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Modelling of the ITER reference divertor plasma
Journal of Nuclear Materials 415 (2011) S492–S496
Contents lists available at ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Modelling of the ITER reference divertor plasma H.D. Pacher a,⇑,1, A.S. Kukushkin b, G.W. Pacher c, V. Kotov d, D. Reiter d a
INRS-EMT, Varennes, Québec, Canada ITER Organization, Cadarache, France c Hydro-Québec, Varennes, Québec, Canada d FZ Jülich, Jülich, Germany b
a r t i c l e
i n f o
Article history: Available online 16 November 2010
a b s t r a c t ITER divertor and edge modelling with the ITER B2-EIRENE code including neutral–neutral and molecule– ion collisions has been applied to the ITER reference divertor plasma with the reference low-li equilibrium. The scaling of key parameters such as peak power load, DT influx to the core, helium density and influx with SOL power, edge/core fuelling ratio, pumping speed and ratio electron/ion power ratio has been determined and ASTRA core plasma modelling consistent with these edge boundary conditions have been carried out to determine the operating window. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The ITER divertor geometry resulting from the 2007 ITER design review has a lower and slightly smaller dome than the previous geometry, permits a wider scrape-off layer, and increases flexibility by accommodating a wider range of magnetic equilibria. In initial modelling [1], the evolution of key parameters of DT plasma was studied as the configuration was changed in several steps from the previous geometry to the current reference (F57, see Fig. 1 of [1]). The B2-Eirene code package SOLPS4.3 (see [1]) used here employs a two-dimensional fluid description for the ions and electrons of the edge plasma, and a three-dimensional Monte-Carlo neutral treatment, taking into account neutral–neutral and molecule–ion collisions. The plasma consists of D (representing both D and T isotopes) ions, atoms, and molecules, as well as of He and C ions and atoms. Only carbon targets are studied. The particle absorption at the duct entrance in the PFR is adjusted to provide the specified ‘‘engineering’’ pumping speed [2]. For the standard case, the total power entering the SOL is 100 MW, the helium outflow from the core corresponds to 600 MW of fusion power, and this ratio is varied in the simulations. In previous work, scaling relations were established [3–7] to link the key parameters of the edge plasma resulting from the simulation to various input parameters including power, size, and pumping speed. Several of these variations, notably the SOL power variation, were carried out only with a less accurate linear neutral model and with the previous reference geometry. In the present paper, a detailed study ⇑ Corresponding author. Address: INRS-EMT, 1650 boul. Lionel Boulet, Varennes, QC, Canada J3X 1S2. Tel.: +1 450 929 8241; fax: +1 450 929 8102. E-mail address: [email protected] (H.D. Pacher). 1 Presenting author. 0022-3115/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2010.10.083
using a non-linear neutral model [10] and the current reference geometry is carried out to update the scaling by investigating the effect of varying the SOL power, the ratio Pe/Pi of this power carried by electrons and ions, the pumping speed, the throughput, and the ratio of core fuelling to gas puffing. Core modelling is carried out with the ASTRA code version containing the integrated core-pedestal (ICPS) model where consistency of the core and edge parameters is ensured by applying the scaling relations from the edge modelling as boundary conditions to the core model [8]. In previous work, an ITER operating window was defined and its variation with power, density, and pumping speed among others was determined with this model [8]. In this paper, the model is updated with the revised scaling relations and the operating window is determined. Several previous studies have not yet been updated with the current model and geometry, notably the size variation [6], the effect of carbon re-erosion [5] and the effect of neon seeding [7] so that these effects are not yet reflected in the scaling.
2. Edge/SOL simulation results and scaling of edge parameters In [3–7], the key point in scaling the edge parameters had been identified to be the full detachment of the inner divertor, then characterised by the drop of the peak electron temperature along the inner divertor to a low value of the order of several eV. This was found to be related to the average neutral pressure along the interface to the private flux region, pn; all the quantities were expressed as functions of normalised pressure l, proportional to pn and equal to 1 at detachment. The maximum inner divertor temperature is however not a robust characterisation; it is local and varies with the main plasma species. Instead, following experimental practice on many tokamaks, we now characterise detachment
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H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
Fig. 1. (a) Normalised ion saturation current on inner and outer divertor targets versus l (Eqs. (1) or (3) and text), (b) peak power load on divertor plate versus average neutral pressure along private flux region interface pn for SOL powers from 50 to 140 MW. Comment: solid symbols and bold lines indicate simulation results, open symbols and light lines scaled values from Eqs. (1)–(12).
as a condition on the rollover of the total ion saturation current (nearly equal to the ion flux) of either divertor. For SOL powers of 50–140 MW, Fig. 1a plots the ion fluxes (ion saturation currents), normalised by a function of the power, versus an updated normalised pressure l defined by
l¼
0:165pn P0:34 #
for P# ¼ PSOL ðMWÞ=100
ð1Þ
Fig. 1a shows that this definition results in a very good superposition of the saturation currents at the maximum and well beyond for all powers. Scans of pumping speed, ratio of core to edge fuelling, and Pe/Pi show no explicit dependence on these quantities. The point l = 1 is chosen to correspond to the point for which the ion saturation current falls to 80% of its maximum for the inner divertor as the pressure is increased. The maximum ion saturation current is at the same point for both divertors but it rolls over more slowly for the outer divertor (Fig. 1a). The maximum temperature along the inner divertor plate is 1.5 eV for l = 1.5 for all these cases. A non-linear Monte-Carlo model for the neutral particle transport, which takes into account neutral–neutral and molecule–ion collisions with up-to-date molecular kinetics [4], is used in these simulations. With this non-linear neutral model, pn is no longer exactly proportional to the ratio of throughput to pumping speed as it was for older calculations with a linear neutral model; it is now modified by a weak function of power and l, reflecting a variation of the pressure drop from the PFR entrance to the pump duct with these quantities:
pn ¼
CDT
27Sn
P0:29 l0:27 #
ð2Þ
where CDT = CDT_puff + CDT_core is the DT throughput in Pam3/s and Sn = Seng/56 is the normalised engineering pumping speed at the pump duct entrance in m3/s (for which the maximum is 75). Combining Eqs. (1) and (2), l therefore becomes, in terms of the external input parameters,
l¼
0:79 CDT 0:79 0:5 P0:63 ¼ P# 164Sn # 164Sn
CDT
ð3Þ 2
Fig. 1b shows the peak power at the divertor plates [MW/m ] as a function of pn [Pa] for the power variation. For l < 1, it is well described by 0:6 qpk ¼ 6:66P1:5 # l
ð4Þ
(the values from the simulation and of the fit are shown for each point).
Fig. 2 shows that the variation of the neutral influx across the separatrix [Pam3/s] for both the power variation and the relative variation of core and edge fuelling (there is also a dependence on pumping speed, not shown), is well fitted by
CDT
sep
¼ 0:033ðCDT
puff
þ 0:25CDT
1:4 0:38 core ÞP # Sn
l0:2
ð5Þ
i.e. neutrals produced by recycling of particles resulting from core fuelling produce a much lower influx than the same number of particles resulting from gas puffing. This is a consequence of geometry (recycling mainly from the targets toward the private flux region versus puffing outboard) and temperature effects (colder neutrals from puffing, local temperature reduction in front of the gas puff leading to better penetration) and different relative importance of the various processes involved (transport through the SOL, recycling at the divertor, direct penetration from the outside) for the core fuelled and gas puffed particles. The details are not important because, as a result of the opacity to neutrals of the ITER SOL, the neutral influx across the separatrix is small, 3.3% of the gas puff plus 0.8% of the core fuelling for typical parameters as shown by Eq. (5). Fig. 3 shows the helium density [1020 m3] at the separatrix for the power and pumping speed variations; it is well fitted by:
nHe
sep
0:93 1:5 ¼ 0:0045f He P 0:6 l # Sn
ð6Þ
The numerical values in Eq. (6) and Fig. 3 correspond to a helium production rate consistent with Q = 5 and 30% core radiation. For other values they should be multiplied by fHe = 1.05 Pa/PSOL, where Pa is the total alpha particle power. (Other simulations, not presented here, for different He production rates showed that helium concentrations are sufficiently small to permit scaling the helium-related quantities linearly with production rate). As expected, the helium density varies almost inversely as the pumping speed. It falls off strongly with increasing normalised pressure l, and increases with SOL power as P0:6 # . Fig. 4a shows that the helium neutral flux [Pam3/s] across the separatrix for the power variation (pumping speed variation not shown) is fitted by
CHe
sep
1:07 1:45 ¼ 0:046f He P0:6 l # Sn
ð7Þ
The He flux varies almost inversely as the pumping speed and falls off strongly with increasing normalised pressure l, but decreases with SOL power as P 0:6 # . At higher pressures (corresponding to l greater than 1 to 2), the strong l dependence is lost. (The power variation is not apparent in the fit values in the figure because power and l variations practically compensate each other).
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H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
Fig. 2. Neutral DT flux across separatrix versus pn (a) for SOL powers from 50 to 140 MW, (b) for core fuelling from 17 to 100 Pam3/s, variable throughput (Eq. (2)). See Comment to Fig. 1.
Fig. 3. Helium density at separatrix versus pn (a) for SOL powers from 50 to 140 MW, (b) for normalized pumping speed Sn (defined after Eq. (2)) from 0.5 to 2. See Comment to Fig. 1.
Fig. 4. (a) Neutral He flux across separatrix, (b) DT density at separatrix, versus pn for SOL powers from 50 to 140 MW. See Comment to Fig. 1.
The average DT density [1020 m3] at the separatrix (Fig. 4b) saturates as pressure increases; its variation with l cannot easily be expressed by a simple function. It is fitted by:
nDT
sep
0:07 CDT puff þ 1:15CDT core ¼ 0:34P0:7 # Sn CDT ( ) 0:75 jlog10 ðlÞj2:5 10 for l < 1 l0:03 for l P 1
ð8Þ
where the l dependence should be viewed as an interpolation function only (as input to the core modelling). Core fuelling produces a somewhat higher density at the separatrix than gas puffing. There is practically no explicit throughput or pumping speed dependence, but there is a strong power dependence. The average electron and ion temperatures [eV] at the separatrix are well fitted by the relatively weak dependences
Te
sep
0:05 0:05 ¼ 163P0:33 l # ðP e =P i Þ
ð9Þ
H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
S495
Fig. 5. ITER operating diagram in the plane (a) heating power Paux [MW] versus volume averaged density [1020 m3] (b) and (c) Q versus alpha particle power Palpha [MW]. Contours shown are: (a) Palpha, (b) l (Eq. (3)), and (c) DT throughput. Limiting curves (see text) are Q = 5, LH transition, Palpha_max, edge limit l = 0.8, and Paux = 73; area outside limits is indicated in pink or white. For Sn = 1.06 and qpk 6 10 MW/m2.
Ti
sep
0:13 ¼ 280P0:31 # ðP e =P i Þ
CDT
puff
þ 1:08CDT
CDT
core
l0:09
ð10Þ
The average carbon ion density at the separatrix nC_sep [1020 m3] (the average charge state is found to be 5.5) has a maximum value near l = 0.5 of
nC
sep
0:12 ¼ 0:008P0:3 # Sn
ð11Þ
This value, taken to be constant with l, is used in the core simulations. In addition, the impurity radiation [MW] in the edge/divertor is well represented by: 0:23 Pimp ¼ 50P1:0 # l
ð12Þ
3. Operating diagram The scaling relations Eqs. (1)–(11) in Section 2 are input quantities to the core plasma simulations with the integrated model described in [8]. Since direct coupling of the core and edge regions is impractical because of the different timescales involved, in this model outer boundary conditions for the core are applied as obtained from SOL/divertor runs, and inner boundary conditions from the core simulations are applied to determine the state of the SOL/ divertor at the interface between the core-pedestal (1.5D code ASTRA), and SOL-divertor (2D code B2-Eirene) regions, thus assuring consistency of operating conditions in both regions (Section 2.2 of [8]). Inputs to the ASTRA simulations from the SOL/divertor scaling include separatrix DT, He, and C densities, separatrix ion and electron temperatures, and separatrix inward neutral DT and He fluxes. Outputs from ASTRA (inputs to the scaling relations) are the power transported across the separatrix by electrons and by ions, the fusion power, and the DT flux into the SOL. (The He ion flux is the sum of production by fusion and neutral influx). The control parameters for the core simulation are the core fuelling flux CDT_core, the gas puff flux into the vessel CDT_n_puff, and the additional heating power Paux. The separatrix density and neutral influx are also transmitted to Astra for any impurities included in the 2-D modelling. For the simulations presented here, this is the intrinsic carbon impurity self-consistently produced by plasma interaction with the divertor plates. The model of [8] calculates the edge pedestal width and height and has been shown there to be consistent with experimental scalings of pressure at the top of the pedestal. Following [8], we deter-
mine the operating window characterising ITER performance in the space Q-Palpha. The window is determined by five limits: (1) Q = 5, (2) the LH back transition, taken equal to the forward transition to provide a margin, (3) Palpha_max, the high-density low-temperature limit beyond which higher density no longer yields higher fusion power, (4) the edge pressure limit l = 0.8 representing detachment (Section 2) with a 20% margin replacing the Greenwald limit, and (5) the available external power Paux. (The beta limit is not attained here [8].) The edge limit was previously [8] taken to be l = 0.8, which was then identified to be fsat_n = 0.9 (l0.43, from the previous scaling of nDT_sep, [7,8]). This identification is no longer appropriate in view of the saturation of nDT_sep with l seen in Fig. 4b, Eq. (8). We therefore maintain the same edge limit l = 0.8 as before, maintaining the same 20% margin in pressure to detachment, but no longer identify it with fsat_n. The calculations are performed as a function of auxiliary power Paux, with core fuelling CDT_core adjusted to give the desired density at every point. The operating window corresponds to a qpk, which is given by the equations of Section 2 and adjusted to remain below a desired value, i.e. qpk 6 10 MW/m2, by additional gas puffing CDT_puff where required. The helium concentration is found to be low for the conditions studied, and therefore He is uncritical and has little effect on the operating window. These core plasma simulations have been carried out using Sn equal to 0.26, 0.53, and 1.06 – only 1.06 is shown. Fig. 5 shows the resulting operating diagram in the Paux hni plane (Fig. 5a) and the Q Pa plane (Fig. 5b and c). The same limitations of the operational window as in [8] are considered (see the explanation above); however, the Palpha_max lies outside the other limits for the case shown and does not play a role here. Fig. 5 shows that Q = 20 and Pa = 150 MW can be attained in ITER at qpk 6 10 MW/m2 without additional impurity seeding, at a throughput of CDT 6 180 Pam3/s, an engineering pumping speed Seng = 60 m3/s, and PSOL = 120 MW. Higher SOL powers than these (which might in any case be too high for the technical capabilities of the device) are more difficult to attain at qpk 6 10 MW/m2 in the simulations resulting from the present scaling than in the previous ones ([7,8]). Indeed, combining Eqs. (3) and (4), the maximum SOL power which can be attained for a limiting lmax (0.8 in Fig. 5) and qpk = 10 MW/m2 is
PSOL
1
max
1:5 ¼ 131l0:4 ¼ 100ð1:5l0:6 max Þ max
ð13Þ
In contrast, the former scaling of [7,8] can be rewritten for the present parameters as:
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H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
lformer ¼ qpk
former
CDT
246Sn
P0:87 ff0:8 #
ð14Þ
1:17 0:6 ¼ 4P1:26 # lformer ff
ð15Þ
(where ff0:8 is a correction of the order of 10%), which yields
PSOL
max former
¼ 100ð2:5l1:17 max
1 1:26
former Þ
¼ 207l0:93 max
former
ð16Þ
Since pumping speed Sn does not appear in Eqs. (13) and (16) (neither l in Eq. (1) nor qpk in Eq. (4) depend explicitly on it) and helium is uncritical (the He concentration is small), changing the pumping speed by a factor less than about four does not affect the operating window significantly. (Decreasing it by a factor of four or more would have an effect, because then the throughput from core fuelling leads to an increase of l by Eq. (3), i.e. the l = 0.8 limit is reached at lower density and alpha power). 4. Conclusions B2-EIRENE edge/divertor modelling of DT plasmas in the ITER reference geometry has been carried out. A robust definition of detachment based on the rollover of the ion saturation current in the simulations has been implemented for the first time and used to fit the key parameters of simulations in terms of the normalised pressure, SOL power, helium production, electron-ion power ratio, pumping speed, and core and edge fuelling. The fits then become inputs to core modelling and to estimates of fuelling requirements [9]. Significant differences to previous simulations [3–7] exist, particularly in the SOL power dependences, which had not previously been examined with the non-linear neutral model [10]. With this improved neutral model, the peak power load increases more rapidly with SOL power (power 1.5 rather than 1.26, Eqs. (4) and (15)) than previously. In addition, the peak power load falls off less strongly with neutral pressure (power 0.6 rather than 1.17, Eqs. (4) and (15)), as a consequence of the com-
bination of more flexible divertor geometry and equilibrium inherent in the current reference configuration. Indeed, examination of Figs. 1 and 3 of [1] would indicate that the transition from a steep to a more shallow neutral pressure dependence of the peak power load is neither due to low internal inductance li nor to a wider grid, but is correlated with the larger separatrix-dome distance required by shape and scenario flexibility. As a consequence of these differing dependencies, DT gas puffing is now less successful than in [8] in extending the operating window at constant peak power load to higher SOL powers (Eqs. (13) and (16)). Nevertheless, core plasma calculations with consistent core–edge conditions from the scalings derived here shows that an operating window with Q = 20 and Pa = 150 MW can be attained in ITER at qpk 6 10 MW/m2 without additional impurity seeding, at a throughput of CDT 6 180 Pam3/s, an engineering pumping speed Seng = 60 m3/s, and PSOL = 120 MW. The helium concentration remains uncritically low at about 3%. Future work should therefore concentrate on revisiting impurity seeding if extension of the operating window to larger fusion powers than above were required. References [1] A.S. Kukushkin, H.D. Pacher, A. Loarte, V. Kotov, D. Reiter, V. Komarov, M. Merola, G.W. Pacher, Nucl. Fusion 49 (2009) 075008 (7pp). [2] A.S. Kukushkin, H.D. Pacher, V. Kotov, D. Reiter, D.P. Coster, G.W. Pacher, J. Nucl. Mat. 363–365 (2007) 308–313. [3] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, G. Janeschitz, J. Nucl. Mat. 313–316C (2003) 657–663. [4] A.S. Kukushkin, H.D. Pacher, G.W. Pacher, G. Janeschitz, D.P. Coster, A. Loarte, D. Reiter, Nucl. Fusion 43 (2003) 716. [5] A.S. Kukushkin, H.D. Pacher, D.P. Coster, G.W. Pacher, D. Reiter, J. Nucl. Mat. 337–339 (2005) 50. [6] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, G. Janeschitz, D.P. Coster, V. Kotov, D. Reiter, J. Nucl. Mater. 363–365 (2007) 400. [7] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, V. Kotov, D. Reiter, D.P. Coster, G. Janeschitz, J. Nucl. Mater. 390–391 (2009) 259. [8] G.W. Pacher, H.D. Pacher, G. Janeschitz, A.S. Kukushkin, Nucl. Fusion 48 (2008) 105003 (26pp). [9] A.S. Kukushkin, A.R. Polevoi, H.D. Pacher, G.W. Pacher, R.A. Pitts, this conference. [10] D. Reiter, V. Kotov, P. Börner, et al., J. Nucl. Mater. 363–365 (2007) 649.
Contents lists available at ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Modelling of the ITER reference divertor plasma H.D. Pacher a,⇑,1, A.S. Kukushkin b, G.W. Pacher c, V. Kotov d, D. Reiter d a
INRS-EMT, Varennes, Québec, Canada ITER Organization, Cadarache, France c Hydro-Québec, Varennes, Québec, Canada d FZ Jülich, Jülich, Germany b
a r t i c l e
i n f o
Article history: Available online 16 November 2010
a b s t r a c t ITER divertor and edge modelling with the ITER B2-EIRENE code including neutral–neutral and molecule– ion collisions has been applied to the ITER reference divertor plasma with the reference low-li equilibrium. The scaling of key parameters such as peak power load, DT influx to the core, helium density and influx with SOL power, edge/core fuelling ratio, pumping speed and ratio electron/ion power ratio has been determined and ASTRA core plasma modelling consistent with these edge boundary conditions have been carried out to determine the operating window. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The ITER divertor geometry resulting from the 2007 ITER design review has a lower and slightly smaller dome than the previous geometry, permits a wider scrape-off layer, and increases flexibility by accommodating a wider range of magnetic equilibria. In initial modelling [1], the evolution of key parameters of DT plasma was studied as the configuration was changed in several steps from the previous geometry to the current reference (F57, see Fig. 1 of [1]). The B2-Eirene code package SOLPS4.3 (see [1]) used here employs a two-dimensional fluid description for the ions and electrons of the edge plasma, and a three-dimensional Monte-Carlo neutral treatment, taking into account neutral–neutral and molecule–ion collisions. The plasma consists of D (representing both D and T isotopes) ions, atoms, and molecules, as well as of He and C ions and atoms. Only carbon targets are studied. The particle absorption at the duct entrance in the PFR is adjusted to provide the specified ‘‘engineering’’ pumping speed [2]. For the standard case, the total power entering the SOL is 100 MW, the helium outflow from the core corresponds to 600 MW of fusion power, and this ratio is varied in the simulations. In previous work, scaling relations were established [3–7] to link the key parameters of the edge plasma resulting from the simulation to various input parameters including power, size, and pumping speed. Several of these variations, notably the SOL power variation, were carried out only with a less accurate linear neutral model and with the previous reference geometry. In the present paper, a detailed study ⇑ Corresponding author. Address: INRS-EMT, 1650 boul. Lionel Boulet, Varennes, QC, Canada J3X 1S2. Tel.: +1 450 929 8241; fax: +1 450 929 8102. E-mail address: [email protected] (H.D. Pacher). 1 Presenting author. 0022-3115/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2010.10.083
using a non-linear neutral model [10] and the current reference geometry is carried out to update the scaling by investigating the effect of varying the SOL power, the ratio Pe/Pi of this power carried by electrons and ions, the pumping speed, the throughput, and the ratio of core fuelling to gas puffing. Core modelling is carried out with the ASTRA code version containing the integrated core-pedestal (ICPS) model where consistency of the core and edge parameters is ensured by applying the scaling relations from the edge modelling as boundary conditions to the core model [8]. In previous work, an ITER operating window was defined and its variation with power, density, and pumping speed among others was determined with this model [8]. In this paper, the model is updated with the revised scaling relations and the operating window is determined. Several previous studies have not yet been updated with the current model and geometry, notably the size variation [6], the effect of carbon re-erosion [5] and the effect of neon seeding [7] so that these effects are not yet reflected in the scaling.
2. Edge/SOL simulation results and scaling of edge parameters In [3–7], the key point in scaling the edge parameters had been identified to be the full detachment of the inner divertor, then characterised by the drop of the peak electron temperature along the inner divertor to a low value of the order of several eV. This was found to be related to the average neutral pressure along the interface to the private flux region, pn; all the quantities were expressed as functions of normalised pressure l, proportional to pn and equal to 1 at detachment. The maximum inner divertor temperature is however not a robust characterisation; it is local and varies with the main plasma species. Instead, following experimental practice on many tokamaks, we now characterise detachment
S493
H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
Fig. 1. (a) Normalised ion saturation current on inner and outer divertor targets versus l (Eqs. (1) or (3) and text), (b) peak power load on divertor plate versus average neutral pressure along private flux region interface pn for SOL powers from 50 to 140 MW. Comment: solid symbols and bold lines indicate simulation results, open symbols and light lines scaled values from Eqs. (1)–(12).
as a condition on the rollover of the total ion saturation current (nearly equal to the ion flux) of either divertor. For SOL powers of 50–140 MW, Fig. 1a plots the ion fluxes (ion saturation currents), normalised by a function of the power, versus an updated normalised pressure l defined by
l¼
0:165pn P0:34 #
for P# ¼ PSOL ðMWÞ=100
ð1Þ
Fig. 1a shows that this definition results in a very good superposition of the saturation currents at the maximum and well beyond for all powers. Scans of pumping speed, ratio of core to edge fuelling, and Pe/Pi show no explicit dependence on these quantities. The point l = 1 is chosen to correspond to the point for which the ion saturation current falls to 80% of its maximum for the inner divertor as the pressure is increased. The maximum ion saturation current is at the same point for both divertors but it rolls over more slowly for the outer divertor (Fig. 1a). The maximum temperature along the inner divertor plate is 1.5 eV for l = 1.5 for all these cases. A non-linear Monte-Carlo model for the neutral particle transport, which takes into account neutral–neutral and molecule–ion collisions with up-to-date molecular kinetics [4], is used in these simulations. With this non-linear neutral model, pn is no longer exactly proportional to the ratio of throughput to pumping speed as it was for older calculations with a linear neutral model; it is now modified by a weak function of power and l, reflecting a variation of the pressure drop from the PFR entrance to the pump duct with these quantities:
pn ¼
CDT
27Sn
P0:29 l0:27 #
ð2Þ
where CDT = CDT_puff + CDT_core is the DT throughput in Pam3/s and Sn = Seng/56 is the normalised engineering pumping speed at the pump duct entrance in m3/s (for which the maximum is 75). Combining Eqs. (1) and (2), l therefore becomes, in terms of the external input parameters,
l¼
0:79 CDT 0:79 0:5 P0:63 ¼ P# 164Sn # 164Sn
CDT
ð3Þ 2
Fig. 1b shows the peak power at the divertor plates [MW/m ] as a function of pn [Pa] for the power variation. For l < 1, it is well described by 0:6 qpk ¼ 6:66P1:5 # l
ð4Þ
(the values from the simulation and of the fit are shown for each point).
Fig. 2 shows that the variation of the neutral influx across the separatrix [Pam3/s] for both the power variation and the relative variation of core and edge fuelling (there is also a dependence on pumping speed, not shown), is well fitted by
CDT
sep
¼ 0:033ðCDT
puff
þ 0:25CDT
1:4 0:38 core ÞP # Sn
l0:2
ð5Þ
i.e. neutrals produced by recycling of particles resulting from core fuelling produce a much lower influx than the same number of particles resulting from gas puffing. This is a consequence of geometry (recycling mainly from the targets toward the private flux region versus puffing outboard) and temperature effects (colder neutrals from puffing, local temperature reduction in front of the gas puff leading to better penetration) and different relative importance of the various processes involved (transport through the SOL, recycling at the divertor, direct penetration from the outside) for the core fuelled and gas puffed particles. The details are not important because, as a result of the opacity to neutrals of the ITER SOL, the neutral influx across the separatrix is small, 3.3% of the gas puff plus 0.8% of the core fuelling for typical parameters as shown by Eq. (5). Fig. 3 shows the helium density [1020 m3] at the separatrix for the power and pumping speed variations; it is well fitted by:
nHe
sep
0:93 1:5 ¼ 0:0045f He P 0:6 l # Sn
ð6Þ
The numerical values in Eq. (6) and Fig. 3 correspond to a helium production rate consistent with Q = 5 and 30% core radiation. For other values they should be multiplied by fHe = 1.05 Pa/PSOL, where Pa is the total alpha particle power. (Other simulations, not presented here, for different He production rates showed that helium concentrations are sufficiently small to permit scaling the helium-related quantities linearly with production rate). As expected, the helium density varies almost inversely as the pumping speed. It falls off strongly with increasing normalised pressure l, and increases with SOL power as P0:6 # . Fig. 4a shows that the helium neutral flux [Pam3/s] across the separatrix for the power variation (pumping speed variation not shown) is fitted by
CHe
sep
1:07 1:45 ¼ 0:046f He P0:6 l # Sn
ð7Þ
The He flux varies almost inversely as the pumping speed and falls off strongly with increasing normalised pressure l, but decreases with SOL power as P 0:6 # . At higher pressures (corresponding to l greater than 1 to 2), the strong l dependence is lost. (The power variation is not apparent in the fit values in the figure because power and l variations practically compensate each other).
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Fig. 2. Neutral DT flux across separatrix versus pn (a) for SOL powers from 50 to 140 MW, (b) for core fuelling from 17 to 100 Pam3/s, variable throughput (Eq. (2)). See Comment to Fig. 1.
Fig. 3. Helium density at separatrix versus pn (a) for SOL powers from 50 to 140 MW, (b) for normalized pumping speed Sn (defined after Eq. (2)) from 0.5 to 2. See Comment to Fig. 1.
Fig. 4. (a) Neutral He flux across separatrix, (b) DT density at separatrix, versus pn for SOL powers from 50 to 140 MW. See Comment to Fig. 1.
The average DT density [1020 m3] at the separatrix (Fig. 4b) saturates as pressure increases; its variation with l cannot easily be expressed by a simple function. It is fitted by:
nDT
sep
0:07 CDT puff þ 1:15CDT core ¼ 0:34P0:7 # Sn CDT ( ) 0:75 jlog10 ðlÞj2:5 10 for l < 1 l0:03 for l P 1
ð8Þ
where the l dependence should be viewed as an interpolation function only (as input to the core modelling). Core fuelling produces a somewhat higher density at the separatrix than gas puffing. There is practically no explicit throughput or pumping speed dependence, but there is a strong power dependence. The average electron and ion temperatures [eV] at the separatrix are well fitted by the relatively weak dependences
Te
sep
0:05 0:05 ¼ 163P0:33 l # ðP e =P i Þ
ð9Þ
H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
S495
Fig. 5. ITER operating diagram in the plane (a) heating power Paux [MW] versus volume averaged density [1020 m3] (b) and (c) Q versus alpha particle power Palpha [MW]. Contours shown are: (a) Palpha, (b) l (Eq. (3)), and (c) DT throughput. Limiting curves (see text) are Q = 5, LH transition, Palpha_max, edge limit l = 0.8, and Paux = 73; area outside limits is indicated in pink or white. For Sn = 1.06 and qpk 6 10 MW/m2.
Ti
sep
0:13 ¼ 280P0:31 # ðP e =P i Þ
CDT
puff
þ 1:08CDT
CDT
core
l0:09
ð10Þ
The average carbon ion density at the separatrix nC_sep [1020 m3] (the average charge state is found to be 5.5) has a maximum value near l = 0.5 of
nC
sep
0:12 ¼ 0:008P0:3 # Sn
ð11Þ
This value, taken to be constant with l, is used in the core simulations. In addition, the impurity radiation [MW] in the edge/divertor is well represented by: 0:23 Pimp ¼ 50P1:0 # l
ð12Þ
3. Operating diagram The scaling relations Eqs. (1)–(11) in Section 2 are input quantities to the core plasma simulations with the integrated model described in [8]. Since direct coupling of the core and edge regions is impractical because of the different timescales involved, in this model outer boundary conditions for the core are applied as obtained from SOL/divertor runs, and inner boundary conditions from the core simulations are applied to determine the state of the SOL/ divertor at the interface between the core-pedestal (1.5D code ASTRA), and SOL-divertor (2D code B2-Eirene) regions, thus assuring consistency of operating conditions in both regions (Section 2.2 of [8]). Inputs to the ASTRA simulations from the SOL/divertor scaling include separatrix DT, He, and C densities, separatrix ion and electron temperatures, and separatrix inward neutral DT and He fluxes. Outputs from ASTRA (inputs to the scaling relations) are the power transported across the separatrix by electrons and by ions, the fusion power, and the DT flux into the SOL. (The He ion flux is the sum of production by fusion and neutral influx). The control parameters for the core simulation are the core fuelling flux CDT_core, the gas puff flux into the vessel CDT_n_puff, and the additional heating power Paux. The separatrix density and neutral influx are also transmitted to Astra for any impurities included in the 2-D modelling. For the simulations presented here, this is the intrinsic carbon impurity self-consistently produced by plasma interaction with the divertor plates. The model of [8] calculates the edge pedestal width and height and has been shown there to be consistent with experimental scalings of pressure at the top of the pedestal. Following [8], we deter-
mine the operating window characterising ITER performance in the space Q-Palpha. The window is determined by five limits: (1) Q = 5, (2) the LH back transition, taken equal to the forward transition to provide a margin, (3) Palpha_max, the high-density low-temperature limit beyond which higher density no longer yields higher fusion power, (4) the edge pressure limit l = 0.8 representing detachment (Section 2) with a 20% margin replacing the Greenwald limit, and (5) the available external power Paux. (The beta limit is not attained here [8].) The edge limit was previously [8] taken to be l = 0.8, which was then identified to be fsat_n = 0.9 (l0.43, from the previous scaling of nDT_sep, [7,8]). This identification is no longer appropriate in view of the saturation of nDT_sep with l seen in Fig. 4b, Eq. (8). We therefore maintain the same edge limit l = 0.8 as before, maintaining the same 20% margin in pressure to detachment, but no longer identify it with fsat_n. The calculations are performed as a function of auxiliary power Paux, with core fuelling CDT_core adjusted to give the desired density at every point. The operating window corresponds to a qpk, which is given by the equations of Section 2 and adjusted to remain below a desired value, i.e. qpk 6 10 MW/m2, by additional gas puffing CDT_puff where required. The helium concentration is found to be low for the conditions studied, and therefore He is uncritical and has little effect on the operating window. These core plasma simulations have been carried out using Sn equal to 0.26, 0.53, and 1.06 – only 1.06 is shown. Fig. 5 shows the resulting operating diagram in the Paux hni plane (Fig. 5a) and the Q Pa plane (Fig. 5b and c). The same limitations of the operational window as in [8] are considered (see the explanation above); however, the Palpha_max lies outside the other limits for the case shown and does not play a role here. Fig. 5 shows that Q = 20 and Pa = 150 MW can be attained in ITER at qpk 6 10 MW/m2 without additional impurity seeding, at a throughput of CDT 6 180 Pam3/s, an engineering pumping speed Seng = 60 m3/s, and PSOL = 120 MW. Higher SOL powers than these (which might in any case be too high for the technical capabilities of the device) are more difficult to attain at qpk 6 10 MW/m2 in the simulations resulting from the present scaling than in the previous ones ([7,8]). Indeed, combining Eqs. (3) and (4), the maximum SOL power which can be attained for a limiting lmax (0.8 in Fig. 5) and qpk = 10 MW/m2 is
PSOL
1
max
1:5 ¼ 131l0:4 ¼ 100ð1:5l0:6 max Þ max
ð13Þ
In contrast, the former scaling of [7,8] can be rewritten for the present parameters as:
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H.D. Pacher et al. / Journal of Nuclear Materials 415 (2011) S492–S496
lformer ¼ qpk
former
CDT
246Sn
P0:87 ff0:8 #
ð14Þ
1:17 0:6 ¼ 4P1:26 # lformer ff
ð15Þ
(where ff0:8 is a correction of the order of 10%), which yields
PSOL
max former
¼ 100ð2:5l1:17 max
1 1:26
former Þ
¼ 207l0:93 max
former
ð16Þ
Since pumping speed Sn does not appear in Eqs. (13) and (16) (neither l in Eq. (1) nor qpk in Eq. (4) depend explicitly on it) and helium is uncritical (the He concentration is small), changing the pumping speed by a factor less than about four does not affect the operating window significantly. (Decreasing it by a factor of four or more would have an effect, because then the throughput from core fuelling leads to an increase of l by Eq. (3), i.e. the l = 0.8 limit is reached at lower density and alpha power). 4. Conclusions B2-EIRENE edge/divertor modelling of DT plasmas in the ITER reference geometry has been carried out. A robust definition of detachment based on the rollover of the ion saturation current in the simulations has been implemented for the first time and used to fit the key parameters of simulations in terms of the normalised pressure, SOL power, helium production, electron-ion power ratio, pumping speed, and core and edge fuelling. The fits then become inputs to core modelling and to estimates of fuelling requirements [9]. Significant differences to previous simulations [3–7] exist, particularly in the SOL power dependences, which had not previously been examined with the non-linear neutral model [10]. With this improved neutral model, the peak power load increases more rapidly with SOL power (power 1.5 rather than 1.26, Eqs. (4) and (15)) than previously. In addition, the peak power load falls off less strongly with neutral pressure (power 0.6 rather than 1.17, Eqs. (4) and (15)), as a consequence of the com-
bination of more flexible divertor geometry and equilibrium inherent in the current reference configuration. Indeed, examination of Figs. 1 and 3 of [1] would indicate that the transition from a steep to a more shallow neutral pressure dependence of the peak power load is neither due to low internal inductance li nor to a wider grid, but is correlated with the larger separatrix-dome distance required by shape and scenario flexibility. As a consequence of these differing dependencies, DT gas puffing is now less successful than in [8] in extending the operating window at constant peak power load to higher SOL powers (Eqs. (13) and (16)). Nevertheless, core plasma calculations with consistent core–edge conditions from the scalings derived here shows that an operating window with Q = 20 and Pa = 150 MW can be attained in ITER at qpk 6 10 MW/m2 without additional impurity seeding, at a throughput of CDT 6 180 Pam3/s, an engineering pumping speed Seng = 60 m3/s, and PSOL = 120 MW. The helium concentration remains uncritically low at about 3%. Future work should therefore concentrate on revisiting impurity seeding if extension of the operating window to larger fusion powers than above were required. References [1] A.S. Kukushkin, H.D. Pacher, A. Loarte, V. Kotov, D. Reiter, V. Komarov, M. Merola, G.W. Pacher, Nucl. Fusion 49 (2009) 075008 (7pp). [2] A.S. Kukushkin, H.D. Pacher, V. Kotov, D. Reiter, D.P. Coster, G.W. Pacher, J. Nucl. Mat. 363–365 (2007) 308–313. [3] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, G. Janeschitz, J. Nucl. Mat. 313–316C (2003) 657–663. [4] A.S. Kukushkin, H.D. Pacher, G.W. Pacher, G. Janeschitz, D.P. Coster, A. Loarte, D. Reiter, Nucl. Fusion 43 (2003) 716. [5] A.S. Kukushkin, H.D. Pacher, D.P. Coster, G.W. Pacher, D. Reiter, J. Nucl. Mat. 337–339 (2005) 50. [6] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, G. Janeschitz, D.P. Coster, V. Kotov, D. Reiter, J. Nucl. Mater. 363–365 (2007) 400. [7] H.D. Pacher, A.S. Kukushkin, G.W. Pacher, V. Kotov, D. Reiter, D.P. Coster, G. Janeschitz, J. Nucl. Mater. 390–391 (2009) 259. [8] G.W. Pacher, H.D. Pacher, G. Janeschitz, A.S. Kukushkin, Nucl. Fusion 48 (2008) 105003 (26pp). [9] A.S. Kukushkin, A.R. Polevoi, H.D. Pacher, G.W. Pacher, R.A. Pitts, this conference. [10] D. Reiter, V. Kotov, P. Börner, et al., J. Nucl. Mater. 363–365 (2007) 649.