Realistic characterization of chirping instabilities in ...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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-_
stabilizing destabilizing (makes integralsign flip)
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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
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stabilizing destabilizing (makes integralsign flip)
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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
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Existence and stability boundaries of solutionsof the cubic equation – bump-on-tail case
Lilley,Breizman andSharapov,PRL2009
Scattering
Drag 7
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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
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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
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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
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Follow from cubicequation
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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
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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):
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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
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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
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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
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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
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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
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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
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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
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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
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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
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TAEinTFTRshot 103101:nochirping observed
Gorelenkov etal,PoP 1999
TAEmode structure
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TAEinTFTRshot 103101:nochirping observed
Drag vs pitch-angle scattering:TAEmode structure
Gorelenkov etal,PoP 1999
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TAEinTFTRshot 103101:nochirping observed
Drag vs pitch-angle scattering: Drag vs pitch-angle scattering+microturbulence:TAEmode structure
Gorelenkov etal,PoP 1999
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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
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Characterization of ararely observed chirping mode inDIII-D
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Characterization of ararely observed chirping mode inDIII-D
TRANSPrun
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Characterization of ararely observed chirping mode inDIII-D
Beforechirping,at900ms:D~1m2/s
TRANSPrun
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Characterization of ararely observed chirping mode inDIII-D
Beforechirping,at900ms:D~1m2/s
Duringchirping,at 960ms:D~0.2m2/s
TRANSPrun
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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
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Non-chirping DIII-Dshot analyzed interms ofpitch-angle scattering and drag
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Inclusion of microturbulence innon-chirpingDIII-Dshots
Microturbulence has asubstantial effect on bringingmodes to the steady region
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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
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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
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Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them
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Conclusions• Diffusive effects suppress chirping structures while slowing down enhances them• Proper modeling needs to account for(multiple) resonance surfaces and mode
structures
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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
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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
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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
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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
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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
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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
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Thank you