The Role of Nonlinear Interactionsin Causing Transitions into ETB Regimes
1I. Cziegler ,2 2 2 2 1A.E. Hubbard , J.W. Hughes , J.H. Irby , J.L. Terry , G.R. Tynan
1University of California, San Diego2Massachusetts Institute of Technology, PSFC
57th Annual Meeting of the APS DPP,Savannah, GA, November 16, 2015
Supported by US DoE, Office of ScienceDE-SC-0008689 and DE-FC02-99ER54512
Motivation: understanding confinement states and transitions
I. Cziegler, APS-DPP, Savannah, GA 2
normalized poloidal flux0.85 0.90 0.95 1.00 1.050
200
400
600
800
1000
1200
1400
0.0
0.5
1.0
1.5
2.0
2.5
3.0ŸH-mode: edge transport barrier of heat and mass
ŸSignificance: – tokamak plasma phases exist – future devices need high pressure
ŸNeed to predict the L-H transition power threshold
Motivation: understanding confinement states and transitions
P[M
W]
L
P [MW]THRESH
10
1
0.1
0.2
0.3
0.50.7
2
3
57
.1 1 10.2 .3 .4 .6 .8 2 3 4 5 7
Devices
AUG
CHS
CMODD3DJETJFT2M
JT60U
MAST
NSTX
Z
Y
I. Cziegler, APS-DPP, Savannah, GA 2
0.8 0.72 0.94 P(MW)=0.049B n Sφ e
(Martin, JoP, 2008)
ŸH-mode: edge transport barrier of heat and mass
ŸSignificance: – tokamak plasma phases exist – future devices need high pressure
ŸNeed to predict the L-H transition power threshold
ŸPresent empirical scaling ignores some real physics (M , v , divertor geometry, etc)i φ
Motivation: understanding confinement states and transitions
I. Cziegler, APS-DPP, Savannah, GA 2
normalized poloidal flux0.85 0.90 0.95 1.00 1.050
200
400
600
800
1000
1200
1400
0.0
0.5
1.0
1.5
2.0
2.5
3.0ŸH-mode: edge transport barrier of heat and mass
ŸSignificance: – tokamak plasma phases exist – future devices need high pressure
ŸNeed to predict the L-H transition power threshold
ŸPresent empirical scaling ignores some real physics (M , v , divertor geometry, etc)i φ
ŸParticularly:Δ
– B× B asymmetry– intermediary states (I-mode, LCO)
Motivation: understanding confinement states and transitions
I. Cziegler, APS-DPP, Savannah, GA 2
normalized poloidal flux0.85 0.90 0.95 1.00 1.050
200
400
600
800
1000
1200
1400
0.0
0.5
1.0
1.5
2.0
2.5
3.0ŸH-mode: edge transport barrier of heat and mass
ŸSignificance: – tokamak plasma phases exist – future devices need high pressure
ŸNeed to predict the L-H transition power threshold
ŸPresent empirical scaling ignores some real physics (M , v , divertor geometry, etc)i φ
Ÿ
ŸPhysics-based model:– identify L-to-H transition trigger mechanism– connect turbulence physics to macroscopic scaling
Particularly:Δ
– B× B asymmetry– intermediary states (I-mode, LCO)
I. Cziegler, APS-DPP, Savannah, GA 3
Outline
ŸBackground:– turbulence-flow interactions in the L-H transition
Δ
– B× B asymmetry: the I-mode and Limit-Cycle-Oscillations
Ÿ
ŸFlow dynamics in the I-mode regime and accessibility
ŸConclusions
Turbulence physics in L-mode leading up to confinement transitions
Study flow dynamics at~5-10ms around L-H transition
...in the region where the edgetransport barrier forms.
ne
(10
20 m
-3)
tLH
+79ms
+9ms
+2ms
-1ms
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
Ψn
Focus on immediate trigger of L-H transition
I. Cziegler, APS-DPP, Savannah, GA 4
Thomson scattering
0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
2.5
P (MW)aux
-20 2∫n dl (10 m )e
Hα (a.u.)
L LH
t (s)
Transient at L-H and the drift-wave/zonal-flow system
I. Cziegler, APS-DPP, Savannah, GA 5
ŸPoloidal flows in the edge are important
ŸParticularly E before r
evolution of the pressure gradient
ŸGradient + electric field lead to feedback loop
Transient at L-H and the drift-wave/zonal-flow system
I. Cziegler, APS-DPP, Savannah, GA 5
ŸPoloidal flows in the edge are important
ŸParticularly E before r
evolution of the pressure gradient
ŸGradient + electric field lead to feedback loop
ŸCandidate for trigger: predator-prey model(Kim,Diamond PRL 2003)(Miki,Diamond PoP 2012)
Gradient Drive
Damping (νii)
ZF Inhibition
L. Schmitz 2014
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Zonal flow production term:
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Zonal flow production term:
1st order condition on turbulence quenching:
Cziegler et al, PPCF 2014/NF 2015
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Zonal flow production term:
ED
D
normalizedtransfer (R )T
turbulentenergy
e density
1st order condition on turbulence quenching:
C-M
od
Exp
erim
ent
Cziegler et al, PPCF 2014/NF 2015
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Zonal flow production term:
ED
D
normalizedtransfer (R )T
turbulentenergy
e density
1st order condition on turbulence quenching:
ŸExperiments pin down sequence of transitionŸnonlinearity leadsŸmean shear flow locks in H-mode
C-M
od
Exp
erim
ent
Cziegler et al, PPCF 2014/NF 2015
Nonlinear transfer explains initial turbulence reductionand generation of poloidal flow
I. Cziegler, APS-DPP, Savannah, GA 6
Turbulent kinetic energy: Large-scale flow energy:
Zonal flow production term:
ED
D
normalizedtransfer (R )T
turbulentenergy
e density
1st order condition on turbulence quenching:
ŸExperiments pin down sequence of transitionŸnonlinearity leadsŸmean shear flow locks in H-mode
C-M
od
Exp
erim
ent
Transfer is localized to narrow region at edge with shear
before
during
ProductionTurbulence power
I. Cziegler, APS-DPP, Savannah, GA 7
Δ
B× B asymmetry: the I-mode and Limit-Cycle-Oscillations
Δ
B× B
Δ
B× B
“Favorable” configuration – for H-mode “Unfavorable” configuration
I. Cziegler, APS-DPP, Savannah, GA 8
ŸDrift active X-pointŸH-mode threshold ~P (Martin)th
ŸBelow threshold: regime, aka “dithers”, aka “I-phase”
towards
LCO
ŸDrift active X-pointŸH-mode threshold ~2 x P (Martin)th
ŸBelow threshold: (at low n )e
away from
I-mode
Δ
B× B asymmetry: the I-mode and Limit-Cycle-Oscillations
Δ
B× B
Δ
B× B
“Favorable” configuration – for H-mode “Unfavorable” configuration
I. Cziegler, APS-DPP, Savannah, GA 8
ŸDrift towards active X-pointŸH-mode threshold ~P (Martin)th
ŸBelow threshold: LCO regime, aka “dithers”, aka “I-phase”
ŸDrift away from active X-pointŸH-mode threshold ~2 x P (Martin)th
ŸBelow threshold: I-mode (at low n )e
Both have further, as yetunexplained sensitivities– (B , n , I )φ e p
I. Cziegler, APS-DPP, Savannah, GA 9
Outline
ŸBackground:– turbulence-flow interactions in the L-H transition
Δ
– B× B asymmetry: the I-mode and Limit-Cycle-Oscillations
ŸFlow dynamics in the I-mode regime and accessibility
ŸConclusions
ŸTurbulence physics in L-mode leading up to confinement transitions
limiter
nozzle
GPI
TCI chords
GPC
TS
Ÿ
sensitive to , Te
Ÿ small toroidal extent (~5cm) allows localization
Ÿ 90 channels cover ~4cm x 3.6cm
Ÿ views coupled to avalanche photodiodes (APD), sampled at 2 MHz
inject D or He, 2
ne
Diagnostic setup
=f=5kHz
50kHz=f=0kHz
3kHz
vr
vθ
88 89 90 91-5
-4
-3
-2
-1 LC
FS
z (c
m)
R (cm)
Ÿ Two-point correla-tion time delay estimate (TDE):
= Δx /τi i d
with a sample
size per point.10μs
Ÿ Averaging in analysis involves
= θ,t
Primary diagnostic:Gas Puff Imaging
I. Cziegler, APS-DPP, Savannah, GA 10
I. Cziegler, APS-DPP, Savannah, GA 11
Bispectral estimate of three-wave coupling with “source” f1 and “target” f freq. resolved
Ohmic L-mode
-40
-20
0
20
40
f 1(k
Hz)
[10 m s ]11 2 -3
0.6
-0.6
-0.3
0.3
0.0
0 40 5010 20 30f (kHz)
f 1(k
Hz)
Strong transfer from all frequencies observed near threshold power
clear transferbtwn flctns in5-40 kHz rangewith aux. powerbelow threshold(favorable geometry)
P =750kWaux
no strongtransfer ofpoloidal flowpower
-40
-20
0
20
40
0 40 5010 20 30
f (kHz)
[10 m s ]11 2 -3
1.6
-1.6
-0.8
0.8
0.0
0 10 20 30 40 50f (kHz)
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
(10
5 s
-1)
Zonal flow
Quantification of kinetic energy transfer from turbulence to ZF
ŸTransfer rates defined as sum of transfer function normalized to flow power
ŸQuantified below some frequency for ZFŸWith strong ICRF heating, clear peak is shownŸOther frequencies average to negative value
Nonlinear drive rate
-40
-20
0
20
40
f 1(k
Hz)
0 40 5010 20 30
f (kHz)
[10 m s ]11 2 -3
1.6
-1.6
-0.8
0.8
0.0
sum
mat
ion
I. Cziegler, APS-DPP, Savannah, GA 12
1.2 MA1.0 MA0.8 MA
T /v
ZFK~
unfav
1.0 1.5 2.00.5P (MW)net
0
1
2
3
(10
s)
5-1
H-mode-favorable more transfer at same heat flux and ne
Δ
B× B:
I. Cziegler, APS-DPP, Savannah, GA 13
Energy transfer rate into ZF
Ÿ
monotonically with Pnet
Transfer into zonal flow increases
radnet
Sensible control parameter is:
1.2 MA 1.2 MA1.0 MA 1.0 MA0.8 MA
T /v
ZFK~
favunfav
1.0 1.5 2.00.5P (MW)net
0
1
2
3
(10
s)
5-1
H-mode-favorable more transfer at same heat flux and ne
Δ
B× B:
I. Cziegler, APS-DPP, Savannah, GA 13
Energy transfer rate into ZF
Ÿ
monotonically with Pnet
Transfer into zonal flow increases
ŸTransfer is ~2x in favorable geom.
radnet
Sensible control parameter is:
1.2 MA 1.2 MA1.0 MA 1.0 MA0.8 MA
favunfav
fav
L-H
thre
shold
unfa
vora
ble
L-H
thre
shold
Ÿ
monotonically with Pnet
ŸSame normalized energy transfer at the H-mode transition in the favorable/unfav. geometry
Ÿ
Transfer into zonal flow increases
ŸTransfer is ~2x in favorable geom.
Consistent with threshold asym-metry: strong ZF are needed
radnet
H-mode-favorable more transfer at same heat flux and ne
Δ
B× B:
I. Cziegler, APS-DPP, Savannah, GA 13
Energy transfer rate into ZF Sensible control parameter is:
2.5 3.00
1
2
3
(10
s)
5-1
1.0 1.5 2.00.5P (MW)net
Is shear or stress the dominant component in transfer?Velocity shear still depends strongly on heat flux
I. Cziegler, APS-DPP, Savannah, GA 14
Reminder: ZF-production
1.0 1.5 2.00.5P (MW)net
0
5
10
15
5 -1poloidal velocity shear 10 s
I. Cziegler, APS-DPP, Savannah, GA 14
Reminder: ZF-production
Is shear or stress the dominant component in transfer?Velocity shear still depends strongly on heat flux
Ÿ
Ÿvelocity shears are clearly distinguishable between favorable/unfavorable
grad-B asymmetry persists
1.0 1.5 2.00.5P (MW)net
0
5
10
15
5 -1poloidal velocity shear 10 s
I. Cziegler, APS-DPP, Savannah, GA 14
Reminder: ZF-production
Is shear or stress the dominant component in transfer?Velocity shear still depends strongly on heat flux
Ÿ
Ÿgrad-B asymmetry
Ÿ
ŸResults seem to point to a minimum shear needed for any Reynolds stress to appear
Ÿconsistent with eddy shearing mechanism of Reynolds stress generation
strong correlation observed
disappears
single correlation → magnetic shear/stress makes little difference in asymmetry
5 -1poloidal velocity shear 10 s
0.5
1.5
0
22
Rey
no
lds
stre
ss (k
m/s
)
05 10 15
1.0
I. Cziegler, APS-DPP, Savannah, GA 15
as a function of
Does shear determine stress? Are other components(e.g. magnetic shear) just as important?
1.2 MA 1.2 MA1.0 MA 1.0 MA0.8 MA
favunfav
1.0 1.5 2.00.5P (MW)net
0
1
2
3
En. tr
ansf
. ra
te to Z
F (
10
s)
5-1
L-H transition L-I transition
Favorable/unfavorable expts go up to different types of transitions
I. Cziegler, APS-DPP, Savannah, GA 16
I. Cziegler, APS-DPP, Savannah, GA 17
Outline
ŸBackground:– turbulence-flow interactions in the L-H transition
Δ
– B× B asymmetry: the I-mode and Limit-Cycle-Oscillations
ŸTurbulence physics in L-mode leading up to confinement transitions
ŸConclusions
ŸFlow dynamics in the I-mode regime and accessibility
I-mode: attractive alternative high confinement regime
I. Cziegler, APS-DPP, Savannah, GA 18
� stationary, high energy �
confinement, low τp, ELM-free regimeexplored on C-Mod, AUG and DIII-D, helping to delineate its operational space
Amanda Hubbard, Invited Tues. Afternoon. KI2.00003
I-mode operating space: C-Mod, DIII-D and AUG
C-Mod
normalized poloidal flux0.85 0.90 0.95 1.00 1.050
200
400
600
800
1000
1200
1400
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ŸOnly regime in Alcator C-Mod with poloidal velocities exhibiting geodesic acoustic modes (GAM)
ŸGAM: n=m=0 electrostatic modes, rotating plasma shells back and forth at moderate frequencies
ŸNonlinear studies show GAM responsible for broad frequency range in weakly coherent mode (WCM)
ŸWhich is responsible for unique quality of transport?
1
2
3
45
TT
ee
(ke
V)
(ke
V)
0.4
0.6
0.8
20
40
60
80100
f (k
Hz)
t (s)
0
100
200
300
400
f (k
Hz)
1101209031
core Te
edge Te
v fluct.θ
density fluct.
I-mode
1.1 1.2 1.3 1.4
GAM
WCM
Characteristic edge fluctuations: WCM and GAM
I. Cziegler, APS-DPP, Savannah, GA 19
L-I transition on a fine time scale dissimilar to L-H
1.116 1.118
vGAM
T (a)e
Dα
beginning of heat pulse
onset of GAM(due to T rise,ν drop)e ii
0.00
0.05
0.10
0.15
0.20
0.5
0.4
0.3
10
8
6
4
2
0
ne~n0
4
2
0
-2
-4
km/s
keV
no significant Dα drop(mass transport L-mode-like)
total turbulence powernot affected
G2
G R -like termT
normalizedtransfer (R )T
ED
D
turbulentenergy
I. Cziegler, APS-DPP, Savannah, GA 20
0
0.5
1.0
1.120 1.122 1.124t (s)
γeff-0.5
-1.0
L-I transition L-H transition
0
5
10
15
Dα
Transfer function in unfavorable dir. indicates GAM/ZF competition
-40
-20
0
20
40
0 40 5010 20 30f (kHz)
fGAM
fGAM
unfavorable
I. Cziegler, APS-DPP, Savannah, GA 21
-40
-20
0
20
40
f 1(k
Hz)
[10 m s ]11 2 -3
0.6
-0.6
-0.3
0.3
0.0
0 40 5010 20 30f (kHz)
-40
-20
0
20
40
0 40 5010 20 30
f (kHz)
[10 m s ]11 2 -3
1.6
-1.6
-0.8
0.8
0.0
T P=v ( 1.6MW)favorable
continuous strip
1.2 MA 1.2 MA1.0 MA 1.0 MA0.8 MA
favunfav
1.0 1.5 2.0 2.5 3.00.5P (MW)net
0
1
2
3Measured nonlinear energy transfer rate (10 s )5 -1
fav
L-H
th
resh
old
un
fav L
-I t
hre
sh
old
un
fav
I-H
th
resh
old
?
I-mode“window”
Can transfer function be “continued” into I-mode?
GAM
Ÿ GAM transfer small but comparable to ZF rates
Ÿ I-H transition is still poorly understood
Ÿ GAM-ZF competition?
Ÿ ZF transfer continues on in the I-mode regime?
Ÿ trend of ZF shows plausible I-H mechanism
I. Cziegler, APS-DPP, Savannah, GA 22
Evidence points to role of GAM damping in I-mode access
1.10 1.15 1.20 1.25 1.30t (s)
0
100
200
300
f (k
Hz)
20
40
60
80100
f (k
Hz)
400
L-I
I-H
0.5 cm < < 2 cm-1 -1
kθn~
0
5
10
15
10
s4
-1
1101209014
Neoclassical estimate:
vθ
~
Ÿ Damping is compared to nonlinear drive (analogous to ZF-production)
Ÿ Demonstrates dependence on T:
Ÿ I-mode threshold shown [Hubbard NF12] to scale as
∝-3/2
∝ ∝1th I p
I. Cziegler, APS-DPP, Savannah, GA 23
0.2
0.4
0.6
0.8 T edge (keV)eCompared to nonlinear drive
Ÿ Threshold is also shown to be sensitive to impurities; which is expected due to:
where is ion impurity collision factor
Ÿ Effective atomic number Z is used as a proxyeff
Ÿ Consistent with difficulty accessing the regime in He plasma
Df
f
I. Cziegler, APS-DPP, Savannah, GA 23
Evidence points to role of GAM damping in I-mode access
Pnet
(MW)
Neoclassical estimate:
1.0
7.5
7.0
6.5
1.5
2.0
2.5
ne19 -2(10 m )
Zave
6
5
4
CORET e
(keV)3.5
4.0
4.5
5.0
5.5
1.2 1.25 1.3
t(s)
pre-I L-mode
Both GAM and edge coherent mode are crucial for I-mode
0
50
100
150
200
250
f (k
Hz)
-6 -4 -2 0 2 4 6k (cm-1)
I-mode S(k,f)
C-Mod:Ÿ
of WCM without GAM)Ÿ I-mode when GAM appearsŸ no GAM in L-mode or H-mode
coherent mode close to I-mode (seed Ÿ GAM are present in L-modeŸ no high frequency coherent
fluctuationŸ I-mode when mode appears
WCM
0
50
100
150
200
250
f (k
Hz)
-6 -4 -2 0 2 4 6k (cm-1)
edge coherentmode
1 10 100f (kHz)
0.0001
0.0010
0.0100
0.1000
P(f
) (A
.U.)
Evolution from L-mode
Ohmicpre-II-mode
df/f=0.1df/f=0.7
AUG: [Manz NF ‘15] Ÿ mid-range frequen-cies are reduced
Ÿ formation of ECM already takes away some power
I. Cziegler, APS-DPP, Savannah, GA 24
Conclusions
ŸMeasurements of edge flow nonlinearities support predator-prey dynamics
ŸReynolds stress is determined by flow shear, showing relatively minor contribution from magnetic effects
ŸTransition dynamics are not analogous in L-H and L-I transition with ZF and GAM phenomena, yet:
ŸKinetic energy transfer in favorable/unfavorable shows a ~2x asymmetry– consistent with strong Zonal Flows being necessary for L-H transition
ŸBoth GAM and edge coherent mode are necessary for I-mode
ŸFine details of GAM damping show correlation with I-mode access
Open Questions
Ÿorigin of WCM - expect coherent mode!
Ÿorigin of the separation of transport channels:- high freq driving mass transport?- reduction of low freq component?
ŸIs the ZF still the trigger in the I-H transition?– need correct ZF measurement in I-mode– compare GAM and ZF transfers
Supplemental material
GAM is responsible for broad frequencyrange of WCM via nonlinear coupling
The weakly coherent mode is predominantly a density fluctuation. Consequently, the relevant
nonlinear transfer process is:
power transferaway from fWCM
WCM
0 50 100 150 200 250 300f1 (kHz)
0
50
100
150
200
250
300
f (k
Hz)
1 10 100 1000f (kHz)
-4
-2
2
4
5 (
10
s-1)
GAM
fGAM
0
0.5
-0.5
0.0
2 2 1/2 | | | |2 | |
Motivation of time-resolved analysis technique
Momentum equation for incompressible fluid
Reynolds decomposition
with : net difference of drive and decorrelation
kinetic energy in turbulence
kinetic energy in low freq flow
Reynolds stress mediated zonal flow production:
Transport terms in the form of energy flux: /
-4 -2 0 2t-t LH (ms)
0
2
4
6
Δv θ
(k
m/s
)v θ
(k
m/s
)
Bt=5.2T Ip=0.8MAP=2.1MW
Bt=5.4-2.8T Ip=0.8MAOhmic
Bt=4.0T Ip=0.8MAP=1.5MW
Time histories are aligned by D light αdrop at t = 0ms
Poloidal velocity excursion is reproduced in all H-mode favorable transitions regardless of
Ÿ the kind of L-H transition
Ÿ magnetic field
Ÿ threshold power
Ÿ geometry
Sample traces
Low frequency flows:from 3 separate experiments
Turbulent velocities
0.826 0.828 0.830 0.832 0.834-10
-5
0
5
10
L-H
t (s)
For a quantitative test of the adequacy of the mechanism to cause the full extent of turbulence quenching, integrate the model Eq.
Nonlinear transfer is the dominant quenching mechanism
0 2 4 6 8∆K (107 m2s-2)
0
10
20
30
I tr
Ÿthe amount that is expected if this is the dominant turbulence quenching mechanism
Ÿ radial structure is considered
Ÿ turbulence is not homogenous poloidally while flow is
Nonlinear energy transfer exceeds
Convergence of bicoherence
ŸGood sign of convergence for both (bicoherence and transfer) estimators
ŸReasonable convergence by ~280 realizations
2ŸCoupling (b ) shows before
drive (T ) is significant oru
Ÿbefore I-mode (typical GAM regime in C-Mod)
0.000 0.002 0.004 0.006 0.008 0.0101/N
0.0
0.5
1.0
1.5
2.0
2.5
B2
ZF
GAM
noise
stat. variance
Critical parameters
1D including the energy-flux-like term
First order condition on collapse: terms taking energy from turbulence into zonal flows must exceed the drive. Written with symbols from the model equations:
turbulence
where transfer terms and the kinetic energy are directly calculated,and can be estimated from the model Eq. in a stationary L-mode as
RT ≡
Full L-H transition requires growth of non-turbulence driven shear, iepressure gradient, to lock in H-mode once turbulence is reduced.