Impact of neutral atoms on plasma turbulence in the tokamak edge region

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1 Impact of neutral atoms on plasma turbulence in the tokamak edge region C. Wersal P. Ricci, F.D. Halpern, R. Jorge, J. Morales, P. Paruta, F. Riva Theory of Fusion Plasmas Joint Varenna-Lausanne International Workshop

2 Physics at the periphery of a fusion plasma Toroidal limiter Limiter Core Edge SOL Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

3 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Plasma Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

4 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Parallel flow in the SOL to the limiter Plasma Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

5 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Parallel flow in the SOL to the limiter Recombination on the limiter Plasma Neutrals Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

6 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Parallel flow in the SOL to the limiter Recombination on the limiter Ionization of neutrals Ionization Density source Energy sink Plasma Neutrals Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

7 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Parallel flow in the SOL to the limiter Recombination on the limiter Ionization of neutrals Ionization Density source Energy sink Plasma Neutrals Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

8 Physics at the periphery of a fusion plasma Toroidal limiter Radial transport due to turbulence Parallel flow in the SOL to the limiter Recombination on the limiter Ionization of neutrals Ionization Recycling Density source Energy sink Plasma Neutrals Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

9 Movie Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 3 / 37

10 The tokamak scrape-off layer (SOL) Heat exhaust Confinement Impurities Fusion ash removal Fueling the plasma (recycling) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 4 / 37

11 1. Modeling the periphery 2. A refined two-point model with neutrals 3. Gas puff fueling simulations Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 5 / 37

12 Modeling the periphery Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

13 Modeling the periphery High plasma collisionality, local Maxwellian Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

14 Modeling the periphery High plasma collisionality, local Maxwellian d/dt ω ci,k 2 k 2 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

15 Modeling the periphery High plasma collisionality, local Maxwellian d/dt ω ci,k 2 k 2 Drift-reduced Braginskii equations n,ω,v e,v,i,t e,t i Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

16 Modeling the periphery High plasma collisionality, local Maxwellian d/dt ω ci,k 2 k 2 Drift-reduced Braginskii equations n,ω,v e,v,i,t e,t i Flux-driven, no separation between equilibrium and fluctuations Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

17 Modeling the periphery High plasma collisionality, local Maxwellian d/dt ω ci,k 2 k 2 Drift-reduced Braginskii equations n,ω,v e,v,i,t e,t i Flux-driven, no separation between equilibrium and fluctuations Kinetic neutral equation Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

18 Modeling the periphery High plasma collisionality, local Maxwellian d/dt ω ci,k 2 k 2 Drift-reduced Braginskii equations n,ω,v e,v,i,t e,t i Flux-driven, no separation between equilibrium and fluctuations Kinetic neutral equation Interplay between plasma outflow from the core, turbulent transport, sheath losses, and recycling Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 6 / 37

19 Fluid plasma model and interaction with neutrals n = ρ 1 [φ,n] + 2 t B [C(pe) nc(φ)] (nv e ) + Dn(n) + Sn+nnν iz nν rec (1) ω t v e t v i t T e t T i t = ρ 1 [φ, ω] v i ω + B2 = ρ 1 [φ,v e ] v e v e + m i m e n j + 2B n C(p) + D nn ω ( ω) n νcx ω (2) ( ν j ) n + φ 1 n pe 0.71 Te = ρ 1 [φ,v i ] v i v i 1 n p + Dv i (v nn i )+ = ρ 1 [φ,t e] v e T e + 4Te 3B [ 1 n C(pe) + 5 C(Te) C(φ) 2 + D Te (T e) + D Te (Te) + S Te + nn n ν iz ( 2 3 E iz T e + me = ρ 1 [φ,t i ] v i T i + 4T [ i 1 3B n C(pe) τ 5 ] 2 C(T i ) C(φ) + D v e (v e )+ nn n (νen + 2ν iz )(v n v e ) n (ν iz + ν cx )(v n v i ) (4) ] [ ] + 2Te n j v e (5) v m e (v e 4 i 3 v n [ + 2T i 3 + D Ti (T i ) + D T i (T i ) + S Ti + nn n (ν iz + ν cx )(T n T i (v n v i )2 ) 2 φ =ω, ρ = ρs/r, f = b 0 f, ω = ω + τ 2 T i, p = n(t e + τt i ) nn me 2 )) + νen n m i (v i v e ) n n v e 3 v e (v n v e )) ] (3) (6) + boundary conditions + kinetic neutral equation Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 7 / 37

20 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

21 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

22 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

23 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

24 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

25 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

26 The density equation n t = ρ 1 [φ,n] + 2 B [C(p e) nc(φ)] (nv e ) (7) + S n + n n ν iz nν rec + D n (n) ExB drift Curvature terms Parallel advection Plasma source from core Interaction with neutrals Perpendicular diffusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 8 / 37

27 The kinetic model of the neutrals One mono-atomic neutral species Krook operators for ionization, charge-exchange, and recombination C. Wersal and P. Ricci 2015 Nucl. Fusion Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 9 / 37

28 The neutral model f n t + v f n x = ν izf n ν cx (f n n n Φ i ) + ν rec n i Φ i (8) ν iz = n e v e σ iz (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Φ i = f i /n i Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

29 The neutral model f n t + v f n x = ν izf n ν cx (f n n n Φ i ) + ν rec n i Φ i (8) ν iz = n e v e σ iz (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Φ i = f i /n i Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

30 The neutral model f n t + v f n x = ν izf n ν cx (f n n n Φ i ) + ν rec n i Φ i (8) ν iz = n e v e σ iz (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Φ i = f i /n i Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

31 The neutral model f n t + v f n x = ν izf n ν cx (f n n n Φ i ) + ν rec n i Φ i (8) ν iz = n e v e σ iz (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Φ i = f i /n i Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

32 The neutral model f n t + v f n x = ν izf n ν cx (f n n n Φ i ) + ν rec n i Φ i (8) ν iz = n e v e σ iz (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (9) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 (10) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

33 Boundary conditions for the neutrals Partial reflection at the limiters Window averaged particle flux conservation at the outer boundary nn nn Z/ ρ s Z/ ρ s R/ ρ s R/ ρ s Gas puffs and neutral background Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 11 / 37

34 Further simplifications Separation of time scales The neutrals time of life is typically shorter than the turbulent time scale T e = 20eV, n 0 = cm 3 τ neutral losses ν 1 eff s τ turbulence R 0 L p /c s s Assume fn / t 0 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 12 / 37

35 Further simplifications Separation of time scales The neutrals time of life is typically shorter than the turbulent time scale T e = 20eV, n 0 = cm 3 τ neutral losses ν 1 eff s τ turbulence R 0 L p /c s s Assume fn / t 0 Plasma anitrosopy The plasma elongation along the field lines is much longer than the typical neutral mean free path Assume f n 0 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 12 / 37

36 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

37 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) v 0 x f n (x,v) (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

38 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) v 0 x f n (x,v) = x 0 dx (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

39 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) v (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

40 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) e 1 x v x dx (ν cx (x )+ν iz (x )) v (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

41 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) v + f w (v)e 1 v x0 dx (ν cx (x )+ν iz (x )) e v 1 x x dx (ν cx (x )+ν iz (x )) (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

42 Solution of neutral eq. with method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v f n x = ν cxn n Φ i (ν iz + ν cx )f n (11) 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) v + f w (v)e 1 v x0 dx (ν cx (x )+ν iz (x )) e v 1 x x dx (ν cx (x )+ν iz (x )) (12) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

43 An equation for the density distribution By imposing f n dv = n n (13) we get a linear integral equation for n n (x) n n (x) = x dx n n (x ) contribution by v < 0 + n w (x) dv ν cx(x )Φ i (x,v) v e d eff ν eff (x x ) v (14) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 14 / 37

44 The GBS code, a tool to simulate SOL turbulence Evolves scalar fields in 3D geometry n,ω,v e,v,i,t e,t i Kinetic neutral physics Limiter geometry Open and closed field-line region Sources S n and S T mimic plasma outflow from the core (Divertor geometry) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 15 / 37

45 Questions that we can address How is the temperature at the limiter related to main plasma parameters? How is the plasma fueled? How do neutrals affect plasma turbulence? SOL width? Heat flux? How do diagnostic gas puffs affect the SOL? Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 16 / 37

46 Questions that we can address How is the temperature at the limiter related to main plasma parameters? How is the plasma fueled? How do neutrals affect plasma turbulence? SOL width? Heat flux? How do diagnostic gas puffs affect the SOL? Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 16 / 37

47 1. Modeling the periphery 2. A refined two-point model with neutrals 3. Gas puff fueling simulations Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 17 / 37

48 The two-point model Relation between upstream and target plasma properties Limiter Target Widely used experimentally for a quick estimate Derived from 1D model along field lines Core Edge SOL Upstream Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 18 / 37

49 The SOL unrolled SOL Main Plasma Main Plasma Limiter Limiter SOL LCFS Limiter Wall Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

50 The SOL unrolled SOL Main Plasma Limiter Main Plasma Limiter Target Upstream SOL LCFS Target Limiter Wall Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

51 The SOL unrolled SOL Main Plasma Limiter Main Plasma Limiter Target s Upstream SOL LCFS Target Limiter Wall Parallel plasma dynamics projected along poloidal coordinate Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

52 The SOL unrolled SOL Main Plasma Main Plasma Limiter Limiter s SOL LCFS Limiter Wall Parallel plasma dynamics projected along poloidal coordinate Plasma and energy outflowing from the core are modeled with prescribed S n and S Q Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

53 The basic two-point model Q = S Q ds = Q cond + Q conv (15) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

54 The basic two-point model Q = S Q ds = Q cond + Q conv (15) Q cond = χ e0 T 5/2 dt e e dz (16) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

55 The basic two-point model Q = S Q ds = Q cond + Q conv (15) Q cond = χ e0 T 5/2 dt e e dz (16) Q conv = c e0 ΓT e (17) Γ = nv = S n ds (18) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

56 The basic two-point model Q = S Q ds = Q cond + Q conv (15) Q cond = χ e0 T 5/2 dt e e dz (16) Q conv = c e0 ΓT e (17) Γ = nv = S n ds (18) Boundary conditions Upstream: dt e /ds = 0 At the limiter: Q L = γ e Γ L T el, γ e 5 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

57 The basic two-point model Q = S Q ds = Q cond + Q conv (15) Q cond = χ e0 T 5/2 dt e e dz (16) Q conv = c e0 ΓT e (17) Γ = nv = S n ds (18) Boundary conditions Upstream: dt e /ds = 0 At the limiter: Q L = γ e Γ L T el, γ e 5 S Q,S n T e,u T e,t Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

58 Simulations with different densities n 0 = cm 3 n 0 = cm 3 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 21 / 37

59 Simulations with different densities n 0 = cm 3 n 0 = cm 3 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 22 / 37

60 Simulations with different densities n 0 = cm 3 n 0 = cm 3 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 22 / 37

61 Poloidal profiles of electron temperature n 0 = cm 3 n 0 = cm 3 Te L 0 L s Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 23 / 37

62 Poloidal profiles of electron temperature n 0 = cm 3 n 0 = cm 3 Te L 0 L s Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 23 / 37

63 Temperature ratio upstream to target 2 basic model 1.8 Te,u/Te,t (tpm) , no n n , no n n x , E iz = T e,u /T e,t (GBS) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 24 / 37

64 A more refined two-point model Obtain an electron heat equation in quasi-steady state 3 2 T n e t n T e 0 (19) t Assume v e, v i, and neglect small terms (e.g., D Te ) Combine perpendicular transport terms into S Q ( 5 2 nv T e ) χ e0 ( T 5/2 e T e ) v (nt e ) (20) = S Q + S neutrals with S neutrals = n n ν iz (T e )E iz and χ e0 = 3/2 nκ e Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 25 / 37

65 Further assumptions and relations v is linear from c s to c s c s = T e,t + T i,t 2T e,t nv = [S n + n n ν iz (T e )]ds n n is decaying exponentially from limiter with λ mfp Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 26 / 37

66 Three external input quantities Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

67 Three external input quantities Perpendicular heat source, S Q GBS cos fit SQ L 0 L s Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

68 Three external input quantities Perpendicular heat source, S Q Perpendicular particle source, S n GBS cos fit Sn L 0 L s Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

69 Three external input quantities Perpendicular heat source, S Q Perpendicular particle source, S n Ionization particle source, S iz Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

70 Three external input quantities Perpendicular heat source, S Q Perpendicular particle source, S n Ionization particle source, S iz S Q,S n,s iz T e,u T e,t Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

71 Temperature ratio upstream to target 2 basic model 2 full model Te,u/Te,t (tpm) , no n n , no n n x , E iz = T e,u /T e,t (GBS) Te,u/Te,t (tpm) , no n n , no n n x , E iz = T e,u /T e,t (GBS) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 28 / 37

72 Questions that we can address How is the temperature at the limiter related to main plasma parameters? How is the plasma fueled? How do neutrals affect plasma turbulence? SOL width? Heat flux? How do diagnostic gas puffs affect the SOL? Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 29 / 37

73 1. Modeling the periphery 2. A refined two-point model with neutrals 3. Gas puff fueling simulations Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 30 / 37

74 Gas puff/fueling simulations Open and closed field lines Various gas puff locations (hfs, bot, lfs, top) Small constant main wall recycling n 0 = cm 3, T 0 = 20eV, q = 3.87, ρ 1 = 500, a 0 = 200ρ s Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 31 / 37

75 Neutral density Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 32 / 37

76 Ionization Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 33 / 37

77 Radial ExB flow outward/inward flow Ballooning outward transport at the low field side Inward fueling at the high field side Robust feature independent of gas puff location Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 34 / 37

78 Questions that we can address How is the temperature at the limiter related to main plasma parameters? How is the plasma fueled? How do neutrals affect plasma turbulence? SOL width? Heat flux? How do diagnostic gas puffs affect the SOL? Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 35 / 37

79 Poloidal ExB flow Poloidal rotation due to radial electric field Shearing of the turbulent eddies Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 36 / 37

80 Conclusions Plasma turbulence at the periphery and interaction with neutrals are crucial issues on the way to fusion electricity GBS is now able to simulate this complex interplay self-consistently Development of a more refined two-point model, in agreement with GBS Initial study of plasma fueling due to ionization and radial flows, and of plasma poloidal rotation. Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 37 / 37

81 Reaction rates - Stangeby Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 1 / 4

82 Reaction rates - openadas σv (m 3 s 1 ) CX ion, n 0 =1e+18 rec, n 0 =1e+18 ion, n 0 =1e+20 rec, n 0 =1e T e,t i (ev) Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 4

83 Timescales T 0 (ev) n 0 (m 3 ) τ turbulence (s) τ nnloss (s) λ mfp (m) 1 1e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-05 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 3 / 4

84 The model in steady state Steady state, f n t = 0, first approach Valid if τ neutral losses < τ turbulence e.g. T e = 20eV, n 0 = m 3 τ neutral losses ν 1 eff s τ turbulence R 0 L p /c s s Otherwise: time dependent model Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 4 / 4

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