CONSIGLIO NAZIONALE DELLE RICERCHE

Transcription

1 CONSIGLIO NAZIONALE DELLE RICERCHE Notes on the Ignitor performance Augusta Airoldi and Giovanna Cenacchi* FP 99/19 Dec 1999 *Centro Ricerche Energia ENEA, Via Don Fiammelli, Bologna Italy ISTITUTO DI FISICA DEL PLASMA "Piero Caldirola" Associazione EURATOM-ENEA-CNR Via R. Cozzi Milano (Italy) Questo rapporto esprime solo l opinione degli autori al momento della redazione 1

2 Notes on the Ignitor performance Augusta Airoldi 1 and Giovanna Cenacchi 1 Istituto di Fisica del Plasma P.Caldirola,EURATOM-ENEA-CNR Association, Milano, Italy Centro Ricerche ENEA, Bologna, Italy Abstract Careful analyses of the parameter evolution of the plasmas that can be produced by the Ignitor machine (B.Coppi et al., MIT Report PTP-96/3, 1996) have been performed along the years under various physical assumptions. The optimal regimes in which full ignition can be achieved correspond approximately to peak densities around 1 1 m -3. This value is well below the Greenwald density limit that is among the most severe conditions to be fulfilled in low magnetic field, larger machines. The high densities allowed by the high toroidal field have been proven to reduce the impurity content, which is another issue heavily affecting the fusion performances. The beneficial effects of the simultaneous increase of the toroidal magnetic field, the plasma current and the particle density are exploited in the project. The dynamic nature of the path to ignition requires simulations based on complex time evolution codes to be used. A specifically updated version [1] of the free-boundary equilibrium-transport code JETTO [] is employed. From a lot of simulations under similar conditions, it was found that fusion power production and ignition depend both on the plasma average density and its radial profile. By considering the optimal density value, the influence of other parameters on the global performance is investigated. While ohmic heating is adequate to reach ignition, even assuming transport diffusion coefficients accounting for energy confinement times close to L-Mode scalings, the ICRH system can be usefully employed to control the evolution of the current density profile and shorten the time needed to ignite. When ignition is not globally achieved, a central ignited core is anyway present.

3 Introduction Some definitions commonly used to indicate the approach to fusion, as power breakeven, ignition condition or ignition margin, Lawson condition [3] and so on, are usually framed in the space ( T, nτ E ), being T,n the plasma temperature and density and τ E the energy confinement time. Notice that they are derived for a plasma with T e =T i. The reference plane ( T, nτ E ) points out that nτ E is the quantity to control; in any case it is necessary to reach a value of nτ m 3. By considering specifically the tokamaks, the plasma E density is bounded by the Greenwald/Murakami limit [4] which is expressed by n I / a B /( Rq). Taking this density limit, the fusion power P = n < σv > E may be written as P ( B / qr) < σ v > E pointing out the dependence on the ratio f 4 B/R. The alternative expression of the fusion power P β B E < σv > / T highlights even a stronger dependence on the magnetic field! This last formulation, moreover, shows the importance of the function < σv >. This function, plotted in Fig.1, exhibits its maximum at T T 14 kev. High magnetic fields assure high densities and high density plasmas guarantee a low impurity content. Actually the mean free path for the ionization of an impurity atom released from the wall depends inversely from the density, so the plasma is efficiently screened from the penetration of impurities. This feature, pointed out by high-field machines as Alcator/Alcator-C at MIT [5] and FTU in Frascati [6] was already established by previous experimental data on other tokamaks [7-8] and recently also on the reversed field pinch RFX in Padua [9]. These considerations, together with those regarding the low poloidal beta assuring the avoidance of MHD instabilities, justify the choice of the high magnetic field in the Ignitor project. The main design parameters are a toroidal field up to 13T, a plasma current up to 1MA, tight aspect ratio (major radius R =1.3m, minor radius a=.47m), considerable elongation (k~1.83) and triangularity (δ ~.4) and flattop from 1s (at 1MA) to 6s (at 1MA) [1]. The f f f f f 3

4 machine is designed to exploit the ignition achievement at the end of the current ramp-up or immediately afterwards in the 1MA scenario. With these operating conditions, the initial phase of the plasma discharge is of paramount importance and requires complex evolutive codes for reliable simulations. Recent and previous evaluations are here summarized in order to compare the importance of different hypotheses on the machine performance. The report is organized as follows. Physics assumptions taken into account are given in Section 1; Section describes both results already obtained and new evaluations with more optimistic assumptions. < σ v> / T 1-4 m 3 /(s kev ) T [kev] Fig. 1 - σv / DT T showing the maximum at T 14keV 4

5 1-Simulation setup The updated version of the equilibrium-transport code JETTO here used has already been detailed in Ref. [1]; the relevant equations are listed in the Appendix. The current rampup and plasma cross-section expansion in a limiter configuration followed by a condition of constant Ip are analyzed for the 1, 11 and 1MA scenarios plotted in Fig.. The current ramp rate is the same for all scenarios. The nominal parameters for the plasma current and the magnetic field here considered refer to simulations starting at t=.3 sec (corresponding to Ip =1MA and Bt =7.6T) and lasting till the ignition attainment in the flattop conditions. The reference to nominal parameters implies the search of ignition conditions with fusion power production not exceeding 1MW. The cross-section expansion is controlled so as to hold q Ψ >3 and to avoid the disruption boundaries in the ( li, qψ ) diagram. This criterion can require to tailor the growth of the plasma poloidal cross-section in a way slightly different from the nominal one. The electron thermal diffusion coefficient is a combination between the Coppi- Mazzucato-Gruber expression and a power depending contribution that accounts for the heating source due to the alpha particles, as detailed in the Appendix. The ion thermal diffusivity has the neoclassical expression plus a small fraction of the electron one. Neoclassical electrical resistivity is adopted. The alpha power produced is modelled as heating source in the equations for electron and ion energy in the form: Sα ( e, i) = ηndnt Eα < σvdt > fe, i( Te ) (1) where n D, n T are the densities of deuterium and tritium ions; E α is the 3.5MeV energy with which alpha particles are born; σv DT the cross section rate for fusion reactions; f e, i the power fraction delivered to electrons or ions, according to the evaluations by Rose [11] and η an efficiency factor taking into account the fraction of confined alpha particles. We have always assumed in our previous evaluations [1,1-15] η =. 95; here we present also results obtained using η = 1. The assumption of neglecting the alpha-particle dynamics is justified 5

6 by the short equilibration times involved in the Ignitor experiment. The working gas is a 5-5 deuterium-tritium mixture. The impurities considered are usually chosen so that at ignition the effective charge is <Z eff > 1.. The syncrotron loss follows the Trubnikov expression [16], with the wall reflectivity, R sync, normally taken to be.8; a comparison will be also made with R sync =. 9. The ignition attainment is pointed out by the unity value of the parameter: f ign = Pα P + P + P + P th rad brems sync () The f ign value is in any case an important marker of the obtained performances. 6

7 MA -T B t Ip t [s] MA - T B t Ip t[s] MA - T B t I p t [s] Fig. - 1, 11,1 MA Scenarios 7

8 -Results There is a range of density values assuring ignition for a choice of design parameters and physics assumptions, as already derived by evaluations in flattop conditions [1] and subsequently confirmed [1,17]. Once singled out, with reference to a specific operation scenario, the optimal density, the analyses refer to other parameters which may affect the ignition achievement. It must just be reminded that the line-averaged density, due to the high current density, is always lower than the Greenwald limit: n = I /( πa ) [1 m -3, MA, m]. (3) e p Notice that this is the most severe condition to be respected in low field machines [18]. The high toroidal field assures also a large margin with respect to the beta limit. a-1ma Scenario The 1MA scenario, designed to exploit the feature of non-stationary conditions for approaching ignition, was carefully considered and discussed in Ref.[1], where attention was paid to the energy confinement time compared with the ITER89P scaling in the form: ITER89 P τ E =. 38Ai k I n19 a R B P (4) [s, MA, 1 19 m -3, m,t,mw] By considering that the ITER Group recently chose another scaling that gives a better fitting of both ohmic and additionally heated discharges [19] and is expressed by: L Mode τ 96 E = A. i k. 64 I n19 a R B P (5) a revisited analysis was performed allowing wider chances. A side comment is that, albeit the new formula has a heavier dependence on the power, there are more favourable dependences on density, current and toroidal field,so on the whole it turns out to be more advantageous for Ignitor. A comparison was also performed [14] by using for the electron thermal diffusivity a combination between Bohm and gyro-bohm scalings []: 8

9 multg cte e e χe α B α gbρ β eb a p p q * r p 1 / = + q (6) e pe whereα B and α gb were calibrated to reproduce similar confinement times (See Fig.3). It must be noted that the energy confinement time evaluated by the code is given by: t CODE E Wtot = P dw / dt tot tot being W tot the total (electrons+ions) energy content and P tot the sum of ohmic, alpha and RF (if present) power content. The dw times near ignition when the energy variation is very fast. tot (7) / dt term in eq.(7) produces high confinement The local electron/ion diffusivities are different for the two models (See Fig.4a-d), but the ignition conditions are only slightly changed. There is indeed only a small difference in the ignition time. The simulations were performed by selecting the optimal density that corresponds to a volume averaged value <ne> 5_ 1 m -3.. The usual R sync =.8 and η =.95 were chosen. Density and temperature profiles at ignition look different (See Fig.5), but the pressure profiles at ignition are very similar as shown in Fig.6. The approach to ignition may be followed by the trajectories of n( ) τ vs T i () in a Lawson plot as in Fig.6. The Lawson condition and the ignition margin are evaluated by using Z eff =1.5. E 9

10 TauE [s] ITER96(BgB) ITER96(Coppi) CODE(BgB) ITER89-P(BgB) CODE(Coppi) ITER89-P-Coppi t [s] Fig.3 - Comparison between the energy confinement time evaluated along the simulation by using the Coppi or the Bohm-gyroBohm model and the values given by the ITER89P and ITER96-LMode scaling. 1

11 3.5 3 dots - Coppi Model diamonds - gb Model.5 m /s 1.5 initial time ρ dots - Coppi Model diamonds - gb Model.5 m /s 1.5 t = 1 s ρ Fig.4 - a) electron (full lines) and ion (dotted lines) thermal diffusivity at the initial time. b) same quantities at t=1s 11

12 dots - Coppi Model diamonds - gb Model m /s t = 3 s ρ m /s final time dots - Coppi Model diamonds - gb Model ρ Fig.4 - c) electron (full lines) and ion (dotted lines) thermal diffusivity at t=3s. d) same quantities at the ignition time. 1

13 1 electron density 1 m dots - Coppi Model diamonds - gb Model 4 final time ρ full lines - electron temperatures dotted lines - ion temperatures kev 15 1 dots - Coppi Model diamonds - gb Model final time ρ Fig.5 - Upper figure shows the density profiles at the ignition time; lower figure the temperature profiles. 13

14 3.5 electron pressure Mpa dots - Coppi Model diamonds - gb Model final time ρ 1 15 [cm -3 s] 1 14 Ignition margin τ n E full blue dot - Coppi Model Lawson condition red open diamond - gb Model kev Fig.6 - Upper figure shows the electron pressure profile at the ignition time; lower figure plots the trajectories in the Lawson plane of n()τ E vs.t e (). 14

15 b - 11MA Scenario The influence of the density growth rate on the attainment of ignition has been analyzed for the 11MA scenario together with the possibility of obtaining better performances by additional heating, during the current rise, foreseen in the machine design [15]. The ignition time is shortened by RF injection and the safety factor profiles may remain completely over unity with a proper choice of the heated region and the injected power. The pressure profiles at ignition maintain similar shapes, as shown in Fig.7. The invariance of the electron pressure profile at ignition was already observed in the 1MA simulations, even based on different transport models, as previously mentioned. Table I shows some quantities evaluated at ignition. Table I Shot A B C D t ign [s] <n e > in 1 m n e ()/<n e > P Ω [MW] P α [MW] P ICRH [MW] 1 1 t ON -t OFF [s] b p T e () [kev] T i () [kev] q() at t ign p e () [MPa] τ E at t ign [s] <n α >/<n DT > [%] The approach to ignition in a Lawson plot is shown in Fig.8. Notice that cases B), C) and D) reach the same final point although their paths differ because of the auxiliary heating. It seems that, under the same global energy confinement time, a consistent plasma pressure characterizes the ignition condition, independently of the path followed. 15

16 3 Mpa A) B) C) D) ρ Fig.7- Pressure profiles at ignition for the cases in Tabel I 1 15 [cm - 3 s] 1 14 Ignition margin τ n E 1 13 Lawson condition Ignition margin A) B) C) D) kev Lawson condition Fig.8.- Evolution of n()τ E vs T i () for cases A), B), C) and D) in Table I. 16

17 c - 11MA Scenario and more favourable assumptions Simulations have been performed by considering the alpha containment parameter η in eq.(1) to be η =1 and the wall reflectivity for the syncrotron losses R sync =. 9. Table II lists the relevant plasma parameters at ignition. Shot 3) benefits also of a slight reduction of the thermal diffusivities due to the choices of the c e, c ei and χ min parameter. Table II Shot 1 3 Alpha Heating Containment η Wall Reflectivity for Sync Emission c e in eq.(a1) c ei in eq.(a16) χ min.4m /s.4m /s.1m /s ignition time 4.99s 4.79s 4.6s Central Electron TemperatureT e () 14.4 kev 13.7 kev 1.8 kev Central Ion Temperature T i () 1.6 kev 1. kev 11.3 kev Central Electron Density n e () 1 1 m m m -3 Central Plasma Pressure p () 4.3 MPa 3.9 MPa 3.6 MPa Average Alpha Density < n α > m m m -3 Plasma Stored Energy W 1.8 MJ 1.5 MJ 1.5 MJ Ohmic Power P OH 1.5 MW 1.6 MW 11. MW Alpha Power P α 4.7 MW 3.5 MW.8 MW Bremsstrahlung Power P brems 4.1 MW 4. MW 4.1 MW Poloidal Beta β p Equilibrium Poloidal β = µ p / B p Toroidal Beta β T 1.16 % 1. % 1.6 % Central Safety Factor q() Edge Safety Factor q Ψ Bootstrap Current I boot.86 MA.85 MA.86 MA Energy Confinement Time τ E.6 s.6 s.6 s Average Effective Charge <Zeff>

18 Figure 9 plots the powers evolution for the cases in Table II. Ignition is reached faster, with lower alpha power; as a consequence the safety factor profile has not time to drop below unity (See Fig.1). 5 P α 15 P OH 1 5 P rad.loss t [s] Fig.9 - Evolution of the ohmic, alpha and total radiation power for the cases in Table II. Crosses mark case 1), full diamonds case ) and open dots case 3) 18

19 ) ) 3) q ρ Fig.1 - Safety factor profiles at ignition time for cases in Table II. 19

20 d - 1MA Scenario By taking all physics assumptions of case3) in Table II the ignition turns out to be achievable also in the case of 1MA. Figure 11 contrasts the power evolution relevant to case 3) and to the1ma simulation. The ignition time is a bit delayed, but the alpha power is even less than in the 11MA case. On the other hand the safety factor profile presents a region where drops below unity (See Fig.1). 5 P α 15 P OH 1 5 Q α t [s] Fig.11 - Ohmic and alpha power (in MW) evolution for the1ma scenario (full squares) and case 3) in Table II (open dots). The parameter Q = P /( P P ) is also shown α α loss α

21 MA 1) 3) q ρ Fig.1- Safety factor profile at ignition for the1ma scenario compared with cases 1) and 3) in Table II 1

22 Concluding remarks The most advanced scenario (1MA) assures ohmic ignition under conservative hypotheses. Ignited plasmas in ohmic conditions are achievable also in the 11MA scenario when favourable thermal confinement is assumed. In these cases the plasma column presents a safety factor maintained over unity. Under less favourable hypotheses it is necessary to rely on the auxiliary heating system to tailor the current density profile and recover the q>1 situation. Notice that the onset of sawtooth instabilities should be prevented firstly by the low poloidal beta and afterwards by the alpha particle population. Anyway a central ignited core is attained even in the presence of degraded confinement conditions [1,1]. In the best assumable conditions (all alphas confined, η = 1., minimum syncrotron losses, R sync =. 9) 1 MA current may guarantee ignited plasmas.

23 References [1] - A. Airoldi and G. Cenacchi, Nucl. Fusion 37 (1997) [] - G. Cenacchi and A.Taroni, JET Report JET-IR(88)3(1988) [3] - J.D. Lawson, Proc. Royal Phys. Soc. 7B (1957) 6. [4] - M. Greenwald, J.L. Terry, S.M. Wolfe, et al., Nucl. Fusion 8 (1988) 199 [5] - M. Greenwald, R.L. Boivin, P. Bonoli, et al., Phys. Plasmas (1995) 38 [6] - M.L. Apicella, G. Apruzzese, M. Borra, et al., Nucl. Fusion 37 (1997) 381 [7] - K. Ida, T. Amano, K. Kawahata, et al., Nucl. Fusion 3 (199) 381 [8] - H.F. Dylla, et al., Journal Nucl. Mat (1987) 48 [9] - L. Carraro, M.E. Puiatti, F. Sattin, et al., Journal Nucl. Mat (1999) 466 [1] - B. Coppi, A. Airoldi, F. Bombarda, et al., MIT (RLE) Report PTP 99/6 September 1999 [11] - D.J. Rose, Nucl. Fusion 9 (1969) 183 [1] - A. Airoldi and G. Cenacchi, Plasma Phys. Control. Fusion 33 (1991) 91. [13] - A. Airoldi and G. Cenacchi, Fusion Technol. 4 (1994) 78. [14]- A. Airoldi and G. Cenacchi, nd Int. Work. on Ignitor, May 5-7,1999,Washington,DC [15] - G. Cenacchi, A. Airoldi and B. Coppi, Proc. 6th EPS Conf. on Control. Fusion and Plasma Physics 1999, ECA 3J (1999) 111 [16] - B.A. Trubnikov, Reviews of Plasma Physics (Ed. M.A. Leontovich) Vol. 7, 345, Consultants Bureau Publ, 1979 [17] - L. Sugiyama, MIT Research Laboratory of Electronics PTP 99/ (1999) [18] - E. Rebhan and U. Vieth, Nucl. Fusion 37 (1997) 51. [19] - T. Takizuka, D. Boucher, G. Bracco, et al., Proc. 16th IAEA Conf. on Fusion Energy 1996, Vol. (1997) 795 [] - M. Erba, T. Aniel, V. Basiuk, et al, Nucl. Fusion 38, (1998) 113 [1] - A. Airoldi, G. Cenacchi and B. Coppi, Bulletin of APS 4 (1997) [] - B. Coppi, P. Detragiache, S. Migliuolo, et al., Proc. 14th IAEA Conf. on Fusion Energy 199, Vol. (1993) 131 [3] - C.S. Chang and F.L. Hinton, Phys. Fluids 3 (198)

24 Appendix By assuming quasi-neutrality, equal temperature for all ion species and coronal equilibrium for impurities, the full set of equilibrium-transport equations is expressed, in SI units, by: χ χ 1 χ π + = µ ( R p' + ff' ) (A1) R Z R R ιρbt ι t 1 ρb T ρ 3 1 ( V' ) 5 / 3 t < E B > fa [( V' ) 5 / 3 p e ] + 1 V' = (A) { } = S e + j E (A3) ρ V' [ q ρ + λt Γ e e e] 3 1 ( V' ) 5 / 3 p ( V' ) 5 / 3 t [ i i ] + 1 V' ρ V' { i [ q ρ + λt Γ i i i] } = i S i (A4) 1 V' t V' n i [ ] + 1 V' [ ρ V' Γ i] = S i (A5) Eq. (A1) is the Grad-Schluter-Shafranov equilibrium equation, eq. (A) the Faraday's equation, eqs (A3)-(A5) represent conservation of energy (electrons and ions) and particles. Here: R, Z are the usual radial and vertical cylindrical coordinates; c is the poloidal magnetic flux; µ the vacuum permeability; p the plasma pressure; f the flux-function of the poloidal plasma current; ι = 1/ q = dχ / dφ the rotational transform; V the volume enclosed by a magnetic surface; <> indicates the flux-surface averaging operator; A =< 1 / R >; ' denotes differentiation with respect to ρ; ρ is the radial flux coordinate defined by ρ = Φ. πb t The subscript e denotes electrons, i primary ions and impurities. Terms S e,i indicate any source or sink, including auxiliary heating, losses due to impurity radiation, bremsstrahlung, syncrotron radiation and equilibration exchange contributions. The boundary value of ι is related to the total plasma current by the condition [] : 4

25 I = V Φ fa < 1 > 8π 3 µ ι R ρ max (A6) The plasma current and the boundary value of T e and T i are imposed functions of time. The initial profiles and the boundary values of p e and p i are determined via T e,t i,n i. The code starts from a MHD equilibrium step with assigned pressure and rotational transform profiles. In the subsequent diffusion step, where a constant geometry is assumed ( V' / t = ) the density profiles are prescribed while the temperatures are consistent with the pressure profile of the first equilibrium computation. In the further equilibrium steps the magnetic surfaces are updated according to the pressure and rotational transform profiles given by the last diffusion step, and adiabatic constraints are imposed: p e,i (V' ) 5/3 = const n i V ' = const (A7) (A8) In the Faraday's equation (A) we take E B = η / / j B (A9) being η / / the neoclassical electrical resistivity. The radial energy flux for electrons (ions of species i) is defined as: q e,i ρ + λt e,i Γ e,i. (A1) The second term represents the thermal convection processes with λ (A1) is the diffusive contribution and is expressed through an anomalous χ e,i : = 5 /. The first term in q e,i ρ = χ e,i n e,i T e,i ρ ρ. (A11) The electron thermal diffusion coefficient χ e is modelled as: P aux χ e = c e χ CMG e + χ UB e ( ). P Ω + P aux χ e CMG is the Coppi-Mazzucato-Gruber formula for general axisymmetric geometry: (A1) χ CMG e = 11 3 / I ( ρ) ln ΛZeff Φ π Va 1 4 / 5 3 / n T A S V e e i / 5 a (A13) (in SI units, with T e in ev). 5

26 Here A i is the primary ion mass in amu, I Φ the toroidal current enclosed by the ρ fluxsurface, S a the plasma cross-sectional area and Z eff the effective charge defined by: Z eff = ( i n i Z i ) / n e. (A14) For each impurity species Z i is evaluated by the coronal model. χ e UB in eq. (A1) represents a contribution based on the ubiquitous modes []: χ UB e = c UB / 1 ρ ρ 1 T T q R n e i e dp di 1 Φ Z eff A i (A15) In order to avoid numerical troubles, in the edge region ( ρ / ρ max >.86) χ e is computed by a parabolic extrapolation. The ion thermal diffusivity is defined as the sum of a neoclassical term [3] and an anomalous one chosen to be a fraction of χ e : χ i = χ i NC + c ei χ e (A16) with c ei usually taken to be.1. A minimum value for both electron and ion thermal diffusivity is prescribed (.4m s 1 ). The radial ion flux, including an outward diffusion and an inward flow, is given for each ion species i by: n Γ = D i ρ + α i p i ρ inwi S ( ρ ) V dv d n ρ i (A17) pl with D p = α i p χ i e (A18) and S the area of the flux surface. For each ion species a time-dependent particle influx can be prescribed. This flux is added to the recycling condition at the boundary for main ions. The electron Γ e is expressed likewise to eq. (A17), being n e = n Z. i i i 6