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Detection of an ion cyclotron instability
Volume 112A, number 5
PHYSICS LETTERS
28 October 1985
D E T E C T I O N OF AN I O N C Y C L O T R O N INSTABILITY
G. LOTHEN and H. SCHLOTER Institut ff4r Experimentalphysik I1, Ruhr-Universiti~t, Bochum, West Germany Received 5 August 1985; accepted for publication 22 August 1985
In the density gradient of adc discharge unstable modes are observed at frequencies 0 < ¢0 < 360ci. Spectral analysis of two probe signals, monitoring the plasma density fluctuations, identifies the modes as an ion cyclotron drift instability and shows active three-wave coupling between several modes of the spectrum.
Mikhailovskii and Timofeev [ 1] were the first to predict unstable waves near the ion cyclotron harmonics. These are driven by the free energy contained in the density gradient. Yamada et al. [2] have observed these ion cyclotron drift waves in the density gradient of an ion beam injected into a Q-machine plasma. Cohen et al. [3] investigated the nonlinear evolution of the ion cyclotron drift waves using a computer simulation, however they did not consider three-wave coupling. There are linear unstable modes near the intersection point of the ion cyclotron (Bemstein) waves 60 ~ n60ci{1 + [(2rr)l/2(1 + Ti/Te)k±Pi]-1} (kip i >~ 1)
(1)
and ion drift waves in an inhomogeneous plasma 60~ 60~(1 + k2h2i) -1
(60~=k±VDi).
(2)
This instability can be understood by considering the coupling between the negative energy ion cyclotron wave and the positiv energy ion drift wave [4]. Instability occurs provided that the ion diamagnetic drift velocity
VDi= (k ri/eB ) Vn/n
(3)
exceeds the perpendicular phase velocity of the ion cyclotron wave. The investigations have been performed using a neon target plasma (positive column of a de discharge with hot cathode; length 120 cm, diameter 10 cm). 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The main plasma parameters have been determined by movable probes [5], n e ~< 3 X 10 16 m -3,
Vn,Jn,, <~0.4 cm -1 ,
6ne/ne<~0.3, T e ~ 1 0 e V , Ti~<0.3eV, B~<36mT, Po ~ 0 . 2 P a . The spectral analysis of two probe signals ~(x, t) monitoring the plasma density fluctuations ~n e [6,7] yields the dispersion of the unstable modes. Truncated time series taken from the probe signals (1-2 ms length) have been digitized and stored by a transient recorder and then transferred to a LSI 1 1/2 microcomputer. The spectra ¢(x, 60) of the probe signals are obtained via a fast Fourier transform (FFT) algorithm. A normalized spectral density has been obtained from the spectra of the probe signals,
~(~, k) =
s ( ~ , k) 0.5 [P11(60) + P22(60)1 '
(4)
where S(60, k) is the spectral density and Pl1(60), P22(60) are the autopowerspectra. The first and second moments of the distribution s'(60, k) are the mean wavenumber/¢(60) and the variance o/~(w) which indicates turbulent broadening of the wave dispersion in k-space. The bispectrum B(601, 6o2) = ( ~ ( x , 601)~(x, 602)~(x, 603)),
(5)
with 601 + 602 + 603 = 0, provides a strong indication for nonlinear three-wave coupling [8]. Fig. 1 shows a typical density fluctuation spectrum. 227
Volume 112A, number 5
PHYSICS LETTERS
, rei, Ampl
number (k r ~ k~). The parallel phase velocity of the waves are beyond wave-particle resonances (kll Vi < co < kllOe). The theoretical dispersion for ion drift waves [eq. (1)] and Bernstein waves [eq. (2)] are plotted in fig. 2 for additional information using the experimental determined parameters
20dB"
lb- 2'0
30
i
4'0
5'0
}
6'0
gO
70
5
f/kHz
Vne/n e ~ 0.3 c m - 1 ,
f/f:
Fig. 1. Spectrum of the low frequency ion density fluctuations. There are modes with frequencies below and above the ion cyclotron frequency (~ci/27r = 20 kHz). Fig. 2 shows the azimuthal wavenumber and the standard deviation of the wavenumber calculated from the normalized spectral density g(co, k) [eq. (4)], which gives a statistical dispersion of the modes. An estimate of the correlation length l c from the standard deviation
l c ~ 2rr/o k yields l c > ] ~ 1 for the large amplitude modes and l c < / ~ ! otherwise. The parallel wavenumber impressed by the machine length is very small compared with the azimuthal wavenumbe r (k II ~ 0.15 c m - 1). The same result is obtained measuring the radial wave-
18
28 October 1985
k T i = 0.3 eV.
The low frequency modes (w < Wci) are harmonics of ion drift waves. The azimuthal phase velocity is in the direction of the ion diamagnetic drift. The frequencies are inversely proportional to the magnetic field (co = B - l ) . The measured dispersion fits the theoretical ion drift dispersion within the experimental errors, except for the most unstable (largest amplitude) modes, the phase velocity of which is two times smaller than the ion diamagnetic drift velocity. This is observed in other experiments, too, and is typical for the nonlinear stage of the modes [7,9,10]. High frequency modes ( ~ > 26%0 are unstable near the intersection point of the ion drift dispersion and the second harmonic of the Bernstein waves. The measured dispersion is in agreement with the theoretical predictions [2,3] which give the most unstable (highest growth rate) ion cyclotron waves for k i p i 10. Thus fig. 2 is a direct verification of an ion cyclo-
k~pi
12
6
/l
'
'
J
ltltl 'L,ll,
o ...... ,JJlJ,ljJllll,,,,IlI, ,l,j,l
,t,, L Jl JllJ,IJ
-12
-18
5
'
1~i
'
i
25
3~5
/*'5
'
5'5
'
65
2
Fig. 2. Azimuthal wavenumber and standard deviation ~( versus frequency; - © - ion drift dispersion; ~ 228
'f/kH:z
% ion cyclotron dispersion.
Volume 112A, number 5
2,1. ~
PHYSICS LETTERS
28 October 1985
-1
16.0
s.aa
I
Fig. 3. Contour plot of the bispectrum (real part);fe i -- 20 kHz. tron drift instability driven by the free energy contained in the density gradient. The saturation of the instability may be due to three-wave coupling between the unstable modes and damped modes. This is evidenced by the existence of sideband modes (usually asymmetric) in the spectrum and is confirmed by the bispectrum (fig. 3). The bispectrum is a measure o f the coherence between three waves which fulfill the frequency selection rule co 1 + co2 + co3 = O. Fig. 2 implies that the wavenumber selection rule is fulfilled, too, The bispectrum o f fig. 3 shows coupling (dark regions) of the most unstable modes ( f = 5 kHz and f - - 45 kHz) with all modes o f the spectrum. This implies that energy transport via three-wave coupling from unstable to damped modes stabilizes the instability. This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 162 Plasmaphysik Bochum/Ji~lich).
References [ 1] A.B. Mikhailovskii and A.V. Timofeev, Soy. Phys, JETP 17 (1963) 626. [21 M. Yamada, S. Sefler, H,W. Hendel and H. Ikezi, Phys. Fluids 20 (1977) 450. [3] B.I. Cohen, N. Maron and G.R. Smith, Phys. Fluids 25 (1982) 821. [4] C.S. Lui and D.K. Bhadra, Phys. Fluids 15 (1972) 2288. [5 ] K. Krauee, G. Liithen and H. Schliiter, Plasma Phys. 22 (1980) 1053. [6] .J.M. Beall, Y.C. Kim and E.J. Powers, J, AppL Phy~ 53 (1982) 3933. [7] L. Schraitz, G. Li~then, G. Derra, G, Derra, G. B6hm and H. S ehliiter, Plasma Phy~, to be published. [8] Y.C. Kim and E.J. Powers, IEEE Trang Plasma Sci PS-7 (1979) 120, [9] H.W. Hendel, B. Coppi, F, Perkins and P.A. Politzer, Phyg Rev. Lett. 18 (1967) 439. [10] M.P. Evrard, A.M. Messiaen, P.E. Vandenplas and G. van Oost, Plasma Phys. 21 (1979) 999.
229
PHYSICS LETTERS
28 October 1985
D E T E C T I O N OF AN I O N C Y C L O T R O N INSTABILITY
G. LOTHEN and H. SCHLOTER Institut ff4r Experimentalphysik I1, Ruhr-Universiti~t, Bochum, West Germany Received 5 August 1985; accepted for publication 22 August 1985
In the density gradient of adc discharge unstable modes are observed at frequencies 0 < ¢0 < 360ci. Spectral analysis of two probe signals, monitoring the plasma density fluctuations, identifies the modes as an ion cyclotron drift instability and shows active three-wave coupling between several modes of the spectrum.
Mikhailovskii and Timofeev [ 1] were the first to predict unstable waves near the ion cyclotron harmonics. These are driven by the free energy contained in the density gradient. Yamada et al. [2] have observed these ion cyclotron drift waves in the density gradient of an ion beam injected into a Q-machine plasma. Cohen et al. [3] investigated the nonlinear evolution of the ion cyclotron drift waves using a computer simulation, however they did not consider three-wave coupling. There are linear unstable modes near the intersection point of the ion cyclotron (Bemstein) waves 60 ~ n60ci{1 + [(2rr)l/2(1 + Ti/Te)k±Pi]-1} (kip i >~ 1)
(1)
and ion drift waves in an inhomogeneous plasma 60~ 60~(1 + k2h2i) -1
(60~=k±VDi).
(2)
This instability can be understood by considering the coupling between the negative energy ion cyclotron wave and the positiv energy ion drift wave [4]. Instability occurs provided that the ion diamagnetic drift velocity
VDi= (k ri/eB ) Vn/n
(3)
exceeds the perpendicular phase velocity of the ion cyclotron wave. The investigations have been performed using a neon target plasma (positive column of a de discharge with hot cathode; length 120 cm, diameter 10 cm). 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The main plasma parameters have been determined by movable probes [5], n e ~< 3 X 10 16 m -3,
Vn,Jn,, <~0.4 cm -1 ,
6ne/ne<~0.3, T e ~ 1 0 e V , Ti~<0.3eV, B~<36mT, Po ~ 0 . 2 P a . The spectral analysis of two probe signals ~(x, t) monitoring the plasma density fluctuations ~n e [6,7] yields the dispersion of the unstable modes. Truncated time series taken from the probe signals (1-2 ms length) have been digitized and stored by a transient recorder and then transferred to a LSI 1 1/2 microcomputer. The spectra ¢(x, 60) of the probe signals are obtained via a fast Fourier transform (FFT) algorithm. A normalized spectral density has been obtained from the spectra of the probe signals,
~(~, k) =
s ( ~ , k) 0.5 [P11(60) + P22(60)1 '
(4)
where S(60, k) is the spectral density and Pl1(60), P22(60) are the autopowerspectra. The first and second moments of the distribution s'(60, k) are the mean wavenumber/¢(60) and the variance o/~(w) which indicates turbulent broadening of the wave dispersion in k-space. The bispectrum B(601, 6o2) = ( ~ ( x , 601)~(x, 602)~(x, 603)),
(5)
with 601 + 602 + 603 = 0, provides a strong indication for nonlinear three-wave coupling [8]. Fig. 1 shows a typical density fluctuation spectrum. 227
Volume 112A, number 5
PHYSICS LETTERS
, rei, Ampl
number (k r ~ k~). The parallel phase velocity of the waves are beyond wave-particle resonances (kll Vi < co < kllOe). The theoretical dispersion for ion drift waves [eq. (1)] and Bernstein waves [eq. (2)] are plotted in fig. 2 for additional information using the experimental determined parameters
20dB"
lb- 2'0
30
i
4'0
5'0
}
6'0
gO
70
5
f/kHz
Vne/n e ~ 0.3 c m - 1 ,
f/f:
Fig. 1. Spectrum of the low frequency ion density fluctuations. There are modes with frequencies below and above the ion cyclotron frequency (~ci/27r = 20 kHz). Fig. 2 shows the azimuthal wavenumber and the standard deviation of the wavenumber calculated from the normalized spectral density g(co, k) [eq. (4)], which gives a statistical dispersion of the modes. An estimate of the correlation length l c from the standard deviation
l c ~ 2rr/o k yields l c > ] ~ 1 for the large amplitude modes and l c < / ~ ! otherwise. The parallel wavenumber impressed by the machine length is very small compared with the azimuthal wavenumbe r (k II ~ 0.15 c m - 1). The same result is obtained measuring the radial wave-
18
28 October 1985
k T i = 0.3 eV.
The low frequency modes (w < Wci) are harmonics of ion drift waves. The azimuthal phase velocity is in the direction of the ion diamagnetic drift. The frequencies are inversely proportional to the magnetic field (co = B - l ) . The measured dispersion fits the theoretical ion drift dispersion within the experimental errors, except for the most unstable (largest amplitude) modes, the phase velocity of which is two times smaller than the ion diamagnetic drift velocity. This is observed in other experiments, too, and is typical for the nonlinear stage of the modes [7,9,10]. High frequency modes ( ~ > 26%0 are unstable near the intersection point of the ion drift dispersion and the second harmonic of the Bernstein waves. The measured dispersion is in agreement with the theoretical predictions [2,3] which give the most unstable (highest growth rate) ion cyclotron waves for k i p i 10. Thus fig. 2 is a direct verification of an ion cyclo-
k~pi
12
6
/l
'
'
J
ltltl 'L,ll,
o ...... ,JJlJ,ljJllll,,,,IlI, ,l,j,l
,t,, L Jl JllJ,IJ
-12
-18
5
'
1~i
'
i
25
3~5
/*'5
'
5'5
'
65
2
Fig. 2. Azimuthal wavenumber and standard deviation ~( versus frequency; - © - ion drift dispersion; ~ 228
'f/kH:z
% ion cyclotron dispersion.
Volume 112A, number 5
2,1. ~
PHYSICS LETTERS
28 October 1985
-1
16.0
s.aa
I
Fig. 3. Contour plot of the bispectrum (real part);fe i -- 20 kHz. tron drift instability driven by the free energy contained in the density gradient. The saturation of the instability may be due to three-wave coupling between the unstable modes and damped modes. This is evidenced by the existence of sideband modes (usually asymmetric) in the spectrum and is confirmed by the bispectrum (fig. 3). The bispectrum is a measure o f the coherence between three waves which fulfill the frequency selection rule co 1 + co2 + co3 = O. Fig. 2 implies that the wavenumber selection rule is fulfilled, too, The bispectrum o f fig. 3 shows coupling (dark regions) of the most unstable modes ( f = 5 kHz and f - - 45 kHz) with all modes o f the spectrum. This implies that energy transport via three-wave coupling from unstable to damped modes stabilizes the instability. This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 162 Plasmaphysik Bochum/Ji~lich).
References [ 1] A.B. Mikhailovskii and A.V. Timofeev, Soy. Phys, JETP 17 (1963) 626. [21 M. Yamada, S. Sefler, H,W. Hendel and H. Ikezi, Phys. Fluids 20 (1977) 450. [3] B.I. Cohen, N. Maron and G.R. Smith, Phys. Fluids 25 (1982) 821. [4] C.S. Lui and D.K. Bhadra, Phys. Fluids 15 (1972) 2288. [5 ] K. Krauee, G. Liithen and H. Schliiter, Plasma Phys. 22 (1980) 1053. [6] .J.M. Beall, Y.C. Kim and E.J. Powers, J, AppL Phy~ 53 (1982) 3933. [7] L. Schraitz, G. Li~then, G. Derra, G, Derra, G. B6hm and H. S ehliiter, Plasma Phy~, to be published. [8] Y.C. Kim and E.J. Powers, IEEE Trang Plasma Sci PS-7 (1979) 120, [9] H.W. Hendel, B. Coppi, F, Perkins and P.A. Politzer, Phyg Rev. Lett. 18 (1967) 439. [10] M.P. Evrard, A.M. Messiaen, P.E. Vandenplas and G. van Oost, Plasma Phys. 21 (1979) 999.
229