NONLINEAR EFFECTS OF A ROTATING MAGNETIC FIELD ERROR PERTURBATION

PLP1192

November 1996

Poster presented at the 38th Annual Meeting

APS Division of Plasma Physics 11-15 November 1996

Denver, Colorado

K.A. Mirus J.e. S prott

Department of Physics University of Wisconsin-Madison

Madison, WI 53706

These PLP Reports are informal and preliminary and as such may contain errors not yet eliminated. They are for private circulation only and are not to be further transmitted without consent of the authors and major professor.

01

Abstract Submitted for the DPP96 Meeting of

The American Physical Society

Sorting Category: 5.1.3 (experimental)

Nonlinear Effects of a Rotating Magnetic Field Error

Perturbation t K.A. MIRUS, J.C. SPROTT, University of Wisconsin­Madison - Periodic perturbations applied to chaotic systems have been illustrated both numerically and experimentally to control the systems' dynamics or change their dimensionality. Such a perturbation has been applied as a rotating n=6 radial magnetic field error to the toroidal gap of the MST reversed-field pinch. Low power experiments done at fixed frequencies of 11 and 22 kHz show a shift ii�' the peak of the power spec­trum of edge magnetic fluctuations, but do not seem to lower their im­measurably high dimension. However, there is sorrie evidence of stochas­tic resonance (the amplification of a weak periodic signal applied to a

nonlinear system in the presence of an optimal amount of noise). Future experiments using greater perturbation amplitudes and different pertur­bation frequencies should provide clearer results.

tThis work was supported by the USDOE.

D Prefer Oral Session [!] Prefer Poster Session

Kevin A. Mirus mirus@juno�physics.wisc.edu

University of Wisconsin-Madison

I Special instructions: Please place in the MST reversed-field pinch poster grouping.

Date submitted: July 10, 1996 Electronic form version 1.1

02

Revised Abstract

Periodic perturbations applied to chaotic systems have been

illustrated both numerically and experimentally to control the

systems' dynamics or change their dimensionality. Such a

perturbation has been applied as a rotating n=6 and n= 1 radial

magnetic field error to the toroidal gap of the MST reversed field

pinch. Low power experiments done at fixed frequencies of 11 and

22 kHz show a shift in the peak of the power spectrum of edge

magnetic fluctuations, but do not seem to lower their immeasurably

high dimension. However, there is some evidence of stochastic

resonance (the amplification of a weak periodic signal applied to a

nonlinear system in the presence of an optimal amount of noise).

Future experiments using greater perturbation amplitudes and

different perturbation frequencies should provide clearer results.

This work was supported by the U.S. Department of Energy.

03

Outline

I. Motivation

II. Experimental Apparatus

I II. General Effects on Mode Rotation and Locking

IV. Nonlinear Effects I: Stochastic Resonance

V . Nonlinear Effects II: Correlation Dimension

VI. Summary

VII. Future Work

I. Motivation

• Motivation from a standard mode rotation point of view:

• Rotating field errors are known to help prevent locking through entrainment of resonant islands to the rotating perturbation. The perturbation essentially serves as an asynchronous induction motor.

• The m= 1, n=6 mode is a dominant mode in the MST. Thus, an n=6 perturbation was initially chosen to be applied.

• An m=O, n= 1 mode is dominant at the reversal surface, so an n= 1 perturbation has also been applied.

• Motivation from a nonlinear dynamics point of view:

• Control of some simple chaotic systems with small system perturbations has been established.'

• The " induction motor" provides a good starting point to search for similar effects in the MST .

05

II. Experimental Apparatus

• The n=6 and n= 1 induction motors each consist of two sets of coils which thread the toroidal gap of MST in appropriate locations to yield the desired mode structure.

• The two coil sets for each induction motor are driven in quadrature with resonant RLC circuits to give a rotating field error (see poster 2S.1 0 for further details) .

• Due to present circuit limitations, no more than -80A P-P has been driven in the induction motors.

06

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I I I. General Effe..cts Qn Mode Rotation and Locking

• The n=6 induction motor see m s to increase the rot ation frequency of the m =l, n=5 ,6,7,8 modes a bit, but it does not increase the n=6 mode amplitude.

• The n= 1 induction motor does not seem to have any effect on the usually locked m=O, n=l mode.

• For a couple ano m alous shots, the pl asma lo cked or unlocked with the application of a 1 0 msec gated n=6 perturbation.

• Su m mary of the overal l occurance of locked shots with the induction motors:

number of pol es

. In

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IV. Nonline�r Effects I: StochastIc Resonance

• Stochastic resonance is of g en erating "coheren t dynamical syst em in the optimum amount of noise.

th e ph enomenon motion" in a

presence of an

• This coherent motion is generally se en in periodically forc ed syst ems with bistable sta t e s or fixed -po int to l i mi t-cycle bifurcations. WJllr..J �

I J I( • Signatures of stochastic resonance are

peaks in th e power spectra of fluctuating quantiti es at the drive frequency and its harmonics, and a peaking in the signal-to­noise ratio (SNR) at an optimum amount of fluctuation noise.

• Stochastic Resonance has been observed in weakly ionized rf plasmas by L. I and J.­M. LiuI:

1 L. I and J.-M. Liu, Phys. Rev. Lett. 74, 3161 (1995).

15

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FIG. 5. The SNR vs V""s at points A. B. and C.

FIG. 3. The' power spectra and time evolutions of probe current at pointB <Yo �,?5;? �YL�ith.�ifferent VIIIIS'

• Power spectra of fluctuations in poloidal field p ick-up co ils at the wall of MST sometimes revealed peaks at t he induction motor driving frequency and its harmonics. Thus, it was supposed that the fluctuation no ise may have been enhanc ing the induction motor drive.

• However, when the SNR for these peaks were plotted against the RMS values of the fl uctuations (Le. the noise), the standard c urve characteristic of stoc hastic resonance was not seen.

16

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STOCHASTIC RESONANCE STUDIES FOR 14-JUL-1996

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v . Nonline�r E{fects. II: CorrelatIon DImenSIon

• The correlation dimension is used to calculate the fractal dimension of chaotic systems. It is the probability that any two points on the trajectory of the system will occupy the same "hypersphere", an d is defined by: I ( n. .L. \' e( . , )) D n (). .(IW\. N� L 1""-\(;-.Al c... :: f.\'fY\ f\I"dO i,i r ... o �r

• C. Watts has claimed that the fractal dimension of MST is immeasurably high.2

• Numerical work suggests that simple per iod ic pertur bations to even h igh-dimensional systems can significantly lower their dimension.

• Such a decrease in the dimensionality of fluctuations in the pickup coil data was not observed f or any of the per tur bation s applied with the induction motor.

2 C. Watts, PhD. Thesis, University of Wisconsin-Madison, 1993.

19

D:'WWD�UTIL'24JU"_5.DAT

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Co�ponent delay = 1 Vector delay = 1 XMax = 54.853388

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-25.81 -15.8 log2 r

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CPU time = 8.165

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VI. Summary

• The rotating field · errors applied to MST by the n=6 and n= 1 induction motors show signs of affecting the plasma mode rotation and locking, but currently lack the power to make a definitive effect.

• MST plasmas do not seem to exhibit stochastic resonance, but a wider range of plasma parameters and driving amplitudes have yet to be explored.

• The perturbations applied so far have not had the effect of decreasing the overall dimensionality of the system.

VII. Future Work

• Increase the power to the n=6 and n= 1 field coils.

• Look for evidence of stochastic resonance under a broader range of plasma parameters and stronger driving (e.g. , the Cold Biased Probe, or plasma guns).

• Examine a broader range of perturbation frequencies and amplitudes in a search to decrease the plasma dimensionality.

23