Propagation and Power Deposition of Electron Cyclotron Waves in Non-Circular Tokamak Plasmas

Li Jing-Chun et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 8, (Part - 5) August 2015, pp.217-223
www.ijera.com 217 | P a g e
The Propagation and Power Deposition of Electron Cyclotron
Waves in Non-Circular HL-2A Tokamak Plasmas
Tang Qing-Yi, Li Jing-Chun, Gong Xue-Yu†, Yan Zheng
(School of Nuclear Science and Technology, University of South China, Hengyang, Hunan, China 421001)
Abstract
By solving the plasma equilibrium equation, ray equations, and quasi-linear Fokker-Planck equation, the ray
trajectories and power deposition of EC wave has been numerically simulated in non-circular HL-2A tokamak
plasma. The results show that shaping effect and temperature profile has little influence on ECRH, while plasma
density affect propagation and power deposition obviously. when the ordinary mode of EC waves are launched
from the mid-plane and low-field-side, ray trajectories are bended as the parallel refractive index increases and
even recurve to the low-field side when the parallel refractive index reaches to a certain value. Single absorption
decreases with increasing both poloidal and toroidal injection angle, and can be 100% when poloidal injection
angle is 180o
and toroidal injection angle is less than 10o
.
Key words: electron cyclotron wave; noncircular cross; ray trajectories
I. Introduction
Electron cyclotron resonance heating and
electron cyclotron current drive (ECRH/ECCD) in
tokamak plasma have both physical and practical
advantages. Firstly, the fact that the power can be
injected as narrow Gaussian beams and gives rise to
highly localized power deposition, makes
ECRH/ECCD an ideal candidate for local MHD
control. Secondly, since the wave may propagate as a
well-defined beam, the wave launcher may be distant
from the plasma and that the coupling is insensitive
to the conditions at the plasma boundary [1].
HL-2A is a medium size tokamak at the
Southwestern Institute of Physics (SWIP), Chengdu,
China. It’s aim is to confirm the ITER physics and
technology basis by combining the known elements
of tokamak plasma physics into integrated operation
scenarios [2]. To explore ECRH/ECCD experiments,
such as current control, MHD control and transport
study, a 3MW ECRH system consists of two
68GHz/0.5MW/1.5s subsystems and four
68GHz/0.5MW/1s subsystems has been developed
and upgraded on it. EC waves are injected into
plasma from lower field side by two launchers which
can be rotated to choose the angle of injection in the
toroidal and poloidal direction between 0o
~30o
. With
the polarizer, EC waves in ordinary and extraordinary
mode were injected into HL-2A at different magnetic
field [3]. So far, series of high powerful ECRH
experiments, including ELM H mode realization,
stabilization of tearing mode, sawtooth control,
suppression of runaway electron and so on, has been
explored [4]. In the study of ECRH, the first step is to
calculate the wave trajectories and then to get the
wave attenuation along wave trajectories, and to
determine the power deposition profile [6-7]. For
convenience, people studied the EC wave trajectories
and absorption in HL-2A plasma under the circular
RESEARCH ARTICLE OPEN ACCESS

cross section approximation [8-9]. However, HL-2A
tokamak have non-circular configuration with
elongation κ = 1.2 and triangularity δ = 0.3. On one
hand, the shaping effect may influence the wave
power deposition. On the other hand, the magnetic
field, temperature and density profile in non-circular
cross section tokamak is different with those in
circular cross section one, they influence the EC
wave propagation and power deposition in plasma.
Therefore, numerical study of EC wave propagation
and power deposition in non-circular cross section
plasma is necessary.
In this paper, the ray trajectories and power
deposition of EC wave has been numerically
simulated in non-circular HL-2A tokamak by solving
the plasma equilibrium equation, ray equations, and
quasi-linear Fokker-Planck equation. Section 2
describes the physical model used here. In section 3,
we report the dependence of ray trajectories and
power deposition on plasma density, injection angle.
A brief conclusion is drawn in section 4.
II. Model
EC ray trajectories are obtained by solving
the following ray equations,
 





0
0
0
0
d
d
;
d
d
D
rD
t
k
D
kD
t
r

(1)
where D0 is the dispersion relation, which is
calculated with a cold plasma approximation and can
be expressed in the following form
11/ 22 2 2 2 2
|| || ||2 2
0 || 2 2 2
||
(1 ) (1 ) 4( (1 ))
( , ) 1 1 1 0
2(1 ) 2(1 ) (1 )
Y N Y N N X
D k N N X
X X Y N



    
                  (2)
Where
  ,,,,
21
0
222
eceeepcep meBmenYX  
N is refraction index, ωp is plasma frequency, ωce is
electron gyrofrequency, ± denote ordinary mode
(O-mode) and extraordinary mode (X-mode)
respectively. The O-mode has a component of
electric field parallel to the back-ground magnetic
field while the X-mode hasn’t.
Since the existence of trapped particles in
tokamak, the electron is described by the bounce
averaged Fokker-Planck equation which can be
written as [10]
0


S
t
f 


(3)
Where cmcpuu //,/ 

 ， f is the
electron distribution function,
e
ba
c SSSS

 
/
,
ba
cS /

is the collisional
flux, eS

is the DC electric field flux, and S

is
the wave induced flux. The collisional operator is
approximated by the linearized truncated operator,
in the relativistic formulation. Transforming
equation (2) to the ),( 0 p coordinate space we
can obtain
   0100
2
0
2
0
0
0
0
2
0
2
0
,1 0
0

  ffIpS
p
p
Sp
t
fp p









(4)
where  01000 ,,,,cos 00
  ffISS p can be
found in ref. [11].Once the stationary solution of Eq.
(3) has been obtained, the RF power absorbed per
unit volume Pabs can be computed as
   00
2
0000
2
dd2,
1


 pppfSpcmnP wtheed
(5)
Plasma equilibrium is obtained by solving
the Grad - Shafranov equation using a direct

variational method based on an energy principle
[12-13]. After obtaining the magnetic field, electron
temperature and density profiles are given in the
following form
( )
( ) 0 0( ) [ ( ) ( )](1 / ) ( )te ti
e i p e i ea ia p pa ea iaT T T T T T T

     
0( ) ( )(1 / ) dn
e p e ea p pa ean n n n
     
(6)
Where p is poloidal magnetic flux and pa edge
magnetic flux. Notation Te0, Tea, ne0, nea are plasma
core temperature, edge temperature, core density, and
edge density. A flat Zeff profile was assumed with Zeff
= 2.5. In all cases, the toroidal magnetic field B0 at
the nominal major radius R0 =1.65m is 2.42T, and the
equilibrium employed corresponding to a total
plasma current of 450kA.
III. Simulation results
Firstly, we solve the plasma equilibrium equation
by using a direct variational method based on energy
principle. Figure 1 is the magnetic field of NSTX
tokamak, which is in good agreement with the result
obtained in ref [12]. The magnetic field of HL-2A is
then calculated with the some code and is shown in
figure 2.The HL-2A equilibrium parameters are given
in table 1.
Table 1. HL-2A equilibrium parameters.
B0(T) Ip(MA) R0(m) a(m) κ δ ne(×1019
/m3
) Te(KeV)
2.42&2.5 1.0 1.65 0.4 1.2 0.3 2.2 3.0
Fig.1 Proﬁles of the poloidal and toroidal induction components against R for Z=0 in NSTX
Fig.2 Proﬁles of the poloidal and toroidal induction components against R for Z=0 in HL-2A
In order to calculate the ray trajectories and
power deposition of EC wave in HL-2A, we map the
magnetic field, density and temperature of every
point to the grid point in toroidal coordinate. A fourth
order Runge-Kutta algorithm is used to solve
equation (1) to get ray trajectories. Equation (4) is

discretized using a nine point stencil, and a large
sparse nine-diagonal lineal system is obtained. The
system is solved by a method of approximate matrix
factorization finally [11]. The results given by the
Grad - Shafranov equation solver and Fokker-Planck
solver have good agreement with those in paper
[10][12]at the some parameters.
Now we turn to the explicit calculations of
ray-tracing. the power and frequency of EC wave
(this paper only consider the EC wave O mode) are:
ne0 = 2.2×1019
m-3
, nea = 0.5×1019
m-3
, Te0 = 3.0keV, Ti0
= 2.8KeV, Ti(e)a = 50eV. Firstly, We start with the
performance of an equatorial launcher, injecting the
EC power from the midplane at an toroidal angle
at  =10o
, where 90o
-Φ is the angle between the
wave-vector and the magnetic field, Φ = arcsinN//, N//
is the parallel index of refraction; and poloidal
angle = 175o
, 180o
, 185o
, respectively, where is
defined as the angle of the central ray with respect to
the tokamak symmetric axis on the plasma cross
section. Figure 3 shows the propagation of ray
trajectories of EC waves launched in different
poloidal injection angle from outboard midplane, and
the power deposition profile.
As already mentioned, there are two main
different factors between circular and non-circular
tokamak, i.e, the shaping effects and profile of
plasma density and temperature. We investigate the
Figure 3. Propagation of ray trajectories and corresponding power deposition profile of EC waves launched in
different poloidal injection angle from outboard midplane of HL-2A with κ = 1.2.
1.1 2.1 3.1
At absorption Peak(r/a0) 0.23755 0.23772 0.23777
Peak of power deposition
density(W/m3
)
1.73380E6 1.73474E6 1.73516E6
Table 1. the absorption peak and the peak of power deposition density at different elongation with = 180o
and
 =20o
.
influence of shaping effect on ray trajectories and
power deposition of EC wave firstly. Table 1 is the
absorption peak and the peak of power deposition
density when the elongation be κ = 1.1, κ = 1.2, κ =
1.3 respectively. It is seen that the distance between
location of power deposition and plasma center
shortens and the power deposition density increases
with increasing elongation, while those changes are
small, which means shaping effect has little influence
on ECRH.
The dependence of ray trajectories and
power deposition on plasma density and temperature
is also investigated. Both the profile of plasma
density and temperature can be changed by the core
Elongation
Term

density, temperature and the parameter αte , αti , αdn .
Figure 1 is the ray trajectories of ECW (a), plasma
refractive index (b) and power deposition of ECW (c)
when the wave is launched at = 185o
and  = 10o
,
and the central electron density is ne0 = 2.0×1019
m-3
,
ne0 = 2.5×1019
m-3
, ne0 = 3.0×1019
m-3
, respectively. We
can see that trajectories turn to low field side and
power deposition decrease with increasing central
plasma density, which is in reasonable agreement
with the experimental results in MTX [14]. This can
be explained with figure 4 (b), larger central plasma
density correspond to larger gradient of plasma
refractive index, leading to larger reduction of power
absorption. To study the influence of density profile,
we set αdn = 0.1, 0.5, 1.0,1.5, respectively, and no
difference is found at those situations. Another useful
information is that the power deposition of top launch
is small compared to the midplane launch, and the
location of deposition peak is near plasma edge.
Obviously, the decrease of deposition here is because
of the trapped particle effect. However, according to
ref [15], we still can expect high heating efficiency at
top injection once the resonance location near the
plasma core through adjusting injection angle.
Fig.4 Ray trajectories of ECW (a), plasma refractive index (b) and power deposition of ECW (c) with different
central electron density
Fig.5 power deposition profile with different central temperature (a). temperature profile versus normalized
poloidal flux with different temperature profile coefficient (b)
1.0dn 5.0dn 1dn 5.1dn
At absorption Peak(r/a0) 0.11976 0.11976 0.11594 0.11976
Peak of power deposition
density(W/m3
)
4.31992E7 4.03509E7 3.73649E7 4.3153E7
Table 2. the absorption peak and the peak of power deposition density at different temperature profile coefficient
with = 180o
and  =10o
coefficient
Term

We studied the influence of plasma temperature
profile on absorption of EC waves. Figure 5 (a) is the
power deposition profile with different central
temperature. The core temperature affect the location
of power deposition while has little influence on the
value of power deposition. Table 2 is the absorption
peak and the peak of power deposition density when
the temperature profile coefficient be αdn = 0.1, 0.5,
1.0,1.5. We can see that both core temperature and
temperature profile has no obvious effect on EC
waves absorption, which is in accordance with the
experimental result in ref [16].
We have also studied the propagation and
power deposition when EC wave is injected from
outboard of the vessel. Figure 5 show ray trajectories
of ECW O mode launched from the LFS in the
poloidal section (a) and equatorial plane (b) , with N//
= 0, N// = 0.574, N// = 0.707, N// = 0.766, N// =0.866.
When N// =0, EC wave propagate in straight line;
while N//≠0, ray trajectories turn to low field side
(LHS) and is more bended as the parallel refractive
index increases, and even recurve to the LHS when
the parallel refractive index reaches to a certain value.
Figure 6 (a) is power deposition profile of EC wave
when = 180o
,  =14o
, 18o
, 22o
,and indicates that
small toroidal injection angle correspond to high
power deposition and small width of power
deposition profile, which means small angle injection
of EC wave would be a useful way not only to heat
plasma but also to suppress MHD instability. Figure
(b) is wave single absorption, which is defined as the
loss of power to total input power when EC wave
propagate through resonance layer at the first time,
versus toroidal injection angle when poloidal
injection angle is 180°, 185°, 190°. Simulation results
show single absorption decreases with increasing
toroidal injection angle, and get maximum when
toroidal injection angle is less than 10°. The width of
power deposition also can be influenced by toroidal
injection angle. One the other hand, single absorption
decreases with increasing poloidal injection angle,
and can be more than 100% when = 180o
,  <10o
.
Thus, both poloidal injection angle and toroidal
injection angle should be as small as possible to get
high wave heating efficiency.
Fig.6 Ray trajectories for O mode launched from the LFS in the poloidal section (a) and equatorial plane (b)
Fig.7 The profile of power deposition of EC wave when the poloidal injection angle is 185° (a) and EC wave
single absorption versus toroidal injection angle when the poloidal injection angle is 180°,185°,190° (b)

IV. Conclusions
The features of electron cyclotron resonance
heating in the HL-2A tokamak have been investigated
using a ray-tracing code. The calculations have been
performed for low-field-side launch or top launch of
the ordinary mode at the first harmonic of the
electron gyrofrequency in the standard magnetic
configuration of HL-2A. The numerical results show
that shaping effect and temperature profile has little
influence on ECRH, while plasma density affect
propagation and power deposition obviously. when
the ordinary mode of EC waves are launched from
the mid-plane and low-field-side, ray trajectories are
bended as the parallel refractive index increases and
even recurve to the low-field side when the parallel
refractive index reaches to a certain value. Single
absorption decreases with increasing both poloidal
and toroidal injection angle, and can be 100% when
poloidal injection angle is 180o
and toroidal injection
angle is less than 10o
. We find that small angle
injection of EC wave would be a useful way not only
to heat plasma but also to suppress MHD instability.
In order to get high wave heating efficiency, both
poloidal injection angle and toroidal injection angle
should be low.
V. Acknowledgements
This work is supported by National Natural
Science Foundation of China (No. 11075073,
No.11205086, No.41104094), Research Fund for the
Doctoral Program of Higher Education of China
(No.20114324110001), the Construct Program of
Fusion and Plasma Physics Innovation Team in
Hunan Province (No.NHXTD03)
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Propagation and Power Deposition of Electron Cyclotron Waves in Non-Circular Tokamak Plasmas

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