Realisticcharacterizationofchirpinginstabilitiesintokamaks

ViníciusDuarte1,2

NikolaiGorelenkov2,Herb Berk3,MarioPodestà2 ,MikeVanZeeland4,DavidPace4,BillHeidbrink5

1Universityof SãoPaulo,Brazil2PrincetonPlasmaPhysics Laboratory

3Universityof Texas,Austin4GeneralAtomics

5Universityof California,Irvine

PPPLTheory Seminar,March 31,20161

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

2

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

2

Frequency chirping• frequencyshiftduetotrappedparticles• doesnot exist without resonant

particles• timescale:~1ms

Frequency sweeping• frequencyshiftduetotime-dependent

backgroundequilibrium• exists without resonant particles• timescale:~100ms

Two types of frequency shiftobserved experimentally

Sharapov etal,PoP 2006 Pinches etal,NF2004

3

Frequency chirping• frequencyshiftduetotrappedparticles• doesnot exist without resonant

particles• timescale:~1ms

Frequency sweeping• frequencyshiftduetotime-dependent

backgroundequilibrium• exists without resonant particles• timescale:~100ms

Two types of frequency shiftobserved experimentally

Sharapov etal,PoP 2006 Pinches etal,NF2004

3

Chirping modes can degradethe confinement ofenergetic particles

GAE/CAEFredrickson etal,PoP 2006.

TAEPodestà etal,NF2012

Up to 40%of injected beam isobserved to be lost inDIII-Dand NSTX

Chirping is ubiquitous inNSTXbut rare inDIII-D.Why??

This presentation focuses onthe conditions forchirpingonset rather than their long-term evolution

4

Phase space holes and clumps inkinetically driven,dissipative systems– reduced bump-on-tail model

• Nonlinear Landaudamping perspective:incomplete phasemixing leadsto small sideband oscillations that may tap freeenergy at the edges of the plateau

• Chirping infrequency may allow foracontinuous interplaybetween the free energy from the distribution function andthe wave dissipation

• Collisions eventually degradethe resonant island plateau,and the process restarts

Vlasovsimulations byLilley and Nyqvist,PRL2014

5

Outline

• Introduction to frequency chirping• Berk-Breizman model:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

Nonlinear dynamicsof driven kinetic systemscloseto thresholdStarting point:kinetic equation pluswave power balance

Assumptions:• Perturbative procedurefor• Truncation at third order due to closeness to marginalstability• Bump-on-tail modalproblem,uniformmode structureCubic equation:lowest-order nonlinear correction to the evolution of mode amplitudeA:

Berk,Breizman and Pekker,PRL1996Lilley,Breizman and Sharapov,PRL2009 6

Nonlinear dynamicsof driven kinetic systemscloseto thresholdStarting point:kinetic equation pluswave power balance

Assumptions:• Perturbative procedurefor• Truncation at third order due to closeness to marginalstability• Bump-on-tail modalproblem,uniformmode structureCubic equation:lowest-order nonlinear correction to the evolution of mode amplitudeA:

Berk,Breizman and Pekker,PRL1996Lilley,Breizman and Sharapov,PRL2009

-_

-_

stabilizing destabilizing (makes integralsign flip)

6

Nonlinear dynamicsof driven kinetic systemscloseto thresholdStarting point:kinetic equation pluswave power balance

Assumptions:• Perturbative procedurefor• Truncation at third order due to closeness to marginalstability• Bump-on-tail modalproblem,uniformmode structureCubic equation:lowest-order nonlinear correction to the evolution of mode amplitudeA:

• If nonlinearity is weak:linearstability,solution saturates at alow level and fmerely flattens(systemnot allowed to further evolvenonlinearly).

• If solution of cubic equation explodes:systementers astrong nonlinear phase with largemode amplitudeand can be driven unstable (precursorof chirping modes).

Berk,Breizman and Pekker,PRL1996Lilley,Breizman and Sharapov,PRL2009

-_

-_

stabilizing destabilizing (makes integralsign flip)

6

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

Existence and stability boundaries of solutionsof the cubic equation – bump-on-tail case

Lilley,Breizman andSharapov,PRL2009

Scattering

Drag 7

Existence and stability boundaries of solutionsof the cubic equation – bump-on-tail case

Lilley,Breizman andSharapov,PRL2009

We want to comparethispredictionwithmodesobserved inrealdischarges.

How?

Scattering

Drag

Stable steady solutioncannot exist withoutscattering

7

Mode structure identification

• NOVAcode:finds linear,idealmode structures• Itskinetic postprocessor NOVA-Kcomputesresonance surfaces and provides damping and

lineargrowth rates.Phase space and bounce averages arenecessary to calculate effectivecollisional coefficients

• NOVA’s mode structures arecomparedwith NSTXreflectometer measurements (fluiddisplacement timesthe localdensity gradient is equivalent to the density fluctuation).InDIII-DECEis used

8Podestà etal,NF2012

Chirping interms of effective collisional coefficients forrealistic resonances and mode structures

Pitch-angle scattering:leadsto loss of correlation(loss of phase information from one bounce toanother)Drag (slowing down):coherently movesstructuresdowninvelocity

Bump-on-tailmodeling is not enough toresolvethe regions incollisionalspace thatallows forchirping modes

Missing physics inthe simplified theoreticalprediction:mode structure, (multiple)resonancesurfaces and phase-space and bounce averages

Experimentalobservations:Red diamonds:chirping was observedGreendots:nochirping observed

9

Follow from cubicequation

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realistic mode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

Phase space integration

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

Phase space integration Eigenstructure information:

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

Phase space integration Eigenstructure information: Resonance surfaces:

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

Phase space integration Eigenstructure information: Resonance surfaces:

>0:steady solution is guaranteed<0:chirping may happen

10

Generalization:cubic equation with collisional coefficientsvarying along resonances and particle orbits

Action-angle formalism forthe generalproblem,with asimilarperturbative approachemployed before,leadsto the generalizedcriterion forexistence of steady-state solutions (nochirping):

Phase space integration Eigenstructure information: Resonance surfaces:

Criterionwas incorporated into NOVA-K

>0:steady solution is guaranteed<0:chirping may happen

10

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion of microturbulence inthe model• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

Correction to the diffusion coefficient:the inclusion ofelectrostatic microturbulence

• Microturbulence can well exceed pitch-anglescattering at the resonance1

• From GTCgyrokinetic simulations forpassingparticles:2

• Aspitch-angle scattering,microturbulenceacts to destroy phase-space holes and clumps

• Unlike DIII-D and TFTR,transport inNSTXinmostly neoclassical

• Complex interplay between gyroaveraging,field anisotropy and poloidaldrift effectsleadsto non-zeroEPdiffusivity3

1Langand Fu, PoP 20112Zhang,Lin and Chen, PRL20083Estrada-Milaetal,PoP 2005

Ratio of fast ion diffusivity tothermal ion diffusivity2

Pueschel etal,NF2012gives similarmicroturbulence levels 11

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis of modes inTFTR,DIII-Dand NSTX• Conclusions

TAEinTFTRshot 103101:nochirping observed

Gorelenkov etal,PoP 1999

TAEmode structure

12

TAEinTFTRshot 103101:nochirping observed

Drag vs pitch-angle scattering:TAEmode structure

Gorelenkov etal,PoP 1999

12

TAEinTFTRshot 103101:nochirping observed

Drag vs pitch-angle scattering: Drag vs pitch-angle scattering+microturbulence:TAEmode structure

Gorelenkov etal,PoP 1999

12

TAEinTFTRshot 103101:nochirping observed

Drag vs pitch-angle scattering: Drag vs pitch-angle scattering+microturbulence:

Microturbulence has astrong effect on destroying resonant,coherent phase-space structures

TAEmode structure

Gorelenkov etal,PoP 1999

12

Characterization of ararely observed chirping mode inDIII-D

13

Characterization of ararely observed chirping mode inDIII-D

TRANSPrun

13

Characterization of ararely observed chirping mode inDIII-D

Beforechirping,at900ms:D~1m2/s

TRANSPrun

13

Characterization of ararely observed chirping mode inDIII-D

Beforechirping,at900ms:D~1m2/s

Duringchirping,at 960ms:D~0.2m2/s

TRANSPrun

13

Characterization of ararely observed chirping mode inDIII-D

Beforechirping,at900ms:D~1m2/s

Duringchirping,at 960ms:D~0.2m2/s

TRANSPrun

Diffusivity dropdue to L→H

modetransition

Strong rotationshear wasobserved

13

Non-chirping DIII-Dshot analyzed interms ofpitch-angle scattering and drag

14

Inclusion of microturbulence innon-chirpingDIII-Dshots

Microturbulence has asubstantial effect on bringingmodes to the steady region

14

NSTXchirping TAEs

Microturbulence doesnot haveappreciable effects inNSTX

Even if scattering levels is increasedseveral timesthe modes cannotreach the boundary

All caseshave negativevalues forthe criterion,which is consistentwith observation

15

Outline

• Introduction to frequency chirping• Berk-Breizmanmodel:cubic equation formode amplitudeevolution at

early times• Bump-on-tail modeling• Generalization tomulti-dimensional resonances inspace and

realisticmode structure• Inclusion ofmicroturbulence inthemodel• Analysis ofmodes inTFTR,DIII-Dand NSTX• Conclusions

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation• Other stochastic contributions may be due to ripples and mode overlap

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation• Other stochastic contributions may be due to ripples and mode overlap• InDIII-D,chirping has been identified to be linked with L->Hmode transition.

Possibility of chirping control viarotation shear

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation• Other stochastic contributions may be due to ripples and mode overlap• InDIII-D,chirping has been identified to be linked with L->Hmode transition.

Possibility of chirping control viarotation shear

Ongoing work

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation• Other stochastic contributions may be due to ripples and mode overlap• InDIII-D,chirping has been identified to be linked with L->Hmode transition.

Possibility of chirping control viarotation shear

Ongoing work• Refinement of turbulence contribution using Beam Emisson Spectroscopy

16

Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode

structures• If microturbulence is included, the generalized Berk-Breizman model reproduces

well experimentalobservation• Other stochastic contributions may be due to ripples and mode overlap• InDIII-D,chirping has been identified to be linked with L->Hmode transition.

Possibility of chirping control viarotation shear

Ongoing work• Refinement of turbulence contribution using Beam Emisson Spectroscopy• Development of aline-broadened quasilinear diffusion solvercoupled with NOVA

and NOVA-K:chirping criterion is important foridentification of parameter space forquasilinear validity 16

Thank you