculated by B2, not only as a background for neutral transport calculations, but also to determine the ion fluxes to material surfaces. These surface fluxes are used to determine the spatial distribution for the neutral gas recycling source and to scale the fluid source rates that are reported back to B2 for the next iteration of plasma calculations. In this implementation, BC s and radial transport coefficients for B2 are chosen to make the plasma solution agree with measured profiles of electron temperature and density. When a converged solution is found, after several iterations between B2 and EIRENE, then other physical measureables FIG. 1. A detail of the geometries used by B2 and EIRENE the region near the limiter. Thick solid lines indicate surfaces which collect plasma fluxes in B2 or emit primary neutral sources in EIRENE. Inthe B2 geometry, thick dotted lines represent material surfaces where isolating conditions are applied. In the EIRENE geometry, such lines indicate reflecting surfaces that are not used for launching neutral sources. can be compared with code predictions. In particular, EIRENE calculates D emissions that can be compared with measurements. Self-consistent calculations of this type, in which the codes are mutually coupled, aim to be of predictive value for the design of future devices. This is in contrast to schemes in which the neutral code uses the plasma only as a background for stand alone calculations, in which the primary neutral sources are chosen in order to match measurements such as D emission, utilizing the linearity of neutral gas transport contributions from different sources can simply be added until the experimental D profile is reproduced. Such calculations are used for interpretation of measurements in existing devices. Applications of the former coupled scheme have only been made to divertor tokamaks. The present application differs from divertors in at least two important ways: the structure of the scrape-off layer SOL is simpler, and there is a large-area material surface parallel to the magnetic field in contact with the plasma. This different configuration presents a valuable opportunity for model validation. This section provides a general description of how the model has been implemented with ALT-II. An explanation of the solution procedure appears in the Appendix. 1. Geometry The 2-D geometry used by B2 in this implementation includes 33 nested, nonconcentric, circular flux surfaces. The relative positions of these surfaces are taken from a tokamak equilibrium with the magnetic axis assumed to be shifted from the geometric axis by 6 cm. The region between the innermost and outermost of these flux surfaces is resolved into 33 poloidal divisions. A portion of this mesh in the vicinity of the limiter is shown in Fig. 1. The spacing in both radial and poloidal directions is relatively fine near the limiter head. The poloidal spacing near the limiter is not as fine as often seen in divertor modeling; this is allowable due to the low-recycling conditions in front of the limiter. Details of the real geometry of ALT-II have been published previously. 11 The limiter belt head is well suited to the

3 2818 Phys. Plasmas, Vol. 6, No. 7, July 1999 Gray et al. axisymmetric geometry used in this work, although the toroidally discrete pumping structures neutralizer plates, scoops beneath the belt cannot be accurately rendered in this way. Material surfaces are depicted in Fig. 1 as thick lines. Of these, the dotted lines represent isolated surfaces, at which radial plasma fluxes have been set to zero. This has been done because it is commonly supposed that fluxes to surfaces parallel to the magnetic field should be small. The solid thick lines represent material surfaces that accept plasma fluxes. As discussed below, the outermost flux surface has been treated as the wall; this surface accepts crossfield fluxes, as fixed-value BC s have been applied, rather than isolating conditions. Parallel fluxes flow to the pumplimiter neutralizer plates the two surfaces under the limiter head and to the limiter tips. B2 calculates only on an orthogonal mesh plasma does not flow to inclined surfaces so a two-step staircase is used to approximate the shape of each limiter tip no such approximation is needed in EIRENE. A thin, solid line bisecting the limiter in the figure shows where the mesh begins and ends in the poloidal direction the poloidal cut. On the closed flux surfaces, these poloidal ends are also isolated, so no fluxes pass through this surface. Isolating this surface greatly reduces the processor time required for convergence, in comparison to applying periodic BC s that allow fluxes to pass through. 12 In one trial in which the cut was opened periodic BC s applied, the difference in comparison to results with the cut closed was minor Sec. III E. The geometry used by EIRENE includes a 3-D equivalent of the B2 geometry, as well as some additional surfaces. The surfaces to which plasma fluxes flow in B2 limiter tips, neutralizer plates, outer boundary are used in EIRENE as locations to launch neutral particles back into the plasma. The tip sources, however, are not launched precisely from the B2 surfaces where the plasma fluxes are collected, but rather from the curved surfaces shown in Fig. 1, which accurately reproduce the shape of the actual ALT-II tiles. This is important because the angular distribution in velocity space of the neutral particles can have a strong influence on the solution. 3 The velocity distribution and the composition of the sources how many reflected atoms versus how many desorbed molecules are calculated from a surface database produced using the TRIM code. 13 It has been found that a significant fraction typically 20% of the neutral atoms pass through the inner boundary of the B2-modeled region before finally being ionized. In order to provide some radial resolution of the plasma-neutral activity in the inner region, surfaces have been defined that divide this region into shells, as can be seen in Fig. 1. The inner shells are not resolved poloidally. Whereas the plasma background used by EIRENE in the B2-modeled region is taken from B2 output, the plasma parameters in the shells of the inner region are simply prescribed based on measured profiles and are not subject to any plasma model. The surfaces that comprise the limiter and the B2 outer boundary are all reflecting surfaces in EIRENE for neutral particles that strike them. The B2 inner boundary is a transparent surface in EIRENE. In the geometry used in the cases described here, the minor radius of the B2 inner boundary is 40 cm and the outer boundary is at r 50 cm. At r 44.5 cm, the last closed flux surface LCFS is slightly further inside than in reality; the input profiles have been adjusted to account for this. Since the B2 outer boundary is a reflecting surface for neutrals in EIRENE, it acts effectively as the liner, although in reality the liner is at r 55 cm rather than 50 cm. 2. Collisional-radiative model The source rates for D ions and D photons are calculated in EIRENE using a collisional-radiative model CRM for the neutral atoms and molecules. This is necessary since the requirements for coronal equilibrium do not hold in general in the tokamak boundary. The model is discussed elsewhere, 14 and here only a brief description is given. The CRM used in this work incorporates rate coefficients for electron impact processes and lifetimes for spontaneous processes that affect the various quantum states of hydrogen atoms, molecules, and molecular ions. In each cell of EIRENE s geometry, a relative population distribution among numerous quantum states is calculated from the local T e and n e. To calculate the absolute densities of these states, the densities of the ground states are needed, and these are provided by EIRENE s Monte Carlo calculations. The ionization rate coefficients used in the Monte Carlo calculations take into account the distribution among the quantum states in each cell a particle passes through the distribution is determined beforehand, by the plasma conditions alone. The D emission from each cell is included in the output of EIRENE, along with the local production rate of D ions. These two production rates can be used in deducing ion source rates from D measurements. 3. Transport coefficients and boundary conditions Radial plasma transport has been described using anomalous transport coefficients: a particle diffusion coefficient D, a bulk velocity V, and thermal diffusion coefficients i and e the thermal conductivities being given by ri n i and re n e ). These coefficients have been held spatially constant. The radial viscosity r is calculated from a momentum diffusivity r /mn that has been held at 1.0 m 2 /s throughout this work. In all but one of the cases studied, it turns out that a single, uniform radial diffusion coefficient D does not provide satisfactory reproduction of the shape of the measured density profile. The options available to address this problem were to use a spatially varying diffusion coefficient D(r) or to introduce a uniform radial bulk velocity V in the continuity equation. The latter option was chosen because it introduces the fewest new degrees of freedom and allows the coefficients to be chosen in a reasonably simple manner. Surfaces in the B2 mesh at which some sort of condition must be applied are the inner and outer boundaries, the limiter surfaces that are flush with the magnetic flux surfaces, and the limiter surfaces that cross magnetic field lines. As mentioned earlier, the flush limiter surfaces are isolated. At the radially-extended limiter surfaces, sheath BC s are

4 Phys. Plasmas, Vol. 6, No. 7, July 1999 Self-consistent plasma-neutral modeling in tokamak applied. 15 At the inner and outer boundaries, T e, T i, and n are each set to a constant value over the flux surface; the values at the inner boundary are chosen as described in the Appendix, and those at the outer boundary are chosen for profile smoothness. At the inner boundary, v is set to zero, and at the outer boundary, the radial flux of the parallel momentum is prescribed to be zero. 4. Other considerations The radiation occurring in the region modeled by B2 is included artificially as a sink term in the electron energy equation since no impurities are present in the simulations to account for the radiation. In these simulations, the radiation has been treated as poloidally uniform, i.e., as a function of radius only. The primary effect of error in the radiation input would be error in the used to fit the T e profile. In the two NBI cases, beam deposition results from TRANSP 16 modeling of TEXTOR discharges have been used to determine how the influence of the beam on the plasma should be accounted for. The modeled discharges are not the same as those discussed here, but sufficiently similar for the results to be usable. In both cases, the beam energy deposition in the region modeled by B2 was found to be negligible deposition at r 40 cm is 2% of the total heating. Like the energy deposition, the particle deposition of the beam also occurs predominantly inside r 40 cm. The particle deposition rate outside r 40 cm is relatively small, amounting to about one-tenth of the calculated particle flux through the r 40 cm surface i.e., only about 2% of the total particle input, and has not been included in the particle balance for the cases described here. The transport heat flux convection plus conduction through the B2 inner boundary (r 40 cm is calculated based on the heating rate of the discharge and on the radiation profile. Considering the beam-deposition result in which the fraction of beam heating that occurs outside this surface is negligible, the transport flux there has been calculated as the total heating rate minus the power radiated within the surface. This figure is needed for the selection of either or T i (r 40), depending on the case. Pump-limiter action has been simulated by applying a nonunity return probability for plasma particles that strike the pump-limiter neutralizer plates. A return probability of 65% has been used at these surfaces, based on the results of earlier ALT-II pump-limiter modeling. 17 The remaining 35% of the ion flux to these surfaces the pumped flux is reintroduced as a gas feed far from the limiter. The location of the gas feed is a surface in the outer boundary at 75. III. RESULTS A. Reproduction of profiles Radial profiles of n and T e at the outboard midplane are shown for the three Ohmic cases in Fig. 2, overlaid with the measured profiles. The same information is shown for the NBI cases in Fig. 3. The measured density profiles are satisfactorily reproduced in all cases. In case 5, it was not clear FIG. 2. Experimental profiles of electron temperature and density in the Ohmic cases 1,2,3 at the equatorial plane, overlaid with modeling results. Experimental profiles are plotted with symbols, and modeling results with lines. how to connect the interferometer profile and the probe profile; the solution shown represents a best guess. The need for an inward pinch velocity or something more than a constant diffusion coefficient is illustrated in Fig. 4. The solid line shows the solution that has been taken as the best reproduction of the measurements, and the broken line shows a solution that best reproduces the density profile given the constraint that V 0. While the pinch-free solution matches the density values about as well as the accepted solution, the density gradient is not in very good agreement with the measured profile too weak inside the LCFS and too strong outside. A satisfactory fit could also have been achieved without a pinch velocity, by using a diffusion coefficient that increases with radius. The figure also illustrates that the choice of D and V is quite sensitive to the precise shape of the density profile. The measured profile that has been used consists of spliced-together results from two different diagnostics, with different errors, since a full profile is not available from a single diagnostic. No attempt has been made to optimize the fit to the spliced profile by making any adjustments to either part of the profile. Since the real shape of the profile is uncertain, caution should be exercised in ascribing physical significance to the specific values of the

5 2820 Phys. Plasmas, Vol. 6, No. 7, July 1999 Gray et al. FIG. 4. Best fits to the experimental density profile found with and without an inward pinch. FIG. 3. Experimental profiles of electron temperature and density in the beam-heated cases 4,5 at the equatorial plane, overlaid with modeling results. fitting parameters D and V that best fit the spliced-together density profile. The radial transport coefficients that produce the best fits to the measured data are shown in Table I. In the NBI cases, is higher than in the Ohmic cases by a factor of about 4. In the high-density NBI case 5, D is about a factor of 2 higher than in the Ohmic cases, and the inward pinch is slightly faster. In the low-density NBI case 4, the best results were obtained with zero pinch velocity and hence a lower D than in case 5. The measured electron temperature profiles are also satisfactorily reproduced. In the Ohmic cases, the calculated values lie well below the probe measurements outside R 222 cm. At such radii, relatively far out in the SOL, the uncertainty in the probe s T e measurement is relatively large because of the high fluctuation level n rms / n. It was found to be critical that careful attention be paid to the manner in which the lost ion fluxes lost at the pumping targets and outer boundary are re-introduced in the particle balance. This has a strong impact on the heat transport. When the lost fluxes are re-introduced in a gas feed and wall source as described above, convection accounts for roughly one-half of the radial heat flux at the inner boundary. If these neutral sources are not included, then the lost fluxes are returned at the inner boundary, appearing as an ion flux from the core. In this situation, convection accounts for nearly 100% of the radial heat flux at the inner boundary, and it becomes impossible to make the electron temperature profile agree with measurements. Experimentally, the gas feed rate typically is feedback-controlled to maintain the plasma density at a constant level. Although the B2-EIRENE modeling described here is also time-independent, the gas feed rate in the simulations may not necessarily match the experimental value. The difference occurs because, in experiments, the walls often act temporarily as a net source or sink of particles, and the time for equilibration between the particle inventories in the plasma and in the wall is on the order of the duration of a discharge. B. Distribution of D A 2-D representation of the predicted D emissivity for a particular case is shown in Fig. 5. At the lower right ( 45 ), bright areas appear near the top and bottom tips of the limiter head and also just below the limiter head. The emission under the limiter head is not experimentally observable by the existing optical systems. The region covered by the B2 mesh has been broken up more coarsely, for the purpose of interpreting the results, into four basic regions see Fig. 5. The region near the front of the limiter head is referred to as limiter. The region behind the head is referred to as scoops. A region near the gas puff is called gasfeed, and the remainder of the area covered by the B2 mesh is referred to as halo. Further, the region inside the inner boundary of the B2 mesh is referred to as inner. The D radiation that occurs in the scoops region is a significant fraction of the total. In the Ohmic cases, it amounts to about 9% of the D from the entire plasma. Since this fraction is significant, and since the scoop D emission is not measured experimentally, care must be taken

6 Phys. Plasmas, Vol. 6, No. 7, July 1999 Self-consistent plasma-neutral modeling in tokamak FIG. 5. A 2-D representation of the D emissivity predicted by B2-EIRENE for case 3. For analysis of the code results, the B2 geometry is broken down into four regions. in comparing the estimated total D emission in the experiments to the results of the simulations. A comparison is shown in Fig. 6. The experimental values shown in the figure are the sums of the observed D, excluding the gas feed. The B2-EIRENE values shown are the sums over the limiter, halo, and inner regions. The dependence on density and heating agrees with experiments, but there is a difference in magnitude between the simulations and experiments of about a factor of two. Representative error bars are included in the figure to indicate the level of uncertainty in the results. The uncertainty in the modeling results is determined, in part, by the uncertainty in the input density profile on the open flux surfaces i.e., the probe profile ; another source of uncertainty is discussed in Sec. III E. In one of the cases #3, an additional run was carried out with 90% return probability at the neutralizer plates rather than the usual 65%, to test the sensitivity of the results to this factor. The higher return probability reduced pumping results in 21% higher particle flux to the neutralizer plates. The flux increase to the surfaces that produce observable D limiter tips plus wall is more modest, however, only 6%. C. Radial variations The radial distribution of the predicted D emission, shown for case 3 in Fig. 7, exhibits characteristics that are typical of all the cases. These are: a strong peak near the LCFS, a smaller peak in the scoop radii due to activity under the limiter head, and a small but non-negligible emission inside the B2 inner boundary (r 40 cm. The figure shows D emissivity integrated in the poloidal and toroidal directions. In the B2 calculation region (r cm, the line labeled total indicates the full poloidal integral, while the plot symbols show contributions from the various regions. The peaks are due to activity near the limiter squares and crosses, while the profiles in the halo and gasfeed regions are smoother. The fraction of D emission occurring inside r 40 cm is about 20% in all cases but #5, in which it is only 4%. It has not been possible to experimentally measure these radial profiles for comparison with the simulations. The fraction of D occurring in the scoops region is about 9% in the Ohmic cases but higher in the NBI cases: 15% in case 4 and 24% in case 5. D. Factors for D interpretation In experiments, it is common to use measurements of D emission to deduce the deuterium recycling flux. The FIG. 6. The predicted D emission is lower than the measured emission. Error bars on the highest-density points indicate the levels of uncertainty. FIG. 7. A radial profile of D emission as predicted by B2-EIRENE.

7 2822 Phys. Plasmas, Vol. 6, No. 7, July 1999 Gray et al. emitted photons-per-second is multiplied by a factor F ip ionizations per photon to deduce an ionization rate; F ip is taken from CRM. This technique is particularly attractive when F ip does not depend too strongly on n e and T e, i.e., when F ip can be treated as a spatial invariant, so that the ionization rate can be deduced without detailed knowledge of the n e, T e, and D emissivity profiles. In CRM of atomic hydrogen, F ip has been found to have a satisfactorily weak n e and T e dependence in the ranges typical of tokamak boundary plasmas so long as T e 20 ev. 18 On the basis of these atomic results, values of F ip in the range of are commonly used in the experimental literature. 6 In hydrogen recycling from the carbon wall surfaces typical of presentday tokamaks, however, much of the hydrogen is introduced in molecular form, and molecular processes need to be incorporated in the CRM used to calculate F ip, a is done in the present work. Molecular processes can alter F ip considerably, for example by direct production of D ions that do not appear first as atoms that can produce D (e D 2 e D D ). Using the atomic-molecular CRM, values of F ip as high as 28 have been found in this work case 5. A major goal of this modeling effort is to determine ions-per-photon factors F ip that can be used to deduce D source rates from D measurements. The approach used here is to provide spatially-averaged ratios for the plasma in three different regions, in each of which the D emission is measured separately. These three D sources are the source in front of the limiter face, the source in the halo, and the source from the gas feed. Since there is no measurement of the D emission behind the limiter, an F ip for that region is not useful. The averaging has been done by dividing the radially integrated particle source also integrated poloidally over the relevant region by the similarly integrated D emission. The relatively small ionization and emission inside r 40 cm have been included in these ratios by making the approximation that the poloidal variation of these quantities in the inner region is the same as that of the radially integrated quantities in the B2-modeled region (40 r 50 cm, scoops region not included. The averaged factors F ip are shown in Fig. 8, plotted versus line-averaged density. For the Ohmic cases, the factors for a given view are essentially the same from case to case, i.e., the factor is independent of the density of the discharge. The different views have different factors, due to differing relevances of molecular processes. The low-density NBI case 4 has the same factors F ip as the Ohmic cases, but the high-density case 5 has higher factors. The former fact is consistent with well-known calculations 18 that the factor is independent of electron temperature as long as the temperature is sufficiently high, and the latter fact is consistent with calculations that the ratio increases with electron density. These results are interpreted as meaning that the heating level of a TEXTOR discharge is unimportant in determining F ip, but that it is necessary to account for a deviation from constant factors if the lineaveraged density is above cm 3. For the gasfeed view, a constant value can be used for all densities. Although the predicted total D emission is significantly lower than the measured emission, it is important to note that FIG. 8. Ions-per-photon ratios appropriate for interpreting D measurements in TEXTOR. this fact does not invalidate the F ip calculated here. Since the neutral model is linear, these factors depend on the plasma parameters which are satisfactorily reproduced and on the composition and distribution of the primary neutral sources, but not on the absolute fluxes. E. Poloidal variations Considering now the variation of the predicted D emission with poloidal angle, it can be seen from Fig. 5 that the emission has a peak near the bottom tip of the limiter head and one near the upper tip. This is typical of all the cases studied. From Fig. 9 it can be seen that the intensities of these two peaks are about equal. Shown in the figure is the D emissivity, integrated over minor radius and in the toroidal direction, as a function of poloidal angle. The emission in the scoops region is excluded, so that the curve can be compared with measurements. FIG. 9. The predicted poloidal profile of D emission has a peak near each of the two limiter tips, with a gap between the peaks. No such gap is observed experimentally.

8 Phys. Plasmas, Vol. 6, No. 7, July 1999 Self-consistent plasma-neutral modeling in tokamak Between the two peaks is a gap in which very little D light is emitted on the front side of the limiter. Such a gap is not observed in the measurements. The presence of the gap is due to the isolating BC on the limiter surface between the tips. This surface parallel to the magnetic field we refer to as the limiter roof. An alternative explanation of the experimentally observed filling-in of the D gap might be that the plasma is not correctly aligned to the limiter. This can be tested by considering a discharge in which the horizontal position was ramped from 1.5 cm inside to 1.5 cm outside the nominally centered position. This produces angles between the magnetic surfaces and the limiter roof of up to 2, approximately. It is found that such shifts do not significantly alter the D distribution seen in front of the limiter. 19 This could in turn be attributed to a vertical misalignment that prevents the flux surfaces from aligning with the roof at any point in the horizontal ramp, but the misalignment would have to be unrealistically large to account for the effect contact between the plasma and a limiter at the top or bottom of the machine would be visible, but none was observed. Thus it appears that large fluxes are drawn to this surface that are parallel or nearly parallel to the magnetic field; thus, replacing the isolating condition there with a condition that allows such fluxes is critical to accurately describing the plasma behavior. The prevention of this roof flux may provide a partial explanation of the difference between the predicted and measured total D emission. Other standard BC s here constant gradient, etc. provide ion fluxes onto the roof that are much too small to fill the gap in the D profile. For one case #3, additional runs have been carried out with the funneling model of Stangeby 20 applied at the roof, but even this model does not provide fluxes strong enough to account for the experimental findings. The resulting D profile remains strongly hollow. This test was run with the poloidal cut Sec. II B 1 in front of the limiter open periodic BC s applied as well as with the cut closed zero poloidal fluxes. The difference between the results with open and closed cuts is minor; opening the cut increases the fluxes to the roof by 20%, with a concomitant reduction in fluxes to the tips, and an overall change in fluxes into the SOL by 4%. There seems to be no nonlinear feedback in the system of combined plasma and neutral equations to explain the peaking of profiles poloidally between the two strikepoints. We have to conclude an anomalously strong funneling action in front of the tangential surface. Much theoretical work is currently devoted to this issue of plasma-surface BC s at parallel surfaces. No generally agreed results seem to be available, and a solution to this problem may have important implications even for divertor machines notably on the main chamber recycling and on the baffle structures. IV. CONCLUSIONS In modeling of plasmas operated on the toroidal belt limiter ALT-II in TEXTOR, using the code package B2- EIRENE, measured radial profiles of density and electron temperature have been reproduced by the use of spatiallyconstant, anomalous transport coefficients. Neutral recycling sources, which are determined by the plasma fluxes, in turn produce source rates for the plasma equations, providing a strict coupling between the plasma and neutral calculations. D emissions are also predicted, for comparison to measurements and for calculation of ions-per-photon ratios F ip for use in interpreting D measurements in experiments. The predicted total D emission differs from measurements by a factor of 2, and the predicted spatial distribution of the emission also differs from measurements. The different distribution is due to isolating boundary conditions on parts of the limiter surface the roof. Large fluxes to this surface are implied by the observed plasma behavior. A BC that self-consistently allows fluxes to material surfaces that are parallel to the magnetic field is critically needed for this type of work. However, models available so far, including the semi-empirical funneling model by Stangeby, still strongly underestimate the role of recycling at tangential surfaces. In simulations such as these, particle fluxes lost at the pumping targets and at the wall must be explicitly reintroduced as neutral fluxes from outside the plasma; otherwise the heat transport calculations will be incorrect. The interpretation of D measurements, in practice, is complicated by variations in electron density and in the composition of the neutral deuterium mix of molecules and atoms. These variations cause the ratio of the local production rate of D ions to that of D photons to depend both on location and on the line-averaged density. The problem of spatial variation of F ip can be solved by using factors appropriately averaged over specific experimentally-observed plasma regions. In the plasmas studied here, these averaged factors ranged from about 17 to 28 for various views and plasma densities the electron temperature was high enough that it did not strongly influence F ip ). The use of constant values of 15 or less, which is common in literature on confinement in tokamaks, could thus lead to significant underestimates of the D source. ACKNOWLEDGMENTS The authors are grateful for the help of P. Börner and T. Küppers in carrying out this work, and wish to thank Dr. J. Ongena for providing the TRANSP modeling results that were used. APPENDIX: PROCEDURE 1. Outline The solution for a given case is found according to the following scheme. Once the measured electron temperature, density, and radiation profiles are known, B2 is run with its internal, analytical diffusive neutral model in order to generate an initial approximation to the 2-D plasma state. In this initial run, it is adequate to set V 0 and use a diffusion coefficient alone, which can be estimated from the scrape-off length of the density profile (D c s 2 /2L, where L is the connection length. Once this initial 2-D plasma has been generated, the iteration between EIRENE and B2 can begin. The values of D,

9 2824 Phys. Plasmas, Vol. 6, No. 7, July 1999 Gray et al. V,, and T i (r 40) are chosen in a manner described in the following section, and the BC s on n and T e are adjusted to optimize the agreement of the calculated profiles at 0 with the measured profiles. The calculation of D, V, and makes use of radial particle fluxes calculated by B2 the calculation of makes use also of heat fluxes. Once these are chosen, several iterations typically five between EIRENE and B2 are run. The resulting B2 solution provides a new set of radial fluxes. From these, a new set of D, V, and is calculated, and this process continues until a converged solution is found. The number of neutral-particle histories in each run of EIRENE was about for case 5 and about for the other cases. The solution is considered to be well-converged when the following criteria are met: 1 the measured density profile is well fit in particular, the calculated profile is consistent with 0, 1, and n 1, which are defined below ; 2 the radial heat flux at the B2 inner boundary agrees with the experimental value; and 3 the transport coefficients D, V, and re-calculated from the output of the latest B2-EIRENE run do not differ significantly from those used as input for that run. Convergence is achieved relatively quickly, due to the low-recycling conditions in the limiter configuration. In contrast, modeling of the high-recycling ITER divertor requires about 100 iterations between B2 and EIRENE to achieve convergence Boundary conditions and transport coefficients In preparing a B2-EIRENE run in this work, the important parameters to be chosen are the values of n, T e, and T i at the inner boundary, in addition to the four constants D, V, i, and e. That is, a total of seven values must be chosen. Consider first the three BC s and the two thermal diffusivities. The selection is based on three quantities taken from measurements, and on two assumptions. The three measurements are the density, electron temperature, and radial transport heat flux. The first assumption, used in all cases, is that i e, i.e., that the heat conduction can be described by a single coefficient,. The second assumption depends on the case. In the two high-density cases 3 and 5, it is assumed that T i T e at the inner boundary. This is justified a posteriori by the fact that the resulting profiles of T e and T i do not diverge strongly from each other near the boundary. In these cases, is chosen so that the sum of the conducted and convected radial heat flux at the inner boundary for ions and electrons together will match the value deduced from measurements. In the lower-density cases, the value of has been based on the value in the relevant high-density case; in the OH cases 1 and 2, was chosen to be proportional to D which in effect was nearly the same as choosing a constant value for all three cases, and in the NBI case 4, the same value used in case 5 was taken. In these lower-density cases 1, 2, and 4, the BC on T i is not set to T e but is chosen to make the transport heat flux at r 40 match the measurements. To determine the remaining two constants, D and V, the one remaining measured quantity is the density gradient dn/dr. In order to determine the two parameters independently, it is necessary to evaluate dn/dr and n) at two different radial locations and make use of calculated particle fluxes. The two locations that have been chosen are the inner boundary and the last closed flux surface. Using subscripts 0 and 1 for these locations, respectively, and defining n/(dn/dr), the particle fluxes are given by 0 D dn Vn 0 n dr 0 0 D 0 V 1 D dn Vn 1 n dr 1 1 D V. 1 From this follows D 1 0 n 1 n and V 1 n 0 n 0 D 0. 0 When a pair (D,V) is calculated using this pair of formulæ, several iterations between B2 and EIRENE are then run, using these values of D and V in B2. The B2 solution provides new calculated fluxes 0 and 1, which are then used to calculate new values of D and V for further iteration. It should be noted that evaluating n and at two surfaces causes the system to be overdetermined. Of the four values, one (n 0 ) was used already to set the density BC at r 40 cm, and three remain (n 1, 0, and 1 ) for calculating the two parameters D and V. The significance of this is that the solution calculated by the code cannot be forced to reproduce all three values if they are chosen arbitrarily; i.e., the three are not truly independent. If the three values are chosen poorly, the calculated density profile typically will be in disagreement with all of them. In order to determine the values n 0, 0, n 1, and 1, it is necessary to first draw through the measured data an expected profile a guess at the profile that will be calculated by the code. After a few attempts and learning what profile shapes the code is capable of producing for various choices of (n 0, n 1, 0, 1 ), it becomes possible to guess with reasonable confidence a solution that will best fit the measured profile. One of the criteria for a satisfactory solution of a given case is internal consistency; the calculated density profile is required to be in good agreement with all three values ( 0, 1, and n 1 ) used in choosing the D and V that produced the profile. 1 P. H. Rebut, Fusion Eng. Des. 30, D. Reiter, H. Kever, G. H. Wolf, M. Baelmans, R. Behrisch, and R. Schneider, Plasma Phys. Controlled Fusion 33, D. Reiter, J. Nucl. Mater , G. H. Wolf and the TEXTOR Team, J. Nucl. Mater. 122&123, D. M. Goebel, R. W. Conn, W. J. Corbett, K. H. Dippel, K. H. Finken, W. B. Gauster, A. Hardtke, J. A. Koski, W. Kohlhaas, R. T. McGrath, M. E. Malinowski, A. Miyahara, R. Moyer, A. Sagara, J. G. Watkins, G. Wolf, the TEXTOR Team, and the ICRH Team, J. Nucl. Mater , D. S. Gray, J. A. Boedo, M. Baelmans, R. W. Conn, R. A. Moyer, K. H. Dippel, K. H. Finken, A. Pospieszczyk, D. Reiter, R. Doerner, D. L. Hillis, G. Mank, G. H. Wolf, and the TEXTOR Team, Nucl. Fusion 38,

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