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Observations on period-doubling phenomena in a basic plasma experiment
Volume 119, number 8
PHYSICS LETFERS A
12 January 1987
OBSERVATIONS ON PERIOD-DOUBLING PHENOMENA IN A BASIC PLASMA EXPERIMENT D. BORA, V.N. RAI and P.K. KAW Plasma Physics Programme, Physical Research Laboratory, Ahmedabad 380 009, India Received 16 July 1986; accepted for publication 10 November 1986
Observations on period-doubling phenomena by exciting lower hybrid and ion cyclotronfluctuations in the presence of rfpower in a plasma experiment are reported. The Feigenbaum number ô is calculated forthe first two bifurcations and foundto be 4.138. Nonlinear nature of the ion cyclotron oscillations gives rise to the interaction between modes that ultimately generate higher harmonics of the initial frequencies.
Recently, the period-doubling cascade as a route to chaos in nonlinear systems has been studied in detail theoretically as well as experimentally [1—5]. Experiments conducted in a variety of nonlinear media such as fluids, semiconductors, particle beams, nonlinear LRC circuits, etc. have demonstrated the period-doubling phenomenon and lend credence to the belief that the onset of chaos in many nonlinear systems can be understood in terms of a universal model. In this letter we report on observations of a plasma experiment in which the period-doubling phenomenon has been observed in nonlinear ion cyclotron waves driven by a radiofrequency field near the lower hybrid frequency. The experiment has been conducted in a toroidal experimental device, BETA (fig. 1). The toroidal vacuum vessel has a major radius of 45 cm. The plasma minor radius is 10 cm and is set by the inner size of a 5 cm wide conducting aperture ring, which alsoprovides the equilibrium. The toroidal magnetic field can take a peak value of 3 kG on the plasma minor axis; it is sinusoidal in shape and has a duration of 8 ms. A hot cathode discharge source is used to produce the plasma during the experiment. Argon plasma with a density of ~e ~ 1O~cm3 at the minor axis and electron temperature of 5—8 eV is produced during the experiment. The base pressure in the device is 2 x 10—6 Torr whereas the working pressure for the argon plasma is between 2 x 10~and l0~ Torr. The ii antennae are in the form of an
6 SETS OF TOROIOAL COIL
FIELC
o C
-
~.YLIP10ED
~NULAR
APE~TupE
~‘CRTICAL
Fig. 1. The top view of BETA (basic experiments in toroidal assembly) is shown. Circles represent flanges oninthe of the machine. Side and bottom flanges are not shown thetop diagram.
inside—outside pair of stainless steel plates placed perpendicular to the toroidal field in the shadow of the aperture (at minor radius r~12 cm in the equatonal plane). A radiofrequency electric field at f~ 8 MHz is applied between the plates. The continuous if source is connected to the antennae through a matching circuit. The if power on the antennae is varied up to 100 W and observations are conducted 411
Volume 119, number 8
PHYSICS LETTERS A
during the 1.5 ms near the peak of the sinusoidal toroidal magnetic field.. Conventional electrostatic probes are used to measure the plasma parameters. The plasma density fluctuation measurements are conducted with the help of if response probes. Eight such probes are placed in one minor cross section at r= 8 cm and separated poloidally from each other by 450~ Two more such probes on radially translating shafts facilitatethe radial measurements. Electron saturation current is registered on various probes to monitor the low-frequency oscillations. Signals f~romdifferent probes are collected simultaneously with the help of a multichannel storage oscilloscope. The stored signals are then digitized and numerically analyzed. Power spectrum analysis is conducted using the FFT method. Data from a single channel is autocorrelated to find the dominant power concentration in different frequency bands. Signals from different probes are cross-correlated to identify oscillations at different frequencies, corresponding to waves with finite azimuthal propagation. To identify coherent oscillations at different frequencies, it is essential to calculate phase and coherence spectra also. The con-
fidence level of coherence is kept 0.85 during the analysis. At the digitizing rates used, the frequency resolution is 1.25 kHz. Furthermore, the maximum frequency with useful information, defined by the Nyquist frequency is 80 kHz. Fig. 2 represents the frequency spectrum in the presence ofvarious levels of if power. Cross correlation between signals from two different probes is conducted and the cross-correlation function P12 is plotted along with the coherence spectra for different if power inputs on the antennae. At very low if powers only the ambient low-frequency fluctuations (~ 4 kHz) are observed. These may be identified as the Rayleigh—Taylor and drift-like oscillations for the BETA plasma. As the if power is increased, coherent oscillations at f~ 25 kHz become distinct. This corresponds to excitation ofion cyclotron in 6SOG). oscillations Ion cyclotron the argon plasma (with BT~ nature of the oscillations was ascertained by varying the ambient toroidal magnetic field. Fig. 3 shows the fluctuation frequency measured as a function of the magnetic field. The solid line corresponds to the calculated ion cyclotron frequency of argon plasma. When the working gas in the experiment was changed 412
12 January 1987
‘~cH—
,
I ui
°~
0 ~
-~
100
200
50-
___________
300 f (KH~I
40-0
500
6510
—
0—c
Lh
Wrt
87w
:ii ~
/
4 0
/
3-0 ~
/
4~
~,
,~
~
~
/
300
400
(orb-unit)
~•o
\~ 00
~
~_._—
100
—~
200
~
_.._L.
______
50
650
H~) —
Fig. 2. Frequency spectra in the presence of various levels of ii power are shown. P
12 are the cross-correlation functionsbetween signals from two different probes and Y2 are the coherence spectra.
from argon to hydrogen, an appropriate frequency change was observed confirming the ion cyclotron nature of the oscillations. Note that the if frequency at 8 MHz is close to the lower hybrid frequency COLH w1,~(1 + w~/w~)1/2 Parametric excitation of ion cyclotron oscillations by lower hybrid fields has been extensively studied in cylindrical plasmas. Our experi60.C 500 40.0
f( KHz) 30,0 20.0 10.0 ______________________ 500 600 700 800 BT (Gauss) .
.
900 1000
Fig.3. Fluctuation frequency as a function of toroidal magnetic field (BT). Solid curve represents the ion cyclotron frequency (J) of argon plasma while the dots correspond to the experimental values.
Volume 119, number 8
PHYSICS LETFERS A
ment is different from previous parametric excitation experiments in two basic features. Firstly, the two if antennae are in the shape of square plates and are fixed in such a way that they do not launch lower hybrid waves into the plasma. Instead, they set up a localized electric field oscillating at 8 MHz perpendicular to the B field (like a finite size dipole pump in parametric theory language). Secondly, the excitation is being done in a plasma with inhomogeneous magnetic fields. In this case, the ion cyclotron oscillation is likely to be in the nature of a global eigenmode with a frequency determined by an appropriately weighted average. Radial measurements conducted with the help oftwo radially placed if probes give the radial wavelength. At 25 kHz, the radial wavelength is measured to be ~ 5 cm which is much larger than the ion Lannor radius p’~ 0.65 cm. Tentatively, we have identified the excited waves as cold plasma ion cyclotron oscillations similar to those in the experiment of Chu et al. [6]. The threshold for instability excitation is determined by lower hybrid wave convection out of the pump region and the neutral damping of ion cyclotron oscillations (y~>( V2/1)f ~). For the’ present experiment, this gives a threshold if field E0~ 3 V cm which is readily exceeded. More details about the properties of the excited modes and the parametric excitation process will be published separately. Before describing the high power results, it is important to verifS’ what changes take place in mean plasma parameters because of the applied if. This is because the oscillation frequencies in a plasma depend on the plasma parameters and any drastic changes in these parameters could directly alter the characteristic oscillation frequencies. We have measured the plasma density and temperature profiles during the experiment in the presenceofif. It is found that the introduction of if power changes the density profile only marginally and that in no case does the change in density at a particular radial position exceed more than 7%. Similarly the change in electron temperature is measured to be within the 10% experimental accuracy of the measurements. As the power on the antennae increases, oscillations at different frequencies become distinct. Fluctuations appear at two different frequencies as the power is increased to 72 W. It is seen that the frequency halves thus doubling the period of oscilla—‘
12 January 1987
w~now
0
40
3~0
I
1’O
P
12
20 ,
~
,~.,
b0~ unit)
r~, ~
~
\J \,
)~I2
to
0.0
00 I
~0’0
200
300 400
500
600 850
Fig.4. Cross correlation P~2and coherence y2 spectra for an rf power level of 110W are shown.
tions. Theexact onset ofthe new fluctuations in terms of the if power is measured. Note that the half-frequency is much lower than the minimum cyclotron frequency in the inhomogeneous toroidal field and cannot arise due to the excitation of local ion cyclotron waves at some other physical location. As the power level is increased further, oscillations at a third frequency corresponding to a second period doubling are observed. Furthermore, at a power level of 100 W five distinct frequencies are observed (fig. 4). The two new frequencies that appear at higher power correspond to excitation at f-~ 2j and f—s 4J~.The generation of these frequencies is a very good mdicator of the nonlinearity of the low-frequency oscillations. Frequencies below ~J., are not easy to identify as they merge with the rather strong low-frequency ambient spectrum of fluctuations existing in the plasma. To measure the Feigenbaum number for perioddoubling bifurcations, we calculate —
~fl=p
(n=l,2,3,...),
n±i n+2
—
n+ i
for the first bifurcation. We find ö1 = 4.138 which is quite close to the universal asymptotic value ô = 4.6642016 of Feigenbaum. It has been difficult for us to identify the next bifurcation. As already mentioned this is due to considerable natural noise at the low-frequency end which makes it difficult to identify new peaks in the region and to accurately 413
Volume 119, number 8
PHYSICS LETTERS A
measure the if power input needed. However, we are in the process ofdeveloping a maximum entropy code which could sharpen the power spectrum peaks and help us in the identification of several bifurcations before the onset of turbulence. In conclusion, we have followed the period-doubling phenomenon in a nonlinear ion cyclotron oscillation driven to finite amplitude by a parametric excitation process with an if near the lower hybrid frequency. From the threshold power for the first two bifurcations, we have calculated a Feigenbaum number, which is in approximate agreement with the universal theoretical value. One point deserving comment is that previous experiments on parametric excitation nearj~1 by lower hydrid pump have not reported this effect. This could be so either (i) because of the fact that the excitation mechanism in the present experiment is different than the previous ones or (ii) because the effect was not carefully stud-
414
12 January 1987
ied and looked for in the previous cylindrical experiments~ We acknowledge the technical assistance of K.S. Sathyanarayana during the experimental work. We thank Dr. Y.C. Saxena for fruitful discussions on data analysis techniques. References [1] M.J. Feigenbaum, Phys. Lett. A 74 (1979) 375. [2] J. Maurer and A.J. Libchaber, J. Phys. (Paris) Lett. 40 (1979) L419. [3] M. Giglio, S. Muzzati and U. Perini, Phys. Rev. Lett. 47 (1981) 243. [4] J.P. Gollub and H.L. Swinney, Phys. Rev. Lett. 35 (1975) [5] 927. J. Testa, J. Perez and C. Jeifries, Phys. Rev. Lett. 48 (1982) 714. [6] T.K. Chu, S. Bernabei and R.W. Motley, Phys. Rev. Lett. 31 (1973) 211.
PHYSICS LETFERS A
12 January 1987
OBSERVATIONS ON PERIOD-DOUBLING PHENOMENA IN A BASIC PLASMA EXPERIMENT D. BORA, V.N. RAI and P.K. KAW Plasma Physics Programme, Physical Research Laboratory, Ahmedabad 380 009, India Received 16 July 1986; accepted for publication 10 November 1986
Observations on period-doubling phenomena by exciting lower hybrid and ion cyclotronfluctuations in the presence of rfpower in a plasma experiment are reported. The Feigenbaum number ô is calculated forthe first two bifurcations and foundto be 4.138. Nonlinear nature of the ion cyclotron oscillations gives rise to the interaction between modes that ultimately generate higher harmonics of the initial frequencies.
Recently, the period-doubling cascade as a route to chaos in nonlinear systems has been studied in detail theoretically as well as experimentally [1—5]. Experiments conducted in a variety of nonlinear media such as fluids, semiconductors, particle beams, nonlinear LRC circuits, etc. have demonstrated the period-doubling phenomenon and lend credence to the belief that the onset of chaos in many nonlinear systems can be understood in terms of a universal model. In this letter we report on observations of a plasma experiment in which the period-doubling phenomenon has been observed in nonlinear ion cyclotron waves driven by a radiofrequency field near the lower hybrid frequency. The experiment has been conducted in a toroidal experimental device, BETA (fig. 1). The toroidal vacuum vessel has a major radius of 45 cm. The plasma minor radius is 10 cm and is set by the inner size of a 5 cm wide conducting aperture ring, which alsoprovides the equilibrium. The toroidal magnetic field can take a peak value of 3 kG on the plasma minor axis; it is sinusoidal in shape and has a duration of 8 ms. A hot cathode discharge source is used to produce the plasma during the experiment. Argon plasma with a density of ~e ~ 1O~cm3 at the minor axis and electron temperature of 5—8 eV is produced during the experiment. The base pressure in the device is 2 x 10—6 Torr whereas the working pressure for the argon plasma is between 2 x 10~and l0~ Torr. The ii antennae are in the form of an
6 SETS OF TOROIOAL COIL
FIELC
o C
-
~.YLIP10ED
~NULAR
APE~TupE
~‘CRTICAL
Fig. 1. The top view of BETA (basic experiments in toroidal assembly) is shown. Circles represent flanges oninthe of the machine. Side and bottom flanges are not shown thetop diagram.
inside—outside pair of stainless steel plates placed perpendicular to the toroidal field in the shadow of the aperture (at minor radius r~12 cm in the equatonal plane). A radiofrequency electric field at f~ 8 MHz is applied between the plates. The continuous if source is connected to the antennae through a matching circuit. The if power on the antennae is varied up to 100 W and observations are conducted 411
Volume 119, number 8
PHYSICS LETTERS A
during the 1.5 ms near the peak of the sinusoidal toroidal magnetic field.. Conventional electrostatic probes are used to measure the plasma parameters. The plasma density fluctuation measurements are conducted with the help of if response probes. Eight such probes are placed in one minor cross section at r= 8 cm and separated poloidally from each other by 450~ Two more such probes on radially translating shafts facilitatethe radial measurements. Electron saturation current is registered on various probes to monitor the low-frequency oscillations. Signals f~romdifferent probes are collected simultaneously with the help of a multichannel storage oscilloscope. The stored signals are then digitized and numerically analyzed. Power spectrum analysis is conducted using the FFT method. Data from a single channel is autocorrelated to find the dominant power concentration in different frequency bands. Signals from different probes are cross-correlated to identify oscillations at different frequencies, corresponding to waves with finite azimuthal propagation. To identify coherent oscillations at different frequencies, it is essential to calculate phase and coherence spectra also. The con-
fidence level of coherence is kept 0.85 during the analysis. At the digitizing rates used, the frequency resolution is 1.25 kHz. Furthermore, the maximum frequency with useful information, defined by the Nyquist frequency is 80 kHz. Fig. 2 represents the frequency spectrum in the presence ofvarious levels of if power. Cross correlation between signals from two different probes is conducted and the cross-correlation function P12 is plotted along with the coherence spectra for different if power inputs on the antennae. At very low if powers only the ambient low-frequency fluctuations (~ 4 kHz) are observed. These may be identified as the Rayleigh—Taylor and drift-like oscillations for the BETA plasma. As the if power is increased, coherent oscillations at f~ 25 kHz become distinct. This corresponds to excitation ofion cyclotron in 6SOG). oscillations Ion cyclotron the argon plasma (with BT~ nature of the oscillations was ascertained by varying the ambient toroidal magnetic field. Fig. 3 shows the fluctuation frequency measured as a function of the magnetic field. The solid line corresponds to the calculated ion cyclotron frequency of argon plasma. When the working gas in the experiment was changed 412
12 January 1987
‘~cH—
,
I ui
°~
0 ~
-~
100
200
50-
___________
300 f (KH~I
40-0
500
6510
—
0—c
Lh
Wrt
87w
:ii ~
/
4 0
/
3-0 ~
/
4~
~,
,~
~
~
/
300
400
(orb-unit)
~•o
\~ 00
~
~_._—
100
—~
200
~
_.._L.
______
50
650
H~) —
Fig. 2. Frequency spectra in the presence of various levels of ii power are shown. P
12 are the cross-correlation functionsbetween signals from two different probes and Y2 are the coherence spectra.
from argon to hydrogen, an appropriate frequency change was observed confirming the ion cyclotron nature of the oscillations. Note that the if frequency at 8 MHz is close to the lower hybrid frequency COLH w1,~(1 + w~/w~)1/2 Parametric excitation of ion cyclotron oscillations by lower hybrid fields has been extensively studied in cylindrical plasmas. Our experi60.C 500 40.0
f( KHz) 30,0 20.0 10.0 ______________________ 500 600 700 800 BT (Gauss) .
.
900 1000
Fig.3. Fluctuation frequency as a function of toroidal magnetic field (BT). Solid curve represents the ion cyclotron frequency (J) of argon plasma while the dots correspond to the experimental values.
Volume 119, number 8
PHYSICS LETFERS A
ment is different from previous parametric excitation experiments in two basic features. Firstly, the two if antennae are in the shape of square plates and are fixed in such a way that they do not launch lower hybrid waves into the plasma. Instead, they set up a localized electric field oscillating at 8 MHz perpendicular to the B field (like a finite size dipole pump in parametric theory language). Secondly, the excitation is being done in a plasma with inhomogeneous magnetic fields. In this case, the ion cyclotron oscillation is likely to be in the nature of a global eigenmode with a frequency determined by an appropriately weighted average. Radial measurements conducted with the help oftwo radially placed if probes give the radial wavelength. At 25 kHz, the radial wavelength is measured to be ~ 5 cm which is much larger than the ion Lannor radius p’~ 0.65 cm. Tentatively, we have identified the excited waves as cold plasma ion cyclotron oscillations similar to those in the experiment of Chu et al. [6]. The threshold for instability excitation is determined by lower hybrid wave convection out of the pump region and the neutral damping of ion cyclotron oscillations (y~>( V2/1)f ~). For the’ present experiment, this gives a threshold if field E0~ 3 V cm which is readily exceeded. More details about the properties of the excited modes and the parametric excitation process will be published separately. Before describing the high power results, it is important to verifS’ what changes take place in mean plasma parameters because of the applied if. This is because the oscillation frequencies in a plasma depend on the plasma parameters and any drastic changes in these parameters could directly alter the characteristic oscillation frequencies. We have measured the plasma density and temperature profiles during the experiment in the presenceofif. It is found that the introduction of if power changes the density profile only marginally and that in no case does the change in density at a particular radial position exceed more than 7%. Similarly the change in electron temperature is measured to be within the 10% experimental accuracy of the measurements. As the power on the antennae increases, oscillations at different frequencies become distinct. Fluctuations appear at two different frequencies as the power is increased to 72 W. It is seen that the frequency halves thus doubling the period of oscilla—‘
12 January 1987
w~now
0
40
3~0
I
1’O
P
12
20 ,
~
,~.,
b0~ unit)
r~, ~
~
\J \,
)~I2
to
0.0
00 I
~0’0
200
300 400
500
600 850
Fig.4. Cross correlation P~2and coherence y2 spectra for an rf power level of 110W are shown.
tions. Theexact onset ofthe new fluctuations in terms of the if power is measured. Note that the half-frequency is much lower than the minimum cyclotron frequency in the inhomogeneous toroidal field and cannot arise due to the excitation of local ion cyclotron waves at some other physical location. As the power level is increased further, oscillations at a third frequency corresponding to a second period doubling are observed. Furthermore, at a power level of 100 W five distinct frequencies are observed (fig. 4). The two new frequencies that appear at higher power correspond to excitation at f-~ 2j and f—s 4J~.The generation of these frequencies is a very good mdicator of the nonlinearity of the low-frequency oscillations. Frequencies below ~J., are not easy to identify as they merge with the rather strong low-frequency ambient spectrum of fluctuations existing in the plasma. To measure the Feigenbaum number for perioddoubling bifurcations, we calculate —
~fl=p
(n=l,2,3,...),
n±i n+2
—
n+ i
for the first bifurcation. We find ö1 = 4.138 which is quite close to the universal asymptotic value ô = 4.6642016 of Feigenbaum. It has been difficult for us to identify the next bifurcation. As already mentioned this is due to considerable natural noise at the low-frequency end which makes it difficult to identify new peaks in the region and to accurately 413
Volume 119, number 8
PHYSICS LETTERS A
measure the if power input needed. However, we are in the process ofdeveloping a maximum entropy code which could sharpen the power spectrum peaks and help us in the identification of several bifurcations before the onset of turbulence. In conclusion, we have followed the period-doubling phenomenon in a nonlinear ion cyclotron oscillation driven to finite amplitude by a parametric excitation process with an if near the lower hybrid frequency. From the threshold power for the first two bifurcations, we have calculated a Feigenbaum number, which is in approximate agreement with the universal theoretical value. One point deserving comment is that previous experiments on parametric excitation nearj~1 by lower hydrid pump have not reported this effect. This could be so either (i) because of the fact that the excitation mechanism in the present experiment is different than the previous ones or (ii) because the effect was not carefully stud-
414
12 January 1987
ied and looked for in the previous cylindrical experiments~ We acknowledge the technical assistance of K.S. Sathyanarayana during the experimental work. We thank Dr. Y.C. Saxena for fruitful discussions on data analysis techniques. References [1] M.J. Feigenbaum, Phys. Lett. A 74 (1979) 375. [2] J. Maurer and A.J. Libchaber, J. Phys. (Paris) Lett. 40 (1979) L419. [3] M. Giglio, S. Muzzati and U. Perini, Phys. Rev. Lett. 47 (1981) 243. [4] J.P. Gollub and H.L. Swinney, Phys. Rev. Lett. 35 (1975) [5] 927. J. Testa, J. Perez and C. Jeifries, Phys. Rev. Lett. 48 (1982) 714. [6] T.K. Chu, S. Bernabei and R.W. Motley, Phys. Rev. Lett. 31 (1973) 211.