2002/02/04US-Japan Workshop on RF Physics and
Profile Control and Steady State Operation using RFKyushu Univ, Chikushi Campus
Analysis of EC Wave Propagationby Beam Tracing Method
A. Fukuyama, S. NishinaDepartment of Nuclear Engineering,
Kyoto University, Kyoto 606-8501, Japan
Contents
• Estimation of Driven Current Width• Beam Tracing Method• Numerical Results• Summary
Motivation
• Current profile control by EC waves◦ Localized profile of driven current◦ Position control by injection angle
• Control of MHD instability◦ Suppression of island growth due to tearing instability◦ Localized current profile is required
• Evaluation of the current profile width◦ Doppler broadening, decay length◦ Finite beam size, focusing◦ Defraction
• Limitation of ray tracing◦ Defraction effect cannot be included
Beam
Magnetic Surface
Beam
Magnetic Surface
Analysis by Ray Tracing for ITER-FEAT• Injection angle dependence
R [m]
Z[m]
0°10°20°
30°
40°
50°
p abs
[arb
.]
ψ1/2
0°10° 20° 30°
0°10°20°
30°
40° 50°
R [m]
Z[m]
0°
10°
20°
30°
ψ1/2
p abs
[arb
.]
40°
R [m]
Z[m]
0°10°20°30°
40°
50°
p abs
[arb
.]
ψ1/2
0°30°
40°
• Current drive efficiency
0.00
0.10
0.20
0.30
0 10 20 30 40 50 60
0510152025303540
γ
IAM (R=6.20, B=5.51, T=20)
θT
θP
• Absorption width (single ray)
0.00
0.10
0.20
0.30
0.40
0 10 20 30 40 50 60
0510152025303540
∆r / a
IAM (R=6.20, B=5.51, T=20)
θT
θP
Propagation of Short-Wavelength Waves
• Ray Tracing (Geometrical Optics)◦Wave length λ � Characteristic scale length L of the medium◦ Plane wave: Beam size d is sufficiently large
— Fresnel condition: L � d2/λ◦ Beam : Diffraction effect determines the beam size d
• Beam Tracing◦ Propagation of beam with finite size
— Spatial evolution of beam size
◦ References— G. V. Pereverzev, in Reviews of Plasma Physics, Vol. 19, p. 1.— A. G. Peeters, Phys. Plasmas 3 (1996) 4386.— G. V. Pereverzev, Phys. Plasmas 4 (1998) 3529.
Ray Tracing Method
• Maxwell equation for wave electric field E e − iωt
∇ × ∇ × E − ω2
c2↔� · E = 0
(Dielectric tensor↔� : Hermite part
↔� H� Anti-Hermite part ↔� A)
• Dispersion relation in a homogeneous plasma:Plane wave : constant wave vector k
K = det[
c2
ω2
(−k2↔I + kk
)+↔� H
]= 0
• Expansion parameterδ =
√cωL� 1
• Eikonal expression of wave electric fieldE(r) = Re
[va(δ2r) e i s(r)
]
• Solvable condition of Maxwell’s equation with eikonal expressiondrdτ
=∂K∂k,
dkdτ
= −∂K∂r
Beam Tracing Method
• Beam size perpendicular to the beam direction: first order in δ• Beam shape : Weber function Hermite polynomial: Hn)
E(r) = Re∑
mn
Cmn(δ2r)ve(δ2r)Hm(δξ1)Hn(δξ2) e i s(r)−φ(r)
◦ Amplitude : Cmn, Polarization : ve, Phase : s(r) + i φ(r)
s(r) = s0(τ) + k0α(τ)[rα − rα0 (τ)] +
12
sαβ[rα − rα0 (τ)][rβ − rβ0(τ)]
φ(τ) =12φαβ[rα − rα0 (τ)][rβ − rβ0(τ)]
◦ Position of beam axis : r0, Wave number on beam axis: k0
◦ Curvature radius of equi-phase surface: Rα = 1λsαα
◦ Beam radius dα =√
2φαα
• Gaussian beam : case with m = 0, n = 0
R1
R2
d1
d2
Beam Propagation Equation
• Solvable condition for Maxwell’s equation with beam fielddrα0dτ
=∂K∂kα
dk0αdτ
= − ∂K∂rα
dsαβdτ
= − ∂2K
∂rα∂rβ− ∂
2K∂rβ∂kγ
sαγ − ∂2K
∂rα∂kγsβγ − ∂
2K∂kγ∂kδ
sαγsβδ +∂2K∂kγ∂kδ
φαγφβδ
dφαβdτ
= −(∂2K∂rα∂kγ
+∂2K∂kγ∂kδ
sαδ
)φβγ −
(∂2K∂rβ∂kγ
+∂2K∂kγ∂kδ
sβδ
)φαγ
• By integrating this set of 18 ordinary differential equations, we obtain trace of thebeam axis, wave number on the beam axis, curvature of equi-phase surface, andbeam size.
• Equation for the wave amplitude Cmn∇ ·
(vg0|Cmn|2
)= −2
(γ|Cmn|2
)
Group velocity: vg0, Damping rate: γ ≡ (v∗e ·↔� A · ve)/(∂K/∂ω)
Beam Tracing in a Uniform Plasma
• 170 GHz, Ordinary Mode, Perpendicular InjectionRc dini = 0.01 m dini = 0.03 m dini = 0.05 m dini = 0.08 m
∞
0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]
2 m0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.000.00 1.0 2.0 3.0 4.0
d [m
]
s [m]
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.000.00 1.0 2.0 3.0 4.0
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]
1 m d [m]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.000.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]0.00 1.0 2.0 3.0 4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00
d [m
]
s [m]
Dependence of Initial Beam Radius, dini
Beam length where d = dmin
necessary condition:
dini >√
Rcλ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 0.02 0.04 0.06 0.08 0.1 0.12
rc0
= 0.5 m
rc0
= 0.5 m
rc0
= 1 m
rc0
= 1 m
rc0
= 2 m
rc0
= 2 m
rc0
= 3 m
rc0
= 3 m
L d,m
in
rd0
Minimum beam radius dmin
dmin ∼ λπdini
Rc
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 0.02 0.04 0.06 0.08 0.1 0.12
rc0
= 0.5 m
rc0
= 0.5 m
rc0
= 1 m
rc0
= 1 m
rc0
= 2 m
rc0
= 2 m
rc0
= 3 m
rc0
= 3 m
r d,m
in
rd0
Beam Tracing in ITER-FEAT Plasma: Rc = 2 m, dini = 0.05 m
θp = 40◦ θp = 50◦ θp = 60◦ θp = 70◦4
2
0
-2
-44 5 6 7 8
4
2
0
-2
-44 5 6 7 8 4 5 6 7 8
4
2
0
-2
-4
4
2
0
-2
-44 5 6 7 8
d [m
]
ψ1/20.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00
0.02
0.04
0.06
d [m
]
0.00
0.02
0.04
0.06
ψ1/20.0 0.2 0.4 0.6 0.8 1.0 1.2
d [m
]
0.00
0.02
0.04
0.06
ψ1/20.0 0.2 0.4 0.6 0.8 1.0 1.2
d [m
]
0.00
0.02
0.04
0.06
ψ1/20.0 0.2 0.4 0.6 0.8 1.0 1.2
0.90 0.92 0.94 0.96 0.98 1.0
ψ1/2
Pab
s [ar
b]
20
40
60
80
00.82 0.84 0.86 0.88 0.90 0.92
20
40
60
80
0
Pab
s [ar
b]
ψ1/20.70 0.72 0.74 0.76 0.78 0.80
60
40
20
0
ψ1/2
Pab
s [ar
b]100
80
60
40
20
00.37 0.39 0.41 0.43 0.45 0.47
ψ1/2
Pab
s [ar
b]
Beam Tracing in ITER-FEAT Plasma
θp = 60◦ θp = 70◦
Rc = 3 m dini [m] Rc = 4 m
0.02
0.04
0.06
0.08
0.10
0.000.0 0.2 0.4 0.6 0.8 1.0 1.2
d [m
]
ψ1/20.70 0.72 0.74 0.76 0.78 0.80
Pab
s [ar
b]
60
40
20
0
ψ1/2
0.050.02
0.04
0.06
0.08
0.10
0.00
d [m
]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
100
80
60
40
20
0
ψ1/2
Pab
s [ar
b]
d [m
]
0.02
0.04
0.06
0.08
0.10
0.000.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
Pab
s [ar
b]
60
40
20
00.70 0.72 0.74 0.76 0.78 0.80
ψ1/2
0.060.02
0.04
0.06
0.08
0.10
0.00
d [m
]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
d [m
]
100
80
60
40
20
0
Pab
s [ar
b]
0.37 0.39 0.41 0.43 0.45 0.47ψ1/2
d [m
]
0.02
0.04
0.06
0.08
0.10
0.000.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
Pab
s [ar
b]
60
40
20
00.70 0.72 0.74 0.76 0.78 0.80
ψ1/2
0.080.02
0.04
0.06
0.08
0.10
0.00
d [m
]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
100
80
60
40
20
0
Pab
s [ar
b]
0.37 0.39 0.41 0.43 0.45 0.47ψ1/2
Beam Tracing in ITER-FEAT Plasma: Rc = 2 m, dini = 0.05 m
θt = 0◦ θt = 10◦ θt = 20◦4
2
0
-2
-44 5 6 7 8
-2.0
-1.0
0.0
1.0
2.0
4.0 5.0 6.0 7.0 8.0-2.0
-1.0
0.0
1.0
2.0
4.0 5.0 6.0 7.0 8.0 4.0 5.0 6.0 7.0 8.0-2.0
-1.0
0.0
1.0
2.0
d [m
]
ψ1/2
0.00
0.02
0.04
0.06
0.08
0.10
0.0 0.2 0.4 0.6 0.8 1.0 1.2d
[m]
0.00
0.02
0.04
0.06
0.08
0.10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
d [m
]
0.00
0.02
0.04
0.06
0.08
0.10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
0
20
40
60
0.82 0.86 0.90 0.94
Pab
s [ar
b]
ψ1/20
10
20
30
Pab
s [ar
b]
0.82 0.86 0.90 0.94
ψ1/20
4
8
12
Pab
s [ar
b]
0.82 0.86 0.90 0.94
ψ1/2
Beam Tracing in ITER-FEAT Plasma: Rc = 2 m, dini = 0.05 m
θt = 0◦ θt = 10◦ θt = 20◦4
2
0
-2
-44 5 6 7 8
-2.0
-1.0
0.0
1.0
2.0
4.0 5.0 6.0 7.0 8.0-2.0
-1.0
0.0
1.0
2.0
4.0 5.0 6.0 7.0 8.0-2.0
-1.0
0.0
1.0
2.0
4.0 5.0 6.0 7.0 8.0
0.00
0.02
0.04
0.06
0.08
0.10
d [m
]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
0.00
0.02
0.04
0.06
0.08
0.10
d [m
]0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
0.00
0.02
0.04
0.06
0.08
0.10
d [m
]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ψ1/2
0
20
40
60
80
0.38 0.42 0.46 0.50
Pab
s [ar
b]
ψ1/20
10
20
30
40
50
Pab
s [ar
b]
0.38 0.42 0.46 0.50
ψ1/20
10
20
30
40
Pab
s [ar
b]
0.38 0.42 0.46 0.50
ψ1/2]
Summary
• Based on the formulation of beam tracing, the wave propagation codeTASK/WR was extended to calculate the spatial evolution of the EC beamsize.
•We have confirmed the diffraction effect and the initial wave front curva-ture dependence of the beam size.
dmin ∼ λπdini
Rc
• In the case of ITER-FEAT (170GHz), initial beam radius of 5cm is re-quired to focus with beam length 3m.
• For toroidally oblique injection, Doppler broadening may mask the effectof diffraction.
• To dos:◦ Coupling with Fokker-Planck Analysis◦ NTM stabilization
