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Global modelling of plasma-wall interaction in reversed field pinches
818
Journal of Nuclear Materials 162-164 (1989) 818-824 North-Holland, Amsterdam
GLOBAL MODELLING OF PLASMA-WALL IN REVERSED FIELD PINCHES M. BAGATIN, Istituto Gas Ionizrati
INTERACTION
S. COSTA and S. ORTOLANI de1 C. N. R., Euratom - ENEA-
CNR Association
V. Gradenigo 6/a,
35131 Padova, Italy
Key words: reversed field pinch, impurity production, recycling The impurity production and deuterium recycling mechanisms in ETA-BETA II and RFX are firstly discussed by means of a simple model applicable to a stationary plasma interacting with the wall. This gives the time constant and the saturation values of the impurity concentration as a function of the boundary temperature and density. If the latter is sufficiently high, the impurity buildup in the main plasma becomes to some extent stabilized by the shielding effect of the edge. A self-consistent global model of the time evolution of an RFP plasma interacting with the wall is then described. The bulk and edge parameters are derived by solving the energy and particle balance equations incorporating some of the basic plasma-surface processes, such as sputtering, backscattering and desorption. The application of the model to ETA-BETA II confirms the impurity concentrations of the light and metal impurities as well as the time evolution of the average electron density found experimentally under different conditions. The model is then applied to RFX, a larger RFP experiment under construction, whose wall will be protected by a full graphite armour. The time evolution of the discharge shows that carbon sputtering could increase Z,,, to - 4, but without affecting significantly the plasma performance.
1. Introduction
2. Experimentalresults of ETA-BETA
In the reversed field pinch (RFP) the plasma is ohmically heated and confined in a low toroidal field system. In particular the magnetic field lines in the outer region are almost poloidal with q < 1 [typically
Operation of ETA-BETA II shows, as in other RFP devices, a fast decrease of the electron density after the initial peak corresponding to the ionization of the filling gas. The electron density n, decays rapidly to an approximately steady state value of the order of some lOI m-3, corresponding also to a nearly constant value during the toroidal current sustainement [3]. From recent thermocouple measurements of the heat load on a small limiter graphite tile inserted radially at the plasma edge a decay length X, = 1.5 cm, approximately constant over a wide range of variation of the plasma density, has been determined. When combined with T,, 4 10 eV and X, of a few cm, obtained by Langmuir probes, one finds for the radial e-folding length of the edge density h, = 2 cm, which is consistent with D, = 50 m2/s [5] and with a connection length L = ~a. The typical impurity concentrations are lOI* m-s oxygen and 10” mm3 iron, corresponding to Z,rr - 2.5. Density control could not be efficiently achieved by gas puffing, because of the too low particle confinement time. The most suitable method has proven to be fueling by pellet injection which gives a sustainment of the
qa=(&ik) a/RI. Optimization of stability and confinement also requires operating at high density (about 10” mw3) and with close proximity of a conducting boundary, thus preventing the use of limiters protruding too much into the plasma. Most RFPs have operated in the past with metallic limiters and no limiters although more recently OHTE has operated with a full graphite armour [l] and HBTX1C has experimented various limiter geometries [2]. ETA-BETA II (a = 0.125 m, R = 0.65 m, I < 200 kA) has a stainless steel vacuum liner with no limiters. In this paper we describe: _ the main features of a O-dimensional plasma-wall interaction model; - its application to high and low density discharges obtained in ETA-BETA II; _ the expected performance of RFX (a = 0.5 m, R = 2 m, Z -C 2 MA) operating with a full graphite armour.
0022-3115/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
II
M. Bagatin et al. / Global modeling of plasma-wall interaction
density, at ne= lo*’ rne3, with an oxygen density comparable to that of discharges without pellet, but with negligible iron ~nt~Mtion [4]. Both the density behaviour and the impurity infl~ may be explained as an effect of the interaction of the plasma with the wall, which acts essentially as a toroidal limiter when the plasma is shifted from the geometric minor axis. Indeed, a remarkable decrease of the pumpout effect is observed when the toroidal ~u~b~urn is improved by a vertical field [3]. The ETA-BETA II discharges have been simulated by a global model treating self-consistently the time evolution of the bulk plasma, with mean electron density and temperature n, and T,, interacting with the wall through a boundary layer characterized by edge parameters Tb (we assume Tbi = T& and n b.
3. Buildup of impurities in a stationary plasma The temporal evolution of the average density of an impurity species, n imp, can be described by the equation:
-?(l
-a),
where rP and 7imp are the plasma ion and impurity confinement times; S and &,t, are the impurity yields due to the incident deuterium or impurity atom, respectively; SC is the chemical sputtering yield, which is important for graphite and p is the particle reflection coefficient. S and SimP represent sputtering or desorption yields, according to the dominant mechanism of impurity production. The factor l/[l- (1 - n)St,,,r] accounts for self sputtering of the recycled impurity ions. The factor n describes the ability of the impurity neutrals to penetrate the plasma. It is defined as the ratio between the number of atoms going into the main plasma to the total emitted by the wall. In eq. (1) we neglect the shielding effect on the backscattered neutrals, since their typical energies are much greater than those of sputtered/desorbed neutrals, at least in the energy range in which shielding becomes important. Major uncertainties exist in the evaluation of n, since the behaviour of the impurity atoms released by the wall is not clear. Certainly the value of u is strongly
819
dependent on the plasma edge parameters and the complexity of the problem would require the use of impurity transport codes to evaluate n in a self-consistent manner. However for the present global model, considering also the uncertainties in the sputtering and reflection coefficients, we can accept the approximation given by a simplified picture assuming the existence of magnetic surfaces in the plasma edge. If this is the case a neutral atom will be “screened” when it is ionized in the SOL, where parallel transport along field hues dominates radial transport. Therefore we use the following approximate expression for n [6]: 71= l - Xn/Xio*r
(2)
where Ai, = uO/( n,,So) is the ionization mean free path of the neutral impurity, which depends on the average radial velocity u. and ionization rate So. The ionization rates are calculated according to Lo&s data for carbon and oxygen, to McWhirter (divided by a factor of 5 to make them compatible with those for oxygen and carbon) for iron. Useful indications on the impurity inventory can be drawn by integrating eq. (1) for a stationary plasma, i.e. with Tb and ni constant. If the impurity yields and the backscattering coefficients are thought of as simple functions of Tb only, the impurity content expressed in terms of the fractional concentration cy= n i,/ni increases monotonically with time as: a(t)=aL-(aL-ao)exp(-t/T*),
(3)
where the limiting value aL is given by: 2” iXr.=
S + AS,
1 l-P-VSimp [ 1-(1--n)
Sm,
and the time constant is proportional particle ~nfinement time:
(4) to the impurity
The dimensionless quantities CQ,and r * could then be plotted as a function of Tb, once the values of (D,L) and nt, have been fried The cases of iron and oxygen impurities have beeu considered, as relevant for ETA-BETA II and the case of carbon for RFX. In order to evaluate the sputtering yield of stainless steel, we assume that the interaction with the wall occurs essentially through both charge-exchange and
820
M. Bagatin et al. / Global modeling of plasma-wall interaction
n
15
20 Tb (ev )
5
10
b
,j01gm-3
20
15 T&d
Fig. 1. Iron impurity production by neutral deuterium sputtering for ETA-BETA II as a function of the edge temperature r,, and density n,,. (a) shows the asymptotic value of the impurity fractional concentration (Y,_, (b) the time constant of its accumulation 7* in terms of the impurity confinement time TV,,,,,.71mp/~p = 2 and (D, L) = 20 rn’ s-’ are assumed.
recombination neutrals, as confirmed also in other experiments [7]. In fact, in ETA-BETA II the strong recycling with R - 0.9 [lo], gives a neutral deuterium influx from the wall F, = (n,/~,)a/2 = 10z3 m-‘s-l, if or is some tens of ps during the density plateau. With an energy of the incoming neutrals of the order of some eV, this gives no = Fe/u, = 1019 rnm3, i.e. a value comparable with nb and sufficient to “screen” the wall from ions. The physical sputtering and self-sputtering yields for iron are evaluated by averaging the empirical functions given by ref. [8] over an isotropic and Maxwellian distribution of neutral energies around Tb. The same is done for the value of p, whose energy and angular dependences are taken from ref. [9]. A value of u0 corresponding to 2 eV has been assumed both for sputtered and desorbed particles. In fig. la the obtained asymptotic value of the impurity concentration is reported for three different values of nb. At the lowest edge density, when no screening is present, an unlimited impurity buildup occurs at low T,, whereas for nb = lOi rnp3 some protection of the central plasma column from impurity penetration should occur. To some extent, this fact has been experimentally confirmed on ETA-BETA II by the observation of hollow Z,,, profiles [lo]. The characteristic time for impurity buildup, as can be seen from fig. lb and the corresponding ones for
oxygen and carbon, is many times the particle confinement time and therefore of the same order as the discharge duration, at least in the quasi steady state Oxygen is desorbed and self-desorbed mainly by ion induced processes. Great uncertainty exists in the desorption cross sections, especially at low energy. Their absolute values and their energy dependences are assumed according to the experimental and theoretical review given in ref. [ll], i.e. increasing with increasing energy until values at 100 eV of 0.2 x lo-*’ rn-* and 5 x lo-*’ me2 for deuterium induced and oxygen induced desorption respectively. For an oxygen surface coverage of 2 X 10” m- * the corresponding yields are respectively 0.4 X lo-* and 10-l. The main differences between the iron and oxygen asymptotic contents, shown in fig. 2a, are the absence of vertical asymptotes, indicating a less sensitive dependence of (Ye on Tbr according also to current experimental observations in tokamaks [12]. Light impurities contamination observed also in the other present RFP devices are compatible with the values of (or from fig. 2a at Tb = 10 eV. Fig. 2b shows that the impurity buildup should occur mainly at the beginning of the discharge, when the confinement times are shorter. Carbon impurity has been examined with reference to the RFX experiment, whose first wall will be covered by a full graphite armour. A chemical sputtering coefficient dependence on the wall temperature T, according
821
M. Bagatin et al. / Global modeling of plasma-wall interaction
15
-a
2. Oxygen production by desorption from stainless steel for ETA-BETA II as a function of the edge temperature r,, and density n ,,. The meanings of a,_ and r */rimi, are the same as in fig. 1. ( rimP/rP) = 2 and (D, L) = 20 m3 s-l are assumed.
evaluation [13] has been used; from the same results an extrapolation to the low energy range has been made ( < 50 ev). For the chemical sputtered carbon an energy of 0.5 eV has been assumed. The parameters for carbon buildup, corresponding to T, = 850 K, which maximizes chemical sputtering, are shown in fig. 3. The relative concentration (also at low T,) easily reaches values of the order of some 10%. At greater Tb a high neutral carbon density at the edge should be expected for boundary electron density greater than 10” rne3, which on the other hand should improve the to recent experimental
a 40
50 Ib (oV)
self-protection of either the plasma from contamination and the wall from erosion.
4. Time dependent calculations and discussion The previous considerations apply to a stationary plasma. Furthermore, effects of the impurity contamination on Tb, as well as the effects on the main plasma density of the surface interaction processes are not self-consistently considered. In the present section we briefly outline a global model described elsewhere [14] of energy and particle balance for the main plasma.
‘~“““‘l”“““““’ 10
b 20
30
40
50 &,(eV)
Fig. 3. Carbon wntamina tion for the KFX plasma as a function of the boundary temperature r, and density nb. q, r* and ~~,,,r have the same meanings as in figs. 1 and 2. Values of ( T~,,,~/T,,)= 2 and (D, L) = 7.5 m3 s-’ are assumed. The chemical sputtering yield is evaluated at its maximum, i.e. at a graphite temperature of about 850 K.
M. Bagaiin et al. / Global modeling of plasma-wall interaction
822
The processes considered are ohmic heating and power losses due to bremsstrahlung, ionization and line radiation of impu~ties and transport. The transport loss, due to heat conduction and particle convection is expressed by means of a confinement time or proportional to beta poloidal &. The time evolution of &, as well as that of the plasma current, are taken directly from the experimental results. The light imp~ties, i.e. oxygen and carbon, are treated by a time dependent corona model, whereas iron losses are calculated by a stationary average ion model adding a recycling term. All the recycled impurities are assumed to return to the bulk plasma as neutrals; their concentrations are calculated by eq. (I) referred to neutrals. A simple relation of proportionality is assumed between the edge and bulk parameters. In the present model, the value of n b/rre, chosen equal to 0.2 has some influence only on the simulations of RFX, because of the low values of Tb, as discussed in section 3. If ail the transport loss were associated with particle convection, the boundary temperature should be approximately 1151:
with y of the order of 10 or more. For ETA-BETA
II with rr/rr 2 1, realistic values of Tb can be obtained only for y > 30. This value is in agreement with the experimental results, i.e. with the ratio between the heat load on the limiter tiles and the power which would be dumped by the ion saturation current with T,, < 10 eV. Possible reasons for this enhancement might be both the creation of a SOL dominated by radiation and charge exchange and the presence of run-away electrons or suprathermal ions. The simulations are obviously very sensitive to the global deuterium recycling coefficient R, which determines the global ion and neutral particle balances, and has been expressed by the sum of three contributions: R=P+
Y,+
YR,
desorption, even if locally important. should be globally negligible. The temperature of the wall, T,. is evaluated in the infinite half-space approximation, by using an intensification factor F, of the transport loss (and of particle loss). With F, = 4 and the initial temperature T, = 300 K, T, > 500 K is reached very early in the discharge, remaining thereafter more or less constant. We can assume therefore that the molecular recombination rate also remains constant. If the hydrogen filling density (about 3 x 10” m- ‘) is considered to be uniformly distributed on the wall, it would saturate the stainless steel for several tens of monolayers at 1% atomic fraction concentration. Since the wall may be thought of as being saturated by deuterium at all times, the men&ted flux due to the recombination F, can be considered constant and Y, is given by: Y, = I-arp/n,.
(8)
that is, it is negligible when the wall is subject to the strongest ion fluxes ( rp very low). Although there are essentially no experimental data on ion induced deuterium desorption at very low energies, theoretical arguments seem to indicate that the desorption cross section should remain relatively constant [12]. We have assumed in our model a value of Yr constant for Th greater than a threshold of 1 eV. The experimental values of electron density are well simulated, as shown in figs. 4 to 7 with F, = 5 x 102’ m‘-Z s-1 and Y, = 0.1, which appear in the range of commonly expected values. The absolute values of
n e (1020m‘3 1
i Te(eV)
I
3
(7)
where Yt and Y, are the desorption yields respectively due to ion induced desorption and to thermally activated molecular recomb~ation. Neutral atom thermal desorption is neglected, since the deuterium atom binding energy is about 1 eV, thus requiring several hundred Kelvin to become significant. Indeed, tile insertion perpendicularly to the field lines have shown that such a temperature could be reached only in limited areas. The associated thermal atom
Fig. 4. Measured electron temperature T, (Cl)and the density ne (0) of the bulk plasma compared with simulations in a discharge in normal conditions.
M. Bagatin et al. / Global modeling of plasma-wall interaction
823
(X10)
aFe
1.
TIME(ms)
n 1:5-
Fig. 5. Time evolution of the oxygen and iron impurity fractional concentrations a,, and are and global recycling coefficient of deuterium R, referred to the discharge of fig. 4.
Fig. 7. The evolution of the oxygen and iron impurity fractional concentrations aox and are and global recycling coefficient of deuterium R, referred to the discharge of fig. 6.
oxygen concentration result in both cases about the same, with fractional contents of 5% and 1.5%, acoording to fig. 2a at T, = 10 eV and 4 eV, respectively. The presence of iron should result negligible in the discharge with pellet injection, due to the lower boundary temperature given by the model. The deuterium recycling due to surface recombination is less important in this case, because of the much greater value of n/7,,. The final values of Z,, ( - 2.3 and - 1.2) are in good agreement with the experimental data, while the time evolution of wall surface temperature appears quite similar in both cases.
Considering carbon sputtering, modelled as discussed in section 3, the model has also been applied to verify the RFX expected performance. For deuterium desorption from graphite a simple scheme, which can be roughly considered as a global saturation model, has been applied. The deuterium saturation inventory is thought of as being given by rectangular depth profiles, with depth empirically increasing with Tb and with height (concentration) decreasing with T,, according to ref. [16]. Therefore a maximum fraction of the implanting deuterium flux can be accommodated, whereas the excess is recycled. .5
Te(eV)
a, .4
200
.3 .2
100
.I
0 0
i 0 .5
1.
TIME(ms)
1.5
Fig. 6. Measured electron temperature r, (0) and density ne (0) of the bulk plasma compared with simulations in a discharge with pellet injection.
50
100
150
200 TIMEtms)
Fig. 8. Example of the simulation of the time evolution of the carbon impurity fractional content a, and of the wall temperature T,. An intensification factor equal to 4 has been assumed for the power dumped to the wall. The behaviour of the electron density ne assumes that the graphite is initially clean.
824
M. Bagatin et al. / Global modeling ofplasma-wall The
with
results
respect
reduction graphite
in
are shown to the
to work
the
in fig. 8. The
stainless
“pump-out” as a deutetium
steel due
main
wall to
store
difference
is the the
strong
capacity
of
in the beginning
and as a source when it is saturated. The behaviour of n, refers to an initially “clean” graphite surface. After a few discharges it should be almost saturated by deuterium, so the foreseen n, - 10” me3 could be reached by much lower filling densities, because the recycling coefficient should approach values about 1 since the beginning of the discharge. The expected performances are essentially confirmed [17], with T, reaching - 1 keV and with a strong carbon contamination, leading to Z,,, - 4. As can be seen from fig. 3, its concentration should be nearly independent of T,,. If the latter is of the order of some tens of eV, physical sputtering and self-sputtering should be more important than the chemical sputtering.
References [l] G.L. Jackson [2] A.A. Newton [3] V. Antoni et trolled Fusion p. 532. [4] V. Antoni et trolled Fusion 1988, p. 553.
et al., J. Nucl. Mater. 145-147 (1987) et al., J. Nucl. Mater. 145-147 (1987) al., in: Proc. 14th Europ. Conf. on and Plasma Physics, Vol. II, Madrid,
470. 487. Con1987,
al., in: Proc. 15th Europ. Conf. on Conand Plasma Physics, Vol. II, Dubrovnik,
interaction
]51 V. Antoni et al., in: Proc. 13th Europ. Conf. on Controlled Fusion and Plasma Physics, Vol. I, Schliersee, 1986, p. 337. [61K.F. Alexander et al., Nucl. Fusion 26 (1986) 1575. [71 D.N. Ruzic et al., J. Nucl. Mater. 145-147 (1987) 527. et al., in: Proc. 10th Symp. on Fusion PI J. Bohdansky Technology, Padova, 1978, p. 801. for [91 W. Eckstein and H. Veerbek. in: Data Compendium Plasma-Surface Interactions, Nucl. Fusion Special issue (1984). [lOI V. Antoni et al., Proc. 11th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Kyoto. 1986, paper CN-44/D-11-5. in: Data Compendium for Plasma Surface 1111 E. Taglauer, Interactions, Nucl. Fusion Special Issue (1984). WI K.H. Behringer, J. Nucl. Mater. 145-147 (1987) 145. et al., in: Proc. 14th Europ. Conf. on 1131 J. Bohdansky Controlled Fusion and Plasma Physics, Vol. II, Madrid, 1987. p. 794. in: Int. School of Plasma P41 M. Bagatin and S. Ortolani, Physics, Varenna 1987, Proc. of the Course on Physics of Mirrors, Reversed Field Pinches and Compact Tori, Commission of the European Communities EUR 11335 EN, Vol. III, p. 1079. I151 J. Hugill, Nucl. Fusion 23 (1983) 331. WI W. Moeller and J. Roth, in: Physics of Plasma-Wall Interactions in Controlled Fusion, Eds. D.E. Post, R. Behrisch (Plenum Press, New York, 1986) NATO AS1 Series B: Physics Vol. 131. u71 V. Antoni et al., in: Proc. 13th Europ. Conf. on Controlled Fusion and Plasma Physics, Vol. I, Schliersee, 1986. p. 385.
Journal of Nuclear Materials 162-164 (1989) 818-824 North-Holland, Amsterdam
GLOBAL MODELLING OF PLASMA-WALL IN REVERSED FIELD PINCHES M. BAGATIN, Istituto Gas Ionizrati
INTERACTION
S. COSTA and S. ORTOLANI de1 C. N. R., Euratom - ENEA-
CNR Association
V. Gradenigo 6/a,
35131 Padova, Italy
Key words: reversed field pinch, impurity production, recycling The impurity production and deuterium recycling mechanisms in ETA-BETA II and RFX are firstly discussed by means of a simple model applicable to a stationary plasma interacting with the wall. This gives the time constant and the saturation values of the impurity concentration as a function of the boundary temperature and density. If the latter is sufficiently high, the impurity buildup in the main plasma becomes to some extent stabilized by the shielding effect of the edge. A self-consistent global model of the time evolution of an RFP plasma interacting with the wall is then described. The bulk and edge parameters are derived by solving the energy and particle balance equations incorporating some of the basic plasma-surface processes, such as sputtering, backscattering and desorption. The application of the model to ETA-BETA II confirms the impurity concentrations of the light and metal impurities as well as the time evolution of the average electron density found experimentally under different conditions. The model is then applied to RFX, a larger RFP experiment under construction, whose wall will be protected by a full graphite armour. The time evolution of the discharge shows that carbon sputtering could increase Z,,, to - 4, but without affecting significantly the plasma performance.
1. Introduction
2. Experimentalresults of ETA-BETA
In the reversed field pinch (RFP) the plasma is ohmically heated and confined in a low toroidal field system. In particular the magnetic field lines in the outer region are almost poloidal with q < 1 [typically
Operation of ETA-BETA II shows, as in other RFP devices, a fast decrease of the electron density after the initial peak corresponding to the ionization of the filling gas. The electron density n, decays rapidly to an approximately steady state value of the order of some lOI m-3, corresponding also to a nearly constant value during the toroidal current sustainement [3]. From recent thermocouple measurements of the heat load on a small limiter graphite tile inserted radially at the plasma edge a decay length X, = 1.5 cm, approximately constant over a wide range of variation of the plasma density, has been determined. When combined with T,, 4 10 eV and X, of a few cm, obtained by Langmuir probes, one finds for the radial e-folding length of the edge density h, = 2 cm, which is consistent with D, = 50 m2/s [5] and with a connection length L = ~a. The typical impurity concentrations are lOI* m-s oxygen and 10” mm3 iron, corresponding to Z,rr - 2.5. Density control could not be efficiently achieved by gas puffing, because of the too low particle confinement time. The most suitable method has proven to be fueling by pellet injection which gives a sustainment of the
qa=(&ik) a/RI. Optimization of stability and confinement also requires operating at high density (about 10” mw3) and with close proximity of a conducting boundary, thus preventing the use of limiters protruding too much into the plasma. Most RFPs have operated in the past with metallic limiters and no limiters although more recently OHTE has operated with a full graphite armour [l] and HBTX1C has experimented various limiter geometries [2]. ETA-BETA II (a = 0.125 m, R = 0.65 m, I < 200 kA) has a stainless steel vacuum liner with no limiters. In this paper we describe: _ the main features of a O-dimensional plasma-wall interaction model; - its application to high and low density discharges obtained in ETA-BETA II; _ the expected performance of RFX (a = 0.5 m, R = 2 m, Z -C 2 MA) operating with a full graphite armour.
0022-3115/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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M. Bagatin et al. / Global modeling of plasma-wall interaction
density, at ne= lo*’ rne3, with an oxygen density comparable to that of discharges without pellet, but with negligible iron ~nt~Mtion [4]. Both the density behaviour and the impurity infl~ may be explained as an effect of the interaction of the plasma with the wall, which acts essentially as a toroidal limiter when the plasma is shifted from the geometric minor axis. Indeed, a remarkable decrease of the pumpout effect is observed when the toroidal ~u~b~urn is improved by a vertical field [3]. The ETA-BETA II discharges have been simulated by a global model treating self-consistently the time evolution of the bulk plasma, with mean electron density and temperature n, and T,, interacting with the wall through a boundary layer characterized by edge parameters Tb (we assume Tbi = T& and n b.
3. Buildup of impurities in a stationary plasma The temporal evolution of the average density of an impurity species, n imp, can be described by the equation:
-?(l
-a),
where rP and 7imp are the plasma ion and impurity confinement times; S and &,t, are the impurity yields due to the incident deuterium or impurity atom, respectively; SC is the chemical sputtering yield, which is important for graphite and p is the particle reflection coefficient. S and SimP represent sputtering or desorption yields, according to the dominant mechanism of impurity production. The factor l/[l- (1 - n)St,,,r] accounts for self sputtering of the recycled impurity ions. The factor n describes the ability of the impurity neutrals to penetrate the plasma. It is defined as the ratio between the number of atoms going into the main plasma to the total emitted by the wall. In eq. (1) we neglect the shielding effect on the backscattered neutrals, since their typical energies are much greater than those of sputtered/desorbed neutrals, at least in the energy range in which shielding becomes important. Major uncertainties exist in the evaluation of n, since the behaviour of the impurity atoms released by the wall is not clear. Certainly the value of u is strongly
819
dependent on the plasma edge parameters and the complexity of the problem would require the use of impurity transport codes to evaluate n in a self-consistent manner. However for the present global model, considering also the uncertainties in the sputtering and reflection coefficients, we can accept the approximation given by a simplified picture assuming the existence of magnetic surfaces in the plasma edge. If this is the case a neutral atom will be “screened” when it is ionized in the SOL, where parallel transport along field hues dominates radial transport. Therefore we use the following approximate expression for n [6]: 71= l - Xn/Xio*r
(2)
where Ai, = uO/( n,,So) is the ionization mean free path of the neutral impurity, which depends on the average radial velocity u. and ionization rate So. The ionization rates are calculated according to Lo&s data for carbon and oxygen, to McWhirter (divided by a factor of 5 to make them compatible with those for oxygen and carbon) for iron. Useful indications on the impurity inventory can be drawn by integrating eq. (1) for a stationary plasma, i.e. with Tb and ni constant. If the impurity yields and the backscattering coefficients are thought of as simple functions of Tb only, the impurity content expressed in terms of the fractional concentration cy= n i,/ni increases monotonically with time as: a(t)=aL-(aL-ao)exp(-t/T*),
(3)
where the limiting value aL is given by: 2” iXr.=
S + AS,
1 l-P-VSimp [ 1-(1--n)
Sm,
and the time constant is proportional particle ~nfinement time:
(4) to the impurity
The dimensionless quantities CQ,and r * could then be plotted as a function of Tb, once the values of (D,L) and nt, have been fried The cases of iron and oxygen impurities have beeu considered, as relevant for ETA-BETA II and the case of carbon for RFX. In order to evaluate the sputtering yield of stainless steel, we assume that the interaction with the wall occurs essentially through both charge-exchange and
820
M. Bagatin et al. / Global modeling of plasma-wall interaction
n
15
20 Tb (ev )
5
10
b
,j01gm-3
20
15 T&d
Fig. 1. Iron impurity production by neutral deuterium sputtering for ETA-BETA II as a function of the edge temperature r,, and density n,,. (a) shows the asymptotic value of the impurity fractional concentration (Y,_, (b) the time constant of its accumulation 7* in terms of the impurity confinement time TV,,,,,.71mp/~p = 2 and (D, L) = 20 rn’ s-’ are assumed.
recombination neutrals, as confirmed also in other experiments [7]. In fact, in ETA-BETA II the strong recycling with R - 0.9 [lo], gives a neutral deuterium influx from the wall F, = (n,/~,)a/2 = 10z3 m-‘s-l, if or is some tens of ps during the density plateau. With an energy of the incoming neutrals of the order of some eV, this gives no = Fe/u, = 1019 rnm3, i.e. a value comparable with nb and sufficient to “screen” the wall from ions. The physical sputtering and self-sputtering yields for iron are evaluated by averaging the empirical functions given by ref. [8] over an isotropic and Maxwellian distribution of neutral energies around Tb. The same is done for the value of p, whose energy and angular dependences are taken from ref. [9]. A value of u0 corresponding to 2 eV has been assumed both for sputtered and desorbed particles. In fig. la the obtained asymptotic value of the impurity concentration is reported for three different values of nb. At the lowest edge density, when no screening is present, an unlimited impurity buildup occurs at low T,, whereas for nb = lOi rnp3 some protection of the central plasma column from impurity penetration should occur. To some extent, this fact has been experimentally confirmed on ETA-BETA II by the observation of hollow Z,,, profiles [lo]. The characteristic time for impurity buildup, as can be seen from fig. lb and the corresponding ones for
oxygen and carbon, is many times the particle confinement time and therefore of the same order as the discharge duration, at least in the quasi steady state Oxygen is desorbed and self-desorbed mainly by ion induced processes. Great uncertainty exists in the desorption cross sections, especially at low energy. Their absolute values and their energy dependences are assumed according to the experimental and theoretical review given in ref. [ll], i.e. increasing with increasing energy until values at 100 eV of 0.2 x lo-*’ rn-* and 5 x lo-*’ me2 for deuterium induced and oxygen induced desorption respectively. For an oxygen surface coverage of 2 X 10” m- * the corresponding yields are respectively 0.4 X lo-* and 10-l. The main differences between the iron and oxygen asymptotic contents, shown in fig. 2a, are the absence of vertical asymptotes, indicating a less sensitive dependence of (Ye on Tbr according also to current experimental observations in tokamaks [12]. Light impurities contamination observed also in the other present RFP devices are compatible with the values of (or from fig. 2a at Tb = 10 eV. Fig. 2b shows that the impurity buildup should occur mainly at the beginning of the discharge, when the confinement times are shorter. Carbon impurity has been examined with reference to the RFX experiment, whose first wall will be covered by a full graphite armour. A chemical sputtering coefficient dependence on the wall temperature T, according
821
M. Bagatin et al. / Global modeling of plasma-wall interaction
15
-a
2. Oxygen production by desorption from stainless steel for ETA-BETA II as a function of the edge temperature r,, and density n ,,. The meanings of a,_ and r */rimi, are the same as in fig. 1. ( rimP/rP) = 2 and (D, L) = 20 m3 s-l are assumed.
evaluation [13] has been used; from the same results an extrapolation to the low energy range has been made ( < 50 ev). For the chemical sputtered carbon an energy of 0.5 eV has been assumed. The parameters for carbon buildup, corresponding to T, = 850 K, which maximizes chemical sputtering, are shown in fig. 3. The relative concentration (also at low T,) easily reaches values of the order of some 10%. At greater Tb a high neutral carbon density at the edge should be expected for boundary electron density greater than 10” rne3, which on the other hand should improve the to recent experimental
a 40
50 Ib (oV)
self-protection of either the plasma from contamination and the wall from erosion.
4. Time dependent calculations and discussion The previous considerations apply to a stationary plasma. Furthermore, effects of the impurity contamination on Tb, as well as the effects on the main plasma density of the surface interaction processes are not self-consistently considered. In the present section we briefly outline a global model described elsewhere [14] of energy and particle balance for the main plasma.
‘~“““‘l”“““““’ 10
b 20
30
40
50 &,(eV)
Fig. 3. Carbon wntamina tion for the KFX plasma as a function of the boundary temperature r, and density nb. q, r* and ~~,,,r have the same meanings as in figs. 1 and 2. Values of ( T~,,,~/T,,)= 2 and (D, L) = 7.5 m3 s-’ are assumed. The chemical sputtering yield is evaluated at its maximum, i.e. at a graphite temperature of about 850 K.
M. Bagaiin et al. / Global modeling of plasma-wall interaction
822
The processes considered are ohmic heating and power losses due to bremsstrahlung, ionization and line radiation of impu~ties and transport. The transport loss, due to heat conduction and particle convection is expressed by means of a confinement time or proportional to beta poloidal &. The time evolution of &, as well as that of the plasma current, are taken directly from the experimental results. The light imp~ties, i.e. oxygen and carbon, are treated by a time dependent corona model, whereas iron losses are calculated by a stationary average ion model adding a recycling term. All the recycled impurities are assumed to return to the bulk plasma as neutrals; their concentrations are calculated by eq. (I) referred to neutrals. A simple relation of proportionality is assumed between the edge and bulk parameters. In the present model, the value of n b/rre, chosen equal to 0.2 has some influence only on the simulations of RFX, because of the low values of Tb, as discussed in section 3. If ail the transport loss were associated with particle convection, the boundary temperature should be approximately 1151:
with y of the order of 10 or more. For ETA-BETA
II with rr/rr 2 1, realistic values of Tb can be obtained only for y > 30. This value is in agreement with the experimental results, i.e. with the ratio between the heat load on the limiter tiles and the power which would be dumped by the ion saturation current with T,, < 10 eV. Possible reasons for this enhancement might be both the creation of a SOL dominated by radiation and charge exchange and the presence of run-away electrons or suprathermal ions. The simulations are obviously very sensitive to the global deuterium recycling coefficient R, which determines the global ion and neutral particle balances, and has been expressed by the sum of three contributions: R=P+
Y,+
YR,
desorption, even if locally important. should be globally negligible. The temperature of the wall, T,. is evaluated in the infinite half-space approximation, by using an intensification factor F, of the transport loss (and of particle loss). With F, = 4 and the initial temperature T, = 300 K, T, > 500 K is reached very early in the discharge, remaining thereafter more or less constant. We can assume therefore that the molecular recombination rate also remains constant. If the hydrogen filling density (about 3 x 10” m- ‘) is considered to be uniformly distributed on the wall, it would saturate the stainless steel for several tens of monolayers at 1% atomic fraction concentration. Since the wall may be thought of as being saturated by deuterium at all times, the men&ted flux due to the recombination F, can be considered constant and Y, is given by: Y, = I-arp/n,.
(8)
that is, it is negligible when the wall is subject to the strongest ion fluxes ( rp very low). Although there are essentially no experimental data on ion induced deuterium desorption at very low energies, theoretical arguments seem to indicate that the desorption cross section should remain relatively constant [12]. We have assumed in our model a value of Yr constant for Th greater than a threshold of 1 eV. The experimental values of electron density are well simulated, as shown in figs. 4 to 7 with F, = 5 x 102’ m‘-Z s-1 and Y, = 0.1, which appear in the range of commonly expected values. The absolute values of
n e (1020m‘3 1
i Te(eV)
I
3
(7)
where Yt and Y, are the desorption yields respectively due to ion induced desorption and to thermally activated molecular recomb~ation. Neutral atom thermal desorption is neglected, since the deuterium atom binding energy is about 1 eV, thus requiring several hundred Kelvin to become significant. Indeed, tile insertion perpendicularly to the field lines have shown that such a temperature could be reached only in limited areas. The associated thermal atom
Fig. 4. Measured electron temperature T, (Cl)and the density ne (0) of the bulk plasma compared with simulations in a discharge in normal conditions.
M. Bagatin et al. / Global modeling of plasma-wall interaction
823
(X10)
aFe
1.
TIME(ms)
n 1:5-
Fig. 5. Time evolution of the oxygen and iron impurity fractional concentrations a,, and are and global recycling coefficient of deuterium R, referred to the discharge of fig. 4.
Fig. 7. The evolution of the oxygen and iron impurity fractional concentrations aox and are and global recycling coefficient of deuterium R, referred to the discharge of fig. 6.
oxygen concentration result in both cases about the same, with fractional contents of 5% and 1.5%, acoording to fig. 2a at T, = 10 eV and 4 eV, respectively. The presence of iron should result negligible in the discharge with pellet injection, due to the lower boundary temperature given by the model. The deuterium recycling due to surface recombination is less important in this case, because of the much greater value of n/7,,. The final values of Z,, ( - 2.3 and - 1.2) are in good agreement with the experimental data, while the time evolution of wall surface temperature appears quite similar in both cases.
Considering carbon sputtering, modelled as discussed in section 3, the model has also been applied to verify the RFX expected performance. For deuterium desorption from graphite a simple scheme, which can be roughly considered as a global saturation model, has been applied. The deuterium saturation inventory is thought of as being given by rectangular depth profiles, with depth empirically increasing with Tb and with height (concentration) decreasing with T,, according to ref. [16]. Therefore a maximum fraction of the implanting deuterium flux can be accommodated, whereas the excess is recycled. .5
Te(eV)
a, .4
200
.3 .2
100
.I
0 0
i 0 .5
1.
TIME(ms)
1.5
Fig. 6. Measured electron temperature r, (0) and density ne (0) of the bulk plasma compared with simulations in a discharge with pellet injection.
50
100
150
200 TIMEtms)
Fig. 8. Example of the simulation of the time evolution of the carbon impurity fractional content a, and of the wall temperature T,. An intensification factor equal to 4 has been assumed for the power dumped to the wall. The behaviour of the electron density ne assumes that the graphite is initially clean.
824
M. Bagatin et al. / Global modeling ofplasma-wall The
with
results
respect
reduction graphite
in
are shown to the
to work
the
in fig. 8. The
stainless
“pump-out” as a deutetium
steel due
main
wall to
store
difference
is the the
strong
capacity
of
in the beginning
and as a source when it is saturated. The behaviour of n, refers to an initially “clean” graphite surface. After a few discharges it should be almost saturated by deuterium, so the foreseen n, - 10” me3 could be reached by much lower filling densities, because the recycling coefficient should approach values about 1 since the beginning of the discharge. The expected performances are essentially confirmed [17], with T, reaching - 1 keV and with a strong carbon contamination, leading to Z,,, - 4. As can be seen from fig. 3, its concentration should be nearly independent of T,,. If the latter is of the order of some tens of eV, physical sputtering and self-sputtering should be more important than the chemical sputtering.
References [l] G.L. Jackson [2] A.A. Newton [3] V. Antoni et trolled Fusion p. 532. [4] V. Antoni et trolled Fusion 1988, p. 553.
et al., J. Nucl. Mater. 145-147 (1987) et al., J. Nucl. Mater. 145-147 (1987) al., in: Proc. 14th Europ. Conf. on and Plasma Physics, Vol. II, Madrid,
470. 487. Con1987,
al., in: Proc. 15th Europ. Conf. on Conand Plasma Physics, Vol. II, Dubrovnik,
interaction
]51 V. Antoni et al., in: Proc. 13th Europ. Conf. on Controlled Fusion and Plasma Physics, Vol. I, Schliersee, 1986, p. 337. [61K.F. Alexander et al., Nucl. Fusion 26 (1986) 1575. [71 D.N. Ruzic et al., J. Nucl. Mater. 145-147 (1987) 527. et al., in: Proc. 10th Symp. on Fusion PI J. Bohdansky Technology, Padova, 1978, p. 801. for [91 W. Eckstein and H. Veerbek. in: Data Compendium Plasma-Surface Interactions, Nucl. Fusion Special issue (1984). [lOI V. Antoni et al., Proc. 11th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Kyoto. 1986, paper CN-44/D-11-5. in: Data Compendium for Plasma Surface 1111 E. Taglauer, Interactions, Nucl. Fusion Special Issue (1984). WI K.H. Behringer, J. Nucl. Mater. 145-147 (1987) 145. et al., in: Proc. 14th Europ. Conf. on 1131 J. Bohdansky Controlled Fusion and Plasma Physics, Vol. II, Madrid, 1987. p. 794. in: Int. School of Plasma P41 M. Bagatin and S. Ortolani, Physics, Varenna 1987, Proc. of the Course on Physics of Mirrors, Reversed Field Pinches and Compact Tori, Commission of the European Communities EUR 11335 EN, Vol. III, p. 1079. I151 J. Hugill, Nucl. Fusion 23 (1983) 331. WI W. Moeller and J. Roth, in: Physics of Plasma-Wall Interactions in Controlled Fusion, Eds. D.E. Post, R. Behrisch (Plenum Press, New York, 1986) NATO AS1 Series B: Physics Vol. 131. u71 V. Antoni et al., in: Proc. 13th Europ. Conf. on Controlled Fusion and Plasma Physics, Vol. I, Schliersee, 1986. p. 385.