simulation of scrape-off layer turbulence

Plasma turbulence in the tokamak SOL Paolo Ricci,

F. Halpern, S. Jolliet, J. Loizu, A. Mosetto, I. Furno, B. Labit, F. Riva, C. Wersal

Centre de Recherches en Physique des Plasmas

École Polytechnique Fédérale de Lausanne, Switzerland

The model to study SOL turbulence

The GBS code and its path towards SOL simulations Anatomy of SOL turbulence: from linear instabilities to SOL width and

intrinsic toroidal rotation

SOL channels particles and heat to the wall

Plasma outflowing from the core

Scrape-off Layer

Perpendicular transport

Losses at the vessel

Parallel flow

The SOL – a crucial issue for the entire fusion program

Pwall ∼Psep

Awet∼ Psep

RLp≤ 5 MW m−2

Psep

R= 7

Psep

R= 20

Psep

R= 80− 100

JET ITER DEMO

The key questions

Properties of SOL turbulence

Cou

rtes

y of

R. M

aque

da

nfluc ∼ neq

Lfluc ∼ Leq

Fairly cold magnetized plasma

The GBS code, a tool to simulate SOL turbulence

Braginskii model

Drift-reduced Braginskii equations

Collisional Plasma

(vorticity) similar equations

parallel momentum balance

Solved in 3D geometry, taking into account plasma outflow from the core, turbulent transport, and losses at the vessel

Source ∂n

∂t+ [φ, n] = C(nTe)− nC(φ)−∇(nVe) + S

Te,Ω (Ti Te)

∇2⊥φ = Ω

vi, ve

ne ni

ρ L,ω Ωci

0

1

2

3

4

Totalpotentialdrop

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.2

0.4

0.6

0

1

2

3

4

Totalpotentialdrop

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.2

0.4

0.6

0

1

2

3

4

Totalpotentialdrop

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.2

0.4

0.6

Boundary conditions at the plasma-wall interface

φ

ni − ne

nse

SOL PLASMA

vx

DRIFT-REDUCED MODEL VALID DRIFT APPROXIMATION BREAKS

DRIFT VELOCITY

WA

LL

•  Set of b.c. for all quantities, generalizing Bohm-Chodura

•  Checked agreement with PIC kinetic simulations

BOUNDARY CONDITIONS

VELOCITY

MAGNETIC PRE-SHEATH

DEBYE SHEATH

Code verification, method of manufactured solution

Our model: , unknown

We solve , but

A(f) = 0 f

An(fn) = 0 ?fn − f =

1) we choose , then g

2) we solve: An(gn)− S = 0

Method of manufactured solution:

= gn − gS = A(g)

100 10110 10

10 5

h = !x/!x0 = !y/!y0 = (!t/!t0)2

||!|| !

nTv",iv",e!"

For GBS: ∼ h2

GBS analysis of configurations of increasing complexity

LAPD, UCLA

HelCat, UNM Helimak, UTexas

TORPEX, CRPP

ITER-like SOL Limited

SOL

MotivationThe plasma-wall transitionGBS turbulence simulationsSheath effects on turbulence

Conclusions

The GBS codeExamples of 3D simulations

The GBS code, a tool to simulate open field line turbulence

Developed by steps of increasing complexity


Global, 3D, Flux-driven, Full-n [Ricci et al PPCF 2012]

J. Loizu et al. 13 / 24 The role of the sheath in magnetized plasma fluid turbulence

Limited SOL


LAPD, UCLA


TORPEX, CRPP


SOL


Conclusions







Limited SOL

From linear devices… (role of non-curvature driven modes, DW vs KH)


LAPD, UCLA


TORPEX, CRPP


SOL


Conclusions







Limited SOL

… to the Simple Magnetized Torus… (role of curvature-driven modes and rigorous code validation)


LAPD, UCLA


TORPEX, CRPP


SOL

…to limited SOL


Conclusions







Limited SOL


LAPD, UCLA


TORPEX, CRPP


SOL

…to limited SOL


Conclusions







Limited SOL

… supported by analytical investigations

Tokamak SOL simulations

Simulations contain physics of ballooning modes, drift waves, Kelvin-Helmholtz, blobs, parallel flows, sheath losses…

Losses at the limiter

Radial transport

Flow along B

Plasma outflowing from

the core

φ20 30 40 50 60 70 80 90 100 110

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

20 30 40 50 60 70 80 90 100 1100.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 0.5 0 0.5 110 4

10 2

100

1 0.5 0 0.5 110 4

10 2

100

t

PDF

Jsat

20 30 40 50 60 70 80 90 100 1100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

20 30 40 50 60 70 80 90 100 1100.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 0.5 0 0.5 110 4

10 2

100

1 0.5 0 0.5 110 4

10 2

100

t

PDF

Jsat

The key questions

Three possible saturation mechanisms

Removal of the turbulence drive (gradient removal):

Kelvin – Helmholtz secondary instability:

Suppression due to strong shear flow:

Γr =

p∂φ

∂θ

t

∼ γp

Lpk2r∼ γp

kθ

Turbulent transport with gradient removal saturation

GR hypothesis Nonlocal linear theory, kr ∼

kθ/Lp

θ

Γr

Turbulence saturates when it removes its drive

∂p

∂r∼ ∂p

∂rkrp ∼ p/Lp

∂p

∂t [p,φ]

DGR =Γr

p/Lp∼ γLp

kθ

The key questions

SOL width – operational parameter estimate

Simulations show expected scaling

Balance of perpendicular transport and parallel losses

dΓr

dr∼ L ∼ n0cs

qRBohm’s Introduction

Global model for SOL turbulenceWhat have we learnt so far ?

Conclusions

Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation

Good agreement between theory and simulationsLp predicted using self-consistent procedure

0 40 80 120 1600

40

80

120

160

Lp (simulation)

Lp(theory)

GBS simulations : R = 500–2000, q = 3–6, ν = 0.01–1, β = 0–3× 10−3

F.D. Halpern et al. 15 / 36 Global EM simulations of tokamak SOL turbulence

Lp (simulations)

LGR

p

Lp q

γ

kθ

max

Lp = Lp(R, q, s, ν)Lp(theory)

The key questions

Lp = R1/2[2π(1− αMHD)αd/q]−1/2

2 0 23

2.5

2

1.5

1

0.5

0

s

log 1

0(!)

SOL turbulent regimes RESISTIVE BALLOONING MODE, with EM EFFECTS

INERTIAL DRIFT WAVES

RESISTIVE DRIFT WAVES

log 1

0(ν)

s

Instabilities driving turbulence depends mainly on q, , . sν

TYPICAL LIMITED SOL

OPERATIONAL PARAMETERS

MAJOR RADIUS αd ∼ (R/Lp)

1/4ν−1/2/qαMHD ∼ q2βR/Lp

γ ∼ γb =2R/Lp

RBM

kθ ∼ kb =

1− αMHD

νq2γb

RBM LIMITED SOL: Lp q

γ

kθ

max

Simulations agree with ballooning estimates

0 30 60 90 120 150 1800

30

60

90

120

150

180

R1/2!

2!"d(1! ")!1/2/q"!1/2

Lp(sim

ulation

)q = 3, R = 500q = 6, R = 500q = 4, R = 500q = 4, R = 1000q = 4, R = 2000

R1/2[2π(1− αMHD)αd/q]−1/2

The key questions

Limited SOL transport increases with and

IntroductionGlobal model for SOL turbulence

What have we learnt so far ?Conclusions


Electromagnetic phase space Build dimensionless phase space with full linear system... Verify turbulent saturation theory with GBS simulations

R = 500, βe = 0 to 3× 10−3, ν = 0.01, 0.1, 1, q = 3, 4, 6

!d/q

!

10!3

10!2

10!1

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

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


Lβ

ν

αM

HD

β ν




SOL turbulence : interplay between β, ν, and ω∗

[LaBombard et al., Nucl Fusion (2005), lower-null L-mode discharges]

Important to understand resistive → ideal ballooning mode transition


Maybe related to the density limit?

Coupling with core physics needs be addressed…

αM

HD

αdLaBombard, NF 2005

0.0

0.4

0.8

1.2

0.0 0.2 0.4 0.6 0.8

Lp = R1/2[2π(1− αMHD)αd/q]−1/2

Limited SOL width widens with

CASTOR

TCV

R

Lp = R1/2[2π(1− αMHD)αd/q]−1/2

Good agreement with multi-machine measurements

The ballooning scaling, in SI units:

! !"!# !"!$ !"!% !"&'!

!"!#

!"!$

!"!%

!"&'()*+,-./012340325(6678990:9-.;/6<=.+

Lp 7.22× 10−8q8/7R5/7B−4/7φ T−2/7

e,LCFSn2/7e,LCFS

1 +

Ti,LCFS

Te,LCFS

1/7

The key questions

The ITER start-up – minimizing vessel heat load

page 132 Chapter 6: Effects of the limiter on the SOL equilibrium

We would like to notice that the SOL configuration considered herein is oversimpli-

fied with respect to the experiments (circular magnetic flux surfaces, no magnetic

shear, cold ions, electrostatic, large aspect ratio, etc.). Therefore we do not tar-

get a quantitative comparison with experimental measurements. Yet for the first

time we provide global, flux-driven, full-n, three-dimensional simulations of plasma

turbulence in different limited SOL configurations, with first-principle ballooning

transport and self-consistent sheath boundary conditions.

! !

Figure 6.1.1: Simulated evolution of the plasma boundary in ITER, from the plasma initiationto the X-point formation. With permission from [146].

6.2 Effect on the scrape-off layer width

The peak heat load onto the plasma facing components of tokamak devices depends

on the SOL width [156,151], which results from a balance between plasma injection

from the core region, turbulent transport, and losses to the divertor or limiter [139].

Typically, the operational definition for the SOL width is the scale length λq of the

radial profile of q, the parallel heat flux, at the location of the limiter or divertor.

The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL

• 

•  Is a LFS or HFS limited plasma preferable (Lp larger)?

Pwall ∝ 1/Awet ∝ 1/Lp

3 2 1 0 1 2 30

20

40

60L p /

s0

SOL width larger in HFS limited plasmas


plasma pressure, namely p∂yφ, in a poloidal cross-section and for each limiter config-uration. Clearly the transport is larger on the LFS, although the poloidal structureof the ballooned transport depends on the position of the limiter. A theoreticaldescription of the effect of the limiter on the ballooning structure of the transportwould probably require a careful study of the linear growth of ballooning modes. Infact, linear studies usually assume poloidally symmetric equilibrium profiles [110].However, as discussed in Chapters 4 and 5, this is in general not true. On theother hand, the sheath boundary conditions may play a role in determining thelinear mode structure, as recently revealed in linear simulations of ideal ballooningmodes [106].

(a) (b)

(c) (d)

Figure 6.2.2: Equilibrium pressure profiles in a poloidal cross-section, with the limiter (a) on the

HFS mid-plane, (b) on the LFS mid-plane, (c) on the top, and (d) on the bottom.




(a) (b)

(c) (d)






(a) (b)

(c) (d)






(a) (b)

(c) (d)




Lp

θ

LFS HFS HFS

3 2 1 0 1 2 30

20

40

60L p /

s0

SOL width larger in HFS limited plasmas

Trends explained by ballooning transport and ExB flow Confirms experiments, but effects smaller



(a) (b)

(c) (d)






(a) (b)

(c) (d)




Lp

θ

LFS HFS HFS

Prediction of the ITER start-up phase

Obtained from the ballooning scaling:

! !"!# !"!$ !"!% !"&'!

!"!#

!"!$

!"!%

!"&'

()*+,-./0.123,304567.689

:;7/.80,<1=>?<>=@:AAB*))<C)804,A230/

Lp 7.22× 10−8q8/7R5/7B−4/7φ T−2/7

e,LCFSn2/7e,LCFS

1 +

Ti,LCFS

Te,LCFS

1/7

The key questions

Potential in the SOL set by sheath and electron adiabaticity On the electrostatic potential in the scrape-off-layer of magnetic confinement devices13

Figure 3. Equilibrium profile of the electrostatic potential φ in a poloidal cross-section

as given from GBS simulations (top row), from Eq. (11) (middle row), and from the

widely used estimate φ = ΛT0 (bottom row) with T0 = (T+e +T−e )/2. Here Λ = 3 (left

column), Λ = 6 (middle column), and Λ = 10 (right column).

Typical estimate: at the sheath

to have ambipolar flows,

Our more rigorous treatment, from Ohm’s law

vi = cs ve = cs exp(Λ− eφ/T she )

φ = ΛT she /e 3T sh

e /e

vi = ve

φ = ΛT she /e+ 2.71(Te − T sh

e )/e

Sheath Adiabaticity ΛT she /e

φt

φtheory

θ

The key questions




GBS simulations show intrinsic toroidal rotation

Snapshot Time-average


GBS simulations show intrinsic toroidal rotation Introduction

Global model for SOL turbulenceWhat have we learnt so far ?

Conclusions



Snapshot Time-average


vi





Snapshot Time-average +/-

There is a finite volume-averaged toroidal rotation (∼ 0.3cs)


vi

t< 0

vi

t> 0

vi

t

A model for the SOL intrinsic toroidal rotation Within the drift-reduced Braginskii model:

Time-averaging:

ExB transport:

Time derivative

Parallel convection

Pressure gradient

ExB transport

∂vi∂t

+ vi∇vi + (vE ·∇)vi +1

min∇p = 0

vi∇vi +∇ · Γv +1

min∇p = 0

θ

rϕ

Γv,θ − σϕ

|Bϕ|vi

∂φ

∂r

Γv,r σϕ

|Bϕ|

vi

∂φ

∂θ

t

Lp∼ qRcs

γky∼ −

L2pcsqR

∂vi∂r

γp∼∂xp∂θφ∼p2t

∂vi∂r

∂xp∼∂xp∼ − γ

kθLp

∂vi∂r

Γv,r ∼vi

∂φ

∂θ

t

γvi∼∂xvi∂θφ∼

∂φ

∂θ

2

t

∂vi∂r

Gradient-removal estimate of ExB velocity transport

Γv,r = −DT∂vi∂r

, DT =L2pcsqR

Parallel momentum

Continuity γp ∼ ∂rp∂θφ

Grad removal ∂rp ∼ ∂rp

Lp estimate Lp ∼ qRγ/(cskθ)

γvi ∼ ∂rvi∂θφ

2D equation for the equilibrium flow

with boundary conditions:

Bohm’s criterion

ExB correction

Turbulent driven radial transport,

gradient-removal estimate

Poloidal convection

Parallel convection

Pressure poloidal asymmetry

Sources of toroidal rotation

Coupling with core physics

vise

= ±cs −q

∂φ

∂r

− ∂

∂r

DT

∂vi∂r

+

σϕ

|Bϕ|∂φ

∂r

∂vi∂θ

+ ασθvi∂vi∂θ

+ασθ

min

∂p

∂θ= 0

Our model well describes simulation results…




GBS simulations agree with the theoryvi

tfrom GBS simulations

vi

tfrom Theory

(limiter position → HFS, down, LFS, up)


Model

Simulation

… and experimental trends

• 

•  Typically co-current

•  Can become counter-current by reversing B or divertor position

M 1

Sheath contribution, co-current

Pressure poloidal asymmetry at divertor plates,

due to ballooning transport, direction: depends

Analytical solution, far from limiter:

Core coupling

M = Mse−r/l +

Λ

2α

ρsLT

e−r/LT − σϕ

2

δn

n+

δT

T

1− e−r/l

What are we learning from GBS simulations? •  To use a progressive simulation approach to investigate

plasma turbulence, supported by analytical theory

•  SOL turbulence: –  Saturation mechanism typically given by gradient removal

mechanism –  Turbulent regimes: in limited plasmas, resistive ballooning

modes –  Good agreement of the scaling of the pressure scale length

with multi-machine measurements –  SOL width larger in HFS limited plasmas –  Sheath dynamics and electron adiabaticity set the electrostatic

potential in the SOL –  Toroidal rotation generated by sheath dynamics and pressure

poloidal asymmetry

simulation of scrape-off layer turbulence

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