The role of magnetic equilibria in determining ECE in MAST
J. Preinhaelter1), V. Shevchenko2), M. Valovič2), H. Wilson2),
J. Urban1), P. Pavlo1), L. Vahala3), G. Vahala4)
1) EURATOM/IPP.CR Association, Institute of Plasma Physics, 182 21 Prague, Czech Republic
2) EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK
3) Old Dominion University, Norfolk, VA 23529, USA
4) College of William & Mary, Williamsburg, VA 23185, USA
This work was partially funded by the United Kingdom Engineering and Physical Sciences Research Council and by EURATOM.
ECE and EBW in MAST
Extensive ECE data are available for MAST in the frequency range 16-60GHz
The low magnetic field and high plasma density in a spherical tokamak do not permit the usual radiation of O and X modes from the first five electron cyclotron harmonics
only electron Bernstein modes (unaffected by any density limits) - converted into the electromagnetic waves in the upper hybrid resonance region - can be responsible for the measured radiation
0 0.1 0.2 0.3 0.4 0.5S [m ] Depth of p lasm a slab
0
10
20
30
40
50
60
70
80
90
f [G
Hz]
UHR
Plasm a resonance
4fce 3fce
2fce
fce
#7789, t=240, dev=17 o, long=17 oR_cut-off
L_cut-off
Cutoffs and resonances in MAST cold plasmafor normal incidence
MAST ECE antenna system
1st mirror
2nd mirror(adjustable)
lens
horn
MAST window
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50
z [cm ]
-5-3-1135
r [c
m]
- 5- 3- 1135
Intensity of Gaussian beam
First fla t m iror Second flat m iror
Thin lens
w0 2
w0 1
MAST window
At the 1st waist (w01) the beam is detached from the horn
The 2nd waist (w02) is the projection of the 1st waist by the lens
ECE from MAST is transmitted only by those rays which can penetrate through the window and are not blocked by the vessel wall
3D MAST plasma model
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
R [m ]
0
0.2
0.4
0.6
0.8
T [k
eV
]
0
1
2
3
4
5
6
n [1
019
m-3
]
n used in E C E s im ula tion
n m easured
T extrapo lation in SO L
T m easured
#7685, t=240m s
LCFS-EFIT=SCENE
density, electron temperature and magnetic potential profiles for the shot #7798, time 240ms
UHR = 36.46GHz0.2 0.4 0.6 0.8 1.0 1.2
R [m ]
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Z [m
]
LCFSUHR
A realistic 3D model of the MAST plasma has been developed for the simulation The magnetic field is reconstructed by splining of the two potentials determined by
the EFIT and SCENE codes, assuming toroidal symmetry The temperature and density profile are obtained from the Thomson scattering
measurements, beyond the LCFS exponentially decaying profiles are used
The Gaussian beam is replaced by rays
Intersection of the antenna beam with the LCFS (last closed flux surface) determines the position of the spot
R _C artesian [m ]
# 4958, t=120W aist rays intersectionSpot of rays at LCFS
Beam direction, dev=18o, long=22o
Rays
Y [m ]
Z [
m]
R _C artesian [m ]
# 4958, t=120W aist rays intersectionSpot of rays at LCFS
Beam direction, dev=18o, long=22o
Rays
Plane stratified slab and mode conversion efficiency estimation
Full wave solution of Maxwell’s equations in the cold plasma slab is used for determination of the EBW-X-O conversion efficiency
0.4 0.8 1.2 1.6 2 2.4
R [m ]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Z [
m]
separatrix
ECE antenna
Conversion efficiency using the adaptive finite elements method
The 2nd order ODE’s in the cold plasma model are solved assuming weak collisions
The absorbed power in the UHR equals the power of the converted EBW
Adaptive mesh is refined in regions of large local errors
0 0.02 0.04 0.06 0.08 0.1x [cm ]
-2
-1
0
1
2
3 R e E z
R e E y
Im E z
Im E y
re la tive node density-0 .03 -0.02 -0.01 0 0.01 0.02 0.03
Y_w aist [m ]
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Z_w
aist
[m]
Sample contour map of the conversion efficiency projected to the waist plane
The red dots represent individual rays, the blue line is the projection of the
MAST window rim
Ray-tracing
A ray-tracing code is used to determine the radiation temperature from the Rayleigh-Jeans law
The rays with Z~0 propagate deep into the plasma and are absorbed close to the second electron cyclotron harmonic
2fce
LCFS
UHR
3fce
0.8 0.9 1 1.1 1.2R [m ]
-0 .2
-0 .1
0
0.1
0.2
Z [m
]
#7798, t=240m s, f=36.46G Hz
Radiative temperature and EBW absorption
Ray equations describe the motion of EBW packet the evolution equation for the power has to be integrated simultaneously with the raydP/dt=-2(t)P
Non-local reabsorption of the radiation is described by the radiative transfer equationdP/dt=Pwhich must be solved simultaneously with the ray evolution equation
The emitted power can be expressed by the Rayleigh-Jeans law with Trad instead of local temperature TP ~ Trad
whereTrad=
0 t´)T(t´) exp{-
0 t´t´´)dt´´}dt
´
0 1E-007 2E-007 3E-007 4E-007tim e a lo n g th e ra y [s]
-1
0
1
2
3
4
5
0.0x10 0
2.0x10 -6
4.0x10 -6
6.0x10 -6
8.0x10 -6
1.0x10 -5
f / fce
P abs / P inc
N z
ce) / kzvT
#7798, t=240m s, Z eff=1.362, f=36.46G Hz, ray=1
ECE intensity
The intensity of ECE detected by the antenna can be expressed as
where
- Gaussian weight (w0 is the waist radius)
- conversion efficiency
- Rayleigh-Jeans black body radiation law
- power transmission coefficient of the MAST window
- relative visible area (w is the Gaussian beam radius at the plasma surface)
The integration is taken over the intersection of the waist and the projection of the vessel window rim
2dECE Gauss EBW X O rad window relatI const SW C T C Vw- -= ´ òò
( )2 202 /r w
GaussW e-
=
EBW X OC - -
2TwwindowC
2 20/relatV w w=
Polarization effects
Only linearly polarized waves can be detected
The polarization changes at the mirrors and at the MAST window (slight elliptical polarization)
The polarization has a very weak effect on the final ECE spectrum
0 45 90 135 180Plar iza tion o f incden t wave [deg ]
0
1
2
3
4
I EC
E [
ar.
u.]
f=23.14G Hz
f=36.46GHz
IEC E(36.46) / IEC E(23.14)
# 7798, t=240 , dev=12o, long=12o
E R_Cartesian
C hanges of po larization of the e lectric fie ld in the EC E M AST antenna system
Linerly po larized w ave in the horn m outhLinerly po larized w ave after re flection from the second m irrorL inerly po larized w ave after transm ission throgh M AST w indow
E y
EZ
E R_Cartesian
C hanges of po larization of the e lectric fie ld in the EC E M AST antenna system
Linerly po larized w ave in the horn m outhLinerly po larized w ave after re flection from the second m irrorL inerly po larized w ave after transm ission throgh M AST w indow
During the reflection at a mirror the linear polarization is changed:
Eref=-Einc + 2Nm(Einc.Nm)where Nm is the normal of the mirror
We found that the best fit between the measured and simulated ECE can be obtained for a different beam direction to that which follows from antenna adjustment
Possible explanation: beam direction is determined
with precision
diffraction of beam in a rarefied plasma in SOL
magnetic equilibrium differs from that determined by EFIT
Effects of the beam direction
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
f [GH z]
0
1
2
3
Inte
nsi
ty o
f EC
E [a
r. u
.]
E xperim ent
EC E sim ulation, dev=12, long=12
EC E sim ulation, dev=15, long=15
EC E sim ulation, dev=17, long=17
#7798, t=240
ECE from MAST, shot #7798 time 240ms, reference frequency 23.14GHz
Comparison of ECE simulation for EFIT and SCENE equilibria
#7798 L-mode, ECE simulation fits well to detected signal for L-modes, SCENE and EFIT give similar results
Waves with f<23GHz are converted in SOL where plasma density strongly fluctuates and our model of ECE does not catch this situation properly
15 20 25 30 35 40f [G Hz]
0
0.5
1
1.5
Inte
nsity
of E
CE
[ar.
u.]
EC E signal
EC E sim ulation, EFIT , dev=17o , long=17o
EC E sim ulation, SCENE , dev=17o , long=17o
EC E sim ulation, SCENE , dev=18o, long=17o
#7798, t=240m s
The detected ray is inclined at dev from the equatorial plane upward and the angle between its projection onto the equatorial plane and the vertical plane going through the tokamak axis and the antenna position is long
Contour map of square deviation of intensity of simulated ECE from measured values
from #7798 at t=240ms
1 4 1 5 1 6 1 7 1 8 1 9 2 0
1 4
1 5
1 6
1 7
1 8
1 9
2 0
ECE from MAST, shot #4958, t=120ms
With the appropriate beam angles the agreement with the experiment is good for both SCENE and EFIT equilibria but the decreases of ECE at the beginning of the second and the third EC band are not described well by any of the simulations. This is typical for ELMy H-modes.
14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
f [GH z]
0
0.2
0.4
0.6
0.8
1
1.2
Inte
nsi
ty o
f EC
E [a
r. u
.]
m easured EC E signal
EC E s im ula tion w ith EFIT
a t dev=18o , long= 22o
EC E s im ula tion w ith SCENE
at dev=18o , long= 22o
transm ision coeffic ien t through the vessel w indow
Resonance topology for #8694, t=280ms, ELM-free H-mode Radial profiles of characteristic resonances at beam spot
(dev= long=12o) demonstrate clearly the difference in equilibria
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4R [m]
5
10
15
20
25
30
35
40
45
50
55
60
65
70
f [G
Hz]
UHR
Plasm a resonance
4fce
3fce2fce
fce
#8694, t=280
5fce
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4R [m ]
5
10
15
20
25
30
35
40
45
50
55
60
65
70
f [G
Hz]
UHR
Plasm a resonance
4fce
3fce
2fce
fce#8694, t=0.280s
5fce
SCENE*) EFIT
*) SCENE – Simulation of Self-Consistent Equilibria with Neoclassical EffectsH.R. Wilson: SCENE, UKAEA FUS 271 (1974), Culham, Abingdon, UK`
ELM-free H-mode (#8694) simulation based on EFIT equilibrium
• Simulated and detected signal do not require additional beam aiming adjustment (new antenna calibration works well)
• Magnetic field at UHR predicted by EFIT is too low (periodicity of the detected ECE requires fce=11 GHz, but EFIT gives fce<10 GHz)
• Shapes of the peaks in the simulated EFIT signal in higher bands resemble well the detected signal
15 20 25 30 35 40 45 50 55 60f [G H z]
0
0.5
1
1.5
EC
E in
tens
ity [a
r.u.
]
EC E signalEC E sim ulation, EFIT
5 th harm onicband
4 th harm onicband
3 rd harm onicband
2nd harm onicband
ELM-free H-mode (#8694) simulation based on SCENE equilibrium
Surface currents considered in SCENE enhance magnetic field at UHR, but fce=12 GHz is too high
Shapes of peaks of simulated signal do not correspond to the detected ones.
Only four bands do not correspond to five band in detected signal
15 20 25 30 35 40 45 50 55 60f [G H z]
0
0.5
1
1.5
EC
E in
tens
ity [
ar.u
.]
de tected E C E signa lE C E sim ula tion, SCENE
em issionfrom 4fce
em issionfrom 3fce
em ission from 2fce
Ray-tracing can explain the peaks shapes in EFIT simulation
Detailed evolution of central rays was studied for frequencies slightly below and slightly above the plasma surface electron cyclotron harmonics
0 1E-007 2E-007 3E-007 4E-007 5E-007t [m s]
-1
0
1
2
3
4
5
6
f / fce
P abs / P inc
N ||
ce) / k ||vT
ce) / k ||vT
0 4E-008 8E-008 1.2E-007t [ms]
-2
-1
0
1
2
3
4
5
6
Time development of ray, shot #8694, f=20.84 GHz, N=2. ECE is radiated from the 2nd harmonic. N|| strongly oscillates and the absorption is highly non-local. The absorption on the 3rd harmonic is negligible.
f=49.44 GHz, N=4. Even if f<5fce, waves are emitted from the 5th harmonic. Because the factor |(-5ce)/N||vT| decreases faster then |(-4ce)/N||vT|. |N||| increases monotonically and reaches 1 at the absorption region.
Other support of EFIT adequacy follows from and N||(f), CEBW-O-X(f)
and Trad(f)
Value of N|| at full absorption of central EBW ray, #8694, t=280ms, EFIT.
Waves with f slightly below Nfce at the plasma boundary are absorbed with
| N|| |~1, which is 3 times higher than at the boundary (| N|| |~0.36)
Conversion efficiency CEBW-X-O for the central rays do not depend on frequency
15 20 25 30 35 40 45 50 55 60 65
f [G H z]
-1 .5
-1
-0.5
0
0.5
N||
15 20 25 30 35 40 45 50 55 60 65
f [G H z]
0
0.2
0.4
0.6
0.8
CE
BW
-X-O
Frequency dependence of Trad for the central rays #8694, t=280ms, EFIT
15 20 25 30 35 40 45 50 55 60 65
f [G H z]
0
0.5
1
1.5
T rad
Space dependence of characteristic resonances in MASTWaves are emitted from well of the electron cyclotron resonances
#8694, t=0.280s, dev=12o, long=12o. We depicted situation at the end of EBW ray, when |N|||=1 for waves having f slightly bellow Nfce at the plasma boundary and |N|||=0.36 for waves having f slightly above Nfce at the plasma boundary. Broadening of Nfce is given by the factor 1/(1±3N||vT/c)
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4R [m ]
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
f [G
Hz]
0
0.2
0.4
0.6
0.8
1
Te
[k
eV
]
UHR
Plasma resonance
4fce
3fce
2fce
fce
5fce
Comparison of EFIT and SCENE magnetic field profiles in equatorial plane
SCENE predicts the very high paramagnetic effect so the total magnetic field increases to the plasma center too sharply. As a consequence, the shape of simulated ECE peaks do not fit with detected signal.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
R [m ]
-0 .5
0
0.5
1
1.5
2
2.5
[ T
]
#8694, t=0.280s|B vac|
|B tor| EFIT
B to t EFIT
B Z EFIT
|B tor| SC EN E
B to t SC EN E
B Z SC EN E
Comparison of plasma current profiles produced in EFIT and SCENE equlibria
For ELMy H-mode (#4958) slightly different hollow profiles of toroidal current J are produce both by EFIT and also SCENE equlibria
For L- mode (#7798) we obtain the simple parabolic profiles of Jfrom both EFIT and SCENE very good fit)
For ELM-free H-mode we obtain totally different profiles of plasma current (a simple parabolic profile of J for SCENE and for a hollow profile of Jfor EFIT)
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6R [m ]
-0 .5
0
0.5
1
MA
/m2
J
JZJR
Equatorial profiles of plasma current#8694, t=280ms, EFIT
0.2 0.4 0.6 0.8 1 1.2 1.4R [m ]
-2
-1
0
1
2
3
4
MA
/m2
J
JZ
JR
Equatorial profiles of plasma current
#8694, t=280ms, SCENE
CONCLUSIONS
Current theoretical model incorporates nearly all the details of the MASTECE antenna and plasma model based on experimental data.
For L-mode, agreement between calculated and experimental EBWemission is good.
For ELMy H-mode, agreement is good but model does not explainthe smaller signal at lower frequency part within each harmonic bands.
For ELM-free H-mode, simulation based on EFIT equilibrium agreeswith experiment at higher harmonics while using SCENE equilibriumprovides higher magnetic field at the plasma surface and better
agreement at lower harmonics.
These results show that EBW emission can provide an additional constraint for equilibrium reconstruction.
References
• Shevchenko V et al., 15th RF Power in Plasma, Moran, 2003,edit. C. Forest, AIP 694, 359.
• Laqua H.P., et al., review, 15th RF Power in Plasma, Moran, 2003, edit. C. Forest, AIP 694, 15.
• H.R. Wilson: SCENE, UKAEA FUS 271 (1974), Culham, Abingdon, UK• Preinhaelter J. e al, 15th RF Power in Plasma, Moran, 2003, edit. C. Forest, AIP
694, 388.• Preinhaelter J. e al, Review of Sci. Instr. Vol, 75, No 10, Oct. 2004• Urban J.: Adaptive Finite Elements Method for the Solution of the Maxwell
Equations in an Inhomogeneous Magnetized Plasma, Czech. J. Phys 54 (2004) Suppl. C, C109
• Preinhaelter J., Kopecky V., J. Plasma Phys. 10, 1 (1973), part 1.• Pavlo P., Krlin L., Tluchor Z., Nucl. Fusion 31, 711 (1991). • Goldsmith P.: Quasioptical systems: Gaussian Beam Quasioptical Propagation
and Applications, Wiley-IEEE Press (1997)
3D Ray trajectories, # 8694, t=289ms
f=20.84GHz
#8694, t=280m s, f=20.84G H z
#8694, t=280m s, f=49.44G H z
f=49.44GHz
Adaptive method convergence properties Typical error dependence of the global and local error are shown for
/f=0.001 For common precision requirements (0.005-0.001) the method is fast The error estimates correspond with each other
1000 10000 100000
to ta l num ber of nodes
1E-005
0.0001
0.001
0.01
0.1
1
10
rela
tive
err
or
M axim um normL 2 normConversion effic iency error
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
x [m ]
1000
10000
100000
1000000
no
de d
ens
ity
1E-005
0.0001
0.001
0.01
0.1
1
10
rela
tive
err
or
(max
imum
nor
m)
2 nd ite ra tion (n=1050) node density
2 nd ite ra tion (n=1050) loca l error
5 th ite ra tion (n=3101) node density
5 th ite ra tion (n=3101) loca l e rror
10 th ite ra tion (n=13032) node density
10 th ite ra tion (n=13032) local error
Dependence of various global error estimates on the total number of nodes
the error decreases approx. as ~n-4.5
Evolution of local error and node density with mesh refinement
Effects of the beam direction
Strong dependence of the EBW-X-O conversion efficiency can be expected
The transmission coefficient of the power of the O-mode to the X-mode at the plasma resonance depends on the detected beam direction
21/ 2 222
2
2 2exp 1 ( )
4
incincvac ce z ceyopt
p ce z
k NT N
N
1/ , /vac p critk c n dn dx
1/ 2( /( ))optz ce ceN
Effects of the beam direction
-0 .05 -0 .04 -0 .03 -0 .02 -0 .01 0 0.01 0.02 0.03 0.04 0.05
Y_w aist [m ]
-0 .04
-0 .03
-0 .02
-0 .01
0
0.01
0.02
0.03
0.04
Z_
wa
ist
[m]
-0 .05 -0 .04 -0 .03 -0 .02 -0 .01 0 0.01 0.02 0.03 0.04 0.05
Y_w aist [m ]
-0 .04
-0 .03
-0 .02
-0 .01
0
0.01
0.02
0.03
0.04
Z_
wa
ist
[m]
-0 .05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Y _w aist [m ]
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Z_w
aist
[m]
angles of beam direction 12x12 15x15 17x17
#7789, t=240m s
-0 .05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Y _w aist [m ]
-0 .04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Z_w
aist
[m
]
-0 .05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Y_w aist [m ]
-0 .04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Z_
wa
ist [
m]
-0 .05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Y _w aist [m ]
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Z_w
aist
[m]
f=36.46G H z
f=23.14G H z