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
Drift-reduced Braginskii equations
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
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
Drift-reduced Braginskii equations
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
From linear devices… (role of non-curvature driven modes, DW vs KH)
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
Drift-reduced Braginskii equations
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
… to the Simple Magnetized Torus… (role of curvature-driven modes and rigorous code validation)
GBS analysis of configurations of increasing complexity
LAPD, UCLA
HelCat, UNM Helimak, UTexas
TORPEX, CRPP
ITER-like SOL Limited
SOL
…to 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
Drift-reduced Braginskii equations
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
GBS analysis of configurations of increasing complexity
LAPD, UCLA
HelCat, UNM Helimak, UTexas
TORPEX, CRPP
ITER-like SOL Limited
SOL
…to 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
Drift-reduced Braginskii equations
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
… 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
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
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
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
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)
F.D. Halpern et al. 23 / 36 Global EM simulations of tokamak SOL turbulence
Lβ
ν
αM
HD
β ν
IntroductionGlobal model for SOL turbulence
What have we learnt so far ?Conclusions
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
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
F.D. Halpern et al. 20 / 36 Global EM simulations of tokamak SOL turbulence
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
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
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
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
page 134 Chapter 6: Effects of the limiter on the SOL equilibrium
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.
The role of the sheath in magnetized plasma turbulence and flows Joaquim LOIZU, CRPP/EPFL
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
IntroductionGlobal model for SOL turbulence
What have we learnt so far ?Conclusions
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
GBS simulations show intrinsic toroidal rotation
Snapshot Time-average
F.D. Halpern et al. 31 / 36 Global EM simulations of tokamak SOL turbulence
GBS simulations show intrinsic toroidal rotation Introduction
Global model for SOL turbulenceWhat have we learnt so far ?
Conclusions
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
GBS simulations show intrinsic toroidal rotation
Snapshot Time-average
F.D. Halpern et al. 31 / 36 Global EM simulations of tokamak SOL turbulence
vi
IntroductionGlobal model for SOL turbulence
What have we learnt so far ?Conclusions
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
GBS simulations show intrinsic toroidal rotation
Snapshot Time-average +/-
There is a finite volume-averaged toroidal rotation (∼ 0.3cs)
F.D. Halpern et al. 32 / 36 Global EM simulations of tokamak SOL turbulence
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…
IntroductionGlobal model for SOL turbulence
What have we learnt so far ?Conclusions
Saturation mechanismDominant instabilitiesElectromagnetic effectsScrape-off layer width scalingIntrinsic rotation
GBS simulations agree with the theoryvi
tfrom GBS simulations
vi
tfrom Theory
(limiter position → HFS, down, LFS, up)
F.D. Halpern et al. 34 / 36 Global EM simulations of tokamak SOL turbulence
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
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