Impact of neutral atoms on plasma turbulence in the tokamak edge - - PowerPoint PPT Presentation

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 29.08. - 02.09. 2016

Introduction Model Two-point model Fueling Conclusions

Physics at the periphery of a fusion plasma

◮ Toroidal limiter

Core Edge SOL Limiter

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

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

◮ Density source ◮ Energy sink

Plasma Neutrals Ionization

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

Introduction Model Two-point model Fueling Conclusions

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

◮ Density source ◮ Energy sink

Plasma Neutrals Ionization

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

Introduction Model Two-point model Fueling Conclusions

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

◮ Density source ◮ Energy sink

◮ Recycling

Plasma Neutrals Ionization

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 2 / 37

Introduction Model Two-point model Fueling Conclusions

Movie

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

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

• 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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

◮ d/dt ≪ ωci,k2 ⊥ ≫ k2

• Christoph Wersal - SPC

Neutrals in the turbulent tokamak edge 6 / 37

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

◮ d/dt ≪ ωci,k2 ⊥ ≫ k2

• ◮ Drift-reduced Braginskii equations

n,Ω,ve,v,i,Te,Ti

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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

◮ d/dt ≪ ωci,k2 ⊥ ≫ k2

• ◮ Drift-reduced Braginskii equations

n,Ω,ve,v,i,Te,Ti

◮ Flux-driven, no separation between

equilibrium and fluctuations

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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

◮ d/dt ≪ ωci,k2 ⊥ ≫ k2

• ◮ Drift-reduced Braginskii equations

n,Ω,ve,v,i,Te,Ti

◮ Flux-driven, no separation between

equilibrium and fluctuations

◮ Kinetic neutral equation

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

Introduction Model Two-point model Fueling Conclusions

Modeling the periphery

◮ High plasma collisionality, local

Maxwellian

◮ d/dt ≪ ωci,k2 ⊥ ≫ k2

• ◮ Drift-reduced Braginskii equations

n,Ω,ve,v,i,Te,Ti

◮ 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

Introduction Model Two-point model Fueling Conclusions

Fluid plasma model and interaction with neutrals

∂n ∂t =−ρ−1

⋆

[φ,n]+ 2 B [C(pe)−nC(φ)]−∇(nve)+Dn(n)+Sn+nnνiz −nνrec (1) ∂ ˜ ω ∂t =−ρ−1

⋆

[φ, ˜ ω]−vi ∇ ˜ ω + B2 n ∇j + 2B n C(p)+D ˜

ω ( ˜

ω)− nn n νcx ˜ ω (2) ∂ve ∂t =−ρ−1

⋆

[φ,ve]−ve∇ve + mi me

• ν

j n +∇φ − 1 n ∇pe −0.71∇Te

• +Dve (ve)+ nn

n (νen +2νiz)(vn −ve) (3) ∂vi ∂t =−ρ−1

⋆

[φ,vi ]−vi ∇vi − 1 n ∇p +Dvi (vi )+ nn n (νiz +νcx )(vn −vi ) (4) ∂Te ∂t =−ρ−1

⋆

[φ,Te]−ve∇Te + 4Te 3B 1 n C(pe)+ 5 2 C(Te)−C(φ)

• + 2Te

3 0.71 n ∇j −∇ve

• (5)

+DTe (Te)+D

Te (Te)+STe + nn

n νiz(− 2 3 Eiz −Te + me mi ve(ve − 4 3 vn))+ nn n νen me mi 2 3 ve(vn −ve)) ∂Ti ∂t =−ρ−1

⋆

[φ,Ti ]−vi ∇Ti + 4Ti 3B 1 n C(pe)−τ 5 2 C(Ti )−C(φ)

• + 2Ti

3

• (vi −ve)

∇n n −∇ve

• (6)

+DTi (Ti )+D

Ti (Ti )+STi + nn

n (νiz +νcx )(Tn −Ti + 1 3 (vn −vi )2) ∇2

⊥φ =ω, ρ⋆ = ρs/R, ∇f = b0 ·∇f, ˜

ω = ω +τ∇2

⊥Ti , p = n(Te +τTi )

+ boundary conditions + kinetic neutral equation Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 7 / 37

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

The density equation

∂n ∂t =−ρ−1

⋆ [φ,n]+ 2

B [C(pe)−nC(φ)] −∇(nve) (7) +Sn +nnνiz −nνrec +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

Introduction Model Two-point model Fueling Conclusions

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 55 123014

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

Introduction Model Two-point model Fueling Conclusions

The neutral model

∂fn ∂t + v · ∂fn ∂ x = −νizfn −νcx(fn −nnΦi)+νrecniΦi (8) νiz = neveσiz(ve), νcx = nivrelσcx(vrel) νrec = neveσrec(ve), Φi = fi/ni

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

Introduction Model Two-point model Fueling Conclusions

The neutral model

∂fn ∂t + v · ∂fn ∂ x = −νizfn −νcx(fn −nnΦi)+νrecniΦi (8) νiz = neveσiz(ve), νcx = nivrelσcx(vrel) νrec = neveσrec(ve), Φi = fi/ni

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

Introduction Model Two-point model Fueling Conclusions

The neutral model

∂fn ∂t + v · ∂fn ∂ x = −νizfn −νcx(fn −nnΦi)+νrecniΦi (8) νiz = neveσiz(ve), νcx = nivrelσcx(vrel) νrec = neveσrec(ve), Φi = fi/ni

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

Introduction Model Two-point model Fueling Conclusions

The neutral model

∂fn ∂t + v · ∂fn ∂ x = −νizfn −νcx(fn −nnΦi)+νrecniΦi (8) νiz = neveσiz(ve), νcx = nivrelσcx(vrel) νrec = neveσrec(ve), Φi = fi/ni

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

Introduction Model Two-point model Fueling Conclusions

The neutral model

∂fn ∂t + v · ∂fn ∂ x = −νizfn −νcx(fn −nnΦi)+νrecniΦi (8) νiz = neveσiz(ve), νcx = nivrelσcx(vrel) νrec = neveσrec(ve), Φi = fi/ni Boundary conditions

(v⊥ in respect to the surface; θ between v and normal vector to the surface)

• dv v⊥fn(

xw, v)+ui⊥ni = 0 (9) fn( xw, v) ∝ cos(θ)emv2/2Tw for v⊥ > 0 (10)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 10 / 37

Introduction Model Two-point model Fueling Conclusions

Boundary conditions for the neutrals

◮ Partial reflection at the limiters ◮ Window averaged particle flux conservation at the outer

boundary

−200 −100 100 200 −200 −100 100 200 nn R/ ρs Z/ ρs 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −200 −100 100 200 −200 −100 100 200 nn R/ ρs Z/ ρs 0.02 0.04 0.06 0.08 0.1 0.12 0.14

◮ Gas puffs and neutral background

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 11 / 37

Introduction Model Two-point model Fueling Conclusions

Further simplifications

◮ Separation of time scales

◮ The neutrals’ time of life is typically shorter than the

turbulent time scale

◮ Te = 20eV, n0 = 5·1013cm−3

→ τneutral losses ≈ ν−1

eff ≈ 5·10−7s

→ τturbulence ≈

• R0Lp/cs0 ≈ 2·10−6s

◮ Assume ∂fn/∂t ≈ 0 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 12 / 37

Introduction Model Two-point model Fueling Conclusions

Further simplifications

◮ Separation of time scales

◮ The neutrals’ time of life is typically shorter than the

turbulent time scale

◮ Te = 20eV, n0 = 5·1013cm−3

→ τneutral losses ≈ ν−1

eff ≈ 5·10−7s

→ τturbulence ≈

• R0Lp/cs0 ≈ 2·10−6s

◮ Assume ∂fn/∂t ≈ 0

◮ Plasma anitrosopy

◮ The plasma elongation along the field lines is much longer

than the typical neutral mean free path

◮ Assume ∇fn ≈ 0 Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 12 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v fn(x,v) (12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v fn(x,v) =

x

0 dx′

(12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v x' fn(x,v) =

x

0 dx′ νcx(x′)nn(x′)Φi(x′,v)

v (12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v x' fn(x,v) =

x

0 dx′ νcx(x′)nn(x′)Φi(x′,v)

v e− 1

v

x

x′ dx′′ (νcx(x′′)+νiz(x′′))

(12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v x' fn(x,v) =

x

0 dx′ νcx(x′)nn(x′)Φi(x′,v)

v e− 1

v

x

x′ dx′′ (νcx(x′′)+νiz(x′′))

+fw(v)e− 1

v

x

0 dx′′ (νcx(x′′)+νiz(x′′))

(12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

Solution of neutral eq. with method of characteristics

Example in 1D, no recombination, v > 0 and a wall at x = 0 v ∂fn ∂x = νcxnnΦi−(νiz +νcx)fn (11) x v x' fn(x,v) =

x

0 dx′ νcx(x′)nn(x′)Φi(x′,v)

v e− 1

v

x

x′ dx′′ (νcx(x′′)+νiz(x′′))

+fw(v)e− 1

v

x

0 dx′′ (νcx(x′′)+νiz(x′′))

(12)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 13 / 37

Introduction Model Two-point model Fueling Conclusions

An equation for the density distribution

By imposing

• fn dv = nn

(13) we get a linear integral equation for nn(x) nn(x) =

x

0 dx′ nn(x′)

∞

0 dv νcx(x′)Φi(x′,v)

v e−

deff νeff (x−x′) v

(14) +contribution by v < 0 +nw(x)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 14 / 37

Introduction Model Two-point model Fueling Conclusions

The GBS code, a tool to simulate SOL turbulence

◮ Evolves scalar fields in 3D geometry

n,Ω,ve,v,i,Te,Ti

◮ Kinetic neutral physics ◮ Limiter geometry ◮ Open and closed field-line region ◮ Sources Sn and ST mimic plasma

• utflow from the core

◮ (Divertor geometry)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 15 / 37

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

• 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

Introduction Model Two-point model Fueling Conclusions

The two-point model

Core Edge SOL Limiter Target Upstream

◮ Relation between

upstream and target plasma properties

◮ Widely used experimentally

for a quick estimate

◮ Derived from 1D model

along field lines

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 18 / 37

Introduction Model Two-point model Fueling Conclusions

The SOL unrolled

Main Plasma SOL Limiter Wall LCFS

SOL

Main Plasma Limiter Limiter Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

Introduction Model Two-point model Fueling Conclusions

The SOL unrolled

Main Plasma SOL Limiter Wall LCFS

SOL

Main Plasma Limiter Limiter Upstream T arget T arget Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

Introduction Model Two-point model Fueling Conclusions

The SOL unrolled

Main Plasma SOL Limiter Wall LCFS

SOL

Main Plasma Limiter Limiter Upstream T arget T arget

s

◮ Parallel plasma dynamics projected along poloidal

coordinate

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

Introduction Model Two-point model Fueling Conclusions

The SOL unrolled

Main Plasma SOL Limiter Wall LCFS

SOL

Main Plasma Limiter Limiter

s

◮ Parallel plasma dynamics projected along poloidal

coordinate

◮ Plasma and energy outflowing from the core are modeled

with prescribed Sn and SQ

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 19 / 37

Introduction Model Two-point model Fueling Conclusions

The basic two-point model

Q =

• SQds = Qcond +Qconv

(15)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

Introduction Model Two-point model Fueling Conclusions

The basic two-point model

Q =

• SQds = Qcond +Qconv

(15) Qcond = −χe0T 5/2

e

dTe dz (16)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

Introduction Model Two-point model Fueling Conclusions

The basic two-point model

Q =

• SQds = Qcond +Qconv

(15) Qcond = −χe0T 5/2

e

dTe dz (16) Qconv = ce0ΓTe (17) Γ = nv =

• Snds

(18)

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

Introduction Model Two-point model Fueling Conclusions

The basic two-point model

Q =

• SQds = Qcond +Qconv

(15) Qcond = −χe0T 5/2

e

dTe dz (16) Qconv = ce0ΓTe (17) Γ = nv =

• Snds

(18) Boundary conditions

◮ Upstream: dTe/ds = 0 ◮ At the limiter: QL = γeΓLTeL,

γe ≈ 5

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

Introduction Model Two-point model Fueling Conclusions

The basic two-point model

Q =

• SQds = Qcond +Qconv

(15) Qcond = −χe0T 5/2

e

dTe dz (16) Qconv = ce0ΓTe (17) Γ = nv =

• Snds

(18) Boundary conditions

◮ Upstream: dTe/ds = 0 ◮ At the limiter: QL = γeΓLTeL,

γe ≈ 5

SQ,Sn ⇓ Te,u Te,t

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 20 / 37

Introduction Model Two-point model Fueling Conclusions

Simulations with different densities

n0 = 5·1012cm−3 n0 = 5·1013cm−3

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 21 / 37

Introduction Model Two-point model Fueling Conclusions

Simulations with different densities

n0 = 5·1012cm−3 n0 = 5·1013cm−3

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 22 / 37

Introduction Model Two-point model Fueling Conclusions

Simulations with different densities

n0 = 5·1012cm−3 n0 = 5·1013cm−3

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 22 / 37

Introduction Model Two-point model Fueling Conclusions

Poloidal profiles of electron temperature

• L

L

s

0.2 0.4 0.6 0.8

Te

n0 = 5· 10

12cm 3

n0 = 5· 10

13cm 3

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 23 / 37

Introduction Model Two-point model Fueling Conclusions

Poloidal profiles of electron temperature

• L

L

s

0.2 0.4 0.6 0.8

Te

n0 = 5· 10

12cm 3

n0 = 5· 10

13cm 3

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 23 / 37

Introduction Model Two-point model Fueling Conclusions

Temperature ratio upstream to target

1 1.5 2

Te,u/Te,t (GBS)

1 1.2 1.4 1.6 1.8 2

Te,u/Te,t (tpm) basic model

5 · 1013, no nn 5 · 1013 5 · 1012, no nn 5 · 1012 5 · 1013 20x480 5 · 1013, Eiz = 30

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 24 / 37

Introduction Model Two-point model Fueling Conclusions

A more refined two-point model

◮ Obtain an electron heat equation in quasi-steady state

3 2Te ∂n ∂t + 3 2n∂Te ∂t ≈ 0 (19)

◮ Assume ve, ≈ vi, and neglect small terms (e.g., D⊥Te) ◮ Combine perpendicular transport terms into SQ

∇ 5 2nvTe

• − χe0∇
• T 5/2

e

∇Te

• −v∇(nTe)

(20) = SQ+Sneutrals with Sneutrals = −nnνiz(Te)Eiz and χe0 = 3/2¯ nκe

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 25 / 37

Introduction Model Two-point model Fueling Conclusions

Further assumptions and relations

◮ v is linear from −cs to cs ◮ cs =

• Te,t +Ti,t ≈
• 2Te,t

◮ nv =

[Sn +nnνiz(Te)]ds

◮ nn is decaying exponentially from limiter with λmfp

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 26 / 37

Introduction Model Two-point model Fueling Conclusions

Three external input quantities

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

Introduction Model Two-point model Fueling Conclusions

Three external input quantities

◮ Perpendicular heat source, SQ

• L

L

s

• 0.05

0.05 0.1 0.15

SQ

GBS cos fit

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

Introduction Model Two-point model Fueling Conclusions

Three external input quantities

◮ Perpendicular heat source, SQ ◮ Perpendicular particle source, Sn

• L

L

s

• 0.02

0.02 0.04 0.06 0.08

Sn

GBS cos fit

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

Introduction Model Two-point model Fueling Conclusions

Three external input quantities

◮ Perpendicular heat source, SQ ◮ Perpendicular particle source, Sn ◮ Ionization particle source, Siz

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Introduction Model Two-point model Fueling Conclusions

Three external input quantities

◮ Perpendicular heat source, SQ ◮ Perpendicular particle source, Sn ◮ Ionization particle source, Siz

SQ,Sn,Siz ⇓ Te,u Te,t

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 27 / 37

Introduction Model Two-point model Fueling Conclusions

Temperature ratio upstream to target

1 1.5 2

Te,u/Te,t (GBS)

1 1.2 1.4 1.6 1.8 2

Te,u/Te,t (tpm) basic model

5 · 1013, no nn 5 · 1013 5 · 1012, no nn 5 · 1012 5 · 1013 20x480 5 · 1013, Eiz = 30

1 1.2 1.4 1.6 1.8 2

Te,u/Te,t (GBS)

1 1.2 1.4 1.6 1.8 2

Te,u/Te,t (tpm) full model

5 · 1013, no nn 5 · 1013 5 · 1012, no nn 5 · 1012 5 · 1013 20x480 5 · 1013, Eiz = 30

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Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

• 1. Modeling the periphery
• 2. A refined two-point model with neutrals
• 3. Gas puff fueling simulations

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Introduction Model Two-point model Fueling Conclusions

Gas puff/fueling simulations

◮ Open and closed field lines ◮ Various gas puff locations

(hfs, bot, lfs, top)

◮ Small constant main wall

recycling

◮ n0 = 1013cm−3, T0 = 20eV,

q = 3.87, ρ−1

⋆

= 500, a0 = 200ρs

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Introduction Model Two-point model Fueling Conclusions

Neutral density

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

Introduction Model Two-point model Fueling Conclusions

Ionization

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Introduction Model Two-point model Fueling Conclusions

Radial ExB flow

◮ outward/inward

flow

◮ Ballooning

• utward 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

Introduction Model Two-point model Fueling Conclusions

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

Introduction Model Two-point model Fueling Conclusions

Poloidal ExB flow

◮ Poloidal rotation

due to radial electric field

◮ Shearing of the

turbulent eddies

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Introduction Model Two-point model Fueling Conclusions

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.

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Reaction rates - Stangeby

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Reaction rates - openADAS

10 10

1

10

2

10

−19

10

−18

10

−17

10

−16

10

−15

10

−14

10

−13

Te, Ti (eV) σv (m3s−1) CX ion, n0=1e+18 rec, n0=1e+18 ion, n0=1e+20 rec, n0=1e+20

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Timescales

T0(eV) n0(m−3) τturbulence(s) τnnloss(s) λmfp(m) 1 1e+17 1.0e-05 1.4e-03 2.5e+00 1 1e+18 1.0e-05 1.4e-04 2.5e-01 1 1e+19 1.0e-05 1.4e-05 2.5e-02 1 1e+20 1.0e-05 1.4e-06 2.5e-03 1 1e+21 1.0e-05 1.4e-07 2.5e-04 20 1e+17 2.3e-06 2.6e-04 4.4e-01 20 1e+18 2.3e-06 2.5e-05 4.3e-02 20 1e+19 2.3e-06 2.4e-06 4.1e-03 20 1e+20 2.3e-06 2.2e-07 3.7e-04 20 1e+21 2.3e-06 1.8e-08 3.1e-05 50 1e+17 1.4e-06 1.6e-04 2.8e-01 50 1e+18 1.4e-06 1.6e-05 2.7e-02 50 1e+19 1.4e-06 1.5e-06 2.6e-03 50 1e+20 1.4e-06 1.4e-07 2.4e-04 50 1e+21 1.4e-06 1.2e-08 2.0e-05

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 3 / 4

The model in steady state

Steady state, ∂fn

∂t = 0, first approach ◮ Valid if τneutral losses < τturbulence ◮ e.g. Te = 20eV, n0 = 5·1019m−3

τneutral losses ≈ ν−1

eff ≈ 5·10−7s

τturbulence ≈

• R0Lp/cs0 ≈ 2·10−6s

◮ Otherwise: time dependent model

Christoph Wersal - SPC Neutrals in the turbulent tokamak edge 4 / 4