Plasma turbulence in the tokamak scrape-off layer · Introduction Global model for SOL turbulence Simulation/Experiment Comparison Conclusions Plasma turbulence in the tokamak scrape-o

IntroductionGlobal model for SOL turbulence

Simulation/Experiment ComparisonConclusions

Plasma turbulence in the tokamak scrape-off layer

F.D. Halpern1, S. Jolliet1, B. LaBombard2, J. Loizu1,A. Mosetto1, M. Podesta3, P. Ricci1, F. Riva1, J. Terry2,

C. Wersal1, S. Zweben3

1École Polytecnique Fédérale de LausanneCentre de Recherches en Physique des Plasmas, CH-1015 Lausanne, Suisse

2Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA3Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540, USA

Theory and Modelling SeminarInstitute for Plasmas and Nuclear Fusion

Instituto Superior Técnico20th Oct 2014, Lisbon, Portugal

F.D. Halpern 1 / 40 Plasma turbulence in the tokamak scrape-off layer



Scrape-off layer physics crucial for magnetic fusionHeat load to PFCs, rotation, impurities, L-H transition...

How do we develop 1st principles understanding of SOL dynamics?




Simple problem: inner wall limited (pol. ×-section)

Toroidal limiter (inner-wall limited)

Plasma inflowing from closed field

line region




Ballooning turbulence with kθρs ≈ 0.1 ∼ 1cm−1

δφ/σφ

δn/σ

n−4 −2 0 2 4

−4

−2

0

2

4

−1 −0.5 0 0.5 110−2

10−1

100

Phase(φ,n) / π

k θρ s

10−3 10−2 10−1 100

10−6

10−4

kθρs

fluc.

pow

er (r

el)




Gaussian in near SOL, intermittent in far SOL

−1 −0.5 0 0.5 10

0.5

1

1.5

δn/n

P(δ

n/n)

Near SOL

−1 −0.5 0 0.5 10

0.5

1

1.5

δn/nP

(δn/

n)

Far SOL




Fluctuation level O(1), skewed PDF

0.10.20.30.40.40.5

δn/n

−0.5

0

0.5

1

1.5

Ske

wne

ss

0 0.5 1 1.5 2

0

1

2

3

Kur

tosi

s

r−r0

[cm]




Power balance → exponentially decaying profiles

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

r−r0

[cm]

p/p 0

SimulationExponential fit

Turbulence Sonic flows towards PFCs




Some of the topics we have studied...

X What mechanism sets the turbulence levels?

X What instability drives the perpendicular transport?

X How does the SOL width change with parameters?

X Can we reconcile theory, simulations, and experiments?

X Poloidal limiter geometry → ISTTOK [Rogerio’s MSc work]

X What are the effects of neutrals?

X How is toroidal rotation generated in the SOL? [Loizu, PoP 2014]

× Is SOL transport related to the density limit? [LaBombard, NF 2005/08]

× How is the SOL coupled with the closed flux surface region?[Tamain et al.]




Turbulence levelsDominant instabilitiesScrape-off layer width scaling

A tool to simulate SOL turbulence

Global Braginskii Solver (GBS) [Ricci, PPCF (2012)]

I Drift-reduced Braginskii equationsd/dt � ωci , k2⊥ � k2‖

I Evolves n, φ, V||e , V||i , Te , Ti in 3D

I Global, flux-driven, no separation be-tween equilibrium and fluctuations

I Power balance between plasma outflowfrom the core, turbulent transport, andparallel losses

I Scalable ρ? up to medium size tokamak(e.g. TCV, C-Mod)





Drift-reduced Braginskii equations to describe the SOL

∂n

∂t=−

ρ−1?B

[φ, n] +2

B[nC(Te ) + Te C(n)− nC(φ)]− n∇‖v‖e − v‖e∇‖n

∂ω̃

∂t=−

ρ−1?B

[φ, ω̃]− v‖i∇‖ω̃ +B2

n∇‖j‖ +

2B

nC(p) +

B

3nC(Gi ), ω̃ = ∇

2⊥(φ + τTi )

∂

∂t

(v‖e +

mi

me

βe

2ψ

)=−

ρ−1?B

[φ, v‖e ]− v‖e∇‖v‖e +mi

me

[νj‖/n +∇‖φ−

∇‖pen− 0.71∇‖Te −

2

3n∇‖Ge

]∂v‖i

∂t=−

ρ−1?B

[φ, v‖i ]− v‖i∇‖v‖i −2

3∇‖Gi −

1

n∇‖p

∂Te

∂t=−

ρ−1?B

[φ,Te ]− v‖e∇‖Te +4

3

Te

B

[7

2C(Te ) +

Te

nC(n)− C(φ)

]+

+2

3

{Te

[0.71∇‖v‖i − 1.71∇‖v‖e

]+ 0.71Te (v‖i − v‖e )

∇‖n

n

}+D‖

Te(Te )

∂Ti

∂t=−

ρ−1?B

[φ,Ti ]− v‖i∇‖Ti +4

3

Ti

B

[C(Te ) +

Te

nC(n)− C(φ)

]+

+2

3Ti

(v‖i − v‖e

) ∇‖nn−

2

3Ti∇‖v‖e −

10

3

Ti

BC(Ti ) +D

‖Ti

(Ti )

+Sheath BCs consistent with PIC simulations [Loizu, PoP (2012)]





Parameters, normalizations, coordinates

I Coordinate system: (θ, r , ϕ)→ (poloidal , radial , toroidal)

I Equations expressed in normalized units:

I L⊥ → ρsI L‖ → R

I v → csI t ∼ γ−1 → R/cs

I The dimensionless code parameters are as follows:

I ρ? = ρs/R

I ν = e2nR/(miσ‖cs)

I βe = 2µ0pe/B2

I q ≈ (r/R)Bϕ/BθI Simplified notation in analytical expressions:

I p0 = 〈p〉t , t � γ−1 I Lp = −〈p/∂rp〉t





Poloidal cross sections showing SOL turbulence





Modes saturate due to pressure non-linearity

We observe in simulations [Ricci, PoP (2013)]:

I Mode saturation caused by local pressure non-linearity

∂rp1 ∼ ∂rp0 → p1p0 ∼σrLp

I Radial eddy length σr is mesoscopic [Ricci, PRL (2008)]

σr ≈√Lp/kθ

I Turbulent flux Γ1 dominated by radial E× B convection

Γ1 = ρ−1?

〈p1

∂φ1∂θ

〉





Saturation model yields E× B turbulent flux

Gradient removal hypothesis





Self-consistent prediction of pressure gradient length

In steady state, ∇ · Γ1 balances parallel losses ∼ ∇‖ · (pv‖e), hence

Lp ≈q

cs

(γ

kθ

)max

I Results in iterative scheme to predict Lp self-consistently:

I Compute γ = f ( Lp︸︷︷︸vary

, kθ︸︷︷︸scan

, ρ?, q, ν, ŝ,mi/me︸︷︷︸plasma parameters

)

I Vary Lp until LHS = RHS using secant method





Excellent agreement between theory and simulations

Lp predicted using self-consistent procedure [Halpern, NF (2014)]

0 30 60 90 120 150 1800

30

60

90

120

150

180

Lp (gradient removal)

Lp(sim

ulation)

GBS sims.: ρ−1? = 500–2000, q = 3–6, ν = 0.01–1, β = 0–3× 10−3





Dominant instability depends principally on q, ν, ŝ, Ti/TeI Build instability parameter space using reduced models→ gradient removal theory, linear dispersion relations

I Verify results using GBS non-linear simulations [Mosetto, PoP (2013)]

−2 0 2−3

−2.5

−2

−1.5

−1

−0.5

0

ŝ

log10(ν)

IBMIDW

RDW

RBM

I Which instability drives ⊥ transport?I Inertial/Resistive Ballooning modes/Drift Waves?foobar





Dominant instability depends principally on q, ν, ŝ, Ti/TeI Build instability parameter space using reduced models→ gradient removal theory, linear dispersion relations

I Verify results using GBS non-linear simulations [Mosetto, PoP (2013)]

−2 0 2−3

−2.5

−2

−1.5

−1

−0.5

0

ŝ

log10(ν)

IBMIDW

RDW

RBM

I Which instability drives ⊥ transport?I Inertial/Resistive Ballooning modes/Drift Waves?





Presence of RBMs verified in TCV SOL sims

I (ñ, φ̃) phase difference, joint (ñ, φ̃) pdf [Halpern, NF (2014)]

0.01

0.1

1

kθ

−4

−2

0

2

4

(p1−〈p

1〉)/σp 1

0.01

0.1

1

kθ

−4

−2

0

2

4

(p1−〈p

1〉)/σp 1

−1 −0.5 0 0.5 10.01

0.1

1

P(φ1, p1)/π

kθ

−4 −2 0 2 4−4

−2

0

2

4

(φ1 − 〈φ1〉)/σφ1

(p1−〈p

1〉)/σp 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

ŝ = 0 ŝ = 0

ŝ = 1 ŝ = 1

ŝ = 2 ŝ = 2

Curvature-driven, non-adiabatic mode → RBMs





Addition of finite Ti weakens adiabatic coupling

I Analysis extended to include Ti effects [Mosetto, PoP (submitted)]

I Joint (ñ, φ̃) pdf in GBS sims with τ = 1, τ = 4

RBM component is enhanced by finite Ti





SOL width in RBM regime scales with ρ?, q

I SOL width obtained analytically with RBMs [Halpern, NF 2013/14]:

I Our simple model leads to a dimensionless scaling:

Machine size

Electromagnetic effectsCollisionality vs

connection length





Parallel dynamics physics in agreement with simulations

I Verify saturated RBM theory with GBS EM simulations

I ρ−1? = 500, βe = 0–3× 10−3, ν = 0.01–1, q = 3, 4, 6

αd/q

α

10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

(Contours of Lp given by theory, symbols are GBS simulations)





GBS simulations confirm size-scaling up to TCV sizeCASTOR

TCV





Dimensionless scaling follows GBS simulation data

Comparison carried out over wide range of parameters (ρ?, q, β, ν)

0 30 60 90 120 150 1800

30

60

90

120

150

180





Good agreement with SOL width measurements

Lp ≈ 7.2× 10−8q8/7R5/7B−4/7φ T−2/7e0 n

2/7e0 (1 + Ti/Te)

1/7 [ m ]

0 0.03 0.06 0.09 0.120

0.03

0.06

0.09

0.12Alcator C-MODCOMPASSJ ETTCVTore Supra

Exp. data:G. Arnoux

I. Furno

J.P. Gunn

J. Horacek

M. Kočan

B. Labit

B. LaBombard

C. Silva





Good agreement with SOL width measurements

Lp ≈ 7.2× 10−8q8/7R5/7B−4/7φ T−2/7e0 n

2/7e0 (1 + Ti/Te)

1/7 [ m ]

0 0.03 0.06 0.09 0.120

0.03

0.06

0.09

0.12

ITER start-up prediction

Alcator C-MODCOMPASSJ ETTCVTore Supra

Exp. data:G. Arnoux

I. Furno

J.P. Gunn

J. Horacek

M. Kočan

B. Labit

B. LaBombard

C. Silva





Not so fast...Recent measurements (e.g. JET, C-Mod) show 2 different ∇ scales

Figure : Data: G.Arnoux [NF’12] (left), B.LaBombard (right)

I Carry out detailed simulation/theory/experiment comparison

I Alcator C-Mod, TCV, RFX (in tokamak mode)




Alcator C-Mod tokamakGPI diagnosticSimulation/experiment comparison

An ideal testbed for simulation-experiment comparisonI Inner-wall limited Ohmic C-Mod

discharges [Zweben, PoP (2009)]

I R = 0.67m, a = 0.20m,B = 2.7, 3.8T, κ = 1.2

I Density scan at each value of B

I Characterize C-Mod SOL turbulence using GPI diagnostic,and compare with GBS results

I Low β, no Ti or B̃ diagnostics → simple electrostatic,cold ion model

I δDαDα, pdf moments ,τauto , Lr , Lθ, vr , vθ, P(kθ), P(ω)

Very stringent test!





Simulated pressure profiles show double scale length

I Two-scale fit for pe yields Lp,near ≈ 5mm, Lp,far ≈ 2cm

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

r−r0 [cm]

p/p 0

I Probe measurements from 2009 campaign not precise enough

I From now on, concentrate on characterizing turbulence





Turbulent modes in the near SOLUse joint-pdf and phase between φ, p fluctuations to understandnature of turbulent modes

I φ1, n1 well correlated → almost adiabatic modeI Small phase shift → curvature drive not important

−4 −2 0 2 4−4

−2

0

2

4

(φ−)/σφ

(p−

<p>

)/σ p

−1 −0.5 0 0.5 110

−3

10−2

10−1

100

P(φ,pe)/π

k yρ s

0





Turbulent modes in the far SOLUse joint-pdf and phase between φ, p fluctuations to understandnature of turbulent modes

I Weak correlation between φ1, n1 → non-adiabaticI Phase shift ∼ π/2 → curvature driven ballooning mode

−4 −2 0 2 4−4

−2

0

2

4

(φ−)/σφ

(p−

<p>

)/σ p

−1 −0.5 0 0.5 110

−3

10−2

10−1

100

P(φ,pe)/π

k yρ s

0





Gas-puff imaging of C-Mod SOL

Phantom 710 high-speed camera at 400’000fps [S.Zweben, J.Terry]





δDα/Dα diagnostic for GBSUsing DEGAS modeling of GPI emissivity, model Dα fluctuations

I Emissivity locally parametrized as E ∝ Tαe nβe , use H656 line

I Fluctuations modelled as δDα/Dα ≈ α(Te , ne)T̃e +β(Te , ne)ñ

−0.3

−0.3

−0.2

−0.2

−0.1

−0.1

−0.1

00

0

0.2

0.2

0.2

0.4

0.4

0.4

0.8

0.8

0.8

1

1

1

22

2

55

5

10

10

10

20

20

20

log Te [ eV ]

log n

e [ m

−3 ]

−1 0 1 2 3 4

17

18

19

20

21

22

0

5

10

15

200.1 0.1

0.2

0.2

0.20.3

0.3

0.30.4

0.4

0.40.5

0.5

0.5

0.6

0.6 0.6

0.6

0.7

0.7

0.7

0.7

0.8 0.8

0.8

0.8

0.8

0.9

0.9

0.9

0.9

11

1

1

1.1

log Te [ eV ]

log n

e [ m

−3 ]

−1 0 1 2 3 4

17

18

19

20

21

22

0.2

0.4

0.6

0.8

1

I Simulate finite GPI resolution (3× 3mm + 2.5µs smoothing),B-field tilt respect to sensors (8mm poloidal smoothing)





δDα/Dα synthetic diagnostic results

I Left to right: ñ, δDα/Dα, δDα/Dα (diode), δDα/Dα (full)

θ/π

r−r0 [m]

0 0.01 0.02−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r−r0 [m]

0 0.01 0.02−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r−r0 [m]

0 0.01 0.02−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r−r0 [m]

0 0.01 0.02−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

High kθ modes strongly damped by smoothing





Large δDα/Dα fluctuations, skewed PDF

I δDα/Dα level increases withSOL, ∼ 30% in far SOL

I Skewness ∼ 1→ blobs (?)

I Moment profiles robust withplasma parameters

0

0.1

0.2

0.3

0.4

δDα/

Dα

−1

−0.5

0

0.5

1

1.5

Ske

wne

ss0 0.005 0.01 0.015 0.02 0.025 0.03

−1

0

1

2

3

Kur

tosi

sr−r

0[m]

B=2.7T low nB=2.7T high nB=3.8T low nB=3.8T high n

Quantitative comparison using shaded area (GPI sensors)





Large δDα/Dα fluctuations, skewed PDF

I δDα/Dα level increases withSOL, ∼ 30% in far SOL

I Skewness ∼ 1→ blobs (?)

I Moment profiles robust withplasma parameters

0

0.1

0.2

0.3

0.4

δDα/

Dα

−1

−0.5

0

0.5

1

1.5

Ske

wne

ss0 0.005 0.01 0.015 0.02 0.025 0.03

−1

0

1

2

3

Kur

tosi

sr−r

0[m]

B=2.7T low nB=2.7T high nB=3.8T low nB=3.8T high n

GPI

Quantitative comparison using shaded area (GPI sensors)





GBS agrees with [Zweben PoP 2009] within error bars

I Compare GBS radial/poloidalaverage against GPI data

I Shot-to-shot variation indicatedwith error bars

I GBS gives good match forδDα/Dα and higher moments

I Previous gyrofluid simulationsgave δDα/Dα ≈ 5–10%

0

0.2

0.4

0.6

0.8

δ D

α/D

α

0

0.5

1

1.5

2

Skew

ness

2 2.5 3 3.5 4 4.50

1

2

3

4

Kurt

osis

B [T]

GBS low n

GBS high n

GPI





Typical spatial, temporal turbulent scales give reasonableagreement

I Compute τauto , Lrad , Lpol using2 point correlations functions Cij

Cii (τauto) =1

2

L = 1.66δx√

− lnCij (t = 0)

I Good match for L ∼ 1.5cm,τauto underpredicted by ∼2

0

10

20

30

40

τauto

[µ

s]

0

0.5

1

1.5

2

Lra

d [cm

]

2 2.5 3 3.5 4 4.50

1

2

3

Lpol [

cm

]

B [T]

GBS low n

GBS high n

GPI low n

GPI high n





Propagation velocities

I Obtain vrad , vpol from time lag that maximizes correlationbetween two neighboring points separated by δx → v = δx/τ

I Good agreement in vrad → poloidal mode structureI Large mismatch in vpol → resolution smoothing in GBS data?

2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

vp

ol [

km

/s]

B [T]2 2.5 3 3.5 4 4.5

0

0.5

1

1.5

2

vra

d [km

/s]

B [T]

GBS low n

GBS high n

GPI





Spectral power vs wavenumber of δDα/Dα

I From FFT of δDα/Dα in θ, then average over r , t

I Significant drop at kpol = 125m−1 high k due to smoothing

I Unsmoothed δDα/Dα has same power law scaling as GPI

101

102

103

10−4

10−3

10−2

10−1

100

kpol

[m−1

]

flu

c.

po

we

r (r

el)

GPI

GBSGBS(no sm.)

B=2.7T, low n

Renormalized

101

102

103

10−4

10−3

10−2

10−1

100

kpol

[m−1

]

flu

c.

po

we

r (r

el)

GPI

GBSGBS(no sm.)

B=3.8T, high n

Renormalized





Spectral power vs frequency of δDα/Dα

I From FFT of δDα/Dα in t, then average over t, r = 2±0.2cmI GPI measurements and GBS show same asymptotic behavior

100

101

102

103

10−6

10−4

10−2

100

f [kHz]

fluc. pow

er

(rel)

GPI +k

GPI −k

GBS +k

GBS −k

B=2.7TB=2.7T, low n

Renormalized

100

101

102

103

10−6

10−4

10−2

100

f [kHz]

fluc. pow

er

(rel)

GPI +k

GPI −k

GBS +k

GBS −k

B=3.8T, high n

Renormalized




Summary and outlook

I Towards first principles understanding of SOL width:

X Non-linearly saturated drift/ballooning turbulence

X SOL width scales with ρ?, q, collisionality

X Simple analytical scaling agrees with experimental data

I Detailed comparison between GBS and C-Mod discharges

X Lp, δDα/Dα pdf moments, Lrad , Lpol , vrad , P(ω), P(kpol )× τauto , vpol → under/overpredicted by factor ∼ 2

I Next: 2 Lp’s profile structure using 2014 C-Mod discharges

I More advanced simulation model → Ti , shapingI Mirror langmuir probe → high res. profiles, (n, φ) phase




Thank you for your attention!




Properties of the SOL

I Lfluc ∼ 〈L〉tI nfluc ∼ 〈n〉tI Collisional magnetized plasma

I Low frequency modes ω � ωciI Open field lines




Sheath BCs from kinetic approach [Loizu, PoP (2012)]I COLLISIONAL PRESHEATH (CP)

I Quasi-neutral, IDA holds

I Potential drop ∼ 0.5Te over ∼ LI Ions accelerated to vs = cs sinα

I MAGNETIC PRESHEATH (MP)

I Quasi-neutral, IDA breaks

I Potential drop ∼ 0.5Te over ∼ ρsI Ions accelerated to vs = cs

I DEBYE SHEATH (DS)

I Non-neutral, IDA breaks

I Potential drop ∼ 3Te over ∼ 10λDI Ions accelerated to vs > cs




Extra slides: Summary of the BC

v||i = cs

(1 + θn −

1

2θTe −

2φ

Teθφ

)v||e = cs

(exp (Λ− ηm)−

2φ

Teθφ + 2(θn + θTe )

)∂φ

∂s= −cs

(1 + θn +

1

2θTe

)∂v||i∂s

∂n

∂s= − n

cs

(1 + θn +

1

2θTe

)∂v||i∂s

∂Te∂s' 0

ω = − cos2 α[

(1 + θTe )

(∂v||i∂s

)2+ cs (1 + θn + θTe/2)

∂2v||i∂s2

]

where θA =ρs

2 tanα∂x A

A, and ηm = e(φmpe − φwall )/Te . [Loizu et al PoP 2012]




Resistive ballooning modes destabilized by EM effects

I Starting from reduced MHD, obtain simple dispersion relation

γ2(ν +

βe02

γ

k2⊥

)= 2

R

Lp

(ν +

βe02

γ

k2⊥

)−

k2‖

k2⊥γ

I Neglecting ideal ballooning mode, the resistive branch gives

(γ2 − γ2b

)k2⊥ = −γ

(1− αq2ν

)and we identify γ ∼ γb =

√2R/Lp and kb ∼

√(1− α)/(νγb)/q


IntroductionGlobal model for SOL turbulenceTurbulence levelsDominant instabilitiesScrape-off layer width scaling

Simulation/Experiment ComparisonAlcator C-Mod tokamakGPI diagnosticSimulation/experiment comparison

Conclusions

Documents

Plasma turbulence in the tokamak scrape-off layer · Introduction Global model for SOL turbulence Simulation/Experiment Comparison Conclusions Plasma turbulence in the tokamak scrape-o