Nonlinear phenomena of edge fluctuations in RF range during high harmonic fast wave

heating experiments in the TST-2 spherical tokamakY. Nagashima, T. Oosako, Y. Takase, A. Ejiri, O. Watanabe,

T. Yamaguchi, H. Kobayashi, H. Kurashina, K. Yamada, B.I. An, H. Hayashi, H. Kakuda, T. Sakamoto, K. Hanashima, J. Hiratsuka,

and T. Wakatsuki

2010 US-Japan RF WorkShop,

March 8-10, 2010 General Atomics; La Jolla, San Diego, CA, USA

The University of Tokyo

Content1. Importance of nonlinear analysis of

fluctuations in RF range

2. Introduction to nonlinear spectral analysis: bispectral analysis

3. Part 1: Identical spectra of parametric decay instability during RF pump wave injection

4. Part 2: Fluctuation of the toroidal cross-phase of the pump wave

5. Summary

Direct observation of fluctuations in RF range

High Harmonic fast wave (21 MHz) heating experiments have been performed on TST-2.

Direct observations of fluctuations in RF range suggest existence of wide variety of nonlinear phenomena of RF waves.

We investigated the nonlinear phenomena by the use of linear/nonlinear spectral analysis.

TST-2 spherical tokamak and RF antenna

Quoted from Y. Takase, et al., IAEA FEC 2008

Conditions of the experiment

Quoted from Y. Takase, et al., IAEA FEC 2008

Introduction to nonlinear spectral analysis: bispectral analysis

Bispectral analysisBispectrum: An indicator of mode coupling among different three modes where coupling conditions (ω1+ω2=ω3) are satisfied.

Auto-bispectrum: coupling among the same physical quantitiesCross-bispectrum: coupling among different physical quantities

( ) ( ) ( ) ( )2121*

21, ωωωωωω ZYXBxyz +=

( ) ( ) ( )ωωω ZYX , , are Fourier components with angular frequency ω

Bicoherence and biphaseBicoherence: relative intensity of nonlinear coupling (0~1)

Biphase: complex phase angle of bispectrum

( )( )

( ) ( ) ( ) 221

221

2

2121

2,

,ωωωω

ωωωω

ZYX

Bb

xyzxyz

+=

( ) ( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛=Θ −

21

21121 ,Re

,Imtan,

ωωωω

ωωxyz

xyzxyz B

B

Physical meaning of bispectrum

Third mode is driven by the beat αiefYfYfY −∝)()()( 3*

21

0)()()( 3*

21 →fYfYfYThird mode is independent of the beat

( ) ( ) ( )( ) ( )( ) ( ) ( )

( ) ( ) ( )

α

αθθ

θθ

αθθθθi

i

ii

eAA

efYfYA

fYfYAfY

efYefY

fYfYfY

|| and

, ,random ,||||||

where,

||||

33

21dp3,do3,

213

2133dp

3dp3do

3product by thedriven 3other termby driven 3

21

dp3,do3,

=

++===

=

+=

+=

++

Driven by quadratic nonlinear terms of observables

Driven by terms independent of observables

Bispectral analysis clarifies contribution of quadratic nonlinearity to mode components

Feynman diagram for resonant mode coupling: M. Porkolab and R.P.H. Chang, Rev. Mod. Phys. 50 (1978) 745

Bispectral analysis oｆ test data

Bicoherence is significant when phase relationship among three-waves are constant.

Part 1: Observation of identical

spectra of parametric decay instability

Parametric Decay Instability (PDI) in the TST-2 spherical tokamak (T. Oosako, et al., NF 2009, HHFW(21 MHz), HHFW or IBW, ICQM as candidate）

Combination of three oscillations of the PDI1. Pump wave2. Lower Side-Band (LSB) wave3. Ion Cyclotron Oscillation (ICO)

Identical spectra (three spectral peaks) are observed (bicoherence is also significant) INSIDE and OUTSIDE plasma(T. Yamada, et al., RSI 2007)

Why are the spectra IDENTICAL?

Identical spectra of parametric decay instability

Reflectometry(inside)

Magnetic probe(outside, also edge Langmuir probe)

What are candidate physics behind the observation?

1. PDIs occur at many locations.(process 1)

2. The three oscillations of the PDI propagate identically with constant phase relationship.(process 2, suggested by significant bicoherence)

3. Beat oscillation is generated by pump and LSB waves, and then we observe identical spectra.(process 3)

Data analysis approach1. Frequency of spectral peak (relationship between ICO frequency and local ion cyclotron frequency)---> Discriminating local PDI from other processes

2. Radial profile of correlation between movable and fixed probes---> Testing conservation of wave phase during radial propagation

3. Dependence of auto powers on product of two other powers (new approach)---> Discrimination of beat oscillation from independent (resonant mode) oscillation

Langmuir probe

Radial correlation is measureable with a fixed Bφprobe and a movable electrostatic probe (Φf).

Bφ probes (radially fixed)

Bθ probes (fixed)

Electrostatic probe (radially movable)

Profile of power spectra

Auto-bicoherence of potential

Profile of radial correlation

Δf vs local ion cyclotron frequency

USB：lower sideband

Potential power inside the limiter is larger than that outside the limiter.Pump and ICO lose correlation beyond the limiter.---> Processes 1 and 2 are unlikely to occur in this observation.

Power spectra and radial correlation

Δf=fpump -fLSB

fion cyclotron

Independent and beat components

The nonlinear term of potential fluctuation can be also represented by product of potential.

Independent (resonant mode, etc…)

Lorentz force Ponderomotive force

Assuming , we can express potential fluctuation as

( )∑ ΦΦΛ+×Λ+Φ=Φ2,1

21ivePonderomot21Lorentzmode,33 Bj

Φ∝B

212,1mode,33 ΦΦ+Φ=Φ Ν

The assumption is valid experimentally.

pump LSB

Origin of potential fluctuation

Bispectral power analysis

We can test whether the fluctuation has independent (resonant mode) components or not.

*3212,1

*3mode,3

23 ΦΦΦ+ΦΦ=Φ Ν

Comparing auto-power and bispectral power

mode,3ΦWithout , the comparison forms straight line from the origin.

*321 ΦΦΦ

23Φ

*3212,1

23 ΦΦΦ=Φ Ν

*321 ΦΦΦ

23Φ

*3212,1

*3mode,3

23 ΦΦΦ+ΦΦ=Φ Ν

2mode ,3

23 Φ=Φ

0*321 →ΦΦΦ

Case 1 Case 2 Case 3

The comparison makes no sense.

offset2

mode,3~ Φ

Bispectral power analysis without ensemble average

Plot of ICO looks like straight lines from the origin while plots of pump and LSB show scatter.

Data in SOL plasmaBiphase (similar nonlinear process) ICO

SUBMITTED

pump LSB

pumpLSBICO

Bispectral power analysis with 100 ensemble average

This observation supports that ICO is dominantly driven by the beat of the pump and LSB.

ICO (red) is likely to have linear relationship from the origin.

Biphase (similar nonlinear process)

SUBMITTED

*321 ΦΦΦ

Discussion part 1Spectral peak frequency of ICO is significantly different from local

ion cyclotron frequency.--->ICO is not local ion cyclotron quasi-mode. (rejecting process 1)

Pump wave and ICO lose radial correlation beyond the limiter, while phase information of LSB wave conserves during radial propagation.--->Three oscillations do not propagate in the radial direction with constant phase relationship. (rejecting process 2)

By use of bispectral power analysis, ICO looks like a beat oscillation, while pump and LSB waves may have resonant mode components. --->Pump and LSB waves can propagate in plasmas, while ICO is purely a beat oscillation. (accepting process 3)

Part 2: Origin of Fluctuations in toroidal cross-phase of

RF pump wave

Observation of fluctuating toroidal cross-phase (N//) of RF

pump waveThe N// of RF pump wave is an important parameter to determine accessibility of the pump wave into plasmas.

After launching the pump wave to plasmas, the pump wave could be scattered by low-frequency fluctuation of the plasmas, and N// of the pump wave may be changed [M. ONO, POF].

We observed magnetic fluctuations in RF range just outside the separatrix to investigate interaction between the pump wave and low-frequency (10-100kHz) fluctuation.

Two neighboring MPs

inner wallprobes

HHFWAntenna

8

91011

12

1

2

34 5

6

7

Magnetic Probes (MP)

Arrangement of the MP

Two toroidal magnetic pick up probes (MP 1-12-1, MP 1-12-2) arranged in the toroidal direction (R=0.635 m)

Time evolution of toroidal cross-phase of the pump wave

Toroidal cross-phase is not stationary.

Short time FFT (no ensemble) conditionΔf=250 kHz, Δt=4 μs

Time [ms]

RF injectiontime = 1 ms

Pump wave power (21 MHz)

Toroidal cross-phase [rad/(2π)]

Details of power spectrumHorizontal axis [MHz]

Power spectrum

Low-frequency fluctuation（10-300 kHz）

21MHｚpump

detail

pumpPDI lower-sideband

More detail (linear plot)

More detail (log plot)

pump ”SIDEBAND” by low-frequency fluctuation?

Significant asymmetry of spectrum around the pump wave frequency was observed.

PDI Upper sideband

Toroidal cross-phase in low-frequency part of spectrum

Frequency [Hz]

Toroidal cross-phase in low-frequency (10-300 kHz) ~ 0.1-0.3Corresponds to n~7-20

Power spectrum

Squared coherence

Toroidal cross-phase [rad/(2π)]

104 107

Details of the cross-phase

The cross-phase of “SIDEBAND” is shifted relative to that of the pump wave.

Auto-power (MP 1-12-1)Horizontal unit: Frequency [MHz]

Squared coherence

The pump wave with toroidal mode n~10 is launched from the antenna strap.

Observed cross-phase of the pump wave|n|~9.4-12.2

Good agreement within experimental error

Auto-power (MP 1-12-2)

Cross-phase（rad/(2π)）

21 MHz

Phase of the pump wave（-0.15±0.02）

“SIDEBAND”

Auto-bicoherence of the MP

Nonlinear couplings in higher-frequency region are stronger than those in lower-frequency region.

Horizontal: Hz

Higher-frequency region

0.1

0.0

Significance level ~ 0.0017

Lower-frequency region

Discussion part 2The cross-phase of the pump wave fluctuates temporally.

Cross-phase analysis with fine frequency resolution shows that toroidal cross-phase of the pump wave is consistent with k//determined by (0, π) phasing at the RF antenna.

The cross-phase of the “SIDEBAND” looks like summation of those of the pump wave and the low-frequency (10-300 kHz) fluctuation. The “SIDEBAND” has significant nonlinear couplings with each other.

Results of the linear/nonlinear analyses does not contradict that the RF wave around pump frequency is anisotropicallyscattered by low-frequency (10-300 kHz) fluctuations (ωRF+ω10-300kHz=ω”SIDEBAND”, kRF+k10-300kHz=k”SIDEBAND”).

The scatter of may affect accessibility of RF wave significantly.

Future directionIt is very difficult to measure RF fluctuation correctly (attenuation, phase delay, nonlinearity of signal response, etc… in electric circuit of diagnostics).

However, these kinds of linear/nonlinear spectral analyses may show many useful information to understand wave propagation/dynamics.

Under CAREFUL INTERPRETATION of analyzed data, we can extract wave physics from large amount of data.

For future direction, higher order nonlinearity should be considered (four wave couplings, frequency shift, etc.).

SummaryWe have investigated nonlinear phenomena of fluctuations in RF range during high harmonic fast wave heating experiments on TST-2.

Linear/nonlinear analyses including bispectral analysis presented important perceptions about wave dynamics in RF range.

By using bispectral power analysis, identical PDI spectra is understood by a picture that propagating pump and LSB waves nonlinearly excite low-frequency beat oscillation in SOL plasma.

Analyses of the cross-phase and auto-bicoherence suggest that the RF wave is anisotropically nonlinearly coupled to low-frequency (10-300 kHz) fluctuations, and N// of the RF wave may be affected.

The analyses may provide a new vision for prediction of RF wave propagation in fusion plasmas.