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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France1
Plasma-wall transition in magnetized plasmasStatic and transient aspects
Giovanni Manfredi
Institut de Physique et Chimie des Matériaux de Strasbourg
Centre National de la Recherche Scientifique
Strasbourg, France
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France2
Motivations
• Magnetically confined fusion plasmas
– Limiters and divertors in tokamaks
– Problems: erosion, sputtering, heat load
Closed magneticsurfaces
Separatrix
Langmuir probe
• Diagnostics in plasmas
– Langmuir probes, retarding field analyzers (RFA), …
– Plasma-probe interaction can lead to significant errors in
the measurements
• Plasma-assisted surface treatment
– Etching, thin-film deposition
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France3
Industrial applications
Fluorescent lamp (“neon tube”)
Plasma screen
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France4
EE
Sheath formation – unmagnetized case
E
λDe
Debye
sheath
Collisional
presheath
λmfp
ni = ne= n0
Density
Debye sheath (DS):
- nonneutral region- width ~ λDe- Bohm criterion: ions velocity at DS edge > Cs
Collisional presheath (CP):
- quasi-neutral region
- width ~ λmfp: mean free path- ion acceleration towards the wall
x
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France5
Debye sheath ― Bohm’s criterion
• Criterion for the stability of the electrostatic Debye sheath (DS).
• What is the minimum velocity at the entrance of the DS sheath?
• Simple fluid model:
0=+ x
)u (n
t
n iii
∂∂
∂∂
x
Φ
m
e
x
uu
t
u
i
ii
i
∂∂
∂∂
∂∂
−=+
)(0
2
2
ei nne
x
Φ −−=
ε∂
∂
=
eBe
Tk
eΦ n(x,t) n exp0
• We look for a stationary solution: ∂/∂t = 0
si
eB cmTk
u ≡≥0Φ
um
Tk
λd x
Φd
i
eB
D
11
20
22
2
−=
• Substituting into Poisson’s equation and linearizing ,
one obtains
2/1
2
00
21
−
−=
um
eΦn
n
i
i
Cold ions: Ti << Te
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France6
Collisional presheath
• Fluid equations for the ions with source term S(x) due to ionization (cold neutrals)
• Assuming quasineutrality:
• We obtain:
• Singularity at M = 1, which defines the Debye sheath edge (DSE)
• At the DSE, the assumption of quasineutrality breaks down
– Need to use full Poisson’s equation
)()( xSundx
dii =
i
i
i
ii
n
xSu
dx
d Φ
m
e
dx
d uu
)(−−=
==
eB
eiTk
ennn
φexp0
M = ui /cs : Mach number
Cold ions: Ti << Te
≈ λmfp
• Example of source term:
• Exact solution:
Boundary cond.
Plasma: x=0 ; M=0
DSE: x=L ; M=1
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France7
• By eliminating S from the full fluid equations, we find:
• At the DS entrance M = 1 and therefore:
Collisional presheath
DSE PlasmaWall
CPDS
• Electric potential on the wall
– Ambipolarity: ion flux = electron flux
=
Assume half-Maxwellian
Ion flux constant in DS
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France8
Ions velocity and density on the wall
• In the DS, conservation of energy holds:
• Also, conservation of ion momentum:
• It follows that:
wallDSE φφφ −=∆
( ) ( )DSE
2DSE
wall
2walli
22φφ e
ume
um i
ii +=+
wall DSE
DSE PlasmaWall
CPDS
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France9
Ion and electron densities, electric potential, and ion velocity at various locations:
Deuterium plasma with Ti = Te
Summary of fluid plasma-wall transition
Kinetic simulations
DSE
Fluid estimations
DSE PlasmaWall
CPDS
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France10
Kinetic modelling I.
Spatial density
Mean velocity
� Phase-space distribution function
= Number of particles in the phase-space volume centered on .
� Velocity moments of the distribution function
Temperature
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France11
Kinetic modelling II.
� Vlasov equation for the ions
Conservation equationin the phase space
� Boltzmann equilibrium for the electrons
⇔ the electrons thermalize much faster than the ions
� Poisson’s equation
Self-consistent system closed by Poisson’s equation ⇒ nonlinearity
Collisions/Ionization ⇒ relaxation to Maxwellian ƒ0(v)ν-1 : typical relaxation time
1D space (x coordinate: normal to the wall) + 3D velocity
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France12
Kinetic modelling III.
• Numerical method: Vlasov Eulerian code
– Meshing of the full phase space (4D for magnetized transition)
– Low level of numerical noise
• Strategy
– Initialize homogeneous Maxwellian distribution for ions: fi= fM (v)
– Let it evolve self-consistently until stationary state appears
– Corollary : no need for very accurate time-stepping technique
• Disparate spatial scales
– λDe << λcoll– Use inhomogeneous grid:
∆x2 ≈ λcoll>> ∆x1
∆x1 ≈ λDe
g(s)
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France13
Unmagnetized plasma-wall transition
Typical case : Te/Ti = 25 ; νννν / ωωωωpi = 10-4
λλλλcoll ≈ 104 λλλλDe
DS
CP
Fluid Bohm’s criterion:
Kinetic Bohm’s criterion:
= cs–2
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France14
Mach number at the DS entrance
• Ordinary Bohm criterion at the DS
entrance: M > 1
– Not always satisfied, particularly for
large collision rates
Mach number at DSE
Te / Ti
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France15
Mach number at the DS entrance
• Role of the distribution tail
– Slow particles that originate from
collisions
– Bohm criterion satisfied by peak
velocity
Ion velocity distribution
DSE–
–
–
–
– – – – – – –
• Ordinary Bohm criterion at the DS
entrance: M > 1
– Not always satisfied, particularly for
large collision rates
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France16
Ion temperature profile
Ion temperature from plasma to wall
Peak at 250 λDe
zoom
Ion velocity distribution from plasma to wall
- - - - - - -
Max Ti: 250 λDe
DSE:19λDeWall
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France17
Experimental validation: ion temperature profile
• Series of temperature measurements in the presheath
Oksuz and Hershkowitz, Plasma Sources Sci. Technol. 14, 201 (2003)
• Experimental conditions: Te/Ti = 25 ; φwall = -30V ; νννν / ωωωωpi = 10-4
Ion temperature profile
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France18
Experimental validation: ion velocity distribution I.
• Low-pressure discharge
• Argon plasma
• LIF measurements (laser-induced
fluorescence)
PAr
= 6 × 10-4 TorrnAr
∼ 2 × 1013 cm-3
ne0 ∼ 109 cm-3
Te0 ∼ 1.8 eV
Ti0 ∼ 0.05 eV
Ti = 1.4 eVTi = 2.26 eV
velocity (km / s)
velocity (km / s) velocity (km / s)
0 2 4 6
velocity (km / s)
-2 -1 0 1 2 3
x = 10 mmx = 5 mm
0 2 4 61 2 3 4 5 6 7
Ti = 0.24 eVTi = 0.86 eV
x = 1 mm x = 3 mm
dis
trib
ution function (A. U
.)dis
trib
ution function (A. U
.)
dis
trib
ution function (A. U
.)dis
trib
ution function (A. U
.)
Bachet et al., Phys. Plasmas 2, 1782 (1995)
Distance from wall
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France19
Experimental validation: ion velocity distribution II.
• Poor agreement with measured ion distributions for x < 10 mm
– No long tail observed near wall
• Large acceleration of ions in Debye sheath (λD ∼ 0.4 mm)
• The ion distribution changes on a scale smaller than the resolution of
the experimental apparatus (~ 2 mm)
-1 0 1 2 3 4 5 6 7
01310 0.4
positions
(mm)
400
0.0
0.2
0.4
0.6
0.8
1.0
velocity (km / s)
fi (v)
Ion distribution at different positions
Ti = 1.4 eVTi = 2.26 eV
velocity (km / s)
velocity (km / s) velocity (km / s)
0 2 4 6
velocity (km / s)-2 -1 0 1 2 3
x = 10 mmx = 5 mm
0 2 4 61 2 3 4 5 6 7
Ti = 0.24 eVTi = 0.86 eV
x = 1 mm x = 3 mm
dis
trib
ution function (A.
U.)
dis
trib
ution function (A.
U.)
dis
trib
ution function (A.
U.)
dis
trib
ution function (A.
U.)
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France20
Experimental validation: ion velocity distribution II.
• Convolution of the ion distribution with some
‘apparatus functions’ (Gaussian or step function)
having width ~ 2 mm (apparatus resolution).
-1 0 1 2 3 4 5 6 7
1310positions
(mm)
1 2 3 4
0.2
0.4
0.6
Apparatus
functions
0
0.0
0.2
0.4
0.6
0.8
1.0
velocity (km / s)
fi (v)
Bachet et al., Phys. Plasmas 2, 1782 (1995)
dis
trib
ution function (A. U
.)dis
trib
ution function (A. U
.)
dis
trib
ution function (A. U
.)dis
trib
ution function (A. U
.)
x = 1 mm
x = 10 mm
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France21
y
x
z
B
Vx
Vy
Bαααα
BVy
ααααVx
Debye sheath ∼ λD
Collisional presheath (CP):- quasi-neutral - width ~ λcoll- ion acceleration along magnetic lines
Collisional presheath ∼ λcoll
Sheath formation in a magnetized plasma
Magnetic presheath (MP):- quasi-neutral - width ~ rL- ion redirection toward the wall
Magnetic presheath ∼ rL
Ordering :
λλλλDe << rL << λλλλcoll
Debye length
Ion Larmorradius
Ion–neutralmean free path
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France22
Magnetized plasma-wall transition: phase-space
αααα = 40° ωωωωci / ωωωωpi = 0.01 Te / Ti =10 νννν / ωωωωpi = 10-3
Wall
Vx
Vz
Vy
Bαααα
PlasmaV//V⊥⊥⊥⊥
X = 0
Vy
B
ααααVx
Debye sheath∼∼∼∼ λλλλD
Magneticpresheath ∼∼∼∼ rL
Collisionnalpresheath ∼∼∼∼ λλλλcoll
plasma
wall
CP
DS
MP
DS edge
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France23
Magnetized Bohm’s criterion
si
ieBx cm
TTk≡
+>
)(v at DS edge
Mach number Debye sheath edge
ωωωω = ωωωωci / ωωωωpi
Bohm’s criterion not satisfied for:
– Large magnetic fields
– Grazing incidence (α small)
Competition between
– Electric field
– Magnetic field
B
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France24
Magnetic presheath (MP) edge: Chodura’s criterion
si
ieB cmTTk
≡+
>)(
v //at MP edge
• Chodura’s criterion not satisfied for:
– Weak magnetic fields (ω << ωpi)
– Large collision rate (υ >> ωpi)
• Competition between:
– B field
– collisions
Parallel Mach number
ωωωω = ωωωωci / ωωωωpi B
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France25
Magnetic presheath width
• Magnetic presheath width ∝ ion Larmor radius ∝ 1/B
• At MP edge, ions start being collected at the wall
Vy B
αααα
Vx
rL
rLcos αααα
Theor. estimate: τ = Te / Ti
MP width
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France26
E X B drift
• The E X B drift is directed along the z direction
• < Vz > and VE coincide in the collisional presheath,
but start diverging in the magnetic presheath
• Guiding-center approach invalid in the MP and
DS
Profile of the velocities Vz and VE
y
x
z
B
E
E X B
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France27
Wall sputtering and erosion
Sputtering yield Y depends on :
• Angle of incidence on the wall: θ• Kinetic energy: Ekin
Incident ion
Ejectedparticle
θ
α = angle of incidence of the magnetic field
F (θ, Ekin) = distribution function in angle/energy variables
ω = ωci / ωpi
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France28
Results: angle of incidence and kinetic energy on the wall
Average angle of incidence
on the wall, <θ>
Average kinetic energy on the wall, <Ekin>
α = angle of incidence of magnetic field
α = angle of incidence of magnetic field
ω = ωci / ωpi
Page 29
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France29
Results: angle of incidence and kinetic energy on the wall
α = angle of incidence of magnetic field
α = angle of incidence of magnetic field
B
αθ v
Vx
Vy
θ > α !
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France30
Sputtering yield on the wall
α = angle of incidence of magnetic field
ω = ωci / ωpi
Page 31
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France31
Conclusion – plasma sheaths
• Kinetic model for ion population – Vlasov code
• Full description of the magnetized plasma-wall transition
– Colllisional presheath, magnetic presheath, Debye sheath
• Experimental validation
• Computed ion distributions allow calculation of:
– Energy and angle of incidence of ions on wall
– Sputtering yield
– Particles and heat fluxes on wall
References
1. F. Valsaque, G. Manfredi, J.P. Gunn, E. Gauthier, Phys. Plasmas 9, 1806 (2002).
2. S. Devaux, G. Manfredi, Phys. Plasmas 13, 083504 (2006).
3. S. Devaux, G. Manfredi, Plasma Phys. Control. Fus. 50, 025009 (2008).
• Next: transient (time-dependent) plasma-wall interactions
– Edge localized modes (ELMs) in tokamaks
– Transient response of sheaths
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France32
Vlasov modelling of Edge Localized Modes (ELMs)
• ITER : high energy heat flux (10 MW / m2) mainly supported by divertor
plates
• « Divertor » configuration
– Specially conceived surface that collects the most energetic particles
– Tungsten, Carbon, …
Closed magneticsurfaces
Separatrix
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France33
Divertor (JET)
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France34
energy density / MJm-2
0.5 1.0 1.5
negligible
erosion
meltingof
tileedges
meltingof the
fulltilesurface
(no droplet
ejection)
dropletejection
and
bridgingof tiles
after50 shots
W
energy density / MJm-2
0.5 1.0 1.5
negligible
erosion
erosionstarts
at PFC corners
PAN fibre
erosionof
flatsurfaces
after100 shot
significant
PAN fibre
erosion
after50 shots
PAN fibre
erosion
after10 shots
CFC
ITER adopted 0.5 MJ/m2 for the maximum allowed ELM energy load in 250 µs
Transient heat load limits in ITER
Courtesy W. Fundamenski
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INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France35
Edge-localized modes (ELMS)
• Violent events located at the tokamak edge
• Burst of energetic plasma particles that cross the
separatrix
• These particles are transported along the direction
parallel to B
• Finally, they hit the divertor targets releasing large
amounts of energy
R
Tped = 0.5 – 5 keV
nped = 0.15 – 15 × 1019 m-3
∆WELM = 0.025 – 2.5 MJ
tELM = 200 µs Post ELM
150 µs
T, n
Page 36
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France36
One-dimensional kinetic modelling of ELMs
2L|| = 60 m
dR
x
S(x)
g(t)
Source spatial and temporal profiles
Page 37
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France37
The “asymptotic-preserving” method
• Poisson’s equation in dimensionless units
• Becomes singular for λ → 0
• It can be shown that Poisson’s equation can be replaced with the equation:
which is not singular.
• The asymptotic-preserving method allows us to use:
– ∆t > 1/ωpe
– ∆x > λD P. Crispel et al., J. Comput. Phys. 223, 208 (2007)
Page 38
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France38
Particles and energy fluxes on divertor targets
Particles flux
Energy flux
Page 39
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France39
Results — instantaneous source
Particle fluxes Energy fluxes
Ions
Electrons
Ions, free streaming
Experiments
)()( :profile temporalSource ttg δ=
Page 40
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France40
Phase-space dynamics
Electrons phase space
Ions phase spaceV
V
X
fluxes
Page 41
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France41
Results — time-distributed source
• Comparison between Vlasov, PIC, and fluid codes
� ELM duration = 200 µs
� No background plasma; no collisions
• Tped = 1.5 keV
• nped = 5e19 m−3
• WELM = 0.4 MJ
Ion energy flux Electron energy flux Total flux
Vlasov
PIC
Fluid
E. Havlickova et al., submitted to PPCF
Page 42
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France42
Scaling for the parallel heat flux
E. Havlickova et al., submitted to PPCF
Page 43
INRIA – AE Fusion, Paris 26-30 September 2011 G. Manfredi, IPCMS, Strasbourg, France43
Conclusion – transient processes
• Modelling of transient events, such as ELMs
• Future extensions:
– Perpendicular dynamics
– Modelling of collisions and other non-ideal processes
– Multi species plasmas (H + D, Ar + Xe)
References
1. G. Manfredi, S. Hirstoaga and S. Devaux, Plasma Phys. Control. Fusion 53, 015012 (2011).
2. E. Havlickova et al., submitted to PPCF