integrated full-wave modeling of icrf heating in tokamak
TRANSCRIPT
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
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
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.
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.
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
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
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
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
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
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 `
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)
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)
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
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
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)
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)
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
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
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
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
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.
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.
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‖
]
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
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
Kernel Functions
• Kernel Function and its integrals
F(100)0 F(100)
1
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.
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
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.