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