Turbulence near the reverse surface measured by microwave imaging

reflectometry on TPE-RX

Shi Zhongbing(石中兵) 1), Y. Nagayama2), S. Yamaguchi2), Y. Hamada2)Y. Hirano3), S. Kiyama3), H. Koguchi3), H. Sakakita3), K. Yambe3)

Y. kogi4), A. Mase 4)

1)The Graduate University for Advanced Studies, Toki, Japan, 2) National Institute for Fusion Science, Toki, Japan 3) National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan4) KASTEC, Kyushu University, Kasuga, Japan

E-mail: Shi.zhongbing@nifs.ac.jp

2

ωne

r/a

Reflectometry

Microwave Imaging Reflectometry (MIR)

• Motivation: 2D imaging of the fluctuation, just like a photograph.

• MIR is based on the radar technique– Microwave is reflected at the cutoff surface,

we can get the phase difference– The phase fluctuation is dominated by the

density fluctuation close to the cutoff surface

– Local measurement• However, the fluctuation is 2D or 3D

which lead to interference at the detector surface

• The optical lens can restore phase front• MIR uses large-lens optical imaging.

– Experimental verification, TEXROR, LHD,...– Simulations (Princeton PPL and KASTEC)

)( cute rnδδφ ∝

Lead to interference

Optical lensDensity fluctuation

Phase fluctuation

δne δφ

Antenna

The density fluctuation can be restored at the detector surface by

the mirror system

3

TPE-RX

• TPE-RX: R / a = 1.72m / 0.45m. • TPE-RX: Reverse field pinch• RFP device is characteristic of a reverse

toroidal field in the outer region.• Bϕ~Bθ , q<<1.• RFP configuration is obtained as a result

of MHD relaxation and sustained by dynamo-effect.

TEP-RX

•Plasma current : Ip ~ 300 kA•Plasma density:

ne~(0.6-1.0)*1e19 m-3

•Pinch parameter: Q = Bpw/<Bt> ~ 1.5-2

•Reversed parameter:F = Btw/<Bt>~-0.16-1

-0.2

0

0.2

0.4

0.6

0.8

1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

Magnetic field profileand q profile

Mag

netic

fiel

d pr

ofile

q profile

r/a

q=(rBtoro)/(RBpolo)

Bpolo

Btoro

Btoro(1)<0

reverse toroidal field

4

Bϕ Bθ

MIR system on TPE-RX

•O-mode is used (Bθ>>Bϕ, ρ>0.5 )

•The cutoff density: nc=0.5x1019m-3

•-Spatial resolution : 3.7cm,-Time resolution: 1μs / 0.5μs.

4X4 antenna

2D(4X4) antenna array

Optical system

Beam splitter

Main mirror

win

dow

5

A>1: hollow profile

A<1: center peak profile

n e(r

)/ne(

0)

Cutoff radius

The cutoff surface is close to the reverse field surface during PPCD.

The large mode numbers near by the reverse surface.

)1)(1)(0()( 44 Arrnrn ee +−=

Electron density profile is obtained by

A is the profile factor

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8

Cut

off s

urfa

cem=1, n=7n=8

n=9

r

Bt(r

)

prob

e (r

=1.0

5)

n e(1

E19m

-3) PPCD

ω

20GHz

The electron density profile is measured using a dual-chord interferometer and the normalized radius are 0.0 and 0.69, respectively.

The typical plasma density profile is flat or hollow. If ne(r)> nc, the cutoff surface is at the plasma edge

MIR

6

PPCD

Pulsed Poloidal Current Drive(PPCD) plasma

•The soft-X-ray intensity increase rapidly during PPCD•The cutoff radius is about 0.9 •The amplitude of the density fluctuation increases during PPCD

Lissajous’ curve of I-Q signalsIQ circle: fluctuation of the cutoffOne circle: 1.5cm

-0 .6

-0 .4

-0 .2

0

0.2

0.4

0.6

-1 .5 -1 -0 .5 0 0 .5 1 1 .5

52973IQ 25 .504 -25.556m s

Q(a

.u.)

I(a .u .)

(a)

50us window

IQ: sine and cosine components of density fluctuations

7

1 10 100 1000f(kHz)

10-6

10-5

10-4

P(a.

u.)

Plasma

(25-30ms)

No plasma

(40-45ms)

f-3.2

f-1.8f-0

Power spectrum

The spectrum shows three distinct frequency ranges, each with characteristic power dependence

The power spectrum shows the power law decay like f-2 in the frequency range 70-200 kHz and f-3~4 in the frequency of 300-700 kHz.

The decay index (f-a) of the power spectrum represents a qualitative indication of an energy exchange process between fluctuations at different scales

8

1 8 16 23 30k(m-1)

0

50

100

150

200

V(k

ms-1

)

toroidal velocity(SH53330)

At low k (k<16m-1), phase velocity keeps constant (about 50km/s).

The velocity is about 3 times lower than the ion sound speed.

The phase velocity increase rapidly at high k.

The phase shift has linear tread at f<200kHz, which means a constant phase velocity.

But at f>200kHz, the phase almost doesn't change which means the fluctuation has a constant wavenumber

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 00.0

0.5

1.0co

here

nce

0 200 400 600 800 1000f(kHz)

0.0

0.5

1.0

1.5

phas

e(ra

d)

Noise level

9

toroidal velocity (low k)

45

50

55

60

65

260 265 270 275 280 285

V(km/s)

V(k

m/s

)

Ip(kA)

45

50

55

60

65

0.7 0.75 0.8 0.85 0.9 0.95

V(km/s)

V(k

m/s

)

ρ

45

50

55

60

65

0.5 0.6 0.7 0.8 0.9 1 1.1

V(km/s)

V(k

m/s

)

Nea(1019m-3)

Radial scanning by changing plasma density.

The cutoff moves to outside with the increase of density, close to the reverse surface.

The phase velocity sharply decrease at rho~0.9 which suggests a velocity shear around the reverse region.

0

20

40

60

80

0.6 0.7 0.8 0.9 1

|Vϕ|

(km

/s)

ρ

MIR (ne) GPI (Dα)

Doppler (C4+)

10

Phase-frequency spectrum

-1.0 -0.5 0.0 0.5 1.0

phase shift

0

200

400

600

f(kH

z)

0 1 2 3 4

- 1

0

1

2

-1

0

1

2

-1.0 -0.5 0.0 0.5 1.0

phase shift

0

200

400

600

f(kH

z)

0 1 2 3 4

- 1

0

1

2

-1

0

1

2

The fluctuation is dominated by f < 200kHz and k<20/m.

The spectrum shows linearly trend which means a constant phase velocity at low k fluctuation.

It becomes saturated at high k fluctuation.

The fluctuations propagate in the electron drift direction.

Toroidal Poloidal

1 2

3

Bt

BpB

ion

electron

Ip

11

Toroidal coherence length

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15

Coherence length (53330, ρ=0.8)

30kHz80kHz130kHz180kHz230kHz

280kHz330kHz380kHz430kHz480kHz

γ

Δx(cm)

k=5m-1

k=20m-1

The toroidal correlation length is defined as the separation for which the coherence has decreased to 1/e.

The toroidal correlation length has a negative correlation with the wavenumber (k)

The correlation length is about 10-30cm for low k fluctuation.

The correlation length is about 3cm for high k fluctuation.

The fluctuations at different scales have different coherence length, which suggest multi-wave in PPCD plasma.

The correlation length is obtained by averaging cross-correlation over different pairs of channels

12

Maximum Entropy Method (MEM)

Reference: Jae S. Lim and Naveed A. Malik. IEEE trans. on acoustics, speech, and signal proc., Vol. Assp-29, No.3 June, 1981.

Iteratively solve for aij:

The definition of entropy is given as:

Rx is the Fourier transform of the cross-correlation function.

Generally, the detector size is smaller than the correlation length.

It is the similar as a filter used in the cross-correlation matrix.

where, aij is the autoregressive filter coefficients to be estimated.

The correlation matrix is obtained by averaging cross-correlation over different pairs of channels

No. of cross-correlation

13

Difference

γ

x

FFT window

w

k

∝1/w

2D FFT

γ

x

w

k

∝1/w

MEM

Size of image in plasma is limited by narrow port.

coherence does not go to zero

broadening is due to finite beam width

Image is extrapolated using high resolution 2D power spectrum estimation techniques

Max Entropy extends correlation measurements outside measured region

Max Entropy method FFT method

14

power spectrum

0 2 4 6

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4

- 0 . 2

0 . 0

0 . 2

-0.2

0.0

0.2

Real part of cross-correlation Imag. part of cross-correlation

-1.0 -0.5 0.0 0.5 1.0

kx

-1.0

-0.5

0.0

0.5

1.0

k y

0 1 2 3 4

2

4

6

8

1 0

2

4

6

8

10

Traditional FFT method

•The spectrum only shows a broadening pattern by FFT.

•The high k fluctuation is affected by the low k.

•FFT method failed to resolve the spectral peaks

-1.0 -0.5 0.0 0.5 1.0

kx

-1.0

-0.5

0.0

0.5

1.0

k y

0 1 2 3 4

- 5 . 5

- 5 . 0

- 4 . 5

- 4 . 0

- 3 . 5

-5.5

-5.0

-4.5

-4.0

-3.5

Lim & Malik MEM

15

-1.0 -0.5 0.0 0.5 1.0

kx

-1.0

-0.5

0.0

0.5

1.0

k y

0 1 2 3 4

- 6

- 4

-6

-4

Shot, 53330, PPCD, Rcut=0.75-0.9, f=290±20kHz

m=-1±1m=0

m=9±4 n=4±2

n=30±2n=-28±1

n=-95±3

m=-17±1

toroidal

Pol

oida

l

Power spectrum estimated by Lim & Malik MEM

•Measurement ranges:

|k| < 85m-1,

|m|/|n| < 34/146.

•The k is normalized by 0.037/π.

•Multi- modes are obtained by MEM. Note that some of them might be caused by the aliasing or the power is too low.

•The high toroidal k is poloidally elongated due to too less channels in poloidal direction

sidelobe?

16

-1.0 -0.5 0.0 0.5 1.0

kx

-1.0

-0.5

0.0

0.5

1.0

k y

0 1 2 3 4

- 6

- 4

-6

-4

Shot, 53330, PPCD, Rcut=0.75-0.9, f=290±20kHz

m=-1±1m=0

m=9±4

n=4±2

n=30±2n=-28±1

n=-95±3

m=-17±1

toroidal

Pol

oida

l

Inside of reverse surface

Reflective signal can be significantly influenced by the waves away from the cutoff layer, whose amplitude is sufficiently large. the spatial resolution becomes poor.

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8

Cut

off s

urfa

cem=1, n=7n=8

n=9

r

Bt(r

)

prob

e (r

=1.0

5)

n e(1

E19m

-3)

ω

20GHz

Bp

Bt

Outside of reverse surface

17

Fluctuation structure

Skewness (S) and Kurtosis (K):

3~/~

~/~

224

2/323

−><>=<

><>=<

xxK

xxS

The skewness and kurtosis are used to describe the asymmetry and peak degrees of a distribution with respect to its mean value, respectively.

For a Gaussian random distribution, S=K=0.

There is no interaction of ambient turbulence if S=K=0 for all frequency components.

A non-Gaussian tail means the existence of coherent structures or intermittency of ambient turbulence.

-0.3-0.2-0.1

00.10.20.3

00.20.40.60.811.2

0 200 400 600 800 1000

53330, PPCD, 20-33ms

SkewnessKurtosisSk

ewne

ss Kurtosis

f(kHz)

The fluctuations at different frequencies are identified by the band pass filter (BPF)

The fluctuation shows positive bursts at low frequency (f<150kHz) and negative bursts at high frequency (f>150kHz).

The dominant burst frequency is <600kHz

18

0 200 400 600 800 1000

f1(kHz)

-1000

-500

0

500

1000

f 2(kH

z)

0 1 2 3 4

1 . 9 3 × 1 0 - 8

9 . 5 4 × 1 0 - 6

1 . 9 1 × 1 0 - 5

2 . 8 6 × 1 0 - 5

3 . 8 1 × 1 0 - 5

1.93×10-8

9.54×10-6

1.91×10-5

2.86×10-5

3.81×10-5

Three wave coupling

sum of bicoherenceSimilar trends appear in the bicoherence and

kurtosis.

The kurtosis and skewness have reverse trends.

The bicoherence has a broad profile between 200-500kHz, which represents the existence of nonlinear coupling (k~20m-1).

The bicoherence has stronger dependence on the skewness and the kurtosis.

The local peaks of bicoherence are about 300kHz, 400kHz and 800kHz.

The local peak of skewness is about 300kHz while the local peak of kurtosis is about 180kHz.

The different local peaks suggest existence of nonlinear multi-wave interaction.-0.3

-0.2-0.1

00.10.20.3

00.20.40.60.811.2

0 200 400 600 800 1000

53330, PPCD, 20-33ms

SkewnessKurtosisS

kew

ness K

urtosis

f(kHz)

0

0.1

0.2

0.3

0.4

0.5

0 200 400 600 800 1000

b2

100*

b2

f(kHz)

19

Summary

Two-dimensional electron density fluctuation have been measured by microwave imaging reflectometry (MIR) on TPE-RX.

The measurements confirm several important properties of plasma edge turbulence, such as velocity shear, k and intermittency.

The phase velocity is about 50km/s at low k fluctuations, which propagate in the electron drift direction.

The fluctuation structures are quantified by skewness and kurtosis.

The power spectrum is estimated by the Lim & Malik maximum entropy method (MEM), which shows the existence of multi-modes close to reverse region.

The m=0 and m=1 modes has a reverse toroidal propagation direction

The different local peaks of bicoherence, skewness and kurtosis suggest nonlinear multi-wave interactions.

20

Thank you for your attention !

21

Skilling MEM

-1.0 -0.5 0.0 0.5 1.0

kx

-1.0

-0.5

0.0

0.5

1.0

k y

0 1 2 3 4

- 6

- 4

-6

-4Lim & Malik MEM

f~290kHz

Malik MEM has sidelobeproblem and convergence problem. It gives spurious peaks and distorts the spectrum sometimes.

Skilling MEM shows a continuous and a more reliable spectrum. But it fails to solvethe low k modes.

Small difference in the toroidal direction, but large difference in the poloidaldirection.

22

Quasi-single helicity (QSH) plasma

Ns = 2 at 35-47ms. The dominate mode is m=1,

n=6 during QSH stateThe cutoff radius is about 0.75

during QSH

00.0050.01

0.0150.02

0.0250.03 nAmp m1/ 5

nAmp m1/ 6nAmp m1/ 7nAmp m1/ 8nAmp m1/ 9

nAm

p m

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Nea(r=0)Nea(r=0.69)

rc/a

0.01 0.02 0.03 0.04 0.05 0.06

n e(1019

m-3

)

rc /a

time (s)

12345678 Shot 52754

NsBtm1 6-15NsBtm0 1-10

Ns

12

' '

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

∑= n

n n

ns W

WN

The quality of the QSH state can be estimated by

where, Wn is the energy of the (m=0 or 1) mode.

Ns=1 SH stateNs<3 QSH state Ns>3 MH state

MH QSH

23

Fluctuation structure (QSH)

0

10

20

30

40

0

10

20

30

40

0 100 200 300 400 500

K_QSH

K_MH

K_Q

SH K

_MH

f(kHz)

-2

-1.5

-1

-0.5

0

-2

-1.5

-1

-0.5

0

0 100 200 300 400 500

S_QSH

S_MH

S_Q

SH S

_MH

f(kHz)

0 2 0 4 0 6 0

0 .7 00 .8 00 .9 01 .0 0

QSHMH

3 0 .0 3 0 .5 3 1 .0 3 1 .5t (m s)

0 .60 .81 .0

MIR

(a.u

.)

MH4 0 .0 4 0 .5 4 1 .0 4 1 .5

t (m s)

0 .60 .81 .0

MIR

(a.u

.)

QSH

The fluctuations versus frequency are identified by the band pass filter

QSH shows a more intermittent behavior (around 200-400kHz)