Spatiotemporal control and synchronization of flute modes and drift ...

PHYSICS OF PLASMAS 13, 052509 共2006兲

Spatiotemporal control and synchronization of flute modes and drift waves in a magnetized plasma column F. Brochard,a兲 G. Bonhomme, E. Gravier, S. Oldenbürger, and M. Philipp LPMIA, UMR 7040 du CNRS, Université Henri Poincaré, BP 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France

共Received 5 January 2006; accepted 31 March 2006; published online 12 May 2006兲 An open-loop spatiotemporal synchronization method is applied to flute modes in a cylindrical magnetized plasma. It is demonstrated that synchronization can be achieved only if the exciter signal rotates in the same direction as the propagating mode. Moreover, the efficiency of the synchronization is shown to depend on the radial properties of the instability under consideration. It is also demonstrated that the control disposition can alternatively be used to produce strongly developed turbulence of drift waves or flute instabilities. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2199807兴 I. INTRODUCTION

Magnetized bounded plasmas are subjected to a class of low-frequency electrostatic instabilities. These instabilities are generally believed to be responsible for anomalous crossfield particle transport.1 Indeed the turbulent regimes are seamingly accounting for anomalous transport in the magnetically confined high-temperature plasmas of tokamaks. Therefore, controlling these instabilities could be of a great interest for improving the performance of the magnetic fusion devices. This work is performed in the cylindrical magnetized triple-plasma device MIRABELLE 共Ref. 2兲 which easily exhibits nonlinear instabilities. In fact the characteristics of the unstable waves are very different from the instabilities observed in tokamaks. Indeed, very high mode numbers are excited in tokamaks and thus the plasma slab approximation is valid. On the contrary, the unstable wavelengths in our device are of the order of the plasma radius and global modes are excited. However, the dynamics is of great interest because the system is extended in space, leading to spatiotemporal characteristics, as in a tokamak. In linear magnetized plasma devices, turbulence is often believed to arise from drift waves regimes.3 However, we have shown in recent works4,5 that a strong E ⫻ B plasma rotation due to the mean radial electric field leads to flute modes, especially when a limiter reduces the plasma diameter. In this case, the Kelvin-Helmholtz shear flow instability is at the origin of plasma structures observed in the shadow of the limiter, where the velocity shear is maximum.4 In a similar way, the centrifugal Rayleigh-Taylor instability has recently been assumed to be responsible for spiral structures convecting the plasma outward.5,6 A series of experiments on temporal feedback control of plasma instabilities were carried out. Control of chaos is a first approach, where unstable periodic states are stabilized by tiny adjustments of one or more accessible parameters. The success of this conception was demonstrated for various different nonlinear systems exhibiting only purely temporal a兲

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chaos.7–12 The time-delay autosynchronization method 共TDAS兲 proposed by Pyragas13 allows for an easy experimental implementation. A continuous control signal is applied to the system, i.e., to the dynamical variable. It is elaborated from the difference between the real-time signal and the time-delayed version of this signal. The delay is chosen equal to the period of the targeted unstable periodic orbit. Thus the control signal is closed to zero after control and the energy injected in the device is minimal. Drift wave chaos in MIRABELLE was successfully controlled by TDAS temporal feedback,14,15 but only in drift wave dynamics low-dimensional chaos. Indeed, as drift wave turbulence is basically a spatiotemporal phenomenon, it is barely expected that a purely temporal control technique proves to be efficient and robust. Other stabilization strategies were proposed for controlling spatiotemporal chaos.16–20 On the MIRABELLE device an open-loop control 共synchronization兲 acting in both space and in time was successfully tested.21 Compared to the previous method, the amplitude of the control signal is not reduced after control is achieved. Thus more energy is required. But the control method is spatiotemporal and it turns out that drift wave turbulence is reduced by driving preselected drift modes to the expense of the broadband spectrum.21–23 The next step was to test these methods on a tokamak. Preliminary experiments have been carried out on the CASTOR tokamak as well as in the W7-AS stellarator, where first results were obtained.24–26 In this paper, we present experimental results obtained with the open-loop synchronization method acting on Kelvin-Helmholtz and Rayleigh-Taylor instabilities observed in the linear MIRABELLE device. This paper is organized as follows. First, the experimental setup is described in Sec. II. In Sec. III we present an example of spatiotemporal synchronization of a Kelvin-Helmholtz instability. Contrary to drift waves, flute modes characteristics can strongly evolve with respect to the plasma radius, in part because of the existence of a shear layer.28 For that reason the efficiency of the openloop control method in the plasma depth is presented in Sec. IV, in the case of a Rayleigh-Taylor instability. In Sec. V, we show that the open-loop control method can be used to re-

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© 2006 American Institute of Physics

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FIG. 1. Scheme of the MIRABELLE plasma device, showing the location of the octupole exciter.

cover the Ruelle-Takens route to turbulence by synchronizing regular drift waves and flute modes. Last, the main results are summarized in Sec. VI. II. EXPERIMENTAL SETUP

Experiments are carried out in the linear section of the low ␤ triple plasma device MIRABELLE 共Ref. 2兲 displayed in Fig. 1. Plasma is produced by a thermoionic discharge in one of the two source chambers; the other one staying unoperated in the experiments reported in this paper. Confinement is ensured in the central section by 24 coils, which create a uniform magnetic field whose strength can be varied up to 120 mT. Compensation coils, installed on each source chamber, allow null axial B field in front of the cathodes, which avoids filamentary plasma. The base pressure inside the device is of the order of 10−5 mbar and the working pressure in argon is typically 1 − 2 ⫻ 10−4 mbar. A high transparency grid is located at the entrance of the column. Its biasing influences the axial drift of the particles and the radial profiles of density and potential, allowing a dynamic control of the plasma regime. Discharge current, magnetic field strength, pressure, and biasing of an internal tube inside the central section determine the plasma regime as well. We have shown in earlier studies4,5 that without any limiter, only drift waves due to the radial density gradient can be excited in the plasma column. Thus, in order to increase transverse gradients, a metallic diaphragm is inserted at the entrance of the column. The plasma rotation due to the increased radial electric field can then trigger flute modes, especially at low B field, when the drift dispersion scale ␳s = cs / ␻ci is large, or of the same order as the transverse length scale L⬜.5 Spatiotemporal control is achieved with an eight-plates exciter located in the first part of the column. The octupole is made up of eight insulated plates with a length of 30 cm, driven by sinusoidal signals with a preselected phase shift ␾ between each plate 共Fig. 2兲. In this way, rotating electric field with an azimuthal mode mE = 4␾ / ␲ can theoretically be generated, with the restriction mE = 1 − 3 due to the Nyquist’s sampling theorem. However, the width of the plates does not allow for producing mE = 3 with a satisfying spatial regularity, and thus we restrict ourselves to mE = 1 − 2. Note that the sign of ␾

FIG. 2. Scheme of the octupole exciter.

determines the direction of propagation of the exciter field. The question of the nature of the perturbation produced in the plasma by the octupole has been addressed by Schröder et al. in Ref. 21. Experimental results on the control of drift waves have been confronted to simulations based on the Hasegawa-Wakatani equations. On the one hand, it has been found that modelling the octupole by a rotating electric field had no influence on the drift wave dynamics. On the other hand, experimental findings have been fully recovered by modelling the octupole plate assembly by an oscillatory parallel current profile. Thus, it is possible to conclude that the exciter interacts with the plasma via the current drawn to the exciter electrodes. Although this current has not been measured in our experiments, Ref. 23 suggests that the current drawn from the plasma to the electrodes and vice versa is about 10% of the whole plasma current. Two octupoles are used in this study. The first one, of diameter 15 cm, is used for experiments on flute modes, when the limiter 共15 cm in diameter as well兲 is present. As it can be seen in Fig. 3, the octupole is positioned in the edge region of the plasma column, out of the maximum offluctuations which is assumed to

FIG. 3. Radial position of the octupole exciter. Typical density 共squares兲 and plasma potential profiles 共triangles兲, recorded at B = 27 mT, are shown for comparison.

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TABLE I. Plasma parameters in MIRABELLE. Chamber length 共D兲

140 cm

Chamber radius 共a兲

11.5 cm

Density gradient length 共L兲 Hybrid Larmor radius 共␳s兲 Magnetic field 共B兲 Plasma density Electron temperature 共Te兲 Ion temperature 共Ti兲

1–5 cm 0.5–3 cm

Operating pressure 共argon兲 Observed instability frequency Ion cyclotron frequency

10–120 mT 1015 − 1016 m−3 1–5 eV ⬍0.05 eV 1–4 ⫻ 10−4 mbar 2–12 kHz 8–50 kHz

be close to the maximum density and plasma potential gradients.27 A second octupole, of diameter 20 cm, is used for experiments on drift waves in the absence of any limiter. The main parameters of the plasma are listed in Table I. Times series of the floating potential and of the electronic saturation current are obtained from measurements performed with a set of Langmuir probes. Two field-aligned probes distant of 40 cm are used to determine the parallel wave number, and to check that it remains constant during the control experiments. The probe arrangement also includes a 3D movable probe, used to investigate the radial structure of the fluctuations, and an azimuthal array of 32 probes for spatiotemporal measurements. The latter is used to deduce the azimuthal mode number m, at the fixed radius r = 7 cm. In order to be sure that the octupole does not change the nature of the instabilities we are dealing with, a preliminary study, based on the same methods as those exposed in Ref. 5, is performed. Results are qualitatively the same, the transition from flute modes to drift waves being observed at a slightly lower B field 共40 mT instead as 50 mT without octupole兲, as it can be seen in Fig. 4. This is due to the fact that the octupole acts itself as a limiter at low B fields.

FIG. 4. Evolution of the parallel wave number according to B, with the unbiased octupole. The transition from flute modes to drift waves occurs in the range 35–40 mT. The variation between 60 and 70 mT is due to the transition from m = 2 to m = 1.

III. SPATIOTEMPORAL SYNCHRONIZATION OF A KELVIN-HELMHOLTZ INSTABILITY A. Imperfect control: Forcing m = 1

In this section, we present a first example of the synchronization of a Kelvin-Helmholtz instability. At first, a stationary Kelvin-Helmholtz instability is selected. For this series, main plasma parameters are: B = 20 mT, P = 2.4⫻ 10−4 mbar, U f = 17.9 V, Ug = 7 V, and Ud = 54 V, where P is the argon pressure, U f is the voltage applied to the filaments, Ug is the grid voltage, and Ud is the discharge voltage. No axial wave number is recorded, but an E ⫻ B velocity shear is observed which increases the rotation frequency of the fluctuations from ␻ = 3.4⫻ 104 rad s−1 at r = 3 cm to ␻ = 4.4⫻ 104 rad s−1 at r = 7 cm. This unperturbed state is depicted in Fig. 5共a兲. As it can be seen in the spatiotemporal image, reconstructed from measurements performed at r = 7 cm, this state corresponds to m = 3. However, temporal series recorded at r = 5 cm are not very regular, and the Fourier spectrum is obviously noisy, with a small frequency peak at f = 6.5 kHz and

FIG. 5. Spatiotemporal KelvinHelmholtz dynamics: unperturbed case 共a兲, co-rotating field 共b兲, and counter-rotating field 共c兲. The three rows show the spatiotemporal density fluctuations, the floating potential fluctuations, and the frequency power spectrum. Dashed lines on the spatiotemporal image are drawn to guide the eye, and white arrows indicate the number of maxima on a plasma circumference, i.e., the azimuthal mode number m.

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a plateau between 2 and 4 kHz. This discrepancy between the measurements at two distinct radial locations is characteristic of the radial structure of flute modes, as discussed in Ref. 28. This chaotic state is likely to be governed by the interactions between the outer m = 3 mode at f 0 ⬃ 7 kHz and a more centered m = 1 or 2 mode which frequency belongs to the above-mentioned plateau. Thus, we try to synchronize this irregular m = 3 state by applying to the octupole a signal with mE = 1 at the frequency f e = 4 kHz. A driver amplitude of 2 V is chosen. The axial wave number remains null during the experiment, which means that the flute nature of the instability is not affected by the octupole. Figure 5共b兲 shows the result when a co-rotating signal is applied, i.e., when the exciter field propagates in the same direction than the fluctuations. At first sight, synchronization is successful, with a well established m = 1 mode. Nevertheless, it can be noticed that the frequency f 0 corresponding to m = 3 has not disappeared from the Fourier spectrum, and that the broadband frequency underground stays at the same level as in the unperturbed state. Moreover, linear combinations of f 0 and f e appear, e.g., at 10 kHz. In that case, dynamics is obviously rather forced than really controlled by mE = 1. To test if the synchronization is really spatiotemporal, and not purely temporal, a counter-rotating signal is then applied to the octupole. The result for the same frequency and driving amplitude is depicted in Fig. 5共c兲. No clear spatiotemporal pattern or propagation direction can be seen in the spatiotemporal image. However it is obviously different to those in the unperturbed or synchronized states. The time series is more periodic than in the unperturbed case, but its amplitude is strongly irregular. Finally, Fourier spectrum exhibits many narrow peaks which seem to be linear combinations of f 0 and f e. Such a spectrum resembles very much the one of a quasiperiodic state observed, e.g., in the drift waves of flute modes transition to turbulence.28,29 A further analysis is thus needed to understand if nonlinear interactions between the counter-rotating pattern with the Kelvin-Helmholtz mode occur. In a first simple approach, two passband filters are used in order to analyze more precisely the spatiotemporal image 共Fig. 6兲. Choosing the cutoff frequencies accordingly, the original image 关Fig. 6共a兲, which corresponds to the case of Fig. 5共c兲兴 is easily decomposed into two patterns, one corresponding to the excited field with mE = 1 关Fig. 6共b兲兴, and the other one to the unperturbed m = 3 Kelvin-Helmholtz mode 关Fig. 6共c兲兴. Note the opposite slope of each pattern, due to the opposite propagation direction. This figure gives some indication that interactions between mE = 1 and m = 3, if they exist, remain at a low level. In a second approach, the time evolution of the frequency is investigated by means of a wavelet analysis. The wavelet transform of a signal s共t兲 is given by Ws共a, ␶兲 =

冕

s共t兲⌿a共t − ␶兲dt,

FIG. 6. Spatiotemporal Kelvin-Helmholtz dynamics, with a counter-rotating signal: without filtering 共a兲, with a first order passband filtering 共3–5 kHz兲 共b兲, and with a first order passband filtering 共5–7 kHz兲 共c兲.

present work we chose the commonly used Morlet wavelet, which is a sinusoidal oscillation convoluted with a Gaussian: ⌿a共t兲 = 共2a␲兲−1/2 exp关i2␲t/a − 共t/a兲2/2兴.

共2兲

The frequency corresponding to the scale a is given by ␻ = 2␲ / a. The frequency and time resolution corresponding to the wavelet ⌿a are, respectively, ⌬␻ = ␻ / 4 and ⌬t = 2a. The wavelet analysis of the time series shows that the temporal behavior of the system is totally different according to the propagation direction of the excited field 共Fig. 7兲. Whereas frequency of the exciter durably dominates when a corotating signal is applied 关Fig. 7共a兲兴, it is dominated and periodically masked by the frequency of the original m = 3 instability mode when a counter-rotating signal is applied

共1兲

where ⌿a is the wavelet function, which is a function of both time t and scale a. Many different wavelets exist, and in this

FIG. 7. Morlet wavelet transform of the signal recorded with a co-rotating field 共a兲, and with a counter-rotating field 共b兲.

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BW共a1,a2兲 =

FIG. 8. Wavelet bicoherence analysis of the signal recorded with a corotating field 共a兲, and with a counter-rotating field 共b兲. The horizontal structure in the top picture clearly demonstrates phase coupling between the frequency of the exciter, at 4 kHz, and a wide range of low frequencies. On the contrary, this frequency is not implied in any phase coupling phenomena when a counter-rotating signal is applied 共b兲.

冕

Ws*共a, ␶兲Ws共a2, ␶兲d␶ ,

共4兲

where 1 / a = 1 / a1 + 1 / a2. The wavelet bispectrum is a measure of the amount of phase coupling between components of scale lengths a1, a2, and a of the signal s共t兲. More details on the calculations as well as on the theory of wavelet bicoherence can be found in Ref. 30. In the case of a co-rotating signal, the autobicoherence undeniably demonstrates that synchronization is achieved because strong phase coupling occurs between the excited field at 4 kHz and the instability 关Fig. 8共a兲兴, as it could be expected. Although strong phase coupling is seen in the case of a counter-rotating signal as well, it is noticeable that no phase coupling occurs at the frequency of the excited field 关Fig. 8共b兲兴. Thus we can conclude that no phase coupling occurs between the excited field and the instability when a counter-rotating signal is chosen, which means that no synchronization can be achieved with such a counter-rotating field. Although fluctuations are obviously perturbed by the counter-rotating field, it is probably due to the fact that the drivers amplitude is of the same order of magnitude as the plasma potential. Hence the global plasma equilibrium can be affected, and consequently so are the fluctuations. B. Successful control: Synchronizing m = 2

关Fig. 7共b兲兴. The high degree of periodicity in the vertical sidebands of the counter-rotating state is however very intriguing and could suggest some periodic coupling phenomena between the excited field and the instability. To investigate this hypothesis, a wavelet bicoherence analysis is finally used. The squared wavelet bicoherence of the signal s共t兲 is defined as: 关bW共a1,a2兲兴2 =

冋冕

兩BW共a1,a2兲兩2 兩Ws共a1, ␶兲Ws共a2, ␶兲兩 d␶ 2

册冋冕

兩Ws共a, ␶兲兩 d␶ 2

册

which is the normalized wavelet bispectrum defined as

共3兲

In this section, we try to synchronize the irregular m = 3 Kelvin-Helmholtz instability which was considered previously, by applying to the octupole a signal with mE = 2 at the frequency f e = 7 kHz. In general, synchronization on m = 2 is easier to achieve than synchronization on m = 1, which is likely to be due to a higher degree of symmetry of the spatiotemporal pattern. In the following example, the driving amplitude is only 1.2 V, and the axial wave number remains null. Here we consider a particular interesting case, since the frequency of the excited field is very close to the frequency of the pre-existing mode, but the mode numbers are different. In such cases, a transient state can be observed before the synchronization is achieved. This transient state is here

FIG. 9. Spatiotemporal KelvinHelmholtz dynamics: transient state 共a兲, synchronized state with a corotating field 共b兲, and counter-rotating field 共c兲. The figure is arranged as in Fig. 5, and the unperturbed case is the one depicted in Fig. 5共a兲.

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FIG. 10. Time series of the signal 共a兲, and its corresponding Morlet wavelet transform 共b兲, showing the transition from the transient state to the synchronized state.

depicted in Figs. 9共a兲 and 10. The transient state is characterized by a periodic sliding from the original m = 3 to the excited m = 2 mode, as it can be seen in the spatiotemporal image of Fig. 9共a兲. This alternation in the wave number space is concomitant with a temporal modulation in phase and in amplitude of the time series, as it is observed in the case of a periodic pulling state.22,31 The duration of the transient state is about 100–1000 periods of the applied signal. Moreover the modulation is highly regular in spite of a slight decrease of the beat frequency just before the transition to the permanent synchronized state, as it can be seen in Fig. 10. Then m = 3 totally disappears and a perfectly synchronized m = 2 mode can be observed, with a high regularity in both the spatiotemporal pattern and in the temporal series 关Fig. 9共b兲兴. The frequency power spectrum is sharply peaked at f e = 7 kHz and its first harmonic. In opposition to the results obtained with mE = 1, there is a significant decrease of the broadband low-frequency components. Thus we can conclude that an efficient spatiotemporal control is achieved. On the contrary, when a counter-rotating field is applied, no synchronization can be achieved as it can be seen in Fig. 9共c兲, and further analysis leads to the same conclusions as those presented previously for me = 1. IV. EFFICIENCY WITH RESPECT TO THE RADIAL LOCATION

To achieve synchronization, the phase velocity of the driven mode v⌽ = 2␲ f ere / m must be compatible with the dispersion relation of the pre-existing wave. In the case of drift waves, successful synchronization can be easily achieved with a relatively weak exciter signal.21 In contrast to that, when flute modes are considered, synchronization requires generally a stronger exciter signal. In the case of a KelvinHelmholtz instability, a simple explanation to this discrepancy is that whereas drift waves almost propagate like a rigid body, Kelvin-Helmholtz modes generally present a radial ve-

FIG. 11. Detail of the time series and power spectrum showing the radial evolution of the efficiency of the control on a Rayleigh-Taylor instability: without control at r = 3 cm 共a兲 and at r = 5 cm 共b兲, and with a co-rotating field exciting m = 1 at 4 kHz at r = 3 cm 共c兲, and at r = 5 cm 共d兲. Almost no change is seen at r = 3 cm.

locity shear. This results in a dependency of the efficiency of the coupling with the phase velocity of the driven mode on the radius. However, this explanation does not stand for a Rayleigh-Taylor instability, whose rotation velocity is not necessarily sheared. Thus a more detailed analysis is needed. To give an example, we present in Fig. 11 simultaneous measurements recorded at two different radial positions. Discharge parameters for this series are: B = 34 mT, P = 2 ⫻ 10−4 mbar, U f = 17.9 V, Ug = 4 V, and Ud = 54 V. The characterization of the unperturbed state, performed with the same methods as in Ref. 5, leads to the conclusion that it consists of a Rayleigh-Taylor instability. In particular, the axial wave number is found to be null, and no radial velocity shear is recorded. As it can be seen in Figs. 11共a兲 and 11共b兲, time series and power spectra in the unperturbed state exhibit substantial differences according to the radial location, fluctuations being much more regular at r = 5 cm than at r = 3 cm. This discrepancy, which is characteristic of the Rayleigh-Taylor instability, has been discussed in a recent paper.28 A further analysis reveals that this state is slightly dominated by m = 2 at f 2 = 7.4 kHz at r = 5 cm, whereas m = 1 at f 1 = 3.6 kHz is more pronounced at r = 3 cm. To synchronize this state, a co-rotating driving mode mE = 1 with the frequency f e = 4 kHz is chosen. The result at both positions for a driving amplitude V = 3 V is shown in Figs. 11共c兲

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and 11共d兲. Whereas almost no change compared to the unperturbed state is seen at r = 3 cm, quite a successful synchronization is achieved at r = 5 cm. Therefore, it is expected that in the case of a RayleighTaylor instability, where no or only a little velocity shear exists, the radial evolution of the synchronization’s efficiency is likely to be due to the radial properties of the instability. Since the latter exhibits more nonlinear coupling in its inner part,28 it is indeed reasonable to assume that a successful control of the deep plasma requires more energy. V. DESTABILIZATION OF A REGULAR STATE: PRODUCTION OF SPATIOTEMPORAL CHAOS A. Acting on drift waves

Drift waves are known to follow the Ruelle-Takens route to turbulence, in which spatiotemporal chaos and turbulence arise from the nonlinear interactions between two modes with incommensurate frequencies.29 Starting from a regular state, the octupole exciter can be used to recover this transition scenario by adding a new mode into the system. Since the excited mode number mE and its frequency f e are perfectly known, with easily adjustable characteristics, the method can in principle be used in order to produce chaotic or turbulent states with special predetermined features. This could be of particular interest for specific studies, such as the characterization of the transport properties according to the level of turbulence. In this section, the device is operated without any limiter in order to easily produce drift waves,4,5 and the large octupole with a diameter of 20 cm, is used. The main plasma parameters are: B = 34 mT, P = 2 ⫻ 10−4 mbar, U f = 17.8 V, Ug = 8 V, Ud = 28 V, and Ut = 4.7 V, where Ut is an additional voltage applied to the inner tube depicted in Fig. 1. The axial wave number is k储 = 2.1± 0.3 m−1, and remains in this range when the octupole is biased. A slightly quasiperiodic state, depicted in Fig. 12共a兲 is selected as a starting point. The quasiperiodicity can be inferred from the phase portrait, which exhibits a two-torus, but is more obvious in the power spectrum. The latter is dominated by a sharp peak at 3 kHz, corresponding to m = 2, and its first harmonic at 6 kHz. Another peak corresponding to m = 1 is seen at 2 kHz, along with its first harmonic at 4 kHz. Other peaks produced by the nonlinear interactions between m = 1 and m = 2 are seen, respectively, at 5 kHz, 7 kHz, 8 kHz, etc. A transition to spatiotemporal chaos and turbulence is easily achieved when a new low-frequency mode is excited with the octupole. In the example depicted in Fig. 12共b兲, a co-rotating m = 2 mode is excited at the frequency 2 kHz. As it can be seen in this figure, a small driver amplitude of V = 0.6 V is sufficient to destabilize totally the quasiperiodic state. Although the frequency 2 kHz is still present in the system’s dynamics, which is normal since the excited field is continuously applied, time series and the phase portrait are completely disorganized. The power spectrum is certainly of even greater interest, because it undeniably exhibits a significant increase of the broadband frequency components above 3.5 kHz. In addition the spectral index f −␣ of the fluctuations-frequency underground is increased from ␣

FIG. 12. Producing driven drift waves turbulence: without polarization 共a兲, and with a co-rotating field of weak amplitude V = 0.6 V, exciting m = 2 at 2 kHz 共b兲. For each case, a detail of the time series of the floating potential, the phase portrait, and the power spectrum are shown.

= 4.2 to ␣ = 5.5. Although dynamics is still dominated by the exciter in a narrow frequency range, the significant increase of the turbulent level in a quite large frequency domain could be of great interest, e.g., for testing turbulence diagnostics and analysis tools, or other control methods, in laboratory devices where a significant level of turbulence is generally difficult to achieve. Moreover, it should be noted that the level of turbulence can be easily chosen by varying the driving amplitude of the exciter. B. Acting on flute modes

It has been recently demonstrated that flute modes follow the Ruelle-Takens route to turbulence as well.28 Thus, the octupole exciter can be used to produce chaotic or turbulent flute states in the same way as with drift waves. In drift waves transition to turbulence, a particular state referred as mode locking can be observed between the quasiperiodic and the chaotic states.29 Mode locking appears when the dispersion relation allows two modes ma and mb to have frequencies which satisfy f a = nf b, with n = a / b. Very recently, mode locking has been reported in the case of a Rayleigh-Taylor instability,28 but up to now it has never been reported in the case of a Kelvin-Helmholtz instability. However, it is very easy to produce such a state with the octupole. The starting point, not shown here, is very close to the one depicted in Fig. 5共a兲. The discharge parameters are the same,

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Phys. Plasmas 13, 052509 共2006兲

FIG. 13. Producing mode locking with a Kelvin-Helmholtz instability: detail of the time series of the floating potential, phase portrait showing a double loop attractor, and power spectrum.

except for P = 3.2⫻ 10−4 mbar and Ug = 3 V. We can identify an m = 2 mode at f = 6 kHz. Applying a co-rotating signal with mE = 1, f e = 3 kHz, and V = 3 V to the octupole, the mode locking state depicted in Fig. 13 is obtained. Although in this case mode locking is forced, this result suggests that the Kelvin-Helmholtz dispersion relation should allow spontaneous mode locking states. Moreover, such a state could present some interest for control purposes, since it proves that a high spatiotemporal periodicity can be achieved even when more than one fluctuation mode is present. A complete transition to fully developed turbulence can also be produced by following the Ruelle-Takens scenario, as it was demonstrated for drift waves in the previous section. Initially, a regular m = 3 Kelvin-Helmholtz mode with f 0 = 10.2 kHz is selected 关Fig. 14共a兲兴. Then, mE = 1 is excited with the octupole at f e = 4 kHz. A first driving amplitude V = 1 V leads to the perturbed state depicted in Fig. 14共b兲. As it can be seen in the spatiotemporal image, dynamics is still dominated by m = 3, but new peaks due to nonlinear interactions with mE = 1 can be seen in the frequency power spectrum, at frequencies corresponding to linear combinations of f 0 and f e. The time series and the phase portrait are also less regular. When the driving amplitude of the exciter is increased, nonlinear interactions are strengthened and lead to spatiotemporal turbulence, as depicted in Fig. 14共c兲. In the power spectrum subsists only the very narrow frequency peak due to the exciter at f e, while the broadband lowfrequency spectral components are significantly enhanced. VI. CONCLUSIONS

Our study demonstrates that an open-loop method can be used to achieve partial or complete synchronization of flute modes by driving a preselected mode to the expense of the broadband spectrum. A better efficiency is obtained when the excited mode and its frequency are chosen close to a dominant mode of fluctuation. It is also noted that driving symmetrical modes requires less energy. It is shown that phase coupling between the excited wave and the unstable mode

FIG. 14. Producing Kelvin-Helmholtz spatiotemporal chaos by exploiting the Ruelle-Takens transition scenario: without polarization 共a兲, with a corotating field of weak amplitude V = 1 V 共b兲, with a co-rotating field of medium amplitude V = 2 V 共c兲. Detail of the time series of the floating potential, phase portrait, power spectrum, and spatiotemporal image are given for each case.

occurs during the synchronization process. On the contrary, no phase coupling involving the excited field is seen when a counter-rotating signal is applied, and no synchronization can be achieved in this case. Thus, the synchronization is shown to be spatiotemporal. Moreover, measurements of the axial wave number ensure that the nature of the instability under consideration is not altered by the exciter. The characterization of the efficiency of the synchronization at two different radial positions highlights however some weaknesses of the synchronization method, which is not well suited for controlling unstable modes characterized by a high radial dependence. In the case of a KelvinHelmholtz instability, the existence of a strong velocity shear does not allow an efficient coupling with the phase velocity of the driven mode. When a Rayleigh-Taylor instability is considered, synchronization of the deep plasma requires more energy because of the increase of nonlinear couplings at lower radii. The study finally demonstrates that the open-loop method can be used in a different approach, in order to easily produce turbulent states. These are driven by the nonlinear interactions between the pre-existing fluctuations and an excited mode. Increasing the amplitude of the excited mode strengthens the nonlinear interactions, and thereby allows us to obtain strongly developed turbulence.

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