A kinetic neutral atom model for tokamak SOL turbulence simulations
C. Wersal, P. Ricci, F.D. Halpern, F. Riva
Ecole Polytechnique Fédérale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas, CH-1015 Lausanne, Switzerland
The Global Braginskii Solver (GBS) code
SOL operating regimes
I
2 k 2, d/dt ω
Two fluid drift-reduced Braginskii equations [Ricci et al., PPCF 2012], k⊥
ci
k
2
=−
+ [C(pe ) − nC(φ)] − ∇k(nvke ) + Dn (n) + Sn + nn nriz − n2rrec
B
2B
B2
−1
= − ρ? [φ, ω̃] − vki ∇kω̃ + ∇kjk +
C(p) + Dω̃ (ω̃)
n
n
jk
mi
1
[φ,
v
]
−
v
∇
v
+
= − ρ−1
ν
+
∇
φ
−
∇kpe − 0.71∇kTe + Dvke (vke )
ke
ke k ke
k
?
me
n
n
1
−1
= − ρ? [φ, vki ] − vki ∇kvki − ∇kp + Dvki (vki ) + nn (riz + rcx )(vkn − vki )
n 4T
1
0.71
5
2T
e
e
[φ,
T
]
−
v
∇
T
+
= − ρ−1
C(p
)
+
C(T
)
−
C(φ)
+
∇kjk − ∇kvke
e
e
e
ke k e
?
3B n
2
3
n
k
+ DTe (Te ) + DTe (Te ) + STe − nn riz Eiz
∇kn
4Ti 1
5
2Ti
∂Ti
−1
= − ρ? [φ, Ti ] − vki ∇kTi +
C(pe ) − τ C(Ti ) − C(φ) +
(vki − vke )
− ∇kvke
∂t
3B n
2
3
n
k
+ DTi (Ti ) + DTi (Ti ) + STi + nn (riz + rcx )(Tn − Ti + (vkn − vki )2)
∂n
∂t
∂ ω̃
∂t
∂vke
∂t
∂vki
∂t
∂Te
∂t
The tokamak scrape-off layer (SOL):
Open field lines
I Heat exhaust from the main plasma
I Important for impurity transport and fusion
ashes removal
I Fueling the plasma (recycling and gas puffing)
I
Three regimes
ρ−1
? [φ, n]
(6)
(7)
(8)
(9)
(10)
(11)
∇2⊥φ = ω, ρ? = ρs /R, ∇kf = b0 · ∇f , ω̃ = ω + τ ∇2⊥Ti , p = n(Te + τ Ti )
These equations are implemented in GBS, a 3D, flux-driven, global turbulence code with circular
geometry including electromagnetic effects [presented by F.D. Halpern, Monday, I-3]
I System is closed with a set of fluid boundary conditions applicable at the magnetic pre-sheath
entrance where the magnetic field lines intersect the limiter [Loizu et al., PoP 2012]
I Some achievements of GBS (see also http://crpp.epfl.ch/research_theory_plasma_edge):
I
SOL width scaling as a function of dimensionless/engineering plasma parameters
I Origin and nature of intrinsic toroidal plasma rotation
I Non-linear turbulent regimes in the SOL
I Mechanism regulating the equilibrium electrostatic potential
I
Recombination
Ionization
Ionization
Neutrals
Ionization
Neutrals
Neutrals
Turbulent simulations of the SOL including neutrals
Convection limited regime
I Low plasma density
I Long λ
mfp for neutrals
I Ionization in the core
I Heat ≈ particle source
I Q is mainly convective
Conduction limited regime
I High plasma density
I Short λ
mfp
I Ionization close to targets
I k Temperature gradients form
I Q is mainly conductive
Detached regime
I Very high plasma density
I Friction drag important
I Volumetric recombination
I Very low ion and energy flux to
the target
I
First fully consistent simulations with the code GBS and the neutrals model have been performed.
Parameters and normalization: n0 = 1014cm−3, v0 = cs , T0 = 10eV , L⊥ = ρs , ρ−1
? = 500
Te
n
1
100
100
0.8
50
1.4
1.2
50
1
Z
0
0.8
Kinetic equation with Krook operators
∂fn
∂fn
= −νiz fn − νcx (fn − nn Φi ) + νrec ni Φi
+ ~v ·
∂t
∂~x
νiz = ne riz = ne hve σiz (ve )i,
νrec = ne rrec = ne hve σrec (ve )i,
0.6
0
Z
A model for neutral atoms in the SOL
−50
0.4
−50
−100
0.2
−100
(1)
−100
νcx = ni rcx = ni hvrel σcx (vrel )i
Φi = fi /ni
−50
0
R − R0
50
0.6
0.4
−100
100
−50
vkn
log10(nn )
100
50
100
Tn
100
−0.5
0
R − R0
100
1
1.2
−1
50
Boundary conditions conserve particles and respect the Knudsen Cosine Law.
Z
~ v fn (~xw , ~v ) + u n = 0
dv
⊥
i⊥ i
fw (~xw , ~v ) ∝ cos(θ)e
mv 2/2T
w
50
−1.5
50
1
0.5
0.8
−2
0
Z
0
Z
0
Z
(2)
−2.5
0.6
0
−3
−50
for v⊥ > 0
−50
0.4
−50
−3.5
with v⊥ the velocity component perpendicular to the surface, θ the angle between ~v and the normal
vector to the surface, and Tw the temperature of the wall.
−4
−100
−100
n = 0, which is valid if τ
We consider the steady state solution, ∂f
neutral losses < τturbulence.
∂t
Typical SOL parameters: Te = 20eV, n0 = 5 · 1019m−3
−50
0
R − R0
50
−100
−100
100
−100
−50
0
R − R0
50
100
−100
−50
Te
1
1.5
1.5
1
1
0.8
0.6
(3)
n0=0
0.5
n0=1e14
0
0
100
200
y
300
400
0
0
100
200
y
300
400
0
0
100
200
y
300
400
Future work
Investigate the transition from convection to conduction limited regime
I Move towards detachment
I
Divertor geometry
I Radiative detachment in limited geometry
I Mimic the geometry of one divertor leg
I
Z ∞
I
Relax the steady state assumption
Conclusions
Integrating over vz , we assume Φi Maxwellian, according to the use of a fluid model for the plasma.
The discretized form can be solved with standard methods.
nn (~xi ) =Ai,j nn (~xj ) + nn,rec (~xi ) + nn,wall (~xi )
(5)
Z
Rr 0
νcx (~xj ) ∞
1
0r̂ )
~
−
dr
ν
(
x
+r
2D
eff j
Ai,j =dxdy
dv Φi (~xj , v r̂ )e v 0
r
0
n0=5e13
The simulations show clear changes in behavior of plasma density, electron and ion temperatures.
I The underlying physical mechanisms behind these changes (e.g. temperature gradient mainly
because of lack of convection or because of direct sink terms?) are still under investigation.
~ fn =
dv
vdv
dϕ
dvz fn
(4)
0 Z
0
−∞
ZZ
R r 0 00
∞
1
00)
1
0
~
0
0
2D
0
−
dr
ν
(
x
~
eff
=
dx nn (~x )
dv 0 νcx (~x )Φi (~x , ~v )e v 0
+ nn,rec (~x ) + nn,walls (~x )
r
0
nn (~x ) =
0.2
I
An integral equation for the neutral density
The contribution to the neutral distribution function from recombination, as well as the contribution by
recycling at the limiter, can be evaluated directly. The contribution by charge exchange and the one
from neutrals
that are absorbed and re-emitted from the walls can be calculated by imposing
R
~ fn . A linear integral equation for the neutral density is obtained.
nn (~x ) = dv
Z 2π
n0=1e12
0.5
n0=1e13
S(~x , ~v ) = νcx (~x )nn (~x )Φi (~x , ~v ) + νrec (~x )fi (~x , ~v )
νeff (~x ) = νiz (~x ) + νcx (~x )
~x = (x, y , z), ~v = (vx , vy , vz )
Z ∞
0
100
Ti
0.4
Z
50
A scan over different densities has been carried out to investigate the transition from convection to
conduction limited regime.
I Time- and space-averaged poloidal profiles during the quasi-steady-state phase of the five
density-scan simulations:
n
2D assumption
I k k⊥ for plasma structures, but neutrals are not affected by B
k
Iλ
mfp,neutrals Lk,plasma
I fn is evaluated independently on each poloidal plane
0
R − R0
I
−1 ≈ 5 · 10−7s
τneutral losses ≈ νeff
q
τturbulence ≈ R0Lp /cs0 ≈ 2 · 10−6s
The method of characteristics
The formal solution of the kinetic equation is
Z x
Rx
R x 00
0, ~
1
1
00
~
S(
x
v
)
− v x dx 00 νeff (~x 00)
− v x 0 dx νeff (~x )
0
fn (~x , ~v ) =
dx
e x
+ fw (~xw , ~v )e x w
vx
xw
0.2
−0.5
I
First fully turbulent SOL simulations self-consistently coupled to a neutral model.
I
A kinetic equation with Krook operators for ionization, recombination and charge-exchange processes
is solved for the neutral species.
I
First results from the GBS simulations show interesting interplay between neutral and plasma physics.
with ~r = ~xi − ~xj , r = |~r |, r̂ = ~r /r
[email protected]
THEORY OF FUSION PLASMAS, JOINT VARENNA - LAUSANNE INTERNATIONAL WORKSHOP, Villa Monastero,
Varenna, Italy - Sept 1-5th 2014