W.W. (Bill) Heidbrink* UC Irvine Shu Zhou, Xi Chen, Liu Chen, Yubao Zhu University of California, Irvine T. Carter, S. Vincena, S. K. P. Tripathi University

W.W. (Bill) Heidbrink*UC Irvine

Shu Zhou, Xi Chen, Liu Chen, Yubao Zhu University of California, Irvine

T. Carter, S. Vincena, S. K. P. TripathiUniversity of California, Los Angeles

M. Van Zeeland, D. Pace, R. FisherGeneral Atomics

G. Kramer, B. Grierson, R. White, K. Ghantous, N.

GorelenkovPrinceton Plasma Physics Laboratory

E. BassUniversity of California, San Diego

*in collaboration with the DIII-D & LAPD teams, especially:

New Insights into Energetic Ion Transport by Instabilities: The importance of phase

Fast-ion orbits have large excursions from magnetic field lines

Plan viewElevation (80 keV D+ ion in DIII-D)

•Perp. velocity gyromotion

•Parallel velocity follows flux surface

•Curvature & Grad B drifts excursion from flux surface

Parallel ~ v

Drift ~ (vll2 + v

2/2)

Large excursions for large velocities

Complex EP orbits are most simply described using constants of motion

Projection of 80 keV D+ orbits in the DIII-D tokamak

Constants of motion on orbital timescale: energy (W), magnetic moment (), toroidal angular momentum (P)

Roscoe White, Theory of toroidally confined plasmas

Distribution function: f(W,,P)

The wave phase determines the sign of the force

t k r

Resonance occurs when the orbit-averaged phase is constant in time, i.e.,

.cons mathematically, resonance produces a secular term ~ t

Outline

Fishbones Convective resonant transport for kperpρ<<1

Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (kperpρ<<1)

Drift Waves Orbit-averaging for kperpρ>>1

Alfvén Eigenmodes Non-resonant losses for kperpρ~1

Alfvén Eigenmodes “Stiff” transport for many small-amplitude modes with kperpρ~1

Outline






Resonant transport occurs when an aspect of the orbital motion matches the

wave frequencyTime to complete poloidal orbit

Time to complete toroidal orbit

0Ev )( c

vllEll0 (when Ell~0)

Parallel resonance condition: np

Write vd as a Fourier expansion in terms of poloidal angle :

,...2,1l

illeA

Energy exchange resonance condition: n(m+l)

(main energy exchange) Evd

Drift harmonicWave mode #s

Fast-ion Loss Detector (FILD) measures lost trapped ions at off-axis

fishbone burst

DIII-D off-axis fishbone data

•Bright spot for ~80 keV, trapped fast ions that satisfy resonance condition

•Scintillator acts as a magnetic spectrometer to measure energy & pitch of lost fast ions Projection of lost orbit

Heidbrink, Plasma Phys. Cont. Fusion 53 (2011) 085028

Losses have a definite phase relative to the mode

•Particles are expelled in a “beacon” that rotates with the mode

•Caused by Ex Bconvective transport

•Losses occur at the phase that pushes particles outward

Heidbrink, Plasma Phys. Cont. Fusion 53 (2011) 085028

DIII-D off-axis fishbone data

Coherent convective transport occurs for modes that maintain resonance across the

plasma

White, Phys. Fluids 26 (1983) 2958

Calculated Fishbone Loss Orbit•The fishbone was a globally extended, low-frequency mode (kperpρ<<1)

•Low frequency 1st & 2nd adiabatic invariants are conserved

•μ conservation particles that move out (to lower B) lose Wperp

•Main loss mechanism: convective E x B radial transport

Convective phase locked transport “marches” particle across the plasma

•Leftward motion on graph implies outward radial motion

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

2B/BE~

v

Resonant transport drives instability

•Ions that move out lose energy (μ conservation)

•Ions that move in gain energy

•Fast-ion profile is peaked more ions move out than in wave gains energy

• Equivalent explanation: ~ / 0n f P

Heidbrink, Phys. Plasmas 15 (2008) 055501

Outline


(Ions “see” constant phase)





Standard theory: resonances at frequency harmonics

• For n=0 mode, expect resonances when

Energy exchange resonance condition: n(m+l)

Find subharmonic resonances in simulation of large-amplitude EGAM!

• Simulate energetic-particle driven geodesic acoustic mode (EGAM)

• Mode has large electric field

• For small potential, find usual harmonic resonances

• For large amplitudes, subharmonic resonances appear

• Analytic theory explains results

DIII-D Simulation

Kramer, PRL 109 (2012) 035003

0( )t k r ��

Experimental evidence of subharmonic losses exists

• No evidence of subharmonics in instability spectra

• Coherent losses at 1/2 resonance appear when EGAM amplitude is large

DIII-D data

Kramer, PRL 109 (2012) 035003

Outline







Large orbits spatially filter electrostatic turbulence

Fluctuation Amplitude •Potential fluctuations in plane perpendicular to B

•Small-orbit ion stays in phase with wave large E x B kick

•Large-orbit ion sees rapid phase change small E x B kick

Drift wave created by an obstacle in the LAPD

Large orbits spatially filter electrostatic turbulence

•Temporal average over gyromotion spatial filter of the potential

•Gyro-phase averaging scales as:

•First simulation in 1979*

( )oJ k

*Naitou, J. Phys. Soc. Japan 46 (1979) 258

Fluctuation Amplitude

<>

<>

<>

•Other types of orbital motion also phase-

average

Launch a beam of particles. How do they spread in time?

LAPD Data TORPEX Simulation

Review paper on LAPD & TORPEX experiments: Heidbrink, PPCF 54 (2012) 124007

Transport is characterized by an exponent

Gustafson, PoP 19 (2012) 062306

The spread in the particle position W is used to extract a transport exponent:

2W t•For example, since there is no force in the parallel direction, z=(vz) t, so =2 (called “ballistic” or “convective” transport)

“diffusive”

“sub-diffusive”

“super-diffusive”

TORPEX Simulation

Three expected turbulent transport regimes

T. Hauff and F. Jenko, Phys. Pl. 15 (2008)112307

•Initially r=vkickt =2 (convective)

•Wave phase changes some particles pushed back toward initial positions <1 (sub-diffusive)

•Eventually many random kicks random walk with W2 ~ t (normal diffusion)

Experimental Setup: Fast ions orbit through turbulence

23

•Create plasma with electrostatic fluctuations

•Pass Li+ beam through waves

•Scan collector spatially to measure beam spreading

•Measure properties of turbulence

LAPD

S. Zhou, PoP 17 (2010) 092103

Beam spot provides information on radial and parallel transport Collector scans measure beam spreading

Use obstacles to enhance turbulence (LAPD)

25

•Obstacle creates sharp density gradient

•Large fluctuations at obstacle edge

•Control turbulence by biasing obstacle & changing plasma species [Zhou, Phys. Pl. 19 (2012) 012116]

Li Source

Fast-ion Orbit

Cu Obstacle

Den

sity

Flu

ctua

tions

Photograph from end of machine

Model the fields with fluid codes (constrained by measurements) then compute orbits

26

•Magnetic fluctuations are small Assume electrostatic

•Long parallel wavelengths Assume 2D fluctuating fields

•Adjust amplitude of simulated turbulence to match experiment•Apply a Lorentz orbit code in simulated fields

Floating Potential Cross-spectrum

Use the resistive fluid code BOUT to simulate the microturbulence.Popovich et al, PoP 17, 122312 (2010)

Fast-Ion Transport Decreases with Increasing Fast-Ion Energy

•Axial speed held constant

S. Zhou, PoP 17 (2010) 092103

Turbulent spreading is super-diffusive (2)

•Classical transport is diffusive (~1)

W2

Data

S. Zhou, PoP 17 (2010) 092103

Simulation Result

Test-particle simulation in a BOUT simulated wave field agrees well with

experiment

Data

S. Zhou, PoP 17 (2010) 092103

Energy Scaling of Beam Transport Shows Gyro-Averaging Effect

Gyro-Averaging Effect:

• The effective potential is phase-averaged over the fast ion gyro-orbit

30

Averaged FluctuatingAmplitude

Experimental Data

ik x0k fk J (k(x )) e

Tur

bule

nt t

rans

port

S. Zhou, PoP 17 (2010) 092103

31

Li Source

Fast-ion Orbit

Cu Obstacle

Den

sity

Flu

ctua

tions

Use annular obstacle to vary the turbulence

•Fixed gyroradius

•Vary correlation length Lcorr & scale length of dominant modes Ls

S. Zhou, PoP 18 (2011) 082104

Different Transport Levels are Observed in 3 Typical Background Turbulence Cases

Helium Vbias=0VLcorr=23cmLs=2.6cm

n/n=0.55δ

Neon Vbias=75VLcorr=19cmLs=6.3cm

n/n=0.35δ


n/n=0.53δ

A

B

C

(A)

(B)

(C)

32DistanceS. Zhou, PoP 18 (2011) 082104

A simple model explains the dependence on Lcorr and Ls

( )( )

( , , ) sin( )r r

am

m

r t m t e

20

0

o Wave potential (amplitude) modeled by:

ik xk 0 fk

(r, ) e J (k )

o Gyro averaging is applied along an off-axis orbit:

o Larger Ls: Gyro-averaged increases with increasing potential scale length

o Gyro-averaged increases for waves with more modes

S. Zhou, PoP 18 (2011) 082104

Large scale size Ls reduces gyro-averaging; Short correlation length Lcorr reduces phase-averaging

Helium Vbias=0V

Lcorr=23cmLs=2.6cm

Neon Vbias=75V

Lcorr=19cmLs=6.3cm


A

B

C

(B)

(C)

(B)

(C)

(A)

(A)

34

Simple Model

Sub-Diffusive Regime is Observed when Fast Ion Time-of-Flight Exceeds Wave Half Period

Con

vect

ive

Sub

-diff

usiv

e

• Simulation uses measured time-dependent wave fields

• Flat-part of curve occurs when dominant mode changes by 1800 pushing ions the opposite way

S. Zhou, PoP 18 (2011) 082104

Conclusion on Fast Ion Transport in Electrostatic Turbulent Waves in the LAPD

In experiment with plate obstacle:

o Fast ion transport decreases with increasing fast ion energy

(more phase averaging) S. Zhou et al., Phys. Plasmas 17, 092103 (2010)

In experiment with annulus obstacle:

o Waves with larger spatial scale size cause more fast-ion transport

o Turbulent waves cause more fast-ion transport than coherent waves

(less phase averaging) S. Zhou et al., Phys. Plasmas 18, 082104 (2011)

Beam diffusivity versus time

o Transport is convective when fast ion time-of-flight << wave period

o Transport is sub-diffusive when fast ion time-of-flight exceeds half the

wave period (phase reversal pushes ions back)

S. Zhou et al., Phys. Plasmas 18, 082104 (2011) 36

Outline







Perform an analogous experiment on DIII-D

• Neutral beams are the fast-ion source

• FILD is the detector

• Alfvén waves with kperpρ~1 are the fluctuationsArrange the orbit

so it passes close to FILD

Plan view of DIII-D

Xi Chen, Phys. Rev. Lett. 110 (2013) 065004

Alfvén eigenmodes deflect fast ions to the scintillator after one bounce

orbit• The contours

show a calculated mode structure

• Unperturbed and perturbed orbits are shown

Elevation


Loss signal oscillates at the Alfvén eigenmode frequency

Xi Chen, PRL 110 (2013) 065004

•Enhanced losses only occur when unperturbed orbit passes close to the detector

•Can infer the radial “kick” from the size of the coherent FILD fluctuations

Displacement is linearly proportional to mode amplitude

•Ions with correct phase are pushed out

•Consistent with ballistic transport

•Non-resonant ions are lost


Enhanced prompt losses are an important new effect

•Powerful diagnostic technique quantifies transport in well-defined orbit

•Losses are concentrated spatially possibility of wall damage

•Non-resonant lost ions do not recover their energy additional instability drive?


TAE

RSAE

Difference

2nd RSAE

2nd TAE

Sum

Nonlinear interactions for multiple Alfvén eigenmodes

Xi Chen, (2013) in preparation

Fluctuations

Losses

•Each mode alters the phase of the ion at the other mode:

k r ��

•This generates fluctuations in the losses at the sum (ω1+ω2) & difference (ω1-ω2) frequencies

The zeroth-order adiabatic invariant μ0=Wperp/B is not conserved in this

process

Kramer (2013) in preparation

•For kperpρ~1, there is a correction to μ even for and ω<<Ωi

•Ion gets “kick” on one side but not other

•Applies for vllδBperp and vperpδBll too

•The calculated shift in μ is ~ 5%

/ 1B B

The zeroth-order adiabatic invariant μ0=Wperp/B is not conserved in this

process

Kramer (2013) in preparation

•Full-orbit SPIRAL* simulation calculates a jump in μ0 when ion traverses mode

•Calculated FILD oscillation in good agreement with experiment

•Analytical calculation:

•Similar deviations found for kinetic Alfvén waves in full-orbit simulations of astrophysical turbulence [Chandran, Ap. J. 720 (2010) 503]

20 1 0 0( )[( ( ) 1) ]t

o

v Ad dv b J k

dt dt B

*Kramer, PPCF 55 (2013) 025013

Outline







Many small amplitude Alfven eigenmodes flatten the fast-ion profile

Radial Te profile during beam injection into DIII-D

Radial fast-ion profile

Heidbrink, PRL 99 (2007) 245002

Van Zeeland, PRL 97 (2006) 135001

These plasmas have an enormous number of resonances

Calculated energy change due to a single harmonic in a DIII-D plasma

•Colors indicate energy exchange

•Each pair is from one p of the resonance condition

•Each toroidal mode is composed of multiple poloidal harmonics hundreds of important resonances

White, Plasma Phys. Cont. Fusion 52 (2010) 045012

Many small-amplitude resonances appreciable transport

White, Plasma Phys. Cont. Fusion 52 (2010) 045012

Partial island overlap of some of the resonances •Although the individual

island widths are small, stochastic transport still occurs flattened profile consistent with experiment

•Recent work: efficient algorithm to calculate profile for situations with numerous small-amplitude modes White, Comm. Nonlinear Science Numerical Simulation 17 (2012) 2200

What I thought (until recently) ...

Major goal of Energetic Particle research: Predict fast-ion transport in ITER (and other future

devices) • Given the fields, we can calculate fast-ion

transport but we have to know the mode amplitude & spectra

• The mode spectra is very hard to predict (extremely complicated nonlinear physics)

Our recent results with off-axis beam injection suggests there may be an easier way ...

q • RSAEs are typically weak or not observed during discharges with only off-axis beams

• Consistent with weaker fast ion gradient near qmin

On-Axis Injection Off-Axis Injection

ECE ECE

RSAEs

Representative Profiles

Use off-axis beams to alter the spatial gradient that drives Alfvén eigenmodes

Heidbrink, Nucl. Fusion (2013) submitted

• Different combinations of on- & off-axis beams at ~ constant power

• Amplitudes summed for channels near qmin

• Time-averaged mode amplitude depends on fast-ion gradient

Near qmin

Stability trends consistently observed


Classical prediction

• FIDA diagnostic measures profiles

• Profiles differ later in discharge (when AEs are weak)

• Strong fast-ion transport by AE instabilities makes profiles similar for all cases

• Suggests a “critical-gradient” model can describe transport in this regime

Actual profiles are nearly identical for all beam combinations!


*K. Ghantous et al, Phys. Plasmas 19 (2012) 092511

•Infinitely “stiff” transport

•Ion redistribution expands unstable region

A simple critical-gradient model explains some features of fast-ion transport in this

regime

Initial profile

Relaxed profile

Linear threshold

Initial gradient

A simple critical-gradient model explains some features of fast-ion transport in this

regime

•Application of this model to these plasmas gives qualitative agreement with experiment

can use linear physics to predict profiles in ITER


Conclusions: The importance of phaseFishbones Convective resonant transport

because ions “see” constant phase (kperpρ<<1)

Energetic-particle GAM Large amplitude modifies the phase and produces fractional resonances

Drift Waves Phase-averaging reduces transport when kperpρ>>1

Prompt Alfvén Eigenmode Losses

•Non-resonant particles are pushed across loss boundaries for the proper phase

•Nonlinear perturbations to the phase produce sum & difference frequencies in the loss spectrum

Alfvén Eigenmodes Many resonances scramble phases, producing a diffusive regime with stiff fast-ion transport

• is conserved for modes with ω<<Ωi and δB/B<<1

•Resonant transport is more important than non-resonant transport

•The resonance condition is ω=nωpre+(m+l)ωbounce with [n,m,l] integers

•To predict the alpha transport in ITER you must be able to predict the amplitude of Alfvén eigenmodes

Four “truths” that aren’t quite true

/W B (there is a O(10%) correction for kperpρ~1)

(not near a loss boundary!)

(fractional resonances for large amplitude)

(not if the transport is stiff)

Backup Slides

Cartoon of a field line that scatters the pitch angle

2 2( )z z x y y x

dv dt v v b v b

dt T

•In a static magnetic field, energy is conserved a change in μ is a pitch-angle scattering event

•Simple Cartesian model for the gyro-averaged change in parallel energy in an Alfvén wave:

To get an effect, the field must

•have kperpρ~1

•be asymmetric relative to the gyro-orbit

the vxδby term is non-zero for this field line

Cross-field correlation function for Isat

400eV

600eV

800eV

1000eV

Broadband Drift Waves Induced at the Obstacle Edge

S. Zhou, PoP 17 (2010) 092103

• Resonance condition, Ωnp = n ω + p ωθ – ω = 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/Ω

np)

-5

-6

-7

-8

-9

-10

P

ci [

MeV

]

Draw curves in phase space to see resonances

Pinches, Nucl. Fusion 46 (2006) S904

Documents

W.W. (Bill) Heidbrink* UC Irvine Shu Zhou, Xi Chen, Liu Chen, Yubao Zhu University of California, Irvine T. Carter, S. Vincena, S. K. P. Tripathi University