Plasma-wall transition in magnetized plasmas · PDF filePlasma-wall transition in magnetized plasmas ... Paris 26-30 September2011 2 G. Manfredi, IPCMS, Strasbourg, France Motivations

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


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


Industrial applications

Fluorescent lamp (“neon tube”)

Plasma screen


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


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


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


• 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


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


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


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


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


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)


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


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


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


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


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


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


.)

dis

trib


.)dis

trib


.)

Bachet et al., Phys. Plasmas 2, 1782 (1995)

Distance from wall


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.)


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


.)dis

trib


.)

dis

trib


.)dis

trib


.)

x = 1 mm

x = 10 mm


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


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


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


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


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


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


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


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


ω = ωci / ωpi


Results: angle of incidence and kinetic energy on the wall



B

αθ v

Vx

Vy

θ > α !


Sputtering yield on the wall


ω = ωci / ωpi


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


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


Divertor (JET)


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


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


One-dimensional kinetic modelling of ELMs

2L|| = 60 m

dR

x

S(x)

g(t)

Source spatial and temporal profiles


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)


Particles and energy fluxes on divertor targets

Particles flux

Energy flux


Results — instantaneous source

Particle fluxes Energy fluxes

Ions

Electrons

Ions, free streaming

Experiments

)()( :profile temporalSource ttg δ=


Phase-space dynamics

Electrons phase space

Ions phase spaceV

V

X

fluxes


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


Scaling for the parallel heat flux

E. Havlickova et al., submitted to PPCF


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

Documents

Plasma-wall transition in magnetized plasmas · PDF filePlasma-wall transition in magnetized plasmas ... Paris 26-30 September2011 2 G. Manfredi, IPCMS, Strasbourg, France Motivations