alfvén instabilities driven by energetic particles in toroidal...

UCLA 1/12/08

Alfvén Instabilities Driven by EnergeticParticles in Toroidal Magnetic Fusion

Configurations

W.W. (Bill) Heidbrink

Calculated AlfvénEigenmode structure in ITER

Satellite measurements of Alfvénwaves that propagate from themagnetosphere to the ionosphere

VanZeeland

Sigsbee, Geophys. Res. Lett. 25 (1998) 2077

Magnetosphere

Fre

qFr

eq

Am

pl.

Am

pl. Ionosphere

Time

Time

UCLA 1/12/08

Instabilities Driven by Energetic Particles areof both scientific and practical interest

V2

R0V2

Damage

•Carbon coats DIII-D mirrors whenescaping fast ions ablate thegraphite wall1

•Transport of fast ions by Alfvénwaves onto unconfined orbits causea vacuum leak in TFTR2

Losses of charged fusionproducts must becontrolled in a reactor!1Duong, Nucl. Fusion 33 (1993) 749.

2White, Phys. Pl. 2 (1995) 2871.

Beam injection into the DIII-D tokamak

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Source Angle Energy Distribution

Neutral beams Beam angle Slowing down

Fusion products Isotropic Slowing down

Ion cyclotron Perpendicular Boltzmann

Electron cycl. Perpendicular Boltzmann

Runaway Parallel Flat

Sources of Energetic Particles

UCLA 1/12/08

Representative Fast-ion Diagnostics

•Fusion Products Neutrons,gammas, escaping chargedparticles

•Fast-ion Loss Escaping ions hitdetector

•Neutrals Escaping neutrals

•Charge-exchangespectroscopy Photons emittedby ions that gain an electronfrom a neutral beam

•Equilibrium Spatial profile,velocity-space average

UCLA 1/12/08

•Signal is the integral over

velocity space of W*F

•Averages are best for

general assessment (e.g.,

“Are fast ions lost?”)

•Local diagnostics address

detailed physics mechanism

(e.g., “Which class of

particles resonate with the

wave?”)

Fast-ion diagnostics are sensitive todifferent parts of velocity space

Heidbrink, PPCF 49 (2007) 1457.

UCLA 1/12/08

Representative Instability Diagnostics

•Density

-- Beam Emission Spectroscopy (BES)

-- Far Infrared Scattering

-- CO2 Interferometers

-- Microwave Reflectometers

•Electron TemperatureElectron Cyclotron Emission(ECE)

•Magnetic field probes

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Mode Identification -- frequency & parametric dependencies -- eigenfunction -- polarizationLinear Stability -- Energetic particle drive (resonant particles, velocity-space anisotropy or spatial gradient) -- Plasma dampingNonlinear Dynamics -- Amplitude, eigenfunction, & frequency evolution -- Fast-ion transportControl -- Alter linear stability -- Exploit instability to improve performance -- Affect nonlinear dynamics

Key Issues for any Energetic

Particle Instability

Bandwidth and sensitivity

Antenna studies of normal modes

Spatial resolutionWhat fluctuates?

UCLA 1/12/08

Outline

1. Alfvén Gap Modes

2. Energetic Particles(EP)

3. Energetic ParticleModes (EPM)

4. NonlinearDynamics

5. Prospects forControl

Pinches,Ph.D. Thesis

UCLA 1/12/08

Shear Alfvén Waves are transverse electromagneticwaves that propagate along the magnetic field

•Dispersionless: = kll vA

•Alfvén speed: vA = B/(μ0nimi)1/2

•Ell tiny for

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Periodic variation of the magnetic field producesperiodic variations in N for shear Alfvén waves

Zhang, Phys. Plasmas 14 (2007)

Periodic Mirror Field in the LAPD

Periodic variation in B Periodic variation in vA Periodic variation inindex of refraction N

Frequency gap that isproportional to N

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Periodic index of refraction a frequency gap

Lord Rayleigh, Phil. Mag. (1887)

“Perhaps the simplest case … is that of astretched string, periodically loaded,and propagating transverse vibrations.…If, then, the wavelength of a train ofprogressive waves be approximatelyequal to the double interval betweenthe beads, the partial reflexions from thevarious loads will all concur in phase,and the result must be a powerfulaggregate reflexion, even though theeffect of an individual load may beinsignificant.”

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The propagation gap occurs at the Braggfrequency & its width is proportional to N

Wikipedia, “Fiber Bragg grating”

•Destructive interferencebetween counterpropagating waves

•Bragg frequency: f=v/2

• f/f ~ N/N

frequency gap for shear Alfvén waves

• f = vA/ 2 , where is thedistance between fieldmaxima along the field line

• f ~ B/B

n1 n2

n3

n3n2

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Frequency gaps are unavoidable in atoroidal confinement device

Field lines in a tokamak

BmaxBmin

R

a

•For single-particleconfinement, field lines rotate.

•Definition: One poloidal transitoccurs for every q toroidaltransits (q is the “safety factor”)

•B ~ 1/R

• B ~ a/R

•Distance between maxima is = q (2 R) so fgap = vA/4 qR

Periodicity constraint on the wavevector: ~ ei (n m )

•n “toroidal mode number”

•m “poloidal mode number”

• and toroidal and poloidal angles

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Frequency Gaps and the Alfvén Continuumdepend on position

Gap caused by toroidicity1

1based on Fu & VanDam, Phys. Fl. B1 (1989) 1949

•Centered at Bragg frequency vA/qR

•Function of position through vA & q

•Gap proportional to r/R

•If no toroidicity, continuumwaves would satisfy = kll vA withkll ~ |n - m/q|

•Counter-propagating wavescause frequency gap

•Coupling avoids frequencycrossing (waves mix)

•Crossings occur at manypositions

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All periodic variations introduce frequency gaps

Shear Alfvén wave continuuain an actual stellarator

Spong, Phys. Plasmas 10 (2003) 3217

BAE “beta” compression

TAE “toroidicity” m & m+1

EAE “ellipticity” m & m+2

NAE “noncircular” m & m+3

MAE “mirror” n & n+1

HAE “helicity” both n’s & m’s

T

E

N

H

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Rapid dispersion strongly damps waves inthe continuum

~ d (kll vA) / dr

Radially extended modes inthe continuum gaps are moreeasily excited

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Radially extended Alfvén eigenmodes aremore easily excited

Pinches, Ph.D. Thesis

Continuum Mode Structure

•Imagine exciting a wave withan antenna--how does thesystem respond?

•In continuum, get singularmode structure that is highlydamped (small amplitude)

•Where gap modes exist, theeigenfunction is regular &spatially extended

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Magnetic shear is the “defect” that createsa potential well for Alfvén gap modes

•In photonic crystals,defects localize gapmodes

•The defect createsan extremum in theindex of refraction

Defect

GapMode

Extremumtype

Couplingtype

Alfvén continuum

Magnetic shear(dq/dr) createsextrema

UCLA 1/12/08

An extremum in the continuum can be the“defect”

VanZeeland, PRL 97 (2006) 135001; Phys.Plasmas 14 (2007) 156102.

Gap structure in a DIII-D plasmawith a minimum in the q profile

•Gap modes reside in effectivewaveguide caused by minimum in qprofile

•These gap modes called“Reversed Shear AlfvénEigenmodes” (RSAE)

•Many RSAEs with different n’s

•All near minimum ofmeasured q

•Structure agreesquantitatively with MHDcalculation

Measured Te mode structure in DIII-D

UCLA 1/12/08

fmeas

dampB

fTAE

time

In the toroidal Alfvén Eigenmode (TAE), modecoupling is the “defect” that localizes the mode

Fasoli, Phys. Plasmas 7 (2000) 1816

JET #49167

Using an external antenna to excite an=1 TAE on the JET tokamak •The frequency of the

measured TAE followsthe frequency gap asthe discharge evolves

•Can infer the wavedamping from the widthof the resonance

•Width is larger whenthe eigenfunction“touches” thecontinuum

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GAE

dominated by m = 3 (n=1)TAE

generated by coupling of m = 5,6 (n = 2)

MHD

Code

Soft X-ray

Tomogr.

Predicted spatial structure is observedexperimentally for both types of gap mode

(coupling)(extremum)Data from W7-AS stellarator

Weller, Phys. Plasmas 8 (2001); PRL 72 (1994)

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Shear Alfven wave polarization confirmed

Magnetics data from NSTX spherical tokamak

Fredrickson

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Part 2: Energetic Particles







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Fast-ion orbits have large excursions frommagnetic field lines

Plan view

Elevation (80 keVD+ ion in DIII-D) •Perp. velocity

gyromotion

•Parallel velocity follows flux surface

•Curvature & Grad Bdrifts excursionfrom flux surface

Parallel ~ v

Drift ~ (vll2 + v 2/2)

Large excursions for large velocities

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Complex EP orbits are most simply describedusing constants of motion

Projection of 80 keV D+

orbits in the DIII-D tokamakConstants of motion on orbital timescale:energy (W), magnetic moment (μμ),toroidal angular momentum (P )

Roscoe White, Theory of toroidally confined plasmas

Distribution function: f(W,μ,P )

UCLA 1/12/08

Orbit topology is well understood

Zweben, Nucl. Fusion 40 (2000) 91

Prompt losses of D-T alpha particlesto a scintillator at the bottom of TFTR

Edge loss detector onthe TFTR tokamak

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The drift motion must resonate with a waveharmonic to exchange net energy

Time tocompletepoloidal orbit

Time tocompletetoroidal orbit

Tct

e tomplete

vllEll 0 (transverse polarization)

Parallel resonance condition: = n + p

Write vd as a Fourier expansionin terms of poloidal angle :

Energy exchange resonance condition: n + (m+l) =0

(main energyexchange)

Drift harmonicWave mode #s

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• Resonance condition, np = n – p – = 0

n=4, p = 1

n=6, p = 2

n=3, p = 1

n=5, p = 2

n=6, p = 3n=7, p = 3

Prompt losses

E [MeV]4.5 5.0 5.5 6.0 6.5 7.0 7.5

Calculated resonances with observed TAEs during RF ion heating in JET0

-50

-100

-200

-150

-250

Log

(f E

/n

p)

-5

-6

-7

-8

-9

-10

Pc

i [M

eV

]

A typical distribution function has many resonances

Pinches, Nucl. Fusion 46 (2006) S904

UCLA 1/12/08

Alphas, TFTR tokamak4

TAEs

& R

SA

Es

Beam injection, CHS stellarator1

Tremendous variety of resonances are observed

Electron tail, HSX stellarator3

RF tail ions, C-Mod tokamak2

EA

Es, TA

Es, &

RSA

Es

GA

EH

AE &

TAEs

1Yamamoto, PRL 91 (2003) 245001; 2Snipes PoP 12(2005) 056102; 3Brower; 4Nazikian, PRL 78 (1997) 2976

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The spatial gradient of the distributionusually drives instability

•Slope of distribution function atresonances determines whetherparticles damp or drive instability

•If the wave damps0v/

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TAEs in TFTR: avoid energy damping by beamions, use spatial gradient drive by alphas

•Strong beam-iondamping stabilized AEs duringbeam pulse

•Theory1 suggested strategy toobserve alpha-driven TAEs

•Beam damping decreasedfaster than alpha spatial gradientdrive after beam pulse

•TAEs observed2 whentheoretically predicted

Wf /

AEs observed after beaminjection in TFTR D-T plasmas

2Nazikian, PRL 78 (1997) 29761Fu, Phys. Plasma 3 (1996) 4036;Spong, Nucl. Fusion 35 (1995) 1687

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EP drive is maximized for large-n modesthat are spatially extended

Theory1•EP drive increases with n (strongertoroidal asymmetry)

•But mode size shrinks with n

•Weak wave-particle interactionwhen orbit is much larger than themode

Drive maximized when orbitwidth ~ mode size (k EP~ 1)

Large n anticipated in reactors

Most unstable mode numbervs. theory (many tokamaks)2

1Fu, Phys. Fluids B 4 (1992) 3722; 2Heidbrink,Pl. Phys. Cont. Fusion 45 (2003) 983

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Part 3: Energetic Particle Modes (EPM)







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EPMs are a type of “beam mode”

Normal Mode (gap mode)

nEP

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EPMs often sweep rapidly in frequency asdistribution function changes

Simulation withkinetic fast ionsand MHD plasma

Modes observed during intense negativeneutral beam injection into JT-60U

Briguglio, Phys. Pl. 14 (2007) 055904

Shinohara, Nucl. Fusion 41 (2001) 603

Radius

Fre

qu

en

cy

EPM

TAE

μμ

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Part 4: Nonlinear Dynamics







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•Analogy between “bump-on-tail”and fast-ion modes:

velocity-space gradient

configuration-space gradient

•Resonant ions get trapped inwave: they bounce in wavepotential ( b) & scatter out ofresonance ( eff)

•Behavior depends on drive, damp,

b, eff

•Wide variety of scenarios possible

1D analogy to electrostatic wave-particle

trapping describes many phenomena

Berk, Phys. Pl. 6 (1999) 3102

UCLA 1/12/08

Striking Success of Berk-Breizman Model

Pinches, Plasma Phys. Cont.Fusion 46 (2004) S47

Fasoli, PRL 81 (1998) 5564

Chirping TAEs during beam injectioninto the MAST spherical tokamak

Simulation of first chirp

Nonlinear splitting of TAEsdriven by RF tail ions in JET

Fre

qu

en

cy

Fre

qu

en

cy

Time

Time (ms)

Small eff

Appreciable eff

UCLA 1/12/08

Changes in canonical angular momentumcause radial transport

•Magnetic momentconserved ( μ = 0)

•Energy changes lessthan angularmomentum: W/W ~0.1( P /P )

• P (radialtransport)

•Leftward motion ongraph implies outwardradial motion

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Four mechanisms of EP transport aredistinguished

•Leftward motion ongraph implies outwardradial motion

1) Convective lossboundary (~ Br)

2) Convective phaselocked (~ Br)

3) Diffusive transport(~ Br

2)

4) Avalanche(Br threshold)

Convective loss boundary (~ Br, small %)Fluctuations in equilibrium push EPs across lossboundary

2) Convective phase locked (~ Br, large %)EPs stay in phase with wave as they “walk” outof plasma

3) Diffusive transport (~Br2)

Random walk due to multiple resonances

4) Avalanche (Br threshold)

EP transport locally increases gradient,destabilizing new modes

UCLA 1/12/08

Convective transport often observed

García-Muñoz, PRL 99 (2007) submitted

Edge scintillator on Asdex-U tokamak

Image on scintillator screenduring TAEs

Coherent fluctuations in loss signalof RF tail ions at TAE frequencies

•Fast ions cross loss boundaryand hit the scintillator in phasewith the waves

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Both convective and diffusive losses areobserved

Nagaoka (2007)

0 10 20 30 400

10

20

30

40

i (

arb

. u

nits)

B (arb. units)

fast responsefBurst

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Avalanche phenomena observed

Fredrickson, Nucl. Fusion 46 (2006) S926

Magnetics data during beam injectioninto the NSTX spherical tokamak •When n=4 & n=6 TAE bursts

exceed a certain amplitude, alarge burst with many toroidalmode numbers ensues

•Fast-ion transport is muchlarger at avalanche events

UCLA 1/12/08

Quantitative calculations of EP transport areunsuccessful

Radial Te profile duringbeam injection into DIII-D

Radial fast-ion profile

Heidbrink, PRL 99 (2007) in pressVan Zeeland, PRL 97 (2006) 135001

•Measured mode structure agrees well with MHD model

•Input these wave fields into an orbit-following code

•Calculate much less fast-ion transport than observed

•What’s missing?

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Part 5: The Frontier







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Diagnose nonlinear interactions

Crocker, PRL 97 (2006) 045002

•This example shows that theTAEs (100-200 kHz) arenonlinearly modified by a low-frequency (~20 kHz) mode

•Similar analysis of AE wave-wave interactions and wave-particle interactions areneededBicoherence analysis

Filtered reflectometer ne signalduring beam injection into NSTX

UCLA 1/12/08

Recent observations indicate kineticinteraction with the thermal plasma

V2

R0V2

Calculated n=40 RSAE that agreeswith ne measurements on DIII-D

•High-n modes are probably drivenby thermal ions.1

•Alfvén modes driven by low-energybeams.2

•New unstable gap modes fromcoupling of acoustic and Alfvénwaves.3

•Wave damping measurements thatdisagree with fluid plasma models.4

•New treatments of thermal plasmaare needed

1Nazikian, PRL 96 (2006) 105006;Kramer, Phys. Pl. 13 (2006) 056104

2Nazikian, JI1.01; 3Gorelenkov, Phys. Lett. A370/1 (2007) 70; 4Lauber, Phys. Pl. 12 (2005)122501

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Use control tools to alter stability

RSAEs

RSAEs

ECH deposition location is variedrelative to mode location ( qmin)

Deposition near qmin stabilizesbeam-driven RSAEs in DIII-D

Van Zeeland, Plasma Phys. Cont. Fusion 49(2007) submitted

•Localized electron cyclotronheating (ECH) alters stability andconsequent fast-ion transport

•Can we turn off deleteriousmodes in a reactor?

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Use control tools to alter nonlinear dynamics

Maslovsky, Phys. Pl. 10 (2003) 1549

Interchange instability driven by energeticelectrons in the Columbia Dipole •In this experiment, a small

amount of power (50 W)scattered EPs out ofresonance, suppressingfrequency chirping &eliminating large bursts

•Can we use analogoustechniques to eliminatedamaging bursts of lost alphasin a reactor?

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Alfvén Eigenmodes can improve performance

Similar discharges with differing levels of AEactivity during beam injection into DIII-D

Weak AE

ModerateAE

Huge AE

•Three discharges withdifferent levels of modeactivity•Fast-ion redistributionbroadens current profile•Optimal redistributiontriggers an internaltransport barrier muchbetter confinement•How can we exploit AEsin a reactor?

Wong, Nucl. Fusion 45 (2005) 30.

Ti=12 keV

8 keV

7 keV

Huge

ModerateWeak

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Bmax

Bmin

Conclusions

•Periodic variations of the index of refractioncause frequency gaps

•Gap modes exist at extrema of Alfvén continuum

•Use constants of motion to describe EP orbits

•Wave-particle resonance occurs when: n + (m+l) =0

•Instability driven by EP spatial gradient

•EPMs are beam modes (not normal modes ofbackground plasma)

•Berk-Breizman analogy to bump-on-tail problemoften describes nonlinear evolution

•Fast-ion transport not quantitatively understood

•Use thermal transport techniques to understandnonlinear dynamics

•Develop tools to control Alfvén instabilities oreven improve performance

Energy

P

8070 7264 6866

Time [ms]

14

0

12

0

10

0

qy

[]

Fre

qu

en

cy

(kH

z)

RSAEs

RSAEs

UCLA 1/12/08

Acknowledgments* & additional resources

Thanks to all who sent me slides:

Brower (UCLA), Crocker (UCLA), Darrow (PPPL),Fasoli (CRPP), Fredrickson (PPPL), García-Muñoz(IPP), Kramer (PPPL), Mauel (Columbia),Nazikian (PPPL), Pinches (UKAEA), Sharapov(UKAEA), Shinohara (JAEA), Snipes (MIT), Spong(ORNL), Toi (NIFS), VanZeeland (GA), Vlad(Frascati), Weller (IPP), White (PPPL), Vann(York), Vincena (UCLA), Zhang (UCI), Zweben(PPPL)

Special thanks for teaching me theory:

Liu Chen, Boris Breizman, and Sergei Sharapov

*Supported by the U.S. Department of Energy

Clear explanation of basic theory: Firstchapters of Pinches’ Ph.D. thesis,http://www.rzg.mpg.de/~sip/thesis/node1.html

Experimental review through 1999(especially TFTR results): King-Lap Wong,PPCF 41 (1999) R1.

Experimental review of fast ions intokamaks (AE material dated): Heidbrink& Sadler, NF 34 (1994) 535.

Lengthy theoretical review paper: Vlad,Zonca, and Briguglio,http://fusfis.frascati.enea.it/~Vlad/Papers/review_RNC_2.pdf

Differences between burning plasmas &current experiments: Heidbrink, PoP 9(2002) 2113

ITER review: Fasoli et al., NF 47 (2007) S264

Recent theoretical review: Chen & Zonca,NF 47 (2007) S727

alfvén instabilities driven by energetic particles in toroidal...

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