optimization of the plasma response for the control of ...€¦ · 2.0 br vacuup 1e−4 positive...

Paz-Soldan/IAEA/10-2016

Optimization of the Plasma Response for the Control of ELMs with 3D fields By C. Paz-Soldan1

On behalf of R. Nazikian2, R. Moyer3, J. Callen4, T. Evans1, N. Ferraro2, N. Logan2, B. Lyons5, X. Chen1, L. Cui2, D. Eldon1, M. Fenstermacher6, A. Garofalo1, B. Grierson2, R. Groebner1, C. Hegna4, S. Haskey2, M. Lanctot1, C. Lasnier6, T. Luce1, G. McKee4, D. Orlov3, T. Rhodes7, M. Shafer8, P. Snyder1, W. Solomon1, E. Strait1, A. Wingen8, R. Wilcox8

1General Atomics 2Princeton Plasma Physics Laboratory 3University of California-San Diego 4University of Wisconsin-Madison 5Oak Ridge Associated Universities 6Lawrence Livermore National Laboratory 7University of California-Los Angeles 8Oak Ridge National Laboratory Presented at the

IAEA Meeting Kyoto, Japan

October 18, 2016

1 C Paz-Soldan/8-05 Meeting/June2013

by C. Paz-Soldan, N. Logan, M. Lanctot, S. Smith, M. Van Zeeland, J. DeGrassie, K. Burrell, A. Garofalo, B. Grierson, J. Hanson, J. King, C. Petty, M. Shaffer, D. Shiraki, W. Solomon, B. Tobias 8:05 Meeting June 17th 2013

MP 2013-14-01 “Plasma Kink Response to NTV torque”

N.C. Logan/APS DPP/11.2015

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4

Positive reluctance modes are paramagnetic

• Isolate the response to the most positive reluctance eigenmode - Amplifies vacuum field at the wall - Large measured response corresponds to large fields in the plasma as intuitively expected

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

1e−4Total FieldPlasma ResponseVacuum Field

0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5

0.80 0.90 1.00

8/2

N

ELMing

Suppressed

0

-30

-60

30

60

This work was supported by the US Department of Energy under DE-FC02-04ER546981, DE-AC02-09CH114662, DE-FG02-05ER548093, DE-FG02-92ER541394, DE-AC52-07NA273446, DE-FG02-08ER549847, and DE-AC05-00OR227258.

Paz-Soldan/IAEA/10-2016 2

DIII-D demonstrates Edge-Localized Mode (ELM) Control with 3-D Fields in ITER 15 MA Q=10 Conditions

M. Wade et al., Nucl. Fusion 2015

•   ELMs suppressed if 3-D field magnitude meets ITER design criteria

Match ITER shape, I/aB, βN


DIII-D demonstrates Edge-Localized Mode (ELM) Control with 3-D Fields in ITER 15 MA Q=10 Conditions

•   ELMs suppressed if 3-D field magnitude meets ITER design criteria

•   Reducing toroidal rotation causes ELM return

•   Plasma response must be understood to explain effect and optimize ELM control with 3D fields

Match ITER shape, I/aB, βN






1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5

3D Coils

Torque Control w/ NBI

Actuators: •   NBI torque (@ fixed power) •   3D coils (n=2 or n=3)

Control of Plasma Rotation and MHD Mode Spectrum Required to Optimize the Plasma Response for ELM Control






1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5 Actuators: •   NBI torque (@ fixed power) •   3D coils (n=2 or n=3)

Diagnostics: •   High-field side (HFS) magnetics •   Plasma rotation & Er





1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5

0.80 0.90 1.00

8/2

N

ELMing

Suppressed

0

-30

-60

30

60

Magnetic Sensors

Rotation



•   Hypothesis: 3D fields drive resonant field penetration at pedestal top to restrict its width

à Prevents ELM instability

•   Requires co-alignment of: –   3-D field (Resonant Drive)

–   Low ω⟂,e rotation region

–   Resonant surface

–   …at the pedestal top

ΨN 1

q

2

5

4

3

Resonant Surfaces

Potential for field

penetration

0.8

P

0 Low rotation

ω⟂,e

Scr

eeni

ng

Penetration at pedestal top

P. Snyder et al., Phys. Plasmas 2012







1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5 •   Observations validate resonant field penetration as optimization criterion

•   Penetration requires optimized electron rotation profile

•   Resonant drive can be optimized by 2D equilibrium conditions and 3D spectrum







1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5


•   Observations validate resonant field penetration as optimization criterion




Fast Changes in Rotation Profiles and HFS Magnetics are Found at Entry to the ELM Suppressed State

•   Use n=2 field to scan applied spectrum and ease diagnosis

•   Bifurcation into ELM suppression impacts high-field side magnetic response and toroidal rotation

•   Changes occur together on a fast (10 ms) time scale

3 4 5t (s)

0

2HFS n=2

||Bpl (G)

Vtor (km/s)

04080

Dα

TearingDrive

C. Paz-Soldan et al., PRL 2015 R. Nazikian et al., PRL 2015

158115

Scan applied n=2 spectrum


MHD Modeling Shows Magnetic Response Changes Expected Purely from Field Penetration at Pedestal Top

•   Model with resistive single-fluid MHD (M3D-C1)

•   Substitute ELMing and ELM suppressed rotation profiles

0.80 0.90 1.00

7/2

8/2

9/2

N

ELMingSuppressed

0

-30

-60

30

60

E ELMing Suppressed

0.800.90

1.00

7/2

8/2

9/2

N

ELMing

Suppressed

0

-30

-60 30 60

E

B. Lyons et al., PPCF (in review)


0.80 0.90 1.00

7/2

8/2

N

ELMingSuppressed

0.4

0.2

0.0

0.6

1.0

0.8




•   Model predicts significant 8/2 penetration @ suppression –   Pedestal expansion stopped

before ELM stability limit

M3D-C1

0.80 0.90 1.00

7/2

8/2

9/2

N

ELMingSuppressed

0

-30

-60

30

60

E

Pedestal

ELMing Suppressed

ELMing Suppressed

0.800.90

1.00

7/2

8/2

N

ELMing

Suppressed

0.4

0.2

0.0

0.6

1.0

0.8

0.800.90

1.00

7/2

8/2

9/2

N

ELMing

Suppressed

0

-30

-60 30 60

E



ELMingSuppressed

0.4

0.2

0.0

-0.2

-0.41.00.50.0-0.5-1.0

Distance Along HFS Wall (m)

HFS

Plas

ma

Resp

onse

(G/k

A)


Helical pattern at one toroidal phase

HFS Response (G/kA) M3D-C1



•   Model predicts significant 8/2

penetration @ suppression –   Pedestal expansion stopped

before ELM stability limit

•   Model predicts shift in HFS response from penetration –   No effect predicted for LFS

•   What about experiment? 0.80 0.90 1.00

7/2

8/2

N

ELMingSuppressed

0.4

0.2

0.0

0.6

1.0

0.8


ELMing Suppressed

ELMing Suppressed

0.800.90

1.00

7/2

8/2

N

ELMing

Suppressed

0.4

0.2

0.0

0.6

1.0

0.8M3D-C1


•   Consider back-transition from ELM suppressed state –   Before any ELMs appear –   Zoom in on ms timescale

ELMing Suppressed

ñ (a.u.)

Back-transition from ELM Suppression Reveals Rotation and HFS Magnetic Changes on Millisecond Timescale

R. Nazikian et al, NF (in preparation)


ñ (a.u.)


•   Prompt change in turbulent Doppler shift in ms timescale –   Indicates rotation change

zoom


R. Nazikian et al., NF (in preparation)



•   Prompt change in turbulent Doppler shift in ms timescale –   Indicates rotation change

•   HFS structures shift in Z, j immediately (1 ms) after losing ELM suppression

•   Qualitative match to model


R. Nazikian et al., NF (in preparation)

ñ (a.u.)

HFS Mag. vs. Z

HFS Mag. vs. ϕ

zoom







1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5 •   Observations validate resonant field penetration as optimization criterion

•   Penetration requires optimized electron rotation profile –   Torque dependence –   Performance recovery


Paz-Soldan/IAEA/10-2016 17 R. Moyer et al., APS-DPP 2016

n=3 fields •   ELMs are suppressed at a

critical rotation (NBI torque)

Rotation Zero-crossing Model Can Explain Why Elms Only Suppressed Above Critical Value of Rotation



Pedestal ELM

R. Moyer et al., APS-DPP 2016

n=3 fields

ELMing

•   ELMs are suppressed at a critical rotation (NBI torque)

•   In ELMing conditions, rotation zero-crossing is at low ΨN



•   ELMs are suppressed at a critical rotation (NBI torque)

•   In ELMing conditions, rotation zero-crossing is at low ΨN

•   ω⟂,e zero crossing moves out as NBI torque increased

•   Field penetration moves out, constricting pedestal width

Pedestal ELM

R. Moyer et al., APS-DPP 2016

n=3 fields

ELMing Suppressed


Imposed NBI Torque Affects Inner Boundary Condition … but w⟂,e Depends on Local Resonant Torques

•   3D field torque at rational surface key in balance

•   NBI torque can be insufficient to unlock rational surface

NBI

3D


Imposed NBI Torque Effects Inner Boundary Condition … but w⟂,e Depends on Local Resonant Torques

•   3D field torque at rational surface key in balance

•   NBI torque can be insufficient to unlock rational surface

•   Zero-crossing point jumps to next rational surface –   Does not linger in between


•   Once 3D field penetrates can reduce coil current: hysteresis!

•   Confinement recovered before ELMs return @ back-transition

159443

Resonant Torques Can Maintain Locked w⟂,e as 3D Coil Current Reduced – Enabling Confinement Recovery


Resonant Torques Can Maintain Locked w⟂,e as 3D Coil Current Reduced – Enabling Confinement Recovery

•   Once 3D field penetrates can reduce coil current: hysteresis!

•   Confinement recovered before ELMs return @ back-transition

•   Wide variety of pedestal

conditions compatible with static ω⟂,e zero-crossing

•   Gradient driven flows balance toroidal rotation spin up

159443

P (kPa) ω⟂,e (krad/s)

Omega - Perp - E

!E⇥B =Er

RBp(1)

=1

6nc

@pc@ | {z }

carbon diamag

± !(V⇥B)C| {z }

carbon tor, pol

(2)

!(V⇥B)e ⌘ !?,e (3)

= !E⇥B +1

ne

@pe@ | {z }

elec diamag

(4)

= ± !(V⇥B)C| {z }

carbon tor, pol

+1

6nc

@pc@ | {z }

carbon diamag

+1

ne

@pe@ | {z }

elec diamag

(5)

!?,e = !E +1

ne

@pe@

(6)

Omega - Perp - E - Simple

!E⇥B =1

6nc

@pc@ | {z }

!⇤C (carbon diamag)

± !(V⇥B)C| {z }

!carbon tor, pol

(7)

!E⇥B = !�,c ⇥ b̂| {z }toroidal

+!✓,c ⇥ b̂| {z }polodial

+ !⇤c|{z}

diamagnetic

(8)

!E⇥B = !�,c|{z}toroidal

⇥b̂+ !✓,c|{z}polodial

⇥b̂+ !⇤c|{z}

diamagnetic

(9)

!(V⇥B)e ⌘ !?,e (10)

= !E⇥B +1

ne

@pe@ | {z }

elec diamag

(11)

= ± !(V⇥B)C| {z }

carbon tor, pol

+1

6nc

@pc@ | {z }

carbon diamag

+1

ne

@pe@ | {z }

elec diamag

(12)

1







1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3haVing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

Vac

uuP

1e−4



1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br

3la

sPa

1e−4

1.0 1.7 2.45 (P)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

z (P

)

0 3hasing

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

Br


0 10 20 30 40 50 60 70 80EigenmRde Index

−3−2−1

012345

5el

ucta

nce

1e5 •   Observations validate resonant field penetration as optimization criteria


•   Resonant drive can be optimized by 2D equilibrium conditions and 3D spectrum –   Role of beta, collisionality –   3D spectrum optimization


3 4 5t (s)

Dα

TearingDrive

02

HFS n=2||Bpl (G)

0

1

Varying Applied Spectrum Demonstrates Correlation of ELM Suppression with HFS (+ Top/Bottom) Response

Bifu

rca

tion

Bifu

rca

tion

Resonant •   Plasma response during ELM

suppression largest on high-field side (HFS) + top / bottom

•   Low-field side (LFS) uncorrelated with ELM suppression

C. Paz-Soldan et al., PRL 2015

Scan applied n=2 spectrum

158115


0.0

0.5

1.0

1.5

2.0

2.5βN = 1.3, 2.1, 2.8

0

2

4

6

8

10 βN

0 90 180 270 360ΔφUL

data IPEC model

LFS

Resp

onse

(G)

HFS

Res

pons

e (G

)

applied n=2 spectrum (ΔϕUL)

0.0

0.5

1.0

1.5

2.0

2.5βN = 1.3, 2.1, 2.8

0

2

4

6

8

10 βN

0 90 180 270 360ΔφUL

data IPEC model

LFS

Resp

onse

(G)

HFS

Res

pons

e (G

)

•   LFS measurements swamped by pressure driven modes βN scan

Measurements Find LFS Plasma Response Sensitive to bN

C. Paz-Soldan et al., NF 2016


0.0

0.5

1.0

1.5

2.0

2.5βN = 1.3, 2.1, 2.8

0

2

4

6

8

10 βN

0 90 180 270 360ΔφUL

data IPEC model

LFS

Resp

onse

(G)

HFS

Res

pons

e (G

)

applied n=2 spectrum (ΔϕUL)

•   LFS measurements swamped by pressure driven modes

•   Striking invariance of the HFS response to plasma pressure –   MHD modeling agrees

•   HFS sensitive to pedestal effects like field penetration

βN scan

Measurements Find LFS Plasma Response Sensitive to bN, HFS Totally Invariant; Collisionality Has Opposite Effect

C. Paz-Soldan et al., NF 2016


Measurements Find LFS Plasma Response Sensitive to bN, HFS Totally Invariant; Collisionality Has Opposite Effect

0.0

0.5

1.0

1.5

2.0

ν*e

0 90 180 270 360ΔφUL

data IPEC model

LFS

Resp

onse

(G)

HFS

Res

pons

e (G

)

0

2

4

6

8ν = 0.3, 4*e

applied n=2 spectrum (ΔϕUL) C. Paz-Soldan et al., NF 2016

ELM suppression

@ low ν

•   LFS measurements swamped by pressure driven modes

•   Striking invariance of the HFS response to plasma pressure –   MHD modeling agrees

•   HFS sensitive to pedestal effects like field penetration

•   Collisionality reduces HFS only

•   ITER-relevant collisionality needed for right MHD modes

νe scan *


•   Resonant drive @ core surfaces increased by core pressure –   Opposite for edge surfaces

1 1.5 2 2.5βN0

1

2

3

4

J res

(sca

led)

∆φ=0

0.5 0.75 1 1.250

1

2

3

J res

(sca

led)

IPEC

∆φ=0

Jboot /Jboot,exp

9/2(edge)

4/2(core)

9/2(edge)

4/2(core)

IPEC

0 0.5 10

100

200P (kPa)

ρN

model

Increasing Core Pressure Works Against Edge Resonant Coupling


1 1.5 2 2.5βN0

1

2

3

4

J res

(sca

led)

∆φ=0

0.5 0.75 1 1.250

1

2

3

J res

(sca

led)

IPEC

∆φ=0

Jboot /Jboot,exp

9/2(edge)

4/2(core)

9/2(edge)

4/2(core)

IPEC

0 0.5 10

100

200P (kPa)

ρN

0.8 0.9 1.00

50

ρN

J|| (A/cm2)< >100


•   Resonant drive @ edge increases with bootstrap current –   Path for low collisionality to

assist ELM suppression model

model

Increasing Core Pressure Works Against Edge Resonant Coupling … Low Collisionality Bootstrap Helps


0 0.5 10

100

200P (kPa)

ρN

0.8 0.9 1.00

50

ρN

J|| (A/cm2)< >100

Increasing Core Pressure Works Against Edge Resonant Coupling … Low Collisionality Bootstrap Helps

1 1.5 2 2.5βN0

1

2

3

4

J res

(sca

led)

∆φ=0

0.5 0.75 1 1.250

1

2

3

J res

(sca

led)

IPEC

∆φ=0

Jboot /Jboot,exp

9/2(edge)

4/2(core)

9/2(edge)

4/2(core)

IPEC


•   Resonant drive @ edge increases with bootstrap current –   Path for low collisionality to

assist ELM suppression

•   Consistent with magnetic LFS & HFS measurement trends

model

model


−4.0

−3.0

−2.0

−1.0

0.0

Pertu

rbed

En

ergy

(δW

)

−1.0 0.0 1.0 2.0Normalized Reluctance

AmplifyingShield

New Reluctance Basis for Categorizing MHD Modes Demonstrates How to Drive Resonant Field Most Stably

•   Reluctance basis sorts MHD modes by magnitude and sign of the plasma response

N. Logan et al., PoP 2016


−4.0

−3.0

−2.0

−1.0

0.0

Pertu

rbed

En

ergy

(δW

)


AmplifyingShield


Least stable


•   Amplifying modes least stable, beta driven, LFS localized




Least stable

Most stable

external e-flux eigenvectors have been similarly combinedbefore calculating the corresponding perturbed equilibrium.The final field is clearly larger than the applied field in thecase of the positive eigenvalue modes and lower than theapplied field in the case of the negative eigenvalues. This isa clear demonstration of the amplifying/shielding dichotomy.These extrema and the other eigenmodes of large magnitudereluctance are predominantly composed of low magnitudepoloidal mode numbers within the range of the applied fieldsfrom the experimental phasing scan. These longer wave-length modes are also the most likely to be measured byexternal magnetics displaced radially from the plasma (thefield would fall off as Dr!m in a cylinder) as well as the fieldsthought to be of the most physical importance (see, forexample, the dominant modes in Section III).

The poloidal Fourier spectra for these two amplifying andtwo shielding modes are shown explicitly in Fig. 14. Again,the amplification/shielding of the external field (dashed lines)is readily apparent. The modes amplified are concentrated inthe usual kink resonant harmonics (m ! nq95) mentioned inSections II and III, while the shielding modes encompass a

spread of poloidal harmonics from very low to pitch-resonantm. Interestingly, both amplifying and shielding modes containsignificantly low magnitude poloidal harmonics with the op-posite helicity as that of the equilibrium field lines. These con-tributions are purely nonresonant, and the resulting plasmaresponse would thus not be captured by, for example, theRCD modes.

Figure 15 shows that the reluctance eigenmodes efficientlydescribe the observed plasma response for the equilibriummodeled in Sec. II. The plots, like those in Section III, showthe sensor signal from the full perturbed equilibrium and per-turbed equilibria isolating the response to a cumulative numberof eigenmodes. Here, reluctance eigenmodes are used in orderof their eigenvalue magnitude. Unlike Sections III A and III B,the relative error between the full and isolated signals quicklydecreases for both the HFS and LFS sensor arrays. Using only8 reluctance eigenmodes to describe the applied field, theHFS(LFS) signal maxima are matched within 13(18") and3(9)% of their full magnitude. That this convergence shouldtake place for an ideal sensor set follows from the very defini-tion of the reluctance, but Fig. 15 is the first application of thereluctance eigenbasis for describing actual sensor data in a realtokamak with complete experimental geometry.

A. Relation between the reluctance and performanceranked bases

This section has shown that the reluctance succeedswhere previous metrics failed to efficiently describe theplasma response measured by magnetic sensors external tothe plasma. The previously prevalent metrics are, however,still of great relevance to the performance of the plasma inthe presence of 3D perturbations. This section thus detailsthe relationship between the reluctance and the stability andRCD bases.

The reluctance is rigorously related to the stability byEq. (11). In a single mode model, the reluctance and each ofits composite matrices can be reduced to scalars such that,

qL ¼ ! 1þ s

s: (15)

FIG. 13. The e-flux applied on the IPEC control surface using 1 kA n¼ 2 I-coil waveforms with 0" phasing (top) and the isolated components alignedwith the largest two positive eigenvalue modes combined (a) and the largesttwo negative eigenvalue modes combined (b). The perturbed equilibrium e-flux calculated using only the isolated positive (c) and negative (d) eigen-value drives shows amplification and shielding of the driving e-flux,respectively.

FIG. 14. The poloidal spectrum of the largest two positive eigenvalue modescombined (red) and the largest two negative eigenvalue modes combined(blue) isolated from the applied 0" phasing (dashed) and the correspondingperturbed equilibrium (solid) showing amplification and shielding,respectively.

056110-9 Logan et al. Phys. Plasmas 23, 056110 (2016)

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•   Shielding modes most stable, beta insensitive, on HFS+LFS


−4.0

−3.0

−2.0

−1.0

0.0

Pertu

rbed

En

ergy

(δW

)


AmplifyingShield

10-2

10-1

100

101

Res

onan

t D

rive

(a.u

.)

−4.0

−3.0

−2.0

−1.0

0.0

Pertu

rbed

En

ergy

(δW

)


AmplifyingShield

10-2

10-1

100

101

Res

onan

t D

rive

(a.u

.)





•   Shielding modes most stable, beta insensitive, on HFS+LFS

•   Both can drive significant resonant field and control ELMs

•   3D spectrum that couples to only shielding modes shows path to more stable ELM control


−4.0

−3.0

−2.0

−1.0

0.0

Pertu

rbed

En

ergy

(δW

)


AmplifyingShield

10-2

10-1

100

101

Res

onan

t D

rive

(a.u

.)

external e-flux eigenvectors have been similarly combinedbefore calculating the corresponding perturbed equilibrium.The final field is clearly larger than the applied field in thecase of the positive eigenvalue modes and lower than theapplied field in the case of the negative eigenvalues. This isa clear demonstration of the amplifying/shielding dichotomy.These extrema and the other eigenmodes of large magnitudereluctance are predominantly composed of low magnitudepoloidal mode numbers within the range of the applied fieldsfrom the experimental phasing scan. These longer wave-length modes are also the most likely to be measured byexternal magnetics displaced radially from the plasma (thefield would fall off as Dr!m in a cylinder) as well as the fieldsthought to be of the most physical importance (see, forexample, the dominant modes in Section III).

The poloidal Fourier spectra for these two amplifying andtwo shielding modes are shown explicitly in Fig. 14. Again,the amplification/shielding of the external field (dashed lines)is readily apparent. The modes amplified are concentrated inthe usual kink resonant harmonics (m ! nq95) mentioned inSections II and III, while the shielding modes encompass a

spread of poloidal harmonics from very low to pitch-resonantm. Interestingly, both amplifying and shielding modes containsignificantly low magnitude poloidal harmonics with the op-posite helicity as that of the equilibrium field lines. These con-tributions are purely nonresonant, and the resulting plasmaresponse would thus not be captured by, for example, theRCD modes.

Figure 15 shows that the reluctance eigenmodes efficientlydescribe the observed plasma response for the equilibriummodeled in Sec. II. The plots, like those in Section III, showthe sensor signal from the full perturbed equilibrium and per-turbed equilibria isolating the response to a cumulative numberof eigenmodes. Here, reluctance eigenmodes are used in orderof their eigenvalue magnitude. Unlike Sections III A and III B,the relative error between the full and isolated signals quicklydecreases for both the HFS and LFS sensor arrays. Using only8 reluctance eigenmodes to describe the applied field, theHFS(LFS) signal maxima are matched within 13(18") and3(9)% of their full magnitude. That this convergence shouldtake place for an ideal sensor set follows from the very defini-tion of the reluctance, but Fig. 15 is the first application of thereluctance eigenbasis for describing actual sensor data in a realtokamak with complete experimental geometry.

A. Relation between the reluctance and performanceranked bases

This section has shown that the reluctance succeedswhere previous metrics failed to efficiently describe theplasma response measured by magnetic sensors external tothe plasma. The previously prevalent metrics are, however,still of great relevance to the performance of the plasma inthe presence of 3D perturbations. This section thus detailsthe relationship between the reluctance and the stability andRCD bases.

The reluctance is rigorously related to the stability byEq. (11). In a single mode model, the reluctance and each ofits composite matrices can be reduced to scalars such that,

qL ¼ ! 1þ s

s: (15)

FIG. 13. The e-flux applied on the IPEC control surface using 1 kA n¼ 2 I-coil waveforms with 0" phasing (top) and the isolated components alignedwith the largest two positive eigenvalue modes combined (a) and the largesttwo negative eigenvalue modes combined (b). The perturbed equilibrium e-flux calculated using only the isolated positive (c) and negative (d) eigen-value drives shows amplification and shielding of the driving e-flux,respectively.

FIG. 14. The poloidal spectrum of the largest two positive eigenvalue modescombined (red) and the largest two negative eigenvalue modes combined(blue) isolated from the applied 0" phasing (dashed) and the correspondingperturbed equilibrium (solid) showing amplification and shielding,respectively.

056110-9 Logan et al. Phys. Plasmas 23, 056110 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 192.5.166.109 On: Tue, 03 May2016 23:49:55

Least stable

Most stable



•   Consistency of field penetration with access to ELM suppression validates optimization criteria presented

•   Penetration requires optimized electron rotation profile –   Good: Wide optimization space enabling performance recovery –   Bad: Potential torque thresholds require careful extrapolation to ITER

•   Resonant drive optimized by 2D equilibrium and 3D spectrum –   Bootstrap current increases edge resonant drive, core beta does not –   Shielding modes can drive resonant fields without increasing δW


Bonus Slides


•   SS hybrid least-stable n=3 mode is more edge-localized

•   Speculate: broad J-profile and bootstrap causes edge-localization of resonant drive –   Despite positive reluctance /

large LFS signal

•   Ideal MHD modeling over-predicts core/LFS drive by 5x due to beta ~ no wall limit –   Kinetic modeling underway –   HFS sensors blind due to small

spatial size of m ~ 20 structures

0.0 0.2 0.4 0.6 0.8 1.0psi_n0

50

100

150

200 <J||> A/cm2 ITER-Similar (158103.03796)

SS Hybrid (161409.05300)

Comparison to SS Hybrid Case Reveals Different Radial Structure Likely Due to Large Bootstrap Current

0 90 180 270 3603hDsLng (GHg.)

0

1

2

3

4

5

3ODs

PD

5HsS

RnsH

(7/kA

)

1H−4

/)6H)6

0 90 180 270 3603hDsLng (GHg.)

0.0

0.2

0.4

0.6

0.8

1.0

6Lng

OH 0

RGH

2vHr

ODS

1H−3

GDLn5CD5HOuFtDnFH

LFS

IPEC 5x larger

data


Modeling Disagrees on Ability of Pedestal Pressure at Fixed Stored Energy to Increase Resonant Drive

•   MARS-F shows significant effect at pedestal-top

•   IPEC shows weak or counter-effect as βN,ped increases

•   IPEC and MARS-F agreed for Jboot and core βN trends

Stored Energy Constant


Disclaimer

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

optimization of the plasma response for the control of ...€¦ · 2.0 br vacuup 1e−4 positive...

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