Integrated Full-Wave Modeling of ICRF Heating in Tokamak ...
Transcript of Integrated Full-Wave Modeling of ICRF Heating in Tokamak ...
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EPS PPC 2007, Warsaw, PolandD1-005, 2007/07/02
Integrated Full-Wave Modeling
of ICRF Heating in Tokamak Plasmas
A. Fukuyama, S. Murakami, T. Yamamoto, H. Nuga
Department of Nuclear Engineering, Kyoto University,Kyoto 606-8501, Japan
1. Introduction2. Integrated tokamak modeling code: TASk3. Modification of velocity distribution function4. Integral formulation of full wave analysis5. Summary
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Introduction(1)
ICRF Waves in Burning Plasmas
ICRF waves
Fusion reaction
Energetic
ions
Wave-plasma
interaction
Fusion-reaction
crosssection
• Both ICRF waves and fusion reaction generate energetic ions andare affected by the energetic ions.
• In the start up phase of ITER plasmas, the role of ICRF waves isimportant, time-evolving, and sensitive to the plasma conditions
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Introduction(2)
Gyrokinetic Behavior of Energetic Ions
ICRF wavesEnergetic
ions
Non-Maxwellian
velocity distribution
Large orbit size
• Self-consistent analysis of non-Maxwellian velocity distribution func-tion is necessary.
• Finite gyroradius and finite orbit size affect the behavior of ICRFwaves.
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Introduction(3)
Comprehensive Modeling of Burning Plasmas
ICRF wavesEnergetic
ions
Alfvén Eigenmode
Transport
Orbit loss
Bulk plasma
• Energetic ions interact with bulk plasmas through, for example, trans-port processes and orbit loss.
• Alfven eigenmodes may affect the energetic ions themselves.
• Integrated comprehensive modeling of burning plasmas is inevitable.
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Introduction (4)
• Analysis of ICRF waves in burning plasma requires
◦ Full wave analysis
◦ Non-Maxwellian velocity distribution function
◦ Finite gyroradius effect
◦ Integrated modeling
• Integrated approach using the TASK code
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TASK Code
• Transport Analysing System for TokamaK
• Features
◦ A Core of Integrated Modeling Code in BPSI— Modular structure, Unified Standard data interface◦ Various Heating and Current Drive Scheme— Full wave analysis for IC and AW— Ray and beam tracing for EC and LH— 3D Fokker-Planck analysis◦ High Portability◦ Development using CVS◦ Open Source◦ Parallel Processing using MPI Library◦ Extension to Toroidal Helical Plasmas
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Modules of TASK
PL Data Interface Data conversion, Profile databaseEQ 2D Equilibrium Fixed/Free boundary, Toroidal rotationTR 1D Transport Diffusive transport, Transport models
WR 3D Geometr. Optics EC, LH: Ray tracing, Beam tracingWM 3D Full Wave IC, AW: Antenna excitation, EigenmodeFP 3D Fokker-Planck Relativistic, Bounce-averagedDP Wave Dispersion Local dielectric tensor, Arbitrary f (u)LIB Libraries LIB, MTX, MPI
Under DevelopmentTX Transport analysis including plasma rotation and ErEG Gyrokinetic linear stability analysis
Imported from TOPICSEQU Free boundary equilibriumNBI NBI heating
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Modular Structure of TASK
EQ
T R
PL
W R
DP
F ixed-boundary equilibrium
Diffusive transport
IT PA��� ���
le DB
JT -6 0 Exp. Data
Experimental
Database
Simulation DB
T X
F P
W M
W X
Dynamic transport
Kinetic transport
Ray tracing
F ull wave analysis
F ull wave analysis including F LR
Dispersion relation
EG Gyrokinetic linear microinstabilities
EQU
NBI
T O PIC S/ F ree-boundary equilibrium
T O PIC S/ Neutral beam injection
Data
Interface
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Inter-Module Collaboration Interface: TASK/PL
• Role of Module Interface
◦ Data exchange between modules:— Standard dataset: Specify set of data (cf. ITPA profile DB)— Specification of data exchange interface: initialize, set, get◦ Execution control:— Specification of execution control interface:
initialize, setup, exec, visualize, terminate— Uniform user interface: parameter input, graphic output
• Role of data exchange interface: TASK/PL
◦ Keep present status of plasma and device◦ Store history of plasma◦ Save into file and load from file◦ Interface to experimental data base
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Standard Dataset (interim)
Shot dataMachine ID, Shot ID, Model ID
Device data: (Level 1)RR R m Geometrical major radiusRA a m Geometrical minor radiusRB b m Wall radiusBB B T Vacuum toroidal mag. fieldRKAP κ Elongation at boundaryRDLT δ Triangularity at boundaryRIP Ip A Typical plasma current
Equilibrium data: (Level 1)PSI2D ψp(R,Z) Tm2 2D poloidal magnetic fluxPSIT ψt(ρ) Tm2 Toroidal magnetic fluxPSIP ψp(ρ) Tm2 Poloidal magnetic fluxITPSI It(ρ) Tm Poloidal current: 2πBφRIPPSI Ip(ρ) Tm Toroidal currentPPSI p(ρ) MPa Plasma pressureQINV 1/q(ρ) Inverse of safety factor
Metric data1D: V ′(ρ), 〈∇V〉(ρ), · · ·2D: gi j, · · ·3D: gi j, · · ·
Fluid plasma dataNSMAX s Number of particle speciesPA As Atomic massPZ0 Z0s Charge numberPZ Zs Charge state numberPN ns(ρ) m3 Number densityPT Ts(ρ) eV TemperaturePU usφ(ρ) m/s Toroidal rotation velocityQINV 1/q(ρ) Inverse of safety factor
Kinetic plasma dataFP f (p, θp, ρ) momentum dist. fn at θ = 0
Dielectric tensor dataCEPS ↔ε (ρ, χ, ζ) Local dielectric tensor
Full wave field dataCE E(ρ, χ, ζ) V/m Complex wave electric fieldCB B(ρ, χ, ζ) Wb/m2 Complex wave magnetic field
Ray/Beam tracing field dataRRAY R(`) m R of ray at length `ZRAY Z(`) m Z of ray at length `PRAY φ(`) rad φ of ray at length `CERAY E(`) V/m Wave electric field at length `PWRAY P(`) W Wave power at length `DRAY d(`) m Beam radius at length `VRAY u(`) 1/m Beam curvature at length `
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Example: Data Structure and Program Interface
• Data structure: Derived type (Fortran95)
type bpsd_plasmaf_data
real(8) :: pn ! Number density [mˆ-3]
real(8) :: pt ! Temperature [eV]
real(8) :: ptpr ! Parallel temperature [eV]
real(8) :: ptpp ! Perpendicular temperature [eV]
real(8) :: pu ! Parallel flow velocity [m/s]
end type bpsd_plasmaf_data
type bpsd_plasmaf_type
real(8) :: time
integer :: nrmax ! Number of radial points
integer :: nsmax ! Number of particle species
real(8), dimension(:), allocatable :: s
! (rhoˆ2) : normarized toroidal flux
real(8), dimension(:), allocatable :: qinv
! 1/q : inverse of safety factor
type(bpsd_plasmaf_data), dimension(:,:), allocatable :: data
end type bpsd_plasmaf_type
• Program interface
Set data bpsd set data(plasmaf,ierr)
Get data bpsd get data(plasmaf,ierr)
Save data bpsd save data(filename,plasmaf,ierr)Load data bpsd load data(filename,plasmaf,ierr)Plot data bpsd plot data(plasmaf,ierr)
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Examples of sequence in a module
• TR EXEC(dt)
call bpsd get data(plasmaf,ierr)
call bpsd get data(metric1D,ierr)
local data <- plasmaf,metric1D
advance time step dt
plasmaf <- local data
call bpsd set data(plasmaf,ierr)
• EQ CALC
call bpsd get data(plasmaf,ierr)
local data <- plasmaf
calculate equilibrium
update plasmaf
call bpsd set data(plasmaf,ierr)
equ1D,metric1D <- local data
call bpsd set data(equ1D,ierr)
call bpsd set data(metric1D,ierr)
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Wave Dispersion Analysis : TASK/DP
• Various Models of Dielectric Tensor↔ε (ω, k; r):◦ Resistive MHD model◦ Collisional cold plasma model◦ Collisional warm plasma model◦ Kinetic plasma model (Maxwellian, non-relativistic)◦ Kinetic plasma model (Arbitrary f (u), relativistic)◦ Gyro-kinetic plasma model (Maxwellian)
• Numerical Integration in momentum space: Arbitrary f (u)◦ Relativistic Maxwellian◦ Output of TASK/FP: Fokker-Planck code
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Full wave analysis: TASK/WM
•magnetic surface coordinate: (ψ, θ, ϕ)
• Boundary-value problem of Maxwell’s equation
∇ × ∇ × E =ω2
c2↔ε · E + iωµ0 jext
• Kinetic dielectric tensor: ↔ε◦Wave-particle resonance: Z[(ω − nωc)/k‖vth]◦ Finite gyroradius effect: Reductive =⇒ Integral (ongoing)
• Poloidal and toroidal mode expansion
• FDM: =⇒ FEM (onging)
• Eigenmode analysis: Complex eigen frequency which maximizewave amplitude for fixed excitation proportional to electron density
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Fokker-Planck Analysis : TASK/FP
• Fokker-Planck equationfor velocity distribution function f (p‖, p⊥, ψ, t)
∂ f∂t= E( f ) +C( f ) + Q( f ) + L( f )
◦ E( f ): Acceleration term due to DC electric field◦ C( f ): Coulomb collision term◦ Q( f ): Quasi-linear term due to wave-particle resonance◦ L( f ): Spatial diffusion term
• Bounce-averaged: Trapped particle effect, zero banana width
• Relativistic: momentum p, weakly relativistic collision term
• Nonlinear collision: momentum or energy conservation
• Three-dimensional: spatial diffusion (neoclassical, turbulent)
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Self-Consistent Wave Analysis with Modified f (u)
•Modification of velocity distribution from Maxwellian
◦ Energetic ions generated by ICRF waves◦ Alpha particles generated by fusion reaction◦ Fast ions generated by NB injection
• Self-consistent wave analysis including modification of f (u)
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Preliminary Results
• Tail formation by ICRF minority heating
Wave pattern
Quasi-linear Diffusion Momentum Distribution
-10 -8 -6 -4 -2 0 2 4 6 8 100
2
4
6
8
10
PPARA
PPERP
DWPP FMIN = 0.0000E+00 FMAX = 5.7337E+01 STEP = 2.5000E+00
-10 -8 -6 -4 -2 0 2 4 6 8 100
2
4
6
8
10
PPARA
PPERP
F2 FMIN = -1.2931E+01 FMAX = -1.1594E+00
Tail Formation Power deposition
0 20 40 60 80 10010-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
p**2
Pabs with tail
Pabs w/o tail
Pab
s [a
rb. u
nit]
r/a
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Finite Gyroradius Effects in Full Waves Analyses
• Several approaches to describe the finite gyroradius effects.
• Differential operators: k⊥ρ→ iρ∂/∂r⊥◦ This approach cannot be applied to the case k⊥ρ & 1.◦ Extension to the third and higher harmonics is difficult.
• Spectral method: Fourier transform in inhomogeneous direction
◦ This approach can be applied to the case k⊥ρ > 1.◦ All the wave field spectra are coupled with each other.◦ Solving a dense matrix equation requires large computer resources.
• Integral operators:∫ε(x − x′) · E(x′)dx′
◦ This approach can be applied to the case k⊥ρ > 1◦ Correlations are localized within several gyroradii◦ Necessary to solve a large band matrix
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Full Wave AnalysisUsing an Integral Form of Dielectric Tensor
•Maxwell’s equation:
∇ × ∇ × E(r) +ω2
c2
∫↔ε (r, r′) · E(r′)dr = µ0 jext(r)
• Integral form of dielectric tensor: ↔ε (r, r′)◦ Integration along the unperturbed cy-
clotron orbit
• 1D analysis in tokamaks
◦ To confirm the applicability
◦ Similar formulation in the lowest orderof ρ/L
— Sauter O, Vaclavik J, Nucl. Fusion 32(1992) 1455.
s'
t'
t
s0 s
�p
v
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One-Dimensional Analysis (1)
ICRF minoring heating without energetic particles (nH/nD = 0.1)
Differential form Integral form[V/m]
Ex
Ey
[W/m2]
PH
PDPe
[V/m]
Ex
Ey
PH
Pe PD
[W/m2]
R0 = 1.31m
a = 0.35m
B0 = 1.4T
Te0 = 1.5keV
TD0 = 1.5keV
TH0 = 1.5keV
ns0 = 1020m−3
ω/2π = 20MHz
Differential approach is applicable
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One-Dimensional Analysis (2)
ICRF minoring heating with energetic particles (nH/nD = 0.1)
Differential form Integral form[V/m]
Ex
Ey
[W/m2]
Pe
PH
PD
[V/m]
Ex
Ey
[W/m2]
PH
Pe
PD
R0 = 1.31m
a = 0.35m
B0 = 1.4T
Te0 = 1keV
TD0 = 1keV
TH0 = 100keV
ns0 = 1020m−3
ω/2π = 20MHz
Differential approach cannot be applied since k⊥ρi > 1.
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One-Dimensional Analysis (3)
ICRF minoring heating with α-particles (nD : nHe = 0.96 : 0.02)Differential form Integral form
[V/m]
Ex
Ey
[W/m 2]
PePD
PHe
[V/m]
Ex
Ey
[W/m 2]
PD
PHe
Pe
R0 = 3.0m
a = 1.2m
B0 = 3T
Te0 = 10keV
TD0 = 10keV
Tα0 = 3.5MeV
ns0 = 1020m−3
ω/2π = 45MHz
Absorption by α may be over- or under-estimated by differentialapproach.
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3D Formulation
• Coordinates
◦Magnetic coordinate system: (ψ, χ, ζ)◦ Local Cartesian coordinate system: (s, p, b)◦ Fourier expansion: poloidal and toroidal mode numbers, m, n
• Perturbed current
j(r, t) = − qm
∫du qu
∫ ∞−∞
dt′[E(r′, t′) + u′ × B(r′, t′)
]· ∂ f0(u′)
∂u′
•Maxwell distribution function◦ Anisotropic Maxwell distribution with T⊥ and T‖ :
f0(s0, u) = n0
(m
2πT⊥
)3/2 (T⊥T‖
)1/2
exp
[− v2
⊥2v2
T⊥
−v2‖
2v2T‖
]
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Variable Transformations
• Transformation of Integral Variables
◦ Transformation from the velocity spacevariables (v⊥, θ0) to the particle positions′ and the guiding center position s0.
◦ Jacobian: J =∂(v⊥, θ0)∂(s′, s0)
= − ω2c
v⊥ sinωcτ.
s'
t'
t
s0 s
�p
v
◦ Express v⊥ and θ0 by s′ and s0 using τ = t − t′, e.g.,
v⊥ sin(ωcτ + θ0) =ωc
v⊥
s − s′
21
tan 12ωcτ
+ωc
v⊥
(s + s′
2− s0
)tan
12ωcτ
• Integration over τ: Fourier expansion with cyclotron motion
• Integration over v‖: Plasma dispersion function
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Final Form of Induced Current
• Induced current:
·
Jmns (s)
Jmnp (s)
Jmnb (s)
= ∫ds′
∑m′n′
↔σm′n′mn(s, s′) ·
Em′n′s (s′)
Em′n′p (s′)
Em′n′b (s′)
• Electrical conductivity:
↔σm′n′mn(s, s′) = −in0q2
m
∑`
∫ds0
∫ 2π
0dχ0
∫ 2π
0dζ0 exp i
{(m′ − m)χ0 + (n′ − n)ζ0
} ↔H `(s, s′, s0, χ0, ζ0)
•Matrix coefficients:↔H `(s, s′, s0, χ0, ζ0)
◦ Four kinds of Kernel functions including s, s′, s0 and harmonicsnumber `
— The kernel functions are localized within several thermal gyro-radii.
◦ Plasma dispersion function
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Kernel Functions
• Kernel Function and its integrals
F(100)0 F(100)
1
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Status of extension to 3D configuration
• In a homogeneous plasma, usual formula including th Bessel func-tions can be recovered.
• Kernel functions are the same as the 1D case,
• FEM formulation is required for convolution integral.
• Development of the FEM version of TASK/WM is ongoing (al-most complete).
• Integral operator code in 3D configuration is waiting for the FEMversion of TASK/WM.
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Consistent Formulation of Integral Full Wave Analysis
• Full wave analysis for arbitrary velocity distribution function◦ Dielectric tensor:
∇ × ∇ × E(r) − ω2
c2
∫dr0
∫dr′
p′
mγ∂ f0(p′, r0)
∂p′· K1(r, r′, r0) · E(r′) = iωµ0 jext
where r0 is the gyrocenter position.
• Fokker-Planck analysis including finite gyroradius effects◦ Quasi-linear operator∂ f0∂t+
(∂ f0∂p
)E+∂
∂p
∫dr
∫dr′E(r) E(r′) ·K2(r, r′, r0) · ∂ f0(p′, r0, t)
∂p′=
(∂ f0∂p
)col
• The kernels K1 and K2 are closely related and localized in the re-gion |r − r0| . ρ and |r′ − r0| . ρ.
• To be challenged
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Summary
• Comprehensive analyses of ICRF heating in tokama plasmas
◦ Time-evolution of the velocity distribution functions and the finitegyroradius effects have to be consistently included. For this pur-pose, the extension of the integrated code TASK is ongoing.
• Self-consistent analysis including modification of f (p)◦ Full wave analysis with arbitrary velocity distribution function and
Fokker-Planck analysis using full wave field are available. Pre-liminary result of self-consistent analysis was obtained.
• 3D full wave analysis including the finite gyroradius effects:
◦ 1D analysis elucidated the importance of the gyroradius effectsof energetic ions. Formulation was extended to a 2D configura-tion. Implementation is waiting for the FEM version of TASK/WM.