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1DFullWaveAnalysisofParametricDecayInstability(PDI)inAlcator

C‐ModandNSTXGuangyeChen,FrederickJaeger,LeeBerry(ORNL)

JimMyra(Lodestar)PhillipRyan,JohnWilgen,TheodoreBiewer,GregHanson(ORNL)

RFSic‐DacGroup,PPPLRFGroupUS‐JapanRFPhysicsWorkshop，GA,

03/09/2010

1

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PDIandedgeionheaRnginIBWandICRFexperiments

Rost2002,CMOD

Pinsker1993,DIII‐D

Biewer2005,NSTX

antenna freq.

LangmuirProbe Magne/cProbeNBAERD

2G. Chen US-Japan Workshop 03/09/2010

Thomas1995,ToreSupra

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MoRvaRon

•  PDIhasbeenwidelyobservedintokamakICRFandIBWheaRngexperiments(NSTX,ALCATORC‐MOD,DIII‐D,TORESUPRA,TST‐2,HT‐7,etc.).

•  PDIinthescrape‐off‐layer(SOL)canleadtosignificantpowerloss,degradingtherfheaRngefficiencyinthebulkplasma(eg.directIBW).

•  AquanRtaRveunderstandingofPDIisneeded.

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Outline

•  AORSAcode,PDImodelequaRons.•  PDIsimulaRonsforAlcatorC‐ModandNSTX.

•  AfullwavesimulaRonofedgereflectometryonNSTX.

•  Summary.

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OuranalysisisbasedonAll‐ORdersSpectralAlgorithm(AORSA)•  AORSAsolvesMaxwell’sequaRonsusingalinearizedplasma

response.

•  Theplasmacurrentisgivenby

•  WeuseaspectralmethodwithFourierbasisfuncRons.•  IntegralequaRonleadstoadensesetoflinearequaRonswith~500k

linearcomplexequaRons.

5G. Chen US-Japan Workshop 03/09/2010

€

jp (r,t) = dt ' dr'σ (r − r',t − t') ⋅ E(r',t')∫−∞

t

∫

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Anexample:HighHarmonicFastWaveheaRngonNSTX

6G. Chen US-Japan Workshop 03/09/2010

•  An antenna excites fast waves.

•  Linear theory predicts that fast waves mainly heat electrons via Landau damping.

•  To explain the observed edge ion heating, we may have to apply a non-linear theory.

x (R)

y Φ

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€

C1(E0,E1,E2) = −µ*

2(σ1 −σ 2) ⋅

species∑ E2(x) +

µ2

4(σ1 −σ 2) ⋅E1(x)]

species∑

•  Thepumpwaveissolvedindependently,andisassumedconstantintheedge

•  Daughterwavesaresolvedby

•  ParametricCouplinggivenadipole‐pump:

•  Withthepumpgiven,thesystemofequaRonsislinearandcanbesolvedbyadirectsolver.

PDImodelequaRons

€

−c 2

ω02 ∇ ×∇ ×E0( ) + K0 ⋅E0 = −

iε0ω0

Jantenna

€

−c 2

ω12 ∇ ×∇ ×E1( ) + K1 ⋅E1 = C1(E0,E1,E2) −

iε0ω1

Jnoise,1

€

−c 2

ω22 ∇ ×∇ ×E2( ) + K2 ⋅E2 = C2(E0,E1,E2) −

iε0ω2

Jnoise,2€

ω0 =ω1 +ω2

k0 = k1 + k2

7G. Chen US-Japan Workshop 03/09/2010

Slection rules:

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PDIsimulaRonsforAlcatorC‐Mod

f2 53.4 MHz f3 26.6 MHz Te 15 eV Ti 3 eV ne 1 x 1019 m-3 B0 5.3 T rt 0.9 m ky 2000 m-1

40

ExponenRalgrowthofthedaughterfield

€

nφ

Pumpfield(A.U.)

E(IBW)

8G. Chen US-Japan Workshop 03/09/2010

An example.

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Whichwavestosimulate?DaughterwavessaRsfyingthefollowingcondiRonsaretobeexcited:

1.selecRonrules

2.oneofthedaughterwavesisalinearplasmawave.IntheICRFrange,fastwave‐‐>IBW+quasimode(Porkolab,1990).? €

ω0 =ω1 +ω2

k0 = k1 + k2

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SpaRalevoluRonofIBW’sintheSOL

•  –  Slow convergence of simulation for large –  larger coupling for large

•  Damping(left)<<damping(right) -> Ion cyclotron heating

€

k⊥

€

k⊥ (left) << k⊥ (right)

10G. Chen US-Japan Workshop 03/09/2010

Ion harmonic

€

k⊥

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LocalIBWdispersionrelaRon&decaychannels

FAST WAVE

QM IBW

IBW QM

QM IBW

CMODMinorityhea/ngregime

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PDIdispersioncalculaRons

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ω/ωcD ω/ωcD

n ϕ

€

E⊥ = 9kV /m

€

E⊥ = 27kV /m

Growth rate (s-1) Growth rate (s-1)

pump

CMODMinorityhea/ngregime

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PDIthresholdisconsistentwithexperiment

 Linear3DAORSAsimulaRonspredicttypicalfastwaveedgeamplitude=kV/m@0.5MWrfpower. ScalingtrendsforthePDIthresholdbehaviorwithdensity,temperatureareconsistentwithRef.[Rost2002].

ExponenRalgrowth

FastwaveIBW+QM

noise

13G. Chen US-Japan Workshop 03/09/2010

(5,12)

CMODMinorityhea/ngregime

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PDIsimulaRonsforNSTX

•  DaughterIBWamplitudesvs.frequencyatafixedpointintheedge.

•  BothweakandstrongPDIaresimulated.

•  Peaksareseparatedbythecyclotronfrequencyattheprobe().

•  Simulation parameters: –  Te = Ti = 50 eV, –  ne = 2 × 1018 m-3, –  Bϕ = 0.45 T –  ky = 7000 m-1, nϕ = 15

Antennafrequency

Logscale

probemeasurements[Diem,APS04]

€

Ωi

€

Ωi

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PDIdispersioncalculaRonsforNSTXGrowthrate(s‐1)

MOST UNSTABLE MODES

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pump ω/ωcD

n ϕ

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PDIkϕdependenceisconsistentwithedgeionheaRngandreflectometermeasurements

RelaRveAmplitu

deofP

DI

Gen

erated

IBW

PRF=1.2‐1.3MW

€

Bφ = 0.5 T

  Poloidal ion temperature increases with toroidal wave length (Talyor 2010).   Predicted power threshold is consistent with previous reflectometer

measurements (100 to 300kW), (Wilgen 2005).

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Reflectometer and Langmiur probe show similar spectra on NSTX

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Reflectometer RF Spectrum RF Langmuir Probe Spectrum

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FullwavecoldplasmasimulaRonofXmodereflectometry

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Typical NSTX wave numbers: FAST WAVE: 50-100 m-1 IBW: 1000-10000 m-1

Matching condition*: k_pertrub = 2 (k_x)

€

ˆ n =1+ 0.001e(r−rp )2 /Δ2

sin(−ω pt + kp (r − rp ))

*N. Bretz, Phys. Fluids B 4, 2414 (1992) Reflecting layer

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FullwavesimulaRonofXmodereflectometry

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Matching condition*: k_pertrub = 2 (k_x)

€

ˆ n =1+ 0.001e(r−rp )2 /Δ2

sin(−ω pt + kp (r − rp ))

*N. Bretz, Phys. Fluids B 4, 2414 (1992)

Higher order Bragg condition n (k_pertrub) = 2 (k_x) n = 1, 2

Reflectometor seems insensitive to very short wave length signals.

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TheplasmaresponsesoffastwaveandIBWaredifferent•  PlasmadensityfluctuaRonscausedbyrfwavesisevaluatedby

•  AssumenoiselevelIBWdensityfluctuaRonis

•  DensityfluctuaRonofIBWabovethethresholdisabout1000Rmeslargerthanthatcausedbyfastwave.

•  GiventhesameleveloffluctuaRons,thereflectometersignaloffastwaveisabout10,000RmeslowerthanthatofIBW.

Daughterwavesignal≈0.1fastwavesignal.

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€

˜ n p = ˜ f (ω,n0,B0,E∫ )dv

€

0.1%n0PDIstrength DensityfluctuaLon(1/m3)

Pumpwave DaughterIBW

Belowthreshold 8e13 3e15

Abovethreshold 4e14 2e17

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Summary•  WesuccessfullysimulatedFastwave‐>IBW+QMusingtheAORSA

code.•  Themostunstablemodesarefoundtobethewavesexcitedjust

aboveancyclotronharmonic.•  PDIsimulaRonsagreewith

–  the threshold in a minority heating experiment on Cmod, –  the edge ion heating measurements on the kϕ dependence in

NSTX, –  the threshold measurement by reflectometer in NSTX.

•  FullwavereflectometrysimulaRonsshowthatthemeasurementforshort‐wave‐lengthdaughterwavesisnon‐local.

•  FutureworksincludeNSTXreflectormetorandprobedataanalysis,apredicRonforNSTXupgrade,andthecalculaRonofPDIpowerdeposiRon(1Dto2Dand3D).

21G. Chen US-Japan Workshop 03/09/2010