1. EC Radiation Transport in Fusion Reactor- Grade Tokamaks: Parameterization of Power Loss Density Profile, Non-Thermal Profile Effects under ECCD/ECRH conditions K.V. Cherepanov, A.B. Kukushkin, L.K. Kuznetsova, E. Westerhof 2 1 Russian Research Center "Kurchatov Institute“___ Nuclear Fusion Institute 2 FOM-Institute for Plasma Physics Rijnhuizen,Association EURATOM-FOM, Trilateral Euregio Cluster, The Netherlands, www.rijnh.nlwww.rijnh.nl

2. GOAL: Numerical studies of contribution of electron cyclotron radiation (ECR) to local energy balance in toroidal magnetically confined plasmas with hot electrons > 10 keV(*) strong magnetic field B T > 5 T(*) highly reflecting walls R w > 0.5(*) (including the ITER-like conditions). MOTIVATION: Spatial profile of net radiated power density, P EC (r), is peaked in the plasma core and, for D-He 3, D-D tokamaks (T e (0) ~ 70 keV), is a major power loss channel: Tamor S., Fusion Technol. (1983) Kukushkin A.B., 14th IAEA (1992) Albajar F., Bornatici M., Engelmann F., Nucl. Fusion (2002) P EC (r) becomes dominant electron cooling mechanism in DT reactor-grade tokamaks: ASTRA+CYTRAN: ITER steady-state scenarios (T e (0) ~ 30-40 keV) Albajar F., Bornatici M., Cortes G., et al., Nucl. Fusion, 45 (2005) 642

3. Comparison of radial profiles of net ECR power loss for (i) two regimes (*),(**) with bi-maxwellian electrons and (ii) maxwellian background plasma, which give almost the same value of total ECR power loss, for wall reflection coefficient R W = 0.6. Curves (1)-(3): code CYNEQ Curves (4)-(6) : “localized” Trubnikov’s formula. ITER-FEAT T e (0)=25 keV T e (a)= 2 keV n e (0)=10 20 m -3 n e (a)=0.5 n e (0)  n e =10%; T e (hot) = 2T e (  0 ); [Cherepanov K.V., Kukushkin A.B., EPS-2004]

4. Te,0 (keV): 36 43 27 (keV): 17 19 8 Ti,0 (keV): 37 53 23 (keV): 17 21 7.5 I (MA): 9 9 15 R wall : 0.60.6 0.6 Steady-state 1Steady-state 2Inductive Profile of local energy balance in ITER for various scenarios (Albajar, Bornatici et. al., Nuclear Fusion, 2005) Numerical code ASTRA is combined with CYTRAN (red stripe = P EC (r) ). ASTRA+CYTRAN

5. TASK: Parameterization of P EC (r) to be used as a simple simulator during the transport calculations for ITER-like range of parameters METHOD: On the basis of calculations of P EC (r) with the code CYNEQ [Cherepanov K.V., Kukushkin A.B., EPS-2004, IAEA-2004, etc.], we propose further simplification of the well-known fast-routine code CYTRAN [Tamor S., Reps. Science Applications, 1981] with an accent on satisfactory description of P EC (r) in the core.

6. Code CYNEQ (electron CYclotron radiation transport in Non-EQuilibrium hot tokamak plasmas) is based [Kukushkin A.B., JETP Lett., 1992; 14th IAEA PPCF conf., 1992] on extending the escape probability methods developed in the theory of nonlocal transport. The code solves semi-analytically the transport problem; simplifies semi-analytic approach of the code CYTRAN [S. Tamor, Reps. Science Applications (1981)]: hot maxwellian toroidal plasmas of - non-circular cross-section, - not too large aspect ratio, - multiple reflection of waves from the wall, retains CYTRAN’s accuracy of approximating the results of Monte Carlo code SNECTR [S. Tamor, 1978]. CYNEQ was also tested [1992] against well-known benchmark -- numerical results, and respective analytic fit [Trubnikov B.A., Rev. Plasma Phys., vol. 7, 1979], for total ECR power loss for maxwellian electron plasmas of homogeneous profiles of temperature and density.

7. r b – plasma’s boundary V esc (  ) is projection of phase space  esc (with angle- averaged absorption coefficient  ) onto coordinate space. Spatial profile of net radiated power density in CYNEQ Kukushkin A.B., Proc. 24th EPS Conf. on Contr. Fusion & Plasma Phys., Berchtesgaden, 1997, vol. 21A, Part II, p. 849-852. k(  ) - absorption coefficient Q(  ) - emission rate P EC (r) =

8. Simple analytic description of spectral temperature T ECR ( ,K) of outgoing EC radiation (1) K= extraordinary (X) and ordinary (O) waves; T e =T e (  );  =r/a; Fitting formulas [Tamor,1981] for normalized absorption coefficients  K ( ,T e ) are slightly modified to avoid the increasing errors at small temperatures. Characteristic optical thickness  = 6.04  10 3 a/B 0.. a, one-dimensional minor radius in meters; B 0, central magnetic field in Tesla; R WK, wall reflection coefficient for K mode; = (f T cut ( ,K) + (1-f)T e (1)); f=0.6; T cut ( ,K) = T e (  cut ( ,K));  cut ( ,K) - boundary of optically thick core in the radiation’s reduced phase space {frequency, radius}  cut /  B0 = 2 + D K (1-  cut -  Kmin ), D K = (T e (0) + T e (1))/2 { ln(  n e (0))/C K } 2 + A K, A X =0, A O = -1;  Kmin = 0.01; C X = 17.9; C O = 19.7; n e (0) in 10 20 m -3.  cut /  B0 = 2 for  > 1-  Kmin,  cut = 0 for  /  B0 > 2 + D K (1-  Kmin ).

9. Analytic description of power loss radial profile Deviation of spectral temperature of EC radiation from local electron temperature determines the spectral density of the local ECR power loss in maxwellian plasmas. The remaining integration over frequency to evaluate P EC (r) has to be done numerically. C = 3.9 10 -8 MW/m 3, B 0 (Tesla), a(meters) (2)

10. The profiles of plasma were taken close to those for one of ITER regimes predicted by the ASTRA code 1D simulations: (major/minor radius 6.2/2 m, B T = 5.3 T,  = r/a ) Polevoi A.R., Medvedev S.Yu., Mukhovatov S.V., et. al., J Plasma Fusion Res. SERIES, 5 (2002) 82-87. T e (0) = 25 keV, T e (1) = 2 keV, ~ Inductive, ITER scenario 2 T e (0) = 24 keV, T e (1) = 0.3 keV, ~ Steady-state, scenario 4 (analytic, ~ flat P EC (r)) T e (0) = 35 keV, T e (1) = 2 keV, ~ Steady-state (P EC (r) of CYTRAN)

11. __ CYNEQ …. Eq. (1) T e (0) = 25 keV T e (1) = 2 keV R WK = 0.6 X mode, O mode, X+O T ECR (  ) ~  2 T ECR (  )

12. __ CYNEQ …. Eq. (1) T e (0) = 24 keV T e (1) = 0.3 keV R WK = 0.6 X mode, O mode, X+O T ECR (  ) ~  2 T ECR (  )

13. __ CYNEQ …. Eq. (1) T e (0) = 35 keV, T e (1) = 2 keV, R WK = 0.6 X mode, O mode, X+O T ECR (  ) ~  2 T ECR (  )

14. ….. Eq. (2) ___ CYNEQ _ _ Eq. (2) with CYNEQ’s numerical absorption coefficients  T e (0) = 25 keV T e (1) = 2 keV R WK = 0.6 X mode O mode X+O

15. T e (0) = 24 keV T e (1) = 0.3 keV R WK = 0.6 X mode O mode X+O ….. Eq. (2) ___ CYNEQ _ _ Eq. (2) with CYNEQ’s numerical absorption coefficients 

16. T e (0) = 35 keV T e (1) = 2 keV R WK = 0.6 X mode O mode X+O ….. Eq. (2) ___ CYNEQ _ _ Eq. (2) with CYNEQ’s numerical absorption coefficients 

17. Conclusion. Part 1 Analysis of comparing the CYNEQ and CYTRAN calculations enabled us to simplify further the fast routine of CYTRAN and retain reasonable accuracy of describing the radial profile of EC net radiated power, P EC (r), in the region of significant contribution of P EC (r) to the local power balance in fusion reactor- grade tokamaks (first of all, in the central part of the plasma column). Acknowledgments. The work is partly supported by the Russian Federal Agency on Science and Innovations (contract 02.445.11.7505) and the RF grant NSh-9878.2006.2 for Leading Research Schools.

18. ECRH 20 MW, O mode, n=1, f=138 GHz, Perpendicular launch (  =0), equatorial plane Formation of Superthermal Electrons Under ECRH/ECCD in ITER-like Tokamak L.K. Kuznetsova, K.V. Cherepanov, A.B. Kukushkin, E. Westerhof METHOD OF CALCULATION Beam tracing code TORBEAM [1] calculates a power deposition profile P TORBEAM (x) and provides w and DN || on set of flux surfaces. Fokker-Planck code RELAX [2] takes w and DN || from TORBEAM and calculates deposition P RELAX (x) with resonance broadening. Fokker-Planck code RELAX [2] outputs the distribution function ==================================== [1] E. Poli, et al., Comp. Phys. Commun. 136 (2001) 90. [2] E. Westerhof, et al., Rijnhuizen Report RR 92-211 (1992). ITER-like Te(0) = 35 keV

19. CONCLUSIONS. Part 2. 1. The EC absorbed power density may attain ~10 MW/m3, for 20 MW total absorbed power and wave beam focusing in the plasma core. 2. For perpendicular launch (ECRH only), the deviation of the EDF from the Maxwellian is stronger for the thermal part (E kin < T e ), with the effective temperature T ef (E kin ) (defined as the exponential slope of the EDF with respect to energy for a given electron kinetic energy E kin ) exceeding T e by 10-20% and 20-40% for, respectively, T e (0)= 25 keV and 35 keV. 3. For oblique launch (ECCD/ECRH), with an injection angle  ~ 20 , T ef /T e is about twice smaller, but in a substantially broader energy range, up to E kin /m e c 2 ~ 0.5, producing thus a strong enough fraction of superthermal electrons. Also, formation of a plateau on the EDF at higher energies is found (E kin /m e c 2 ~1, T ef /T e ~2-5) for both launch geometries.

20. Influence of ECCD/ECRH-Produced Superthermal Electrons on Transport of Plasma’s Electron Cyclotron Radiation in Tokamak-Reactor A.B. Kukushkin, K.V. Cherepanov, L.K. Kuznetsova, E. Westerhof Wall reflection coefficient R W =0.6 Comparison of radial profiles of net ECR power loss density: (1) maxwellian background plasma; (2) non-maxwellian EDF produced by ECRH; (3) maxwellian EDF with the same relativistic mean electron energy. How the distortions of the electron velocity distribution (EDF) caused by the absorption of external intense ECR, injected into the plasma at low harmonics of the cyclotron frequency (n=1,2) for ECRH and ECCD, may influence the transport of ECR, emitted by the plasma itself at all other harmonics (2<n<15) responsible for formation of the P EC (r) profile in reactor-grade tokamaks. Method of calculation 1. ECRH and ECCD (n=1,2) [1] in ITER-like tokamak (Scenario 2, Te(0)=25 keV; modified steady state, Te(0)=35 keV) code TORBEAM [2] (beam tracing)  [4]  code RELAX [3] (Fokker-Planck, distribution function f e (v,r) )  2. ECR transport (n>2):  code CYNEQ [5] (power loss P EC (r)) ======================================= [1] L. K. Kuznetsova, K. V. Cherepanov, A. B. Kukushkin, E. Westerhof, Formation of superthermal electrons … (EC-14 paper 71). [2] E. Poli, et al., Comp. Phys. Commun. 136 (2001) 90. [3] E. Westerhof, et al., Rijnhuizen Report RR 92-211 (1992). [4] L.K. Kuznetsova, Juelich (2002). [5] Cherepanov K.V., Kukushkin A.B., EPS-2004, IAEA- 2004.

21. CONCLUSIONS. Part 3 1. For the same value of total absorbed EC power, the effect of ECCD/ECRH-produced superthermal electrons on the net ECR power loss density, P EC (r), is stronger for power absorption at larger electron velocities. For equatorial plane launch, the effect is maximal (~20%) for wave beam focusing in the core and toroidal injection angle  ~20 . For perpendicular launch, effect is ~few percents only, similarly to the case when EDF’s distortions are caused exclusively by the ECR emitted by the plasma (“self ECR”). 2. In ITER, total power of self ECR inside the chamber (15 MW for scenario 2, wall reflection coefficient R w =0.6) will be comparable to 20 MW ECCD total power. As far as the ECCD is resulted exclusively from asymmetry of electron velocity distribution in co- and counter-current directions, the impact of self ECR (at high harmonics, of total 15 MW) on ECCD may not be small and has to be evaluated.