Role of Upper Hybrid Resonance and
Diffraction Effects at Electron Cyclotron Heating
.
In Tokamaks
V.Vdovin
RRC Kurchatov Institute, Nuclear Fusion Institute
Abstract. We present modelling results of basic Electron Cyclotron Heating scenarios in tokamaks
performed with newly developed 3D full wave STELEC (stellarator_ECH, tokamaks included as particular
case) code. Code includes all basic wave physics as interference, diffraction, wave tunnelling, mode
conversion at Upper Hybrid resonance to electron Bernstein waves and appropriate boundary conditions.
Code operates in real 3D magnetic geometry and uses massive parallel terabyte computers and firstly
permitted solution of above problem. Modelled are fundamental and second harmonics O-mode and Xmode scenarios in T-10, FTU, DIII-D, JT-60U, RTP tokamaks and in ITER at fundamental harmonic. The
Upper Hybrid resonance plays important role, leading to strong broadening of power deposition profiles at
fundamental at O-mode RF power launch. This is partly supported by experiments on DIII-D, RTP, T-10
and JT-60U. Diffraction effects are investigated at second harmonic and these are shown to be important
even at moderate plasma densities. Code discovered important new effect of remarkable RF power
reflection on second harmonic resonance layer in poloidally oblique EC power launch scenarios, intended
to be used for NTM suppression. The O-X-B scheme for over dense plasma is also explored .
Email of V.Vdovin:
vdov@pike.pike.ru
Introduction
Electron Cyclotron Resonance plasma heating (ECH) and current drive (CD) in
Fusion plasma research in tokamaks and stellarators plays a key role in investigation of
basic wave - plasma interaction physics like electrons heating, local transport coefficients
behaviour, CD and current profile tailoring, Internal Transport Barrier (ITB) creation,
Neoclassical Tearing Modes (NTM) stabilization, etc. The method, initially supposed to
be very local in space, is an essential and very expensive tool in present fusion machines
and ITER Contrary, theoretical and modelling support during last 30 years was
performed by so called very simple geometric optic approach, named as “ray tracing”,
dropping important physics like reflection, interference, diffraction, wave tunnelling,
mode conversion to electron Bernstein waves and appropriate boundary conditions.
Reason for that simple ray tracing approach over all world use was due to non sufficient
power of world computers in the past for ECH method modelling to solve exact problem
in 3D plasma with small scale waves. Respectively planning, interpretation and
prediction of ECH/CD were unsatisfactory ones as shows below full wave 3D code
modelling.
1. STELEC Code
Code [1] solves 3D full wave equation
rotrotE 
4ik 0
c
( jp  j
ext
2
)  k0 E
(1.1)
in flux coordinates with appropriate boundary conditions, in 3D stellarator geometry, on
real VMEC code equilibrium, using poloidal and toroidal harmonic expansions over
respective angles and center difference scheme over radial (psi) coordinate ( k 0 = /c, c speed of light).
In eq.(1.1) jext is an imposed antenna current density and jp is the RF plasma current
being evaluated using the small Larmor radius approximation by procedure starting from
the results of [2], in which the Vlasov equation is solved in plane-stratified geometry,
and assuming that the vector form of jp in this limit holds also in 3D toroidal geometry.
Zero plasma response is given by dielectric components :
11(0)  1 
2
 pe
2 2

3
 
F7 / 2 (1)) 
  ( F5/ 2 (1) 

2
  ce 

12(0)  i
2
 pe
2 2

3
 
F7 / 2 (1)) 
  ( F5/ 2 (1) 

2
  ce 

(1.2)
2
 pe

  F5/ 2 (0)  2 2 F7(2)
/ 2 (0)  ,
2

and finite electron Larmor radius wave induced current is given as
 33(0)  1 











c2 
J 
R      RE   i   eˆb  RE  
4i
eˆb    



  eˆb  RE  i   RE
 
(1.3)







( RE  E  2u g  E  u g , u g    B0 /   B0 in 3D magnetic geometry, with metric
tensor, both given by VMEC) and expressed through Shkarovsky functions Fq (n) [3]:
2
1  pe vTe2 
 

 F7 / 2 (2) 

2
2 
4 ce c 
  2ce 
2
1  pe vTe2 
 

 F7 / 2 (2) 

2
2 
4 ce c 
  2ce 
N // 
k //
,
k0
where parallel wave number is expressed through contravariant
(1.4)
components of equilibrium 3D magnetic fields and toroidal and poloidal modes numbers
n, m [4]:


B
B
k n
m
B
B
Huge matrix resulting after discretization is solved by “Progonka” Algorithm (Gaussian
inversion specialized for three diagonal block system) for parallel processors computer
option, distributing matrices across processors, and uses the ScaLapack library and
message passing interface (MPI) for parallel matrix operations. The individual blocks are
distributed across all processors. The compact storage methods with out of core solver
can also be used for serial code option.
Antenna is located in vacuum layer between plasma and conducting wall and
modeled by current straps. Currents’ generally elliptical polarization is imposed
according to ECH O-mode or X-mode polarization at plasma separatrix, calculated
through the local dispersion relation. Antennae currents generally are corrected by
additional multipliers, requested by Maxwell equations solutions in vacuum layer
surrounding plasma. Faraday toroidal and poloidal screens options are also provided,
when needed, for linear polarization of an antenna.
2. Similarity Laws
Analysis of above equations shows that only combinations of parameters are used
(  pe , ce - plasma and electron cyclotron frequencies, Te - electron temperature)
2
 pe
,
2
ce
,

Te , N // (magnetic equilibrium)
Thus keeping these fixed (similar) ones and providing multi mode regime for excited
k
ECH waves ( N    , 0 - vacuum wave length, a – plasma minor radius)
k0
1
a 
0 ,
2N 
it is possible to model ECH scenarios in large scale fusion machines at reduced
frequencies thus diminishing requirements on computer memory capabilities (important
for ITER like machines).
3. FTU and DIII-D Tokamaks Fundamental Harmonic Modeling
FTU tokamak with major radius R=93 cm uses quasi perpendicular O-mode outside
launch at 140 GHz. Contour plots of total wave electrical field are displayed in Fig.1a.
The axis magnetic field Bo=4.667 T , plasma density 4.5x1019 m-3 (parabolic density
profile, αn=1), Te(0) =9.2 kV (αT=2), Ip=0.72 MA, N//(0)=0.027. Upper Hybrid resonance
2
layer manifests itself, contrary to ray tracing, by bright “mirror” broadly
 2  ce2   pe
FIGURE 1a |E|total in FTU at Ne=4.5x1019 m-3
FIGURE 1b Power dep. Pe
FIGURE 1c 2D Pe
radiating Electron Bernstein Waves (EBW) and slow X-waves to cold ECR side.
Averaged over flux surface power deposition, in Fig.1b, shows peaked power deposition
mainly responding to large electrical fields near UHR layer (X=3 cm). This is confirmed
by 2D power deposition in Fig.1c. Thus RF power is absorbed far away from cold EC
resonance (X=-6cm). This ECH picture summarizes early record ECH Te results from
middle and small tokamaks TM-4, T-10, TFR, RTP etc. obtained at low plasma densities
(~1019 m-3) when distance between ECR and UHR resonances ΔX~R n(0) is small one.
Location of EC resonance at magnetic axis at Bo=5.0 T moves UHR outside
(|real(E_psi)| contours in Fig.2a) with respective off axis power deposition: Fig.2b.
Above “Arbalet-like” wave pictures show also EC X-wave cut off and remarkable
FIGURE 2a |real(E_psi)| in FTU at Bo=5.0 T
FIGURE 2b Pe in FTU at Bo=5.0 T
parallel fields amplitudes (Fig.1a) generated by used O-mode antenna. Doubling plasma
density (9x1019 m-3) leads in FTU to more serious broad power deposition as is shown by
Fig.2c. That effect is much more pronounced at oblique power launch for CD goals due
to larger classical Doppler broadening effect influence. Contour plots of |E_minus| wave
electrical field in DIII-D plasma are displayed in Fig.3a at fundamental harmonic Omode 44.6 GHz launch: N//(0) = 0.227, Ne(0) = 0.481019 m-3, Nes=0.121019 m-3, Te(0)
=2.7 kV (αn=0.2, αT=2), Bo=1.47 T, Ip=0.657 MA. Plasma elongation is κ=1.65,
triangularity δ= 0.5. The EC resonance
is far inside displaced one. Power
deposition peaks in Fig.3b may be
responsible
for
temperature
filamentation. The benefits off centre
inside EC resonance scenario are
supported
by
T-10
experience
(K.Razumova, private communication).
FIGURE 2c Pe in FTU at Ne(0) = 9x1019 m-3
FIGURE 3a |E_minus| in DIII-D
FIGURE 3b Pe in DIII-D at N//(0) = 0.227
Again, power deposition is dominated by UH resonance position due to large amplitudes
of converted waves. One pays attention that RF power is absorbed by fast electrons.
4. Second Harmonic ECH Scenarios in T-10
Majority of present tokamaks and stellarators operate at second harmonic and at for
not so large plasma densities UHR is absent one. Second harmonic X-mode launch in T10: |real(E_psi)|, Fig.4a at N=90 (N//(0)=0.02), F=140 GHz, Ne(0)=4.51019 m-3,
Te(0)=8.7 kV, (αn=1, αT=2), Bo=2.5 T, Ip=300 kA (relativistic effects are included) and
power deposition in Fig.4b. Diffraction effects are more severe at density Ne(0)=81019
m-3 and are given in Fig.5a for |real(E_psi)| and in Fig.5b for 1D and 2D power
depositions.
FIGURE 4a |real(E_psi)|,in T-10 at Ne(0)=4.51019 m-3
FIGURE 4b Pe in T-10 at 2ωce F=140 GHz
FIGURE 5a |real(E_psi)|,in T-10 at Ne(0)=81019 m-3 FIGURE 5b 1D&2D Pe in T-10 at Ne=81019 m-3
At plasma density Ne(0)=91019 m-3 diffraction is more strong one (Figs.6a,b)..
FIGURE 6a |real(E_psi)|,in T-10 at Ne(0)=91019 m-3
FIGURE 6b 1D&2D Pe in T-10 at Ne=91019 m-3
The X-mode cut off near plasma axis appears at density Ne(0)=101019 m-3 : Figs.7a,b.
FIGURE 7a |real(E_psi)| in T-10 at Ne(0)=1020 m-3
FIGURE 7b 1D&2D Pe in T-10 at Ne(0)=1020 m-3
5. Second Harmonic ECH Scenarios in DIII-D at Oblique Launch
Second harmonic X-mode oblique 4 MW launch in DIII-D: plasma elongation κ=1.65,
triangularity δ= 0.5, N=920 (N//(0)=0.29), F=89.2 GHz, Te0=7.2 kV, Ne(0)=1.21019
m-3 , Ip=720 kA, q(0)=0.82, q(a)=3.5 is displayed in Figs.8a,b: by |real(E_minus)| and
|real(E_psi|) fields components
FIGURE 8a |real(E_minus) |in DIII-D at 2ωce
FIGURE 8b |real(E_psi)| in DIII-D at 2ωce
Radial power depositions calculated by STELEC and GENRAY-GA (R.Prater [6]) for
elliptically polarized X-antennae are given by Figs.9a,b. One can see that STELEC gives
up to 3 times broader power deposition than predicts ray tracing code.
FIGURE 9a Pe(ρ) in DIII-D calculated by STELEC FIGURE 9b Pe(ρ) in DIII-D due to GENRAY-GA
6. Fundamental Harmonic Scenario at JT-60U
Modelled was JT-60U regime with fundamental harmonic O-mode equatorial 4 MW
launch in plasma with Ro=3.5 m, elongation κ=1.25, triangularity δ= 0.34, N=360
(N//(0)=0.11), F=44.6 GHz, Te0=10.8 kV, Ne(0)=4.81018 m-3 , (αn=0.2, αT=2),
Bo=1.41 T, Ip=634 kA, q(0)=1.41, q(a)=4.4 is displayed in Figs.10a,b: by |real(E_minus)|
One can see High Field Side EC resonance is at X=-40cm and UH resonances at
X=-5cm. Power deposition is in Fig.10b again indicates crucial role of UH resonance in
FIGURE 10a |real(E_minus)| in JT-60
FIGURE 10.b Pe(ρ) in JT-60U calculated by STELEC
broadening of EC power localization.
6. Fundamental Harmonic Scenario in ITER-like Machines
Modelled was ITER regime for NTM stabilization, relying on above similarity laws,
with fundamental harmonic O-mode equatorial 20 MW launch into plasma with Ro=6.2
m, elongation κ=1.85, triangularity δ=0.45, N=380 (N//(0)=0.13), F=22.6 GHz, Te0=25.8
kV, Ne(0)=1.21018 m-3 , (αn=0.2, αT=2), Bo=0.646 T, Ip=1.27 MA, q(0)=0.88, q(a)=4.2.
Classical Doppler broadening was assumed. Results are displayed in Figs.11a,b: by
|real(E_minus)| and Pe(r). One can see HFS EC resonance at X=-117cm and UH
resonances at X=-62cm. Power deposition is in Fig.11b again indicates crucial role of UH
resonance in broadening of EC power localization, leading to large deposited power
outside of supposed NTM localization.
FIGURE 11a |real(E_minus)| in ITER FIGURE 11b Power deposition Pe(ρ) in ITER
7. OXB Scenarios For Over Dense Plasma
These schemes were theoretically proposed in frame of 1D geometry to exploit O-mode
conversion to X-mode with subsequent its conversion to EB waves (O-X-B scheme)
which have no density cut off and can propagate at plasma densities larger than cut off
densities for e.m. EC waves. We modeled this OXB scheme at fundamental harmonic in
FIGURE 12a |real(E_psi)| in Tore Supra
FIGURE 12b Pe(ρ) in TS at Ne(0) = 1.7 Ncrit
Tore Supra tokamak with Ro=2.38 m, a=0.61 m, N=80 (N//(0)=0.125), F=12.85 GHz,
Te0=9.5 kV, Ne(0)=61017 m-3 , thus Ne(0) = 1.7 Ncrit , (αn=2.5, αT=0.1), Bo=0.37 T,
Ip=103 kA, q(0)=1.35, q(a)=3.65. STELEC code found that really O-X-B 2D scheme
operates in regime with far inside EC resonance location as demonstrated by Fig.12a for
|real(E_psi)| contour plots and by Fig.12b for radial power deposition Pe(r).
However we did not succeed at second harmonic O-mode launch in DIII-D and Tore
Supra. More work is needed.
8. Fundamental harmonic ECH power deposition in RTP
The RTP tokamak (major radius R=72 cm) [8] used oblique O-mode fundamental
harmonic outside launch at 60 GHz in Co-Counter ECCD scenarios. Contour plots of
|real(Eψ )| wave electrical field, 2D and 1D power depositions are displayed in Fig.13.
The axis magnetic field is Bo=1.98 T , central plasma density 1.65x1019 m-3 (parabolic
density profile), Te(0) =2.9 kV (αT=2), Ip=40 kA, N//(0)=0.35. One sees that EB waves
play crucial role in creation of EC power deposition profile. Classical Doppler effects
FIGURE 13 |real(Eψ
)| and Power depositions: 1D Pe (ρ) and 2D Pe (ρ,θ) in RTP
dominate in this RTP scenario. Counter CD scenario with N//(0)= -0.35 reveals
significant amount of RF power being deposited broadly near half of plasma minor
radius, consistent with reduced counter CD efficiencies observed in RTP experiments [8].
9. Discussion
Thus full wave code discovered crucially new physics result at fundamental EC
harmonic: Upper Hybrid Resonance appearance at scenarios with O-mode radiating
antenna, contrary to ray tracing (based on local dispersion relation) philosophy. This
UHR layer fires practically as a hole, occupying remarkable part of plasma and radiates
large amplitude Electron Bernstein Waves and Slow X-waves. These strong waves
deposit dominant part of RF input power at LFS to energetic part of electron distribution
with account to relativistic and classic Doppler effects. At low plasma densities in small
and middle size fusion machines (with small major radius R) a space separation between
cold EC resonance and UH resonance, being proportional to Ne•R, is small one and RF
power, being absorbed near UHR position in area of its intersection with equatorial plane,
is absorbed sufficiently locally thus recovering majority of best electron heating world
results for near on axis ECR used scenarios in tokamaks and stellarators.
However, in large machines like JT-60U, TS, ITER at increased densities, required by
fusion gain, that resonances pair is well separated and STELEC modeling manifests loss
of EC power localization at fundamental harmonic. Fortunately, at second harmonic,
typically, UHR is absent in interesting for fusion scenarios and localization still works.
There are interesting experimental confirmations of UHR appearance at fundamental
O-mode RF power launch: 1) Similar EC heating efficiencies at fundamental harmonic in
DIII-D for O-mode and X-mode launches were reported [6] and recently confirmed again
[7]. Nice explanation argued to X-mode reflection from X’s cut off and conversion to Omode during reflection from the walls. Full wave code treats this “conversion” effect
automatically and knows something about O or X modes only through antenna
polarization. 2) Experimental evidence of UHR existence in RTP tokamak: at outside 60
GHz O-mode launch they observed two orders rise of X-mode signal in central receiving
inside chamber horn vs position of ECR layer and central electron density, when UHR
according to simple formula, appears at plasma core [8]. They stressed sawtooth
modulation of X-signal, thus arguing that signal was coming from plasma core.
3) The measured by quasi perpendicular oriented antenna radiation temperatures in RFP
MST plasma with Ip=250 kA, <Ne>=4 1018 m-3 for X and O polarizations are comparable
ones and attribute to conversion efficiencies from EBW to O-mode and X-mode [7].
They argue that reciprocal principle is valid and both types of plasma heating may be
applied at fundamental harmonic.
4) Transformation of wave polarization due to 2D plasma non uniformity [9]:
- first cylindrical model of theoretical and experimental insight to the problem of wave
polarization conversion due to axial plasma density gradient or to wave incident power
axial localization. These [9] results are directly relevant to ECH in stellarators and
mirrors. So natural is our similar idea of transformation wave polarization due to
POLOIDAL non uniformity of toroidal plasmas in tokamaks to understand code’s news.
5) The WEGA stellarator electron heating with O- mode fundamental resonance out of
plasma [5] is qualitatively confirmed by STELEC modeling.
Conclusions
In toroidal bounded plasmas O-mode and X-mode are coupled ones through space
inhomogenity and boundary conditions. This modes coupling effect in toroidal plasmas
is especially strong one at fundamental harmonic scenarios due to singular layer created
by UHR resonance. This effect leads to strong broadening of EC power deposition at
fundamental and must be accounted through full wave codes. In second harmonic
scenarios the modes coupling is more weak one. Ray tracing/bi-tracing technique may be
still used, neglecting by X and O-modes coupling, for power deposition evaluations, with
corrections from full wave codes at high densities for self consistent account of modes
coupling and diffraction effects. Full wave codes are powerful tools and must be used,
when available, for fundamental harmonic scenarios in small, large and ITER machines
REFERENCIES
1. V.Vdovin Electron Cyclotron Heating modelling in tokamaks with full wave 3D code, XXXIII International
Conf. on Plasma Phys. and Contr. Fusion, 13 – 17 February 2006, Zvenigorod, Moscow, Russia
2. M. Brambilla Plasma Physics Controlled Fusion 31 (1989), 723
3. I. Shkarovsky, J. Plasma Phys. 35, part2, 319-331 (1986)
4. V.Vdovin, T.Watari, A.Fukuyama, Proc.ICPP 96 (Nagoya 1996),1070; NIFS- 469
5. Y.Y. Podoba “Radio frequency heating on the WEGA stellarator”, Ph. D. These
Ernst-Moritz-Arndt-University Greifswald, 2006
6. R. Prater et al, Proc. of the EPS25 (Praha) Vol. 22C (European Physical Society, 1998) p. 1410
7. R. Prater (private communication).
8. A.Donne, A.Oomens, EC-9, Proc. of 9th Joint Workshop on ECE and ECH, Borrego Springs, California, 23-26
January 1995, 645-650, F.Schuller et al,. Proc. of 4th Int. Workshop Strong Microwaves in Plasma, NN Russia, 1999
9. D.Falconer, A.Messian, P.Vandenplas Physics of Fluids, 17 (1974), 1566-1573