OXB-converted E BWs in Tokamaks, Spherical Tokamaks, Stellarators and RFPs
Introduction – Power exhaust in tokamaks€¦ · ON THE INFLUENCE OF TURBULENT FLUCTUATIONS ON...
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ON THE INFLUENCE OF TURBULENT FLUCTUATIONS ON AM/PMI DATA FOR EDGE PLASMA MODELING Y. Marandet 1 , F. Guzmán 1 , R. Guirlet 2 , P. Tamain 2 , H. Bufferand 2 , G. Ciraolo 2 , Ph. Ghendrih 2 and J. Rosato 2 1 PIIM, CNRS, Aix-Marseille Université, Marseille 2 IRFM-CEA, Cadarache Introduction – Power exhaust in tokamaks 18/12/14 In steady state, the fusion energy has to be extracted ITER : P fus = 500MW 400 MW neutrons > 500 m 2 100 MW α particles ~ 1 m 2 + 50 MW heating (Q=10) Divertor Spreading the energy is mandatory Transport codes used to address these issues Turbulence averaged out : Turbulent transport coefficients = free parameters Effect of averaging on AM and PWI physics ? Outline 18/12/14 AIEA meeting, Daejeon 2014 3 1) Derivation of transport equations 2) Transport of neutral particles in a turbulent plasma 3) Ionization balance in turbulent plasmas 4) Conclusions and Perspectives Can we capture the effects discussed in 2) and 3) using “fluctuation dressed” AM/PWI data ? 1- Derivation of transport equations 18/12/14 AIEA meeting, Daejeon 2014 4 TOKAM3X P. Tamain et al. SolEdge2D H. Bufferand et al. TOKAM2D Y. Sarazin and Ph. Ghendrih, PoP 1998 Coarse graining level averaging no AM&PWI data yet D a ,χ a, … 18/12/14 HDR Y. Marandet 5 Averaging issues in transport codes 18/12/14 AIEA meeting, Daejeon 2014 5 Mean turbulent fluxes and their origin n = 〈n〉 + δn v = 〈v〉 + δv ∂n ∂t + ∇ · (nv)= S Averaging issues in transport codes 18/12/14 AIEA meeting, Daejeon 2014 6 Mean turbulent fluxes and their origin ∂〈n〉 ∂t + ∇ · [〈n〉〈v〉 + 〈δn δv〉]= 〈S〉 Issues associated to averaging of source terms 〈S(T e )〉 =S(〈T e 〉) 〈n z n e σv〉 = 〈n z 〉〈n e 〉〈 σv〉 Parametric nonlinearities Statistical nonlinearities i) Size of the effects ? ii) How to take this in account ? Γ turb = −D a ∇〈n〉 How to express in terms of <n> ? « Gradient diffusion hypothesis » Γ turb = 〈δnδv〉 Main features of SOL fluctuations 18/12/14 AIEA meeting, Daejeon 2014 7 ∼ 1 The density PDF can be fitted to a gamma distribution 10 -5 10 -4 10 -3 10 -2 10 -1 Density PDF 15 10 5 0 n/n 0 TOKAM2D Gamma Lognormal W (n)= 1 Γ(β)α β n β-1 exp - n α R = σ 1/2 〈n〉 = 1 √ β 〈n(r,t)〉 = α(r)β(r) Correlation time Correlation length τ c ≃ 10μs λ ≃ 1cm Similar distribution for T e Use this information to build stochastic models for fluctuations 2) Transport of neutral particles in turbulent plasmas 18/12/14 AIEA meeting, Daejeon 2014 8 250 200 150 100 50 0 Y/ρ s 240 220 200 180 160 140 X/ρ s 0.04 0.03 0.02 0.01 Neutral and Plasma density 250 200 150 100 50 0 Y/ρ s 240 220 200 180 160 140 X/ρ s 15 10 5 Plasma density [a.u.] Plasma density [a.u.] Atom density [a.u.]
Transcript of Introduction – Power exhaust in tokamaks€¦ · ON THE INFLUENCE OF TURBULENT FLUCTUATIONS ON...
ON THE INFLUENCE OF TURBULENT FLUCTUATIONS ON AM/PMI DATA FOR
EDGE PLASMA MODELING
Y. Marandet1, F. Guzmán1, R. Guirlet2, P. Tamain2, H. Bufferand2,
G. Ciraolo2, Ph. Ghendrih2 and J. Rosato2
1 PIIM, CNRS, Aix-Marseille Université, Marseille
2 IRFM-CEA, Cadarache
Introduction – Power exhaust in tokamaks
18/12/14
! In steady state, the fusion energy has to be extracted
ITER : Pfus = 500MW
400 MW neutrons > 500 m2
100 MW α particles ~ 1 m2
+ 50 MW heating (Q=10)
Divertor
Spreading the energy is mandatory
! Transport codes used to address these issues
Turbulence averaged out :
Turbulent transport coefficients = free parameters
Effect of averaging on AM and PWI physics ?
Outline
18/12/14 AIEA meeting, Daejeon 2014
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1) Derivation of transport equations
2) Transport of neutral particles in a turbulent plasma
3) Ionization balance in turbulent plasmas
4) Conclusions and Perspectives
Can we capture the effects discussed in 2) and 3)
using “fluctuation dressed” AM/PWI data ?
1- Derivation of transport equations
18/12/14 AIEA meeting, Daejeon 2014
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TOKAM3X
P. Tamain et al. SolEdge2D
H. Bufferand et al. TOKAM2D
Y. Sarazin and Ph. Ghendrih, PoP 1998
Coarse graining level
averaging
no AM&PWI data yet
Da,χa, …
18/12/14 HDR Y. Marandet
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Averaging issues in transport codes
18/12/14 AIEA meeting, Daejeon 2014
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! Mean turbulent fluxes and their origin
n = 〈n〉+ δn
v = 〈v〉+ δv
∂n
∂t+∇ · (nv) = S
Averaging issues in transport codes
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! Mean turbulent fluxes and their origin
∂〈n〉
∂t+∇ · [〈n〉〈v〉+ 〈δn δv〉] = 〈S〉
! Issues associated to averaging of source terms
〈S(Te)〉#=S(〈Te〉)
〈nzneσv〉 #= 〈nz〉〈ne〉〈σv〉
Parametric nonlinearities
Statistical nonlinearities
i) Size of the effects ? ii) How to take this in account ?
Γturb = −Da∇〈n〉
! How to express in terms of <n> ?
« Gradient diffusion hypothesis »
Γturb = 〈δnδv〉
Main features of SOL fluctuations
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∼ 1
! The density PDF can be fitted to a gamma distribution
10-5
10-4
10-3
10-2
10-1
De
nsity P
DF
151050n/n0
TOKAM2D Gamma Lognormal
W (n) =1
Γ(β)αβnβ−1 exp
(
−
n
α
)
R =σ1/2
〈n〉 =1√β
〈n(r, t)〉 = α(r)β(r)
! Correlation time Correlation length τc ≃ 10µs λ ≃ 1cm
! Similar distribution for Te
Use this information to build stochastic models for fluctuations
2) Transport of neutral particles in turbulent plasmas
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250
200
150
100
50
0
Y/ρ
s
240220200180160140
X/ρs
0.04
0.03
0.02
0.01
Ne
utra
l an
d P
lasm
a d
en
sity
250
200
150
100
50
0
Y/ρ
s
240220200180160140
X/ρs
15
10
5
Pla
sm
a d
en
sity
[a.u
.]
Plasma density [a.u.] Atom density [a.u.]
Considerations on time and spatial scales
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250
200
150
100
50
0
Y/ρ
s
240220200180160140
X/ρs
15
10
5
Pla
sm
a d
en
sity
[a.u
.]
! The fluctuations are often frozen during the neutral lifespan
3.5 cm
10-1
100
101
102
103
Io
nis
atio
n m
ea
n fre
e p
ath
(cm
)
101
102
103
Te (eV)
n=1018
m-3
n=1020
m-3
E0=1 eV
E0=10 eV
! Atoms’ mean free paths can be large compared to λ
Exemples of sampled density fields
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0
L
0 L
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
x1
01
3
0
L
0 L
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
x10
13
0
L
0 L
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
x10
13
0
L
0 L
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
x1
013
λ/L=10-3 λ/L=0.025
λ/L=0.15 λ/L=2
Multivariate generalization of the gamma distribution
Y. Marandet et al. PPCF 2011
Density fluctuations reduce the stopping power
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10-3
10-2
10-1
100
⟨n
0⟩
(a.u
.)
86420
|r-rw|/l
Turbulence free a= 0.45 a = 2.7 a = 9 a = ∞
R=0.8, adiabatic
a =
λ
ℓ
Why is it so ?
Jensen’s inequality on
convex functions:
〈e−x〉 ≥ e−〈x〉
! The effect of density fluctuations strongly increases with
! Substantial for high relative fluctuations levels only (>30%)
a
“Fluctuations dressed” ionization rate coefficient
! How to implement these effects in transport codes ?
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A possibility: fluctuations dressed rate coefficients
〈Sn〉 = −̟(s)〈n0〉 = −〈n0〉〈n〉[σv]eff (s)
18/12/14 HDR Y. Marandet
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1.5
1.0
0.5
0.0
ϖS
,F /⟨ν⟩
43210
|r-rw|/l
⟨ν⟩
ϖF, ϖ
S, aL=1
ϖF
, ϖS
, aL=10
R=0.8
(A. Mekkaoui et al., PoP 2012)
Szegö Limits
(Y. Marandet et al., PPCF 2011)
“Fluctuations dressed” ionization rate coefficient
! How to implement these effects in transport codes ?
A possibility: fluctuations dressed rate coefficients
Examples in realistic conditions
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1015
1016
1017
1018
⟨N
D2⟩
(a
.u.)
43210
|r-rw| (cm)
no fluctuations
τ = 1 µs
τ = 10 µs
τ = 50 µs
adiabatic
⟨Ne(rw)⟩ = 5×1018
m-3
! D2 more affected than D because mean free path shorter
! Time dependent effects taken into account
(Y. Marandet et al., IAEA FEC Conference 2012)
2.5x105
2.0
1.5
1.0
ϖS
,F
6420
|r-rw| (cm)
no fluctuations
ϖF, ϖ
S, τ = 1 µs
ϖF, ϖ
S, τ = 10 µs
ϖF, ϖ
S, τ = 50 µs
Role of the boundary conditions
! So far: Γ0 constant; what happens for recycling ?
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τR!= 10-12 s
D+ D
Backscattering
Γ0(t) = ΓD+(t)
! Effect on mean neutral particle penetration
τR = ?
D+ D2
Molecule desorption
if τR ≫ τcΓ0 = 〈ΓD+(t)〉
3- Effects of fluctuations on ionization equilibrium
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Time scales in the system
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100
101
102
103
104
105
106
eig
en
va
lue
s -λ
i a
nd
-M
ii e
ntr
ies [s
-1]
1 10 100
Te (eV)
diag. entries eigenvalues
1 2 3 4 5 6 7
dN
dt= MN N = {nC0 , . . . nC6+}
No separation of scales valid over the whole temperature range
〈N〉 ?
40
30
20
10 T
e [e
V]
200150100500t [µs]
⟨Te⟩ = 15 eV
R = 0.9
ν=105s
-1
Kubo Andersen process
Discussion of the results
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1.2
1.0
0.8
0.6
0.4
0.2
0.0
Ab
un
da
nce
s
10 100 Te (eV)
CIII CIV CV CVI
ν =105 s
-1
ne=5x1012
cm-3
CVII
! Ionization stages shift towards lower temperatures
1.0
0.8
0.6
0.4
0.2
0.0
Ab
un
da
nce
s
10-2 10
0 102 10
4 106 10
8
ν (s-1
)
fCV R=0 %
fCV R=50%
fCVI R=0 %
fCVI R=50 %
Te=50 eV
ν=105s
-1Adiabatic
Diabatic
! The system is in the diabatic regime
F. Catoire et al., Phys. Rev. A (2010)
〈M〉〈N〉 = 0
Effective ionization/recombination coefficients
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10-20
10-19
10-18
10-17
10-16
10-15
⟨σ
rcv⟩
(m
3s
-1)
1 10 100⟨Te⟩ (eV)
ne=5x1012
m-3
no fluctuations with fluctuation, R=50%
10-18
10-17
10-16
10-15
10-14
10-13
10-12
⟨σ
iov⟩
(m
3s
-1)
1 10 100⟨Te⟩ (eV)
no fluctuations with fluctuation, R=50%
ne=5x10
12 m
-3
〈σv〉 =
∫ +∞
0
dTeW (Te)σv(Te)
! Increased ionization at low temperature
[ as an non-maxw. rate coeff.]
Going beyond simple analytical models
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! Impurities introduced as passive scalars in TOKAM2D
! Compare sputtering yields to those obtained in mean
fields calculations (first wall erosion) [on-going]
! Address statistical correlations between ne,Te,nz
! Take transport into account: Γturb = −Da∇〈n〉
F. Guzman et al., J. Nucl. Mater. in press
Discussion of first results
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! Transport plays an essential role and weakens source averaging
effects: transport to regions where rate coeff. are flat …
Nesep = 1019 m-3 Ne
sep = 1020 m-3
18/12/14 HDR Y. Marandet
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Discussion of first results
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! Transport plays an essential role and weakens source averaging
effects: transport to regions where rate coeff. are flat …
Nesep = 1019 m-3 Ne
sep = 1020 m-3
! Calculation of effective data may be more difficult
σveff = 〈σv〉+〈δneσv〉
〈ne〉+
〈δnzσv〉
〈nz〉+
〈δneδnzσv〉
〈ne〉〈nz〉
Good news, but W in divertor conditions might be more problematic
0 1 2 3 4 5 6 7 8 9 10 1110
−2
10−1
100
distance to the wall (cm)
fractional abundances
W
W+
W2+
W3+
W4+
W5+
W6+
0 2 4 6 8 10−2
−1
0
1
2
3x 10
15
distance to the wall (cm)
So
urc
e t
erm
s (
cm
−3s
−1)
W+
W2+
W3+
W4+
Conclusions
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! Evaluation of the strength of averaging effects on AM (and PMI)
data needed to assess related modelling uncertainties
! In all cases, substantial effects only if fluctuation level > 30%
! Density fluctuations tend to reduce the screening efficiency of
the plasma in particular for molecules and impurities atoms ! Temperature fluctuations shift the local ionization balance
towards lower temperatures
! In both these simplified cases, the effects can be accounted for
by dressing AM data with fluctuations (when necessary !)
! More complex cases are likely to require either more information
on fluctuations and/or further approximations
! To do so, we used stochastic models for fluctuations
