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.