1D Full Wave Analysis of Parametric Decay Instability (PDI ... · 1D Full Wave Analysis of...
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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
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PDIandedgeionheaRnginIBWandICRFexperiments
Rost2002,CMOD
Pinsker1993,DIII‐D
Biewer2005,NSTX
antenna freq.
LangmuirProbe Magne/cProbeNBAERD
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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.
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jp (r,t) = dt ' dr'σ (r − r',t − t') ⋅ E(r',t')∫−∞
t
∫
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Anexample:HighHarmonicFastWaveheaRngonNSTX
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• 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
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−c 2
ω02 ∇ ×∇ ×E0( ) + K0 ⋅E0 = −
iε0ω0
Jantenna
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−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
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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
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nφ
Pumpfield(A.U.)
E(IBW)
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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
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k⊥
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k⊥ (left) << k⊥ (right)
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Ion harmonic
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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
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E⊥ = 9kV /m
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E⊥ = 27kV /m
Growth rate (s-1) Growth rate (s-1)
pump
CMODMinorityhea/ngregime
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PDIthresholdisconsistentwithexperiment
Linear3DAORSAsimulaRonspredicttypicalfastwaveedgeamplitude=kV/[email protected]. ScalingtrendsforthePDIthresholdbehaviorwithdensity,temperatureareconsistentwithRef.[Rost2002].
ExponenRalgrowth
FastwaveIBW+QM
noise
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(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
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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)
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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)
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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).
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