The Role of Nonlinear Interactions in Causing Transitions ... · ŸH-mode: edge transport barrier of heat and mass ŸSignificance: – tokamak plasma phases exist – future devices

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


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


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



ŸPresent empirical scaling ignores some real physics (M , v , divertor geometry, etc)i φ




200

400

600

800

1000

1200

1400

0.0

0.5

1.0

1.5

2.0

2.5





ŸParticularly:Δ

– B× B asymmetry– intermediary states (I-mode, LCO)




200

400

600

800

1000

1200

1400

0.0

0.5

1.0

1.5

2.0

2.5





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)


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


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


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


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


Turbulent kinetic energy: Large-scale flow energy:




Zonal flow production term:





1st order condition on turbulence quenching:

Cziegler et al, PPCF 2014/NF 2015





ED

D

normalizedtransfer (R )T

turbulentenergy

e density


C-M

od

Exp

erim

ent






ED

D


turbulentenergy

e density


ŸExperiments pin down sequence of transitionŸnonlinearity leadsŸmean shear flow locks in H-mode

C-M

od

Exp

erim

ent






ED

D


turbulentenergy

e density


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


Δ

B× B asymmetry: the I-mode and Limit-Cycle-Oscillations

Δ

B× B

Δ

B× B

“Favorable” configuration – for H-mode “Unfavorable” configuration


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


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


Outline


Δ



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



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


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:


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


Δ

B× B:


Energy transfer rate into ZF

Ÿ



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


ŸSame normalized energy transfer at the H-mode transition in the favorable/unfav. geometry

Ÿ


ŸTransfer is ~2x in favorable geom.

Consistent with threshold asym-metry: strong ZF are needed

radnet


Δ

B× B:


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


Reminder: ZF-production

1.0 1.5 2.00.5P (MW)net

0

5

10

15

5 -1poloidal velocity shear 10 s




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





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


0.5

1.5

0

22

Rey

no

lds

stre

ss (k

m/s

)

05 10 15

1.0


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



Outline


Δ


ŸTurbulence physics in L-mode leading up to confinement transitions

ŸConclusions


I-mode: attractive alternative high confinement regime


� 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


200

400

600

800

1000

1200

1400

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0.5

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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


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


ED

D

turbulentenergy


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


-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


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


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


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


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.

Documents

The Role of Nonlinear Interactions in Causing Transitions ... · ŸH-mode: edge transport barrier of heat and mass ŸSignificance: – tokamak plasma phases exist – future devices