Brochard, F.

Grulke, O.

Klinger, T.

Naulin, V.

Windisch, T.

English

MPG.PuRe

Spatiotemporal investigation of nonlinear coupling and energy transfersin drift wave turbulenceF. Brochard1, T. Windisch1, O. Grulke1, T. Klinger1, and V. Naulin21 Max Planck Institut for Plasma Physics, EURATOM Association, Greifswald, Germany2 Risø National Laboratory, Roskilde, DenmarkSeveral recent studies have shown that intermittent coherent structures are a common featureof edge magnetized plasmas [1]. Such structures, large compared to typical turbulent scales,can significantly contribute to the high cross-field particle and energy transport in fusion exper-iments. In this contribution we will focus on the particular issue of energy transfers in drift waveturbulence. Since a net transfer of energy must be accompanied by a finite phase coherence be-tween the involved structures, a bicoherence analysis, which provides a direct measurement ofnonlinear phase coupling, is often used to characterize it [2, 3]. Although the physical processhappens in k and not in ω space [4], experimental diagnostics usually suffer from a very lim-ited spatial resolution. Thus, frequency-based bicoherence techniques are generally performedalong with attempts to connect k and ω representations [2]. In this contribution we will presentresults obtained from a direct wavenumber-based bicoherence analysis, performed on measure-ments from an azimuthal array of 64 probes, in the linear magnetized helicon device VINETA[5]. In order to refine the description of spatiotemporal intermittent behaviour, a wavelet bico-herence [6] is used in both k and ω space. The autobicoherence is a normalized function, whichis close to unity when three waves satisfy the resonance condition ω1 + ω2 = ω , k1 + k2 = k,and Φ1 +Φ2 = Φ+const. The k-autobicoherence reads:[bW (a1,a2)]2 =|BW (a1,a2)|2[∫|Ws(a1,X)Ws(a2,τ)|2dX ][∫|Ws(a,X)|2dX ], (1)where BW is the wavelet bispectrum defined as:BW (a1,a2) =∫W ∗s (a,X)Ws(a1,X)Ws(a2,X)dX . (2)In these equations, Ws(a,X) is the spatial wavelet transform of the signal s(x), and a, a1 and a2are wavelet scale lengths, which satisfy the relation 1/a = 1/a1 +1/a2. In the present analysis,a Morlet wavelet is used.In the following we focus on the analysis of the weakly turbulent state depicted in Fig. 1.The spatiotemporal plot reveals irregular structures, non-periodic in time. From the wavelet k-power spectrum, various mode numbers are found to co-exist or alternate in time. The dynamicsis dominated by mode numbers m = 4-6 (Fig. 2). Averaging the wavelet ω-spectrum over the33rd EPS Conference on Plasma Phys. Rome, 19 - 23 June 2006 ECA Vol.30I, P-2.120  (2006)Figure 1: Spatiotemporal density fluctuations of drift waves turbulence, measured with an az-imuthal 64 probe array. Discharge parameters are B=69 mT, PAr=0.3 Pa, PRF=2.6 kW.Figure 2: Temporal evolution of the modenumber for the regime depicted in Fig 1.Figure 3: Temporal evolution of the fre-quency for the regime depicted in Fig. 1.64 probes, a similar time evolution is found in the frequency domain, with a main frequencyin the range 3-4 kHz (Fig. 3). Since several modes and frequencies exist simultaneously, thebicoherence will be used in order to detect whether phase coupling occurs between wave triplets.Working with the whole data set leads to an average bicoherence, which is unlikely to givemuch information regarding the turbulent state under investigation (see e.g. Ref [6]). The totalk-bicoherence is obtained by integrating the autobicoherence over all scales at each time instant.Its numerical value is not fundamental, since it depends on the chosen calculation grid.Fig. 4 clearly shows strong temporal variations of the total bicoherence. We notice that thelocations of the maxima and minima of the total bicoherence follow an evolution very similar tothe temporal evolution of the frequency, Fig. 3, and to a less extent to the evolution of the modenumber, Fig. 2. The dashed line in Fig. 4 represents the total statistical noise due to the waveletdecomposition. It is calculated using Eq.(7) of Ref. [6], which is an upper bound. Thus, a larger33rd EPS 2006; F.Brochard et al. : Spatiotemporal investigation of nonlinear coupling and energy transfers in ... 2 of 4bicoherence level can be regarded as a reliable signature of phase coupling.0 1 2 3 4 5 665707580859095100Total bicoherence vs. timeTime (ms)(bk )2Figure 4: Time evolution of the total k-bicoherence. The dashed line indicatesthe maximum noise level.In the following we focus on a smaller time win-dow from t = 1 to 2 ms in order to investigate howthe bicoherence level is increased at this time in-terval (similar results are found as well, e.g., in theinterval 3.4-4 ms). In this interval, a first analysisbased on the averaged f− and k− spectra shows theexistence of phase coupling in both spectra (Fig. 5).More precisely, the k-bicoherence shows couplingsbetween m = 1 and m = 2−6 (horizontal structure),between m = 6 and m = 1− 3 (vertical structure),and between m = 2 and m = 3. The f -bicoherenceshows couplings between 4kHz and 2, 3, 6, and 8kHz, and also between 2kHz and lower fre-quencies. However, these figures don’t take into account the statistical noise, leading to anoverestimation of the bicoherence level, especially at low scales as it will be shown later.Figure 5: f - and k-autobicoherence for the time interval t = 1−2 ms.To get insight into the dynamics of the coupling, it is necessary to further decompose the timeinterval. In the following, the k-bicoherence is calculated at each time instant, and averaged overshort time intervals. The summed bicoherence at three selected intervals is plotted in Fig. 6,along with the corresponding wavenumber spectra. Although the maximum noise level is veryhigh, especially at small wavenumbers, several bicoherence peaks can be seen exceeding thenoise level. Looking at the first time interval (t1 = 1.3− 1.45 ms), one notices that while thewavenumber spectrum is dominated by mode numbers 4, 5, and 6, the main bicoherence levelis observed at wavenumbers corresponding to m=1, 2, and 3. Although the bicoherence level33rd EPS 2006; F.Brochard et al. : Spatiotemporal investigation of nonlinear coupling and energy transfers in ... 3 of 4does not look significant with respect to the noise level, especially in the case of m=1, it isnoticeable that the amplitudes of these modes are increased in the wavenumber spectrum whencalculated in the following time interval (t2 = 1.45− 1.6 ms). Considering the whole figure,it seems reasonable to conclude that small wavenumbers grow due to an energy transfer fromhigher mode numbers: m = 1 is fed by the interactions between m = 4 and m = 5, and/or betweenm = 5 and m = 6, m = 2 by the interactions between m = 4 and m = 6, according to the phasecoupling criterion k1 + k2 = k. While the spectra broaden, the energy is diffused into a widerange of scales, which is a characteristic of a transition to turbulence.2 4 6 8 10100101102t = 1.3−1.45 msS(k) (a.u.)mode #2 4 6 8 100510t = 1.3−1.45 msmode #(bk (a))22 4 6 8 10100101102t = 1.45−1.6 msS(k) (a.u.)mode #2 4 6 8 100510t = 1.45−1.6 msmode #(bk (a))22 4 6 8 10100101102t = 1.75−1.9 msS(k) (a.u.)mode #2 4 6 8 100510t = 1.75−1.9 msmode #(bk (a))2Figure 6: k−power spectra (left column)and summed bicoherence (right column)at three selected time intervals. Dots indi-cate the maximum statistical noise level.In conclusion, it is worth noting that these resultsare in very good agreement with the hypothesis ofan inverse cascade of energy in drift wave turbu-lence, predicted by theoretical work [7]. Althoughthe present paper focuses on the k-bicoherence,preliminary results show a good agreement withfrequency-based analysis, as well as with a comple-mentary analysis based on 3D fluid simulations. Di-rect cascading will be investigated in further steps.References[1] G. Y. Antar et al., Phys. Rev. Lett. 87, 065001(2001).[2] C. Holland et al., Nucl. Fusion 43, 761 (2003).[3] G. R. Tynan et al., Phys. Plasmas 11, 5195(2004).[4] M. Ramisch et al., New J. Phys. 5, 12.1 (2003).[5] C. M. Franck et al., Phys. Plasmas 10, 323(2003).[6] B. Ph. van Milligen et al., Phys. Rev. Lett. 74,395 (1995).[7] A. Hasegawa and M. Wakatani, Phys. Rev. Lett.50, 682 (1983).33rd EPS 2006; F.Brochard et al. : Spatiotemporal investigation of nonlinear coupling and energy transfers in ... 4 of 4

Spatiotemporal investigation of nonlinear coupling and energy transfers in drift wave turbulence

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Spatiotemporal investigation of nonlinear coupling and energy transfers in drift wave turbulence

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