The causal impact of magnetic fluctuations in slow and fast L-H transitions at TJ-II
B.Ph. van Milligen, T. Estrada, B.A. Carreras, E. Ascasíbar, C. Hidalgo, I. Pastor, J.M. Fontdecaba, R. Balbín, TJ-II Team
TThe causal impact of magnetic fluctuations in slow and fast L–Htransitions at TJ-II
B.Ph. van Milligen , T. Estrada , B.A. Carreras , E. Ascas´ıbar , C.Hidalgo , I. Pastor , J.M. Fontdecaba , R. Balb´ın and the TJ-II Team Laboratorio Nacional de Fusion, CIEMAT,Avda. Complutense 40, 28040 Madrid, Spain BACV Solutions, 110 Mohawk Road,Oak Ridge, Tennessee 37830, USA and Instituto Espa˜nol de Oceanograf´ıa, Centro Oceanogr´afico de Baleares,Muelle de Poniente s/n, 07015 Palma de Mallorca, Spain (Dated: October 19, 2018)
Abstract
This work focuses on the relationship between L–H (or L–I) transitions and MHD activity inthe low magnetic shear TJ-II stellarator. It is shown that the presence of a low order rationalsurface in the plasma edge (gradient) region lowers the threshold density for H–mode access. MHDactivity is systematically suppressed near the confinement transition.We apply a causality detection technique (based on the Transfer Entropy) to study the relationbetween magnetic oscillations and locally measured plasma rotation velocity (related to ZonalFlows). For this purpose, we study a large number of discharges in two magnetic configurations,corresponding to ‘fast’ and ‘slow’ transitions. With the ‘slow’ transitions, the developing ZonalFlow prior to the transition is associated with the gradual reduction of magnetic oscillations. Thetransition itself is marked by a strong spike of ‘information transfer’ from magnetic to velocityoscillations, suggesting that the magnetic drive may play a role in setting up the final shearedflow responsible for the H–mode transport barrier. Similar observations were made for the ‘fast’transitions. Thus, it is shown that magnetic oscillations associated with rational surfaces playan important and active role in confinement transitions, so that electromagnetic effects should beincluded in any complete transition model. a r X i v : . [ phy s i c s . p l a s m - ph ] J un . INTRODUCTION The physics of the Low to High (L–H) confinement transition is a topic of fundamentalimportance for the operation of fusion experiments and a future fusion reactor. Since thediscovery of the H mode in 1982, the fusion community has devoted an immense effortto improve the understanding of this remarkable phenomenon [1], and much progress hasbeen made, although many mysteries remain. For example, it is known that the L–Htransition is affected by magnetic perturbations, although the detailed physics of this processis still unclear. The external application of 3D magnetic field perturbations (above a certainminimum value) is generally found to raise the L–H power threshold (e.g., at NSTX [2], DIII-D [3], ASDEX [4], MAST [5]) and to selectively increase particle transport, while hardlyaffecting energy transport (e.g., at NSTX [6], DIII-D [7]).Like externally applied magnetic perturbations, internal MHD instabilities are usuallyconsidered unfavorable for improved confinement [8], although simultaneously, it is widelyrecognized that they may trigger the formation of transport barriers near rational sur-faces [9, 10]. Theoretically, the development of Zonal Flows may be facilitated in zones oflow Neoclassical shear viscosity near low-order rational surfaces [11]. The interaction be-tween MHD modes associated with rational surfaces and Zonal Flows has been addressed in anumber of theoretical works [12–19]. In summary, under specific circumstances, MHD activ-ity associated with rational surfaces might in fact stimulate the development of Zonal Flowsvia the Magnetic Reynolds Stress, leading to the formation of transport barriers [20, 21].MHD activity is often modified sharply at the L–H transition, and usually low frequencyMHD modes are suppressed at the transition (e.g., ASDEX [22], W7-AS [23]), althoughthey are also sometimes triggered by it (LHD [24–26]). Interestingly, MHD fishbone activityhas recently been identified as a trigger of the L–H transition at HL-2A [27]. Similarly,at the Globus-M Spherical tokamak, MHD events were sometimes seen to trigger an L–Htransition [28]. At LHD, an L–H transition was sometimes seen to stabilize MHD modeslocated well inside the plasma [29].Experimentally, this issue is most easily addressed at stellarators, as these devices have abetter control over the magnetic configuration than, e.g., tokamaks. Indeed, the dependenceof confinement on the magnetic configuration is a well-known feature of stellarators [23].Significantly, at W7-AS a dependence of the access to the H-mode on edge iota values was2bserved [23].This work will specifically address the issue of the interaction of MHD activity and the L–H transition at the TJ-II stellarator. In order to study whether the modification of the MHDactivity at the L–H transition is mediated by the modification of profiles or the consequenceof a direct interaction between magnetic fluctuations and Zonal Flows, we will make use ofa causality detection technique that was recently introduced in the field of plasma physics,based on the Transfer Entropy [30].The Low to High confinement (L–H) transition at TJ-II is a ‘soft’ transition in the sensethat the confinement improvement factor is small, and yet it possesses the typical featuresof any H-mode (rapid drop of H α emission, reduction of fluctuation amplitudes, formationof a sheared flow layer, ELM-like bursts) [31–34]. Since the first observation of the H-mode at TJ-II, many experiments have been performed to explore its features. The presentpaper will summarize some of this work with special focus on the interaction between themagnetic configuration (rotational transform and Magneto-HydroDynamic or MHD modeactivity), L–H transitions and the associated Zonal Flow (ZF). The currentless, flexible,low magnetic shear stellarator TJ-II is ideally suited to study this issue due to completeexternal control of the magnetic configuration and, in particular, the capacity to modify therotational transform profile significantly.The structure of this paper is as follows. In Section II, we will analyze the evolutionof some global plasma parameters for a large database of L–H transitions. In Section III,we analyze the interactions between magnetic fluctuations and Zonal Flows during L–Htransitions. Finally, in Section IV, we discuss the results, while in Section V we draw someconclusions. 3 I. STATISTICS OF L–H TRANSITIONS AT TJ-II
The present work summarizes experiments performed between April 2008 and May 2012.In this period, the H-mode was observed in several hundred discharges, in a range of magneticconfigurations.The TJ-II vacuum magnetic geometry is completely determined by the currents flowingin four external coil sets, meaning that each configuration is fully specified by four numbers.However, the magnetic field is normalized to 0.95 T on the magnetic axis at the ECRHinjection point in order to guarantee central absorption of the ECR heating power, so onlythree independent numbers are left, and these are compounded into a label to identify eachconfiguration [35]. At TJ-II, the normalized pressure (cid:104) β (cid:105) is generally low, even in dischargeswith Neutral Beam Injection (NBI), and currents flowing inside the plasma are generallyquite small (unless explicitly driven), so that the actual magnetic configuration is typicallyrather close to the vacuum magnetic configuration [36].The experiments have been carried out in pure NBI heated plasmas (line averaged plasmadensity (cid:104) n e (cid:105) = 2 − × m − , central electron temperature T e = 300 −
400 eV). The NBIinput heating power is kept constant at about 500 kW during the discharge but the fractionof NBI absorbed power – taking into account shine through, CX and ion losses, as estimatedusing the FAFNER2 code [37] – increases from 55 to 70% as the plasma density rises.As noted in the introduction, the L–H transition has a general experimental signature,resulting in a characteristic response of some experimental time traces [31–34]. In this work,we specifically define this transition time point as the time at which the amplitude of thedensity fluctuations measured by the Doppler reflectometer drops sharply. As shown inRef. [32], this time is more precise than, e.g., the time needed for the formation of theshear layer, which may take several ms. This procedure typically allows identifying the L–Htransition time with a precision of about 1 ms.Fig. 1 shows the breakdown of observed L–H transitions according to magnetic configu-ration in a database of 218 discharges. Fig. 2 shows the vacuum rotational transform of themagnetic configurations considered. For simplicity, the magnetic configurations consideredhere are also identified by the value of the vacuum rotational transform (¯ ι = ι/ π = 1 /q )at ρ = r/a = 2 /
3, which is immediately inward from the gradient region for most dis-charges [36]. 4
20 40 60 80 100100_032_60100_033_61100_034_61100_035_61100_036_62101_038_62100_040_63101_042_64100_043_64100_044_64100_045_65100_049_65100_050_65100_051_650 20 40 60 80 100 N H mode database: configurations
FIG. 1: Configuration breakdown of L–H transition discharges at TJ-II ( N is the number ofdischarges). The right axis specifies the value of ¯ ι at ρ = 2 / Fig. 3a shows the mean line average density at the L–H transition. Fig. 3b shows theposition of the main low order rational surfaces. The figure shows that the line averagedensity at which the L–H transition occurs is lower when a major low-order rational existsin the outer regions of the plasma, suggesting that a low-order rational in the edge regionfacilitates the L–H transition. 5 i o t a / π ρ FIG. 2: Vacuum rotational transform of the magnetic configurations considered (the line labelsspecify the first two numbers of the configuration label). The rotational transforms of the mostcommon configurations are drawn as continuous curves, those of the remaining configurations aredashed. Horizontal dashed lines indicate the main low-order rational values. The vertical dashedline indicates the position ρ = 2 / .45 1.5 1.55 1.6 1.65 ι ( ρ =2/3)/2 π n e a t L – H ( m − ) ι ( ρ =2/3)/2 π ρ (a)(b) FIG. 3: Magnetic configuration scan (configurations identified via ¯ ι ( ρ = 2 / ρ = 2 /
3. The vertical dashed linescorrespond to ¯ ι ( ρ = 2 /
3) = 3 / , / , and 5/3. The two small rectangles indicate the approximatemeasurement locations of Doppler Reflectometry, as discussed in the text. . Slow and fast L–H transitions In this work, we will mainly focus on two magnetic configurations with good statisticsand Doppler Reflectometry data (cf. Fig 1, noting that configuration 100 44 64 behaves verysimilar to 100 42 64): • In configuration 100 35 61, with ¯ ι ( ρ = 2 /
3) = 1 . ι = 3 / ρ (cid:39) .
73 in vacuum, cf. Fig. 2. This configuration is characterizedby a ‘slow’ transition. The transition is not straight into the H phase, but rather into anI phase, characterized by Limit Cycle Oscillations (LCOs), as reported elsewhere [21,38] and similar to LCOs reported at other devices [39]. • In configuration 100 42 64, with ¯ ι ( ρ = 2 /
3) = 1 . ι = 8 / ρ (cid:39) .
86 in vacuum, cf. Fig. 2. In this case, the transition is‘fast’ and enters directly into the H phase.Fig. 4 shows the evolution of a specific H α monitor (noting that other H α signals atother toroidal locations behave similarly), averaged over the discharges belonging to eachconfiguration, versus ∆ t , where ∆ t is the time minus the L–H transition time in ms. Inthis and the succeeding graphics, the grey area indicates the shot to shot variation (not theerror). The drop in H α is one of the markers used to identify such transitions.Fig. 5 shows the mean density profiles at the time of the L–H transition, calculated usinga Bayesian profile reconstruction technique described elsewhere [36], and averaged over anumber of discharges. The location of the main low-order rational surface in the edge regionis indicated, showing that the rational surface is located near the ‘foot’ of the gradient regionin both cases. 8 − − − − ∆ t (ms)Config: 101_042_64 HALFAC4 − − − − − ∆ t (ms)Config: 100_035_61 HALFAC4 H α ( a . u . ) H α ( a . u . ) (a)(b) 100_42_64100_35_61 FIG. 4: Mean evolution of H α emission across confinement transitions. (a): configuration 100 42 64(‘fast’ L–H transitions). (b): configuration 100 35 61 (‘slow’ L–I transitions). ρ n e ( m − ) FIG. 5: Mean density profiles at the L–H transition time, calculated using a Bayesian techniqueand averaged over 32 and 26 discharges (for configuration 100 42 64, ‘fast’ L-H transitions, andconfiguration 100 35 61, ‘slow’ L-I transitions, respectively). Arrows indicate the (vacuum) locationof the main low order rational surface in the edge for each configuration.
20 -10 0 10 20 " t (ms) f ( k H z ) -2-1012 -25 -20 -15 -10 -5 0 5 10 15 20 25 " t (ms) M ea n R M S ( a . u . ) -25 -20 -15 -10 -5 0 5 10 15 20 25 " t (ms) M ea n R M S ( a . u . ) -20 -10 0 10 20 " t (ms) f ( k H z ) -2-1012 (a)(c) (b)(d) FIG. 6: Mean evolution of magnetic activity across confinement transitions. (a,b): configuration100 42 64 (‘fast’ L–H transitions, 90 discharges). (c,d): configuration 100 35 61 (‘slow’ L–I tran-sitions, 36 discharges). (a,c): mean RMS of a Mirnov coil. (b,d): mean spectrogram of a Mirnovcoil.
B. Evolution of magnetic activity during L–H transitions
We take the Root Mean Square (RMS) amplitude of a specific Mirnov coil signal as aglobal measure for MHD activity. We calculate the RMS of the coil signal using 0.2 msoverlapping time bins. Fig. 6 shows both the RMS evolution of the coil signal and themean spectrum versus time for this coil. In the case of the ‘slow’ transitions, one observes agradual decrease of MHD activity prior to the L–H transition time (as defined in the mannerdescribed above), the decay starting some ten ms beforehand. This early decay with respectto the reference time is also visible, to some degree, in the H α trace shown in Fig. 4b. Inthe case of the ‘fast’ transitions, the drop of RMS is rather sharp and lasts only a few ms,although one could still argue that the decay starts before the transition time. The drop11n RMS of the Mirnov coil signal from the pre-transition peak to the transition point istypically by a factor of about 2. Afterwards, in I or H mode, the RMS amplitude remainslow. The spectra show that the drop of RMS is mainly associated with the suppression oflow frequency ( f <
80 kHz) MHD modes, and not so much with that of high frequency( f >
80 kHz) Alfv´en modes.Using a poloidal set of Mirnov coils [40], it is possible to identify the dominant modenumbers of the low frequency modes before the L–H transition, leading to the result shownin Table I. In each case, the dominant mode number m corresponds to the main low orderrational surface present inside the plasma according to the vacuum rotational transformprofile, cf. Fig. 2. The dominant frequency of the corresponding mode is indicated, andwhen a given mode (8/5 or 3/2) is further inward, the frequency is higher due to a higherrotation velocity. This is in accordance with earlier work, which showed the existence of arotation velocity profile that increases from the edge inward [35]. TABLE I: Dominant mode numbers prior to L–H transitionConfiguration ¯ ι ( ρ = 2 /
3) Dominant mode Lowest rational ρ (rational) Frequencynumber, m (kHz)100 50 65 1.643 3 5/3 0.83 17100 44 64 1.588 5 8/5 0.76 60100 42 64 1.568 5 8/5 0.86 27100 38 62 1.526 2 3/2 0.33 ∼ The decay of magnetic activity associated with confinement transitions (Fig. 6) is ratherstriking. A priori, the origin of this phenomenon is unclear. One might be inclined to thinkthat the decay is related to a gradual evolution of profiles on the transport time scale (ofthe order of 5 ms) associated with the rising density (cf. Fig. 7). However, in the case ofthe ‘fast’ L–H transition (configuration 100 42 64), the decay is sufficiently fast to make itdoubtful that transport effects are playing a significant role. Section III will attempt to shedsome light on the origin of this phenomenon.12 − − − − ∆ t (ms)Config: 101_042_64 Densidad2_ − − − − − ∆ t (ms)Config: 100_035_61 Densidad2_ (a)(b) n e ( m ) n e ( m ) FIG. 7: Mean evolution of the line average density across confinement transitions (units: 10 m − ). (a): configuration 100 42 64 (‘fast’ L–H transitions). (b): configuration 100 35 61 (‘slow’L–I transitions). Dashed red lines indicate a linear extrapolation of the evolution prior to ∆ t = 0. C. Temporal evolution of some global parameters
Fig. 7 shows the evolution of the mean line average density n e as measured by the in-terferometer [36]. It should be borne in mind that the mean temporal evolution away fromthe transition time may be affected by irrelevant events, such as ECRH switch-off or NBIswitch-on for large negative values of ∆ t , and plasma termination for large positive values of∆ t (this argument applies to all graphs of this paper having ∆ t on the abscissa). However,one observes that the time derivative of the density evolution is increased slightly at thetransition, associated with an enhancement of particle confinement.Fig. 8 shows a summary of the evolution of the corrected diamagnetic energy content [41].13 − − − − ∆ t (ms)Config: 101_042_64 W_b4_corr_ − − − − − ∆ t (ms)Config: 100_035_61 W_b4_corr_ W d i a ( a . u . ) W d i a ( a . u . ) (a)(b) 100_42_64100_35_61 FIG. 8: Mean evolution of W dia across confinement transitions. (a): configuration 100 42 64 (‘fast’L–H transitions). (b): configuration 100 35 61 (‘slow’ L–I transitions). Dashed red lines indicatea linear extrapolation of the evolution prior to ∆ t = 0. In both cases, a significant change of slope is visible, coincident (within about a ms) withthe L–H transition time (as defined above). Since the energy confinement time is defined as τ E = W dia / ( P abs − dW dia /dt ), where P abs is the absorbed power, this change of slope impliesa change in energy confinement time. The sharpness of the change in slope implies that ourdefinition of the L–H transition time is indeed reliable to about one ms.Fig. 9 shows the mean evolution of the Thomson Scattering (TS) [42] profiles acrossthe L-H transition in configuration 100 42 64 (the most populated case, corresponding tothe ‘fast’ transition; insufficient data were available for the ‘slow’ transitions to produce asimilar graph). The TJ-II TS diagnostic is a single pulse diagnostic, so the evolution shown14 e (keV) -20 -10 0 10 20 ∆ t (ms) -0.500.5 ρ n e (10 m -3 ) -20 -10 0 10 20 ∆ t (ms) -0.500.5 ρ p e -20 -10 0 10 20 ∆ t (ms) -0.500.5 ρ -20 -15 -10 -5 0 5 10 15 20 ∆ t (ms) N ( T ho m s on ) FIG. 9: Evolution of mean Thomson Scattering data across the L–H transition in configuration100 42 64 (‘fast’ transitions). Top left: T e ; top right: n e ; bottom left: p e ; bottom right: numberof Thomson Scattering profiles per 5 ms bin. is obtained by interpolating between profiles measured in different discharges at differenttimes, each discharge being characterized by slightly different densities and temperatures.At high densities, the temperature tends to drop due to radiation effects [43]. In spite ofthe fact that the discharges are similar but not identical, the mean evolution is rather clear:across the transition, the T e profile is mostly constant or slightly decreasing in amplitude;on the other hand, the n e profile increases significantly in amplitude and width, associatedwith the establishment of the edge transport barrier (in accordance with results reportedpreviously [36]). Thus, the edge transport barrier mainly affects the electron density, andhardly affects the electron temperature.Fig. 10 shows the mean evolution of the core ion temperature, measured by the chargeexchange diagnostic [44] in configuration 100 42 64 (‘fast’ transitions); insufficient data wereavailable for the ‘slow’ transitions to produce a similar graph. The core ion temperatureis shown to decrease slightly after the L–H transition, similar to T e . Thus, the observedincrease of τ E is due to the increase in density.15
20 -15 -10 -5 0 5 10 15 20 ∆ t (ms) h T i ( ) i ( e V ) FIG. 10: The evolution of (cid:104) T i (0) (cid:105) , the average of the core ion temperature T i (0) over 20 discharges,across the L–H transition in configuration 100 42 64 (‘fast’ transitions). The grey area indicatesthe standard deviation of the shot to shot variation (not the error). II. INTERACTION BETWEEN MAGNETIC AND ZONAL FLOW OSCILLA-TIONS
The previous section has shown that the magnetic configuration plays a role in confine-ment transitions at TJ-II (cf. Fig. 3), but it does not clarify the details. It is known thatZonal Flows play a major role in the confinement transitions in general [13], and this is alsothe case at TJ-II [45, 46]. To study the interaction between magnetic fluctuations and ZonalFlows, we will use the signal from a magnetic poloidal field pick-up coil ( ˙ B ) and signalsfrom the Doppler reflectometry system installed at TJ-II. Although a Mirnov coil measuresglobal magnetic activity, its signal is often dominated by low-order modes associated withlow-order rational surfaces, and we exploit this property here. The Doppler reflectometerallows measuring the perpendicular rotation v ⊥ (within the flux surface) of fluctuations attwo specific radial locations [47]. The perpendicular velocity, v ⊥ ( ρ, t ), is calculated withhigh temporal resolution [48].To analyze the mentioned interaction, we apply a causality detection technique [49–51].The Transfer Entropy between signals Y and X quantifies the number of bits by which theprediction of a signal X can be improved by using the time history of not only the signal X itself, but also that of signal Y (Wiener’s ‘quantifiable causality’).Consider two processes X and Y yielding discretely sampled time series data x i and y j .In this work, we use a simplified version of the Transfer Entropy, T Y → X = (cid:88) p ( x n +1 , x n − k , y n − k ) log p ( x n +1 | x n − k , y n − k ) p ( x n +1 | x n − k ) (1)Here, p ( a, b, c ) is a joint probability distribution over the variables a, b and c , while p ( a | b )is a conditional probability distribution, p ( a | b ) = p ( a, b ) /p ( b ). The sum runs over thearguments of the probability distributions (or the corresponding discrete bins when theprobability distributions are constructed using data binning). The number k is the ‘timelag index’ shown in the graphs in the following sections. The construction of the probabilitydistributions is done using ‘course graining’, i.e., a low number of bins (here, m = 5), toobtain statistically significant results. The value of the Transfer Entropy T , expressed inbits, can be compared with the total bit range, log m , equal to the maximum possible valueof T , to help decide whether the transfer entropy is significant or not.A simple way of estimating the statistical significance of the Transfer Entropy is bycalculating T for two random (noise) signals. Fig. 11 shows the Transfer Entropy calculated17 N -4 -3 -2 -1 T FIG. 11: Transfer Entropy for two random Gaussian signals, as a function of the number of samplesof the signals, N (with m = 5). Each point is calculated as the average over 100 independentrealizations, and the error bar indicates the variation of the result. This average value can be takenas the statistical significance level of the Transfer Entropy. The red dashed line is proportional to1 /N . for two such random signals, with a Gaussian distribution, each with a number of samplesequal to N . It can be seen that the value of the Transfer Entropy (averaged over 100equivalent realizations) drops proportionally to 1 /N .In interpreting the Transfer Entropy, it should be noted that it is a non-linear quantifierof information transfer that can help clarifying which fluctuating variables influence whichothers. In this sense, it is fundamentally different from the (linear) cross correlation, whichonly measures signal similarity. For example, the cross correlation is maximal for two iden-tical signals ( X = Y ), whereas the Transfer Entropy is exactly zero for two identical signals(as no information is gained by using the second, identical signal to help predicting thebehavior of the first).As with all methods for causality detection, an important caveat is due. The methodonly detects the information transfer between measured variables. If the net informationflow suggests a causal link between two such variables, this may either be due to a direct18ause/effect relation, or due to the presence of a third, undetected variable that can mediateor influence the information flow.Fig. 12 (a) shows the linear correlation between v ⊥ and a Mirnov coil signal, ˙ B θ , inconfiguration 100 35 61 (‘slow’ transitions). The correlation has been calculated using over-lapping time windows with a length of 0.5 ms each. Before the transition, a regular structureis visible, mainly reflecting the presence of an MHD mode with a frequency of slightly morethan 10 kHz. Immediately after the transition, lasting slightly more than 5 ms, a relativelystrong correlation is visible as a diagonal striped feature.Fig. 12 (b) shows the Transfer Entropy, calculated with m = 5, using overlapping timewindows of 0.5 ms, for a lag of k = 500, equivalent to 50 µ s. The 0.5 µ s time windowcorresponds to N = 5000 data points, so the statistical error level of the Transfer Entropyis (cid:39) .
01 (cf. Fig. 11). As noted above, the theoretical maximum of T equals log m = 2 . v ⊥ on a time scale of 50 µ s. Note the coincidence of this time window with the correlationsignature seen in Fig. 12 (top).Fig. 13 shows similar results for configuration 100 42 64 (‘fast’ transitions). Again, avertically striped structure appears in the correlation at the transition. The Transfer Entropypeaks sharply at the transition, the dominant direction of the interaction being from ˙ B to v ⊥ (i.e., ˙ B ‘leads’).To obtain a more general view of the interaction between magnetic fluctuations (mea-sured by the Mirnov pick-up coil) and the perpendicular flow velocity (associated with ZonalFlows), we analyze Doppler Reflectometry data from a series of discharges in which the ra-dial position was varied systematically [52, 53]; the same data have been analyzed beforeto calculate the auto-bicoherence [54]. We do this analysis for the two of the most pop-ulated configurations in the database, namely 100 42 64 (¯ ι ( ρ = 2 /
3) = 1 . ι ( ρ = 2 /
3) = 1 . ι ( ρ = 2 /
3) = 1 .
20 -10 0 10 20 " t (ms) T _ B ! v ? v ? ! _ B -20 -10 0 10 20 ∆ t (ms) -0.2-0.100.10.2 T i m e l a g ( m s ) -0.4-0.200.20.4 T i m e l a g ( m s ) -20 -10 0 10 20 (a)(b) FIG. 12: Discharge 27135, configuration 100 35 61 (¯ ι ( ρ = 2 /
3) = 1 . v ⊥ of Doppler reflectometry channel 1 ( ρ (cid:39) .
77 in the H phase) and Mirnov coil signal,˙ B θ (vertical axis: time delay). (b): Transfer Entropy between ˜ v ⊥ of Doppler reflectometry channel1 and a Mirnov coil signal, ˙ B θ . Settings: m = 5, lag k = 500, or 50 µ s. configuration 100 42 64 (¯ ι ( ρ = 2 /
3) = 1 .
20 -10 0 10 20 " t (ms) T _ B ! v ? v ? ! _ B -20 -10 0 10 20 ∆ t (ms) -0.2-0.100.10.2 T i m e l a g ( m s ) -0.4-0.200.20.4 T i m e l a g ( m s ) -20 -10 0 10 20 (a)(b) FIG. 13: Discharge 23029, configuration 100 42 64 (¯ ι ( ρ = 2 /
3) = 1 . v ⊥ of Doppler reflectometry channel 1 ( ρ (cid:39) .
86 in the H phase) and Mirnov coil signal,˙ B θ (vertical axis: time delay). (b): Transfer Entropy between ˜ v ⊥ of Doppler reflectometry channel1 and a Mirnov coil signal, ˙ B θ . Settings: m = 5, lag k = 500, or 50 µ s. . Spatio-temporal analysis, configuration 100 35 61 (‘slow’ transitions) Fig. 14(a) shows the Transfer Entropy T v ⊥ → σ ( | ˜ n | ) ( m = 5, lag k = 500, or 50 µ s), re-flecting the interaction between the perpendicular flow velocity and the density fluctuationamplitude, both measured by the reflectometer. The statistical error level of the reported T values is 0.01. σ ( | ˜ n | ) is the running RMS of the density fluctuation amplitude signal(calculated using a running time window of 2 µ s), considered a measure of the turbulenceamplitude envelope. The radii calculated for the Doppler Reflectometer channels vary ac-cording to the evolving density profile. The radii reported on the ordinate of the graphswere calculated for density profiles corresponding to ∆ t >
0. The figure shows that theperpendicular velocity mainly has a causal impact (in the restricted sense explained above)on the density fluctuations in a period of about 30 ms after the L–I transition, in a specificradial range. In vacuum, the radial position of the ¯ ι = 3 / ρ (cid:39) . ρ (cid:39) . − . T σ ( | ˜ n | ) → v ⊥ (not shown), is much smaller, in line with the expectations based on a similaranalysis for a simplified predator-prey model [51], in which the dominant direction of theinteraction between turbulence amplitude and sheared flow was found to be from shearedflow to turbulence amplitude. Thus, these results constitute a validation of the analysismethod in the framework of confinement transitions and LCOs.As this interaction occurs mainly after the transition, it is unlikely to be responsible forthe transition. In the following, we will examine the role played by magnetic fluctuations inthe transition.Fig. 14(b) shows the Transfer Entropy T v ⊥ → ˙ B ( m = 5, lag k = 500, or 50 µ s), to study theinteraction between the perpendicular flow velocity and the magnetic fluctuations, measuredby a pickup coil. The Transfer Entropy is now only large in a time period of about 25 mspreceding the transition time. Thus, one may presume that the interaction between v ⊥ and˙ B is associated with the gradual reduction of RMS( ˙ B ) near the transition, as shown inFig. 6. 22e draw attention to the fact that there seem to be two predominant zones of interaction:one at ρ (cid:39) .
76 and one at ρ (cid:39) .
8. To emphasize this fact, a single level contour has beendrawn in the figure (black). It should be noted that the radii in the L phase (∆ t <
0) aresomewhat smaller than indicated on the ordinate axes of the graphs (valid for the I phase,∆ t > ρ L (cid:39) − .
656 + 1 . ρ I , so the actual radii are, respectively, ρ L (cid:39) .
69 and 0 .
76, bracketing the theoretical vacuum position of the rational surface (0 . T ˙ B → v ⊥ ( m = 5, lag k = 500, or 50 µ s). The TransferEntropy exhibits a short burst at the L–I transition time, at the same two radial locationsas in Fig. 14(b). This phenomenon corresponds to the short-lived interaction described inthe preceding section. We note that the direction of the interaction is reversed; here, themagnetic fluctuations are ‘influencing’ the perpendicular fluctuating velocity, precisely at thetransition. The results shown seem to indicate that magnetic fluctuations play an importantrole in confinement transitions.We note that the definition of the ‘transition time’ used to define ∆ t = 0 is based on globaldischarge parameters such as the decay of H α emission, which means that the definition maydeviate from the ‘true’ transition time by about a ms. Thus, it may be that the burst ofTransfer Entropy, T ˙ B → v ⊥ , is a more precise marker of the ‘actual’ transition.23 = 500; T v ⊥ → σ ( | ˜n | ) -20 -10 0 10 20 ∆ t (ms) ρ k = 500; T v ⊥ → ˙B -20 -10 0 10 20 ∆ t (ms) ρ k = 500; T ˙B → v ⊥ -20 -10 0 10 20 ∆ t (ms) ρ (a)(b)(c) FIG. 14: Configuration 100 35 61 (¯ ι ( ρ = 2 /
3) = 1 . T v ⊥ → σ ( | ˜ n | ) . (b)Transfer Entropy T v ⊥ → ˙ B . (c) Transfer Entropy T ˙ B → v ⊥ . . Spatio-temporal analysis, configuration 100 42 64 (‘fast’ transitions) In this configuration, the plasma makes a rapid transition from the L to the H state.Fig. 15(a) shows the Transfer Entropy T v ⊥ → ˙ B ( m = 5, lag k = 500, or 50 µ s), whileFig. 15(b) shows the Transfer Entropy T ˙ B → v ⊥ ( m = 5, lag k = 500, or 50 µ s). The statisticalerror level of the reported T values is 0.01. The radii shown correspond to the H phase. TheTransfer Entropy is sharply concentrated at the L–H transition. The temporal sharpness ofthe interaction between perpendicular velocity and magnetic fluctuations is likely related tothe sharp decay of RMS( ˙ B ) shown in Fig. 6 for this configuration. In vacuum, the radialposition of the ¯ ι = 8 / ρ (cid:39) .
86 (cf. Fig. 2). Probably, the rationalsurface is shifted inward somewhat in the presence of the plasma with a small positive netcurrent, and possibly coincides with the radial position at which the Transfer Entropy isshowing a response ( ρ (cid:39) . − . t (cid:39) = 500; T v ⊥ → ˙B -20 -10 0 10 20 ∆ t (ms) ρ k = 500; T ˙B → v ⊥ -20 -10 0 10 20 ∆ t (ms) ρ (a)(b) FIG. 15: Configuration 100 42 64 (¯ ι ( ρ = 2 /
3) = 1 . T v ⊥ → ˙ B . (b) TransferEntropy T ˙ B → v ⊥ . V. DISCUSSION
In this work, we have shown how various global parameters evolve systematically acrossthe L–H transition at TJ-II in a number of magnetic configurations.First, we have shown that the confinement transitions are affected by the presence ofrational surfaces. The line average density at which the L–H (or L–I) transition is triggeredtends to be lower when a low order rational surface is present in the edge (gradient) region(approximately, 0 . < ρ < . ι = n/m = 3 / / T ), tostudy the interaction between magnetic fluctuations, ˙ B (measured by a poloidal field pick-up coil), and the local perpendicular plasma rotation velocity, v ⊥ (measured by DopplerReflectometry). We obtain a clear and significant interaction between magnetic fluctuationsand the fluctuating velocity. Making a distinction between two magnetic configurations (cor-responding to different low-order rational surfaces in the edge region), we find the following.Configuration 100 35 61 (¯ ι ( ρ = 2 /
3) = 1 . / T v ⊥ → σ ( | ˜ n | ) measures the causal impact of the poloidal velocity (Zonal Flow) on the turbu-lence amplitude envelope, and is large after the transition, during the LCOs reported inearlier work [38], cf. Fig. 14(a). The direction of the dominant interaction was in line withexpectations obtained from a simplified predator-prey model [51].To clarify the interaction between Zonal Flows and magnetic oscillations, we calculatedthe Transfer Entropy T v ⊥ → ˙ B , and found that it is large at two radial positions associatedwith the rational surface and during several tens of ms before the transition, cf. Fig. 14(b).Thus, it seems that the gradual reduction of magnetic fluctuation amplitude prior to thetransition, reported in Fig. 6, can be explained by the gradual development of Zonal Flowsassociated with the rational surface. At the transition itself, however, there is a short burstof interactions in the opposite direction such that T ˙ B → v ⊥ is large, cf. Fig. 14(c), suggesting27hat the magnetic fluctuations play an important role during the L–I transition, when themean sheared flow is formed.In configuration 100 42 64 (¯ ι ( ρ = 2 /
3) = 1 . T v ⊥ → ˙ B and T ˙ B → v ⊥ are both large in a relativelynarrow time window around the transition time, and near the radial position of the rationalsurface, cf. Fig. 15, although the precise sequence of events is hard to follow for these ‘fast’transitions. In any case, it seems the magnetic fluctuations play an essential role in thetransition.It is still unclear why the rational 8 / / . CONCLUSIONS In this work we have analyzed an extensive database of confinement transitions at theTJ-II stellarator. The mean line average density at the L-H (or L–I) transition was found tobe lower when a low-order rational surface was present in the edge region, cf. Fig. 3. Also,it was found that low frequency MHD activity was systematically and strongly suppressednear the transition, cf. Fig. 6. Together, these are clear indications that the presence oflow-order rational surfaces in the region where the Zonal Flow forms (the edge or densitygradient region) affects the confinement transition.We applied a causality detection technique to reveal the detailed interaction between mag-netic fluctuations (measured by a pick-up coil) and the perpendicular flow velocity (measuredby Doppler Reflectometry). We conclude that with ‘slow’ transitions, the developing ZonalFlow prior to the transition is associated with the gradual reduction of magnetic oscillations.Apparently, this reduction of the amplitude of the MHD mode oscillations is a prerequisitefor the confinement transition to occur. At the transition, we observe a strong spike of ‘infor-mation transfer’ from magnetic to velocity oscillations, suggesting that the magnetic drivemay play a role in setting up the final sheared flow, responsible for the transport barrier.Similar observations were made for the ‘fast’ transitions, although the temporal resolutionwas insufficient to clarify the sequence of events fully.This work clearly suggests that magnetic oscillations associated with rational surfacesplay an important and active role in confinement transitions, so that electromagnetic effectsshould be included in any complete transition model [12]. Further work to confirm or studythis issue on other devices is suggested. In this framework, we consider that causalitydetection based on the Transfer Entropy constitutes an indispensable tool.
Acknowledgements
Research sponsored in part by the Ministerio de Econom´ıa y Competitividad of Spainunder project Nrs. ENE2012-30832, ENE2013-48109-P and ENE2015-68206-P. This workhas been carried out within the framework of the EUROfusion Consortium and has receivedfunding from the Euratom research and training programme 2014-2018 under grant agree-ment No 633053. The views and opinions expressed herein do not necessarily reflect those29f the European Commission. 30 eferences [1] F. Wagner. A quarter-century of H-mode studies.
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