Spectral Estimation of Plasma Fluctuations II: Nonstationary Analysis of ELM Spectra

SPECTRAL ESTIMA TION OF PLASMA FLUCTUA TIONS I I: NONST A TIONAR Y ANAL YSIS OF ELM SPECTRA Kurt S. Riedel ∗ , Alexander Sidorenk o ∗ , Norton Bretz † , and Da vid J. Thomson + ∗ Couran t Institute of Mathematical Sciences, New Y ork Univ ersit y , New Y ork, New Y ork 10012-1185 † Princeton Plasma Ph ysics Lab oratory , Princeton, NJ 08544-0451 + A T&T Bell Lab oratories, Murra y Hill, NJ 07974-0636 Abstract Sev eral analysis metho ds for nonstationary fluctuations are describ ed and applied to the edge lo calized mode (ELM) instabilities of limiter H-mo de plas- mas. The microw a v e scattering diagnostic observes p oloidal k θ v alues of 3.3 cm − 1 , a v eraged ov er a 20 cm region at the plasma edge. A short autoregres- siv e filter enhances the nonstationary comp onent of the plasma fluctuations b y removing m uc h of the background lev el of stationary fluctuations. Betw een ELMs, the sp ectrum predominan tly consists of broad-banded 300-700 kHz fluc- tuations propagating in the electron diamagnetic drift direction, indicating the presence of a negativ e electric field near the plasma edge. The time-frequency sp ectrogram is computed with the multiple taper tec hnique. By using the sin- gular v alue decomposition of the sp ectrogram, it is shown that the sp ectrum during the ELM is broader and more symmetric than that of the stationary sp ectrum. The ELM p erio d and the ev olution of the sp ectrum b etw een ELMs v aries from disc harge to disc harge. F or the disc harge under consideration whic h has distinct ELMs with a 1 msec p erio d, the sp ectrum has a maximum in the electron drift direction which relaxes to a near constant v alue in the first half millisecond after the end of the ELM and then gro ws slowly . In contrast, the lev el of the fluctuations in the ion drift direction increases exp onentially b y a factor of eight in the five milliseconds after the ELM. High frequency precur- sors are found whic h o ccur one millisecond b efore the ELMs and propagate in the ion drift direction. These precursors are very short ( ∼ 10 µ secs), coheren t bursts, and they predict the o ccurrence of an ELM with a high success rate. A second detector, measuring fluctuations 20 cm from the plasma edge with k θ v alues of 8.5 cm − 1 , sho ws no precursor activit y . The spectra in the ion drift direction are v ery similar on b oth detectors, while the “electron” spectrum lev el is significantly larger on this second detector. 1 P ACS 52.35, 52.55, 52.70, 06., 2.50 2 I. In tro duction Edge lo calized mo des (ELMs) occur in the H-mo de phase 1 , 2 of a tok amak discharge and are usually asso ciated with p erio dic loss of confinement and transient dep osition of heat on the div ertor/limiter. A v ariety of differen t types of ELMs hav e b een observ ed (and ELM precursors) 1 − 7 , and there remains no widespread consensus on the nature and causes of ELMs. Therefore, it is desirable to understand the physical c haracteristics of ELMs and if p ossible to predict their o ccurrence. In one class of ELM disc harges 1 , 5 , 6 , a magnetohydrodynamic (MHD) precursor is seen prior to the ELM. When the precursor lasts for man y sampling times and consists of a single mo de, straightforw ard analysis is adequate to resolv e the basic features of the ELM precursor. Another class of precursors 2 − 5 , consisting of a transient burst of high frequency fluctuations, has also b een observ ed. F or these short lived bursts, the choice of analysis metho ds can noticeably enhance the resolution of the transien t ev en ts. In this article, w e describ e a num ber of adv anced signal pro cessing techniques and apply these metho ds to a limiter H-mo de discharge from the tok amak fusion test reactor (TFTR) 2 , 7 , 8 . Our goal is to sho w ho w these metho ds give high resolution es- timates of transient plasma fluctuations. In a previous article 9 , w e describ e adv anced sp ectral analysis metho ds whic h giv e highly accurate sp ectral estimates for relativ e short time series. F or the transien t ELM precursors, the t ypical data length is 100 p oin ts so that high resolution estimates are particularly imp ortant. W e examine the evolution of the sp ectrum b et ween ELMs. The sp ectral densit y fluctuates strongly on the millisecond time scale. This indicates that there are only a v ery small n um b er of w av es presen t in a given frequency range at one time. When the sp ectrum is a v eraged o ver a n um b er of milliseconds, the fluctuations can b e treated as a stationary pro cess. The time p erio d b et w een ELMs and the sp ectral evolution b et w een ELMs v aries appreciably b et w een discharges. W e consider a particular TFTR discharge, # 49035, whic h has particularly well defined ELMs. W e caution that other TFTR disc harges sometimes hav e distinctly different b eha vior and that no single pattern can describ e the ELM b eha vior in all disc harges. Our results confirm the findings of Ka y e et al. 3 and McGuire et al. 4 that one or more short bursts of high frequency fluctuations precede the ELM b y a fraction of a millisecond. Our findings giv e a more precise characterization of the TFTR ELM precursors: the precursor bursts o ccur in the 500 kHz range propagating in the ion drift direction, and therefore hav e a sp ectral densit y which is distinctly different from 3 b oth the stationary sp ectrum b etw een ELMs and the ELM sp ectrum itself. Our sampling rate is ten times faster than that in Ref. 3; th us it is understandable that w e observe precursors with one ten th the duration as Kay e et al 3 . W e often use the terms “electron” sp ectrum and “ion” sp ectrum to denote the p oloidal direction of propagation in the electron and ion diamagnetic drift directions resp ectiv ely . After the ELM, the in tensity of the sp ectrum in the electron drift direction grows b y 50–80% during the first half millisecond after the end of an ELM. After the half millisecond, the intensit y of the electron drift sp ectrum saturates at a more or less constan t v alue. In contrast, the intensity of the sp e ctrum in the ion drift dir e ction incr e ases exp onential ly b etwe en ELMs. During the 5-6 mil lise c onds b etwe en ELMs, this “ion ” sp e ctrum gr ows by a factor of eight. When this “ion” sp ectrum grows more slo wly , the onset of the ELM is delay ed. Thus, in this particular disc harge, the ELM onset app ears correlated with the “ion” sp ectrum reaching a critical lev el. In Section I I, w e describ e the TFTR H-mo de micro w av e scattering data set 10 − 12 whic h w e use as an example. In Section I I I, w e review autoregressiv e filters 13 , 14 and sho w how these filters highlight nonstationary phenomena. W e consider time- frequency represen tations of evolutionary sp ectra 15 , 16 in the next three sections. In Section IV, we compute the singular v alue decomp osition of the time-frequency dis- tribution to isolate the sp ectrum during the ELMs. In Section V, we examine the ev olution of the sp ectral densit y b etw een ELMs. In Section VI, we use a high res- olution, prewhitened, m ultitap ered evolutionary sp ectral estimate to examine high frequency precursors to the ELMs. I I. Stationary analysis of the TFTR ELM data set a) Microw a v e scattering data set The TFTR micro w av e transmitter launches a 60 GHz plasma w av e linearly p o- larized in the extraordinary mo de b elow the electron cyclotron frequency 10 − 11 at 112 GHz. The 60 GHz extraordinary mo de plasma wa v e propagates in the p oloidal plane through the plasma to the edge, where plasma densit y fluctuations scatter the in- coming wa ve . Figure 1 displays a ray tracing calculation of the extraordinary mode w a ve path. The center of the b eam path is the central curv e while the outer curv es mark the b eam 1 e p o w er half-width of 5.0 cm. The scattered w a ve is measured by t w o detectors located near the b ottom of the v acuum v essel. A third detector at the top of the vessel, kno wn as detector #1, measures backscattered pow er at | ~ k scat | ∼ 20 cm − 1 , and is not used in this article. 4 W e examine the fluctuation sp ectrum of TFTR dischar ge #49035 as measured by the microw a v e scattering diagnostic 10 − 12 . The detected signal is down-sh ifted from 60 GHz to 1 MHz using standard intermediate frequency (IF) techniques. The time series b egins 4.2 sec into the discharge, and is totally con tained in the H-mo de phase. Our data consist of 524,288 time samples with a uniform sampling rate of 5 MHz o v er the time in terv al. Thus, the fluctuations are recorded ov er a ten th of a second time interv al. The data has b een band-pass filtered with an an ti-aliasing filter with a low-pass filter half-width of 2.5 MHz. F or TFTR discharge #49035, the plasma parameters are the toroidal magnetic field: B t = 4 . 0 T esla; the edge q , q a = 5 . 7; the total curren t, I p = 0 . 9 MA; the line a v erage densit y , ¯ n e = 2 . 5 × 10 13 cm − 3 ; and the absorb ed p ow er, P N B I = 13 MW. The cen tral electron temp erature is appro ximately 6 keV and the cen tral ion temp erature is approximately 21 k eV. In the scattering v olume, the lo cal plasma parameters are ρ i = 0 . 16 cm, ρ S ∼ ρ i q T e /T i = 0.09 cm. W e concen trate on detector #3, whic h measures fluctuations av eraged ov er a 20 cm region at the plasma edge. Figure 1a displays the geometry of the transmitted extraordinary mo de and the receiver antenna pattern. The three curves emerging from the b ottom righ t are the cen ter of the detector line of sight and its inner and outer 1 e p o w er sensitivit y . The ellipses denote the pro duct of the transmitter p o w er profile and the receiv er an tenna profile. Detector #3 measures | ~ k scat − ~ k inc | ∼ 3 . 3 cm − 1 and ~ k is parallel to the p oloidal magnetic field at r a ∼ 1 . 0 ± 0 . 1. Detector #2 measures fluctuations somewhat farther in the plasma interior, at r a ∼ 0 . 75 ± 0 . 1 with | ~ k scat − ~ k inc | ∼ 8 . 5 cm − 1 . Figure 1b gives the corresp onding line of sight information for detector #2. Unless otherwise sp ecified, our analysis is based on detector #3, and detector #2 will b e used for comparison. Unfortunately , w e cannot determine whether the signal differences from detector #2 to #3 are due to detector #2 measuring smaller w av elength turbulence or due to the scattering volume for detector #2 b eing lo cated farther in the in terior of the plasma. ELM activit y is not observed when the center of the scattering volume is lo cated inside r a ∼ 0 . 5. These strongly beam-heated disc harges are rotating toroidally with a v elo cit y pro- file whic h is p eak ed on axis with v φ ( r = 0) ∼ 10 6 − 10 7 cm/sec. A t the edge, the toroidal rotation is typically very small, v φ < 3 × 10 5 cm/sec. Because the fluctu- ations are primarily aligned p erp endicular to the total magnetic field, the observ ed p oloidal Doppler shift is prop ortional to ∆ ω ∼ k θ v φ ( r ) B θ B φ . Thus, only sp ectral shifts of less than 20 kHz may b e due to toroidal fluid motion 2 . The ion and electron drift frequencies are appro ximately 50 kHz at r a ∼ 0 . 9 for ˜ k θ ∼ 3 . 3 cm − 1 . Detailed profile information in the region 0 . 9 ≤ r a ∼ 1 . 0 is not av ailable, how ev er. 5 b) Exploratory analysis W e b egin b y computing the sp ectrum using the multiple sp ectral windo w analysis as describ ed in Refs. 9, 17-19. W e use 20 orthogonal tap ers and then kernel smo oth the estimate o v er a 20 kHz frequency bandwidth. In assuming the plasma fluctuations are stationary , we are av eraging the fluctuation sp ectrum ov er the short ELM bursts and the long quasistationary times b et ween ELMs. Figure 2 displays this “comp osite” sp ectrum whic h we compute using the smo othed multitaper metho d 9 , 17 , 19 . The solid curv e is the estimated sp ectrum from detector #3 and the dashed curve is from detector #2. On detector #3, a frequency shift greater than 1 MHz represents fluctuations whic h are tra veling p oloidally in the electron drift direction in the lab oratory frame. F requency shifts less than 1 MHz correspond to fluctuations mo ving in the ion drift direction in the lab oratory frame. Detector #2 measures k θ fluctuations with the ˜ k θ direction rev ersed in comparison with that of detector #3, and therefore electron drift frequencies are rev ersed with resp ect to 1 MHz on detector #2. T o comp ensate for this reversal of ˜ k θ in detector #2, w e ha ve reflected the frequencies ab out the 1 MHz line: f → 1 MHz – f . The sp ectral peak near 1 MHz is partially coheren t. The broadening of the 1 MHz p eak is b eliev ed to b e due to intense edge fluctuations 9 , 11 , 12 at low k , k ⊥ ∼ 0 . 1 cm − 1 . The detector sensitivity to these lo w k fluctuations is w eak, but the edge fluctuation in tensit y is sufficien tly large that broadening is appreciable. The broadening of the 1 MHz line on detector #3 is larger than that of detector #2 b ecause detector #3 measures fluctuations muc h closer to the plasma edge. The spectrum is asymmetric with a larger sp ectral density on the high frequency/electron drift side, as is t ypical of H-mo de discharges 2 . There is a plateau in the sp ectral den- sit y from 1300 kHz to 1500 kHz. A frequency shift of 300 kHz at k ⊥ ' 3.3 cm − 1 corresp onds to a fluid velocity of ˜ v θ ' 2 π ∆ f ˜ k θ ∼ 6 × 10 5 cm/sec. The edge toroidal rotation v elo cit y is measured b y the CHERS diagnostic 2 , 8 , and is at most B θ B φ × 3 × 10 5 cm/sec ' 3 × 10 4 cm/sec. Th us, the toroidal rotation is not resp onsible for the fre- quency shift. The poloidal fluid rotation v elocity is not measured, and can account for an unkno wn but significan t fraction of the frequency shift. The other p ossible explanation for the asymmetric broadening of the sp ectrum is the presence of plasma fluctuations which are propagating in the electron drift direction with a frequency range of 300-700 kHz. These fluctuations ha ve frequencies fiv e to ten times larger than the electron diamagnetic frequency . W e are unaw are of 6 an y plasma instabilities with frequencies in this range. Th us, we b eliev e the most lik ely cause of the frequency shift is the presence of a strong p oloidal electric field, leading to strong p oloidal fluid motion. W e also measure fluctuations in the ion drift direction in the 500 kHz range. One explanation is that the gradien t in the p oloidal electric field is sufficien tly strong that b oth the ion and electron frequency shifts are caused by the p oloidal electric field which is c hanging signs within the scattering v olume. In DI I I-D 5 , 20 , 21 , H-mo de electric fields are observ ed to b e negativ e near the edge scap e-off llay er and to b ecome p ositiv e in the interior. In discharges such as #49035, we primarily measure the p oloidal comp onen t of the w a v e v ector. In other disc harges where the fluctuations are primarily measured in the radial direction, we find the frequency shift is m uc h less. This supp orts our b elief that the p oloidal electric field is primarily resp onsible for the frequency shift. In Figure 2, the sp ectrum in the ion drift direction de c ays exp onential ly in fr e- quency with the same r ate in b oth dete ctors. T o normalize the amplitudes, we ha v e m ultiplied the sp ectrum of detector #2 b y a factor of ten. W e hav e chosen this normalization such that the amplitudes of the tw o detectors are the same for the frequencies where the sp ectrum of detector #3 b egins to broaden rapidly . This nor- malization is natural because these frequencies constitute the b eginning of the true scattered signal. With this normalization, the “ion” sp ectrum in b oth detectors has the same amplitude while the electron amplitude is three times larger. Since our nor- malization of detector #2 is somewhat arbitrary , the precise ratio is also arbitrary . Nev ertheless, the electron turbulence is stronger on detector #2, and th us the elec- tron turbulence is stronger at r a ∼ 0 . 75 than at r a ∼ 1 . 0, or the electron turbulence is stronger for ˜ k θ ∼ 3 . 3 cm − 1 than for ˜ k θ ∼ 8 . 5 cm − 1 . Figure 3a displa ys the 12 millisecond data segment from t = 2 msec to t = 14 msec. The ELMs o ccur at t = 6.07 and 13.19 msec. W e will show that c haracteristically , ELM precursor bursts are cen tered in the 250 − 650 kHz range roughly 1 msec prior to the ELM. Two of these precursor bursts in the 250 − 650 kHz range o ccur at t = 5.186 and 12.734 msecs. But, it is v ery difficult to identify the 250 − 650 kHz bursts in the ra w data. The ELMs app ear to consist of a series of distinct bursts, and resem ble a sinusoid whic h is modulated at high frequency . W e ha v e tried unsuccessfully to develop a joint time-frequency representation of the ELM burst. I I I. Autoregressiv e filters to enhance nonstationary ev en ts 7 In this section, we assume that the measured time series, { x i } , is related to an underlying basic time series, { z i } , through an autoregressive (AR) pro cess 13 , 14 of order p : x t = p X i =1 α ( p ) i x t − i + z t . (1) { z i } is called the innov ation sequence b ecause it is usually assumed to b e a series indep enden t random p erturbations. The autoregressive mo del introduces time corre- lation into the measured time series through the autoregressive lag parameters, α ( p ) i . W e denote an autoregressiv e filter of length p by AR( p ). Sev eral differen t n umerical methods for determining the autoregressive lag param- eters, α ( p ) i , are given in Refs. 13-14. W e estimate the autoregressiv e parameters b y solving the Y ule-W alk er equations, i.e. minimize the residual sum of squares: ˆ σ 2 z = 1 N − p N X t = p x t − p X i =1 α ( p ) i x t − i ! 2 . (2) The main p oin t of this section is that the autor e gr essive filter appr e ciably enhanc es the nonstationary bursts in the data b e c ause the filter r emoves the stationary p art of the signal. Figure 3a displa ys the raw data and Fig. 3b plots the residuals, { z i } , of the autoregressive filter. In Figure 3b, the ELM amplitude is enhanced relative to the background lev el b ecause the ELM sp ectrum is broader than the am bient spectrum. The t w o precursor bursts in the 250 − 650 kHz range, at t = 5.186 and 12.734 msecs, ha ve b een noticeably enhanced relative to the raw data of Fig. 3a. The large residuals in Fig. 3b last for roughly 60 data points, whic h corresp onds to 11 µ sec. A third precursor in the 250 − 650 kHz range, at t = 5.56 msecs, is less visible in the AR residual plot and can only b e seen in the time-frequency plots of Sec. VI. In Figure 3a, there are tw o large non-ELM bursts, at t = 2.44 and 3.01 msecs, whic h are cen tered in the 1 MHz frequency range. Most of the other fluctuation bursts are also due to changes in the 1 MHz frequency range (shown in Sec. VI). These 1 MHz features are greatly reduced in Fig. 3b due to the AR filter. As a second example to illustrate the p o w er of the AR filter to emphasize nonsta- tionary bursts, w e consider the Ohmic sawtooth TFTR discharge # 50616 which w e studied in Ref. 9. F or this discharge, the microw a v e scattering v olume is cen tered at r a = 0 . 3 and k θ = 4 cm − 1 . Figure 4a displa ys the 65,535 p oin t ra w data. The sawtooth oscillation at 3.86 msec is barely visible. The blip at 9.12 msec is an unkno wn and unrelated even t. Fig. 4b plots the residuals of the autoregressive filter whic h nearly remo v es the 1 MHz p eak. The sawtooth is easily discernable in the filtered data in con trast to the raw data. 8 The v ariance of the raw data in Fig. 4a app ears to b e increasing in time, thereby calling into question the assumption of stationarity . Figure 4b sho ws that the filtered data has no temp oral increase in the v ariance. Thus the increasing v ariance of the ra w data is asso ciated with the 1 MHz IF of the receiv er and not with the broad-banded plasma fluctuations 11 , 12 . The changing amplitude of the 1 MHz p eak o ccurs due to phase c hanges b et w een the scattered signal and the lo cal oscillator of the detection circuit and is not directly related to c hanges in the plasma fluctuations. The autoregressive mo del may also b e used to detect and correct statistical out- liers. W e compare the measured data with the predicted v alue giv en by the AR mo del. When the residual error is many standard deviations large, we mark/plot the outliers for further scrutiny . In robust spectral estimation 22 , 23 , sev ere statistical outliers are replace by their fitted v alues. In robustifying the data, we fo cus on the stationary sp ectrum and discard the transien t burst associated with the ELMs and precursors. Since we are primarily interested in these transient phenomena, w e do not r obustify the data. When b oth pro cesses are stationary , the sp ectra of the measured and inno v ation pro cesses, { x } and { z } , are related b y S x ( f ) = S z ( f )    1 − P p k =1 α ( p ) k e − 2 π ikf    2 = | Γ( f ) | 2 S z ( f ) , (3) where the sp ectral transfer function, Γ( f ), is defined b y Eq. (3). In AR sp ectral anal- ysis, we assume that the autoregressive filter has remov ed all of the time correlation and that { z i } are indep enden t iden tically distributed random v ariables with v ariance, σ 2 z , estimated by Eq. (2). The resulting AR sp ectral estimate is given b y Eq. (3) with S z ( f ) = ˆ σ 2 z . Th us, the AR sp ectral estimate is a rational appro ximation to the actual sp ectral densit y function S ( f ). This type of lo w order sp ectral approximation tends to describ e the bulk characteristics of simple sp ectra well, but has difficulty resolving the fine features of the sp ectrum. Figure 5 sho ws AR(10) and AR(20) fits to the sp ectrum. The AR fitted sp ectra hav e difficulty fitting the spectral plateau betw een 1450-1600 kHz. By AR(20), the reduction in the ro ot mean squared error has virtually saturated as a function of the filter order. Using the the autor e gr essive filter ac c entuates the tr ansient bursts b e c ause it applies the filter, Γ( f ) − 1 , to r emove the ambient sp e ctrum, esp e cial ly the 1 MHz p e ak, and ther efor e ac c entuates the fr e quency c omp onents which ar e only we akly pr esent in the ambient sp e ctrum. Sp ectral prewhitening 13 , 14 w as dev elop ed b y J. T ukey to reduce the sp ectral v ari- ation in the analyzed time series. Prewhitening uses the AR mo del as an initial 9 filter and then uses lo cal, F ourier-based metho ds, such as the smo othed p erio dogram or m ultiple tap er analysis to estimate the spectrum of the residual pro cess, { z } . Prewhitening implicitly assumes that the autoregressive parameters, α ( p ) i , are inde- p enden t of the measured and residual pro cesses. In our precursor analysis in Sec. VI, w e use a prewhitening filter prior to computing the time-frequency distribution. IV. Nonstationary plasma fluctuations during the ELMs When the plasma fluctuations are stationary sto c hastic pro cesses, we can iden tify the time auto cov ariance of the fluctuations with the F ourier transform of the sp ectral densit y S ( f ). In Ref. 9, we compare metho ds of estimating the spectral densit y for stationary plasma fluctuations. When the sp ectral density is c hanging slowly in time, the natural generalization is that of Karhunen pro cesses 14 − 16 : x t = Z 1 / 2 − 1 / 2 A ( f , t ) e if t d Z ( f ) , (4) where d Z ( f ) is a white noise pro cess with indep endent sp ectral increments. When A ( f , t ) evolv es slowly with resp ect to the sampling rate, w e can interpret x t as an appro ximately stationary pro cess. The evolutionary sp ectrum, S ( f , t ) = | A ( f , t ) | 2 , then corresp onds to the instantaneous v alue of the sp ectral densit y . T o estimate the evolutionary sp ectrum, first we compute the lo cal F ourier trans- form, y ( f , t ), of x t using a sliding tap ered (or multitapered) time windo w, and the one p oin t sp ectral estimate is ˆ S ( f , t ) = | y ( f , t ) | 2 , whic h we ev aluate on a time-frequency grid. When the sampling rate is m uc h larger than the c haracteristic time scale, we can improv e on these p oint estimates of the evolutionary sp ectrum b y smo othing the sp ectral density in time and frequency 16 . Ho w ever, w e are not in this limit due to the burstlik e nature of the ELMs and precursors. A go o d mo del for these plasma fluctuations is that the fluctuations consist of a stationary comp onen t and a transient bursting comp onent: S ( f , t ) = ¯ S ( f ) + A e ( t ) S e ( f ) + ˜ S ( f , t ) , (5) where the subscript e denotes “ELM”, and ˜ S ( f , t ) is the residual transient sp ectrum. T o estimate ¯ S ( f ), A e ( t ), and S e ( f ), we begin b y computing a m ultiple tap er estimate of S ( f , t ). T o retain high time-frequency resolution, we use 1000-p oint segmen ts with 8 tap ers ( w ∼ 20 kHz). W e subtract the mean v alues, ¯ S ( f ): S ( f , t ) − ¯ S ( f ), and then estimate A e ( t ) S e ( f ) b y computing the singular v alue decomp osition of ˆ S ( f , t ) − ˆ ¯ S ( f ) = X k λ k A k ( t ) h k ( f ) . (6) 10 The singular v alue decomp osition divides the sp ectrum into its fundamen tal comp onents 24 . Neither A k ( t ) nor h k ( f ) need b e p ositive. W e equate A e ( t ) h e ( f ) with λ 1 A 1 ( t ) S 1 ( f ). In practice, we find that the quasi-coheren t part of the 1 MHz p eak has a differen t time evolution than the rest of the sp ectrum. T o remo v e the effect of the IF frequency , w e band limit the signal to exclude the 1 MHz p eak ± 40 kHz prior to computing the singular v alue decomposition. Figure 6 displa ys the mean sp ectral estimate, where we ha v e used 1000 data p oin ts corresp onding to a 0.4 msec time in terv al with 8 tap ers. The mean sp ectrum is similar to, but broader than, the multitaper estimate in Fig. 2. The broadening occurs b ecause w e hav e low ered the frequency resolution to increase the time resolution. Figure 7 presen ts the first singular time v ector, A 1 ( t ), whic h corresp onds to the ELM bursts. The actual rise time of the ELMs is m uc h sharp er than that display ed in Figure 7. As w e increase the time resolution, the rise time of the ELMs decreases with the time windo w length. Therefore the rise time is not resolved. Figure 8 displa ys the corresp onding first singular frequency vector, whic h is broader and more symmetric than the mean sp ectrum. The broader and more symmetric sp ec- trum was originally observ ed in Ref. 12. Ho w ev er, the time-frequency singular v alue decomp osition allows us to quantify the sp ectrum of a bursting ev en t. S e ( f ), as esti- mated by the singular v alue decomp osition, is essentially a time-weigh ted a verage of the sp ectra of the 14 ELMs with the weigh ting function given by A e ( t ). Our singular v alue decomposition differs from that of Nardone 25 and Zohm et al. 26 b ecause our tw o axes are time and frequency . In con trast, Refs. 25 & 26 ha v e m ultiple measuremen t channels, and compute a singular v alue decomposition with time and space/mo de n umber as the tw o axes. V. Sp ectral ev olution b et w een ELMs W e now examine the temporal evolution of the spectral density betw een ELMs. W e estimate the a v erage growth rate as a function of frequency . Our results are strictly for disc harge # 49035. Other disc harges sometimes hav e significantly differen t sp ectral ev olution, b oth quan titativ e and qualitative. In our initial analysis, we computed the ev olutionary spectrum as a sequence of sp ectral estimates. Due to the broad-banded nature of the plasma turbulence, w e present only the total sp ectral energy in four 300 kHz bands versus time. By in tegrating in frequency , we reduce the v ariance of the estimate and display only the essen tial features of the data. W e then estimate the gro wth rates of the sp ectrum 11 during the time after the end of an ELM. T o estimate ¯ S ( ¯ f , t ) = R ¯ f +150 kH z ¯ f − 150 kH z S ( f , t ) d f , we estimate S ( f ) on a 0.4 msec interv al cen tered at t using a 100 tap er estimate with w = 150 kHz. T o further reduce the v ariance, we av erage l og 10 [ ˆ ¯ S ( ¯ f , t )] ov er all 14 quiescent p erio ds. This a veraging is necessary b ecause the fluctuation level is highly v ariable, indicating that there are only one or t wo wa v es presen t at eac h frequency . W e use the logarithm to reduce the influence of the outliers 9 , 19 . This ELM av eraging is reminiscen t of sa wtooth av eraging in heat pulse propagation 27 . In sawtooth a veraging, the signal is a v eraged directly to reduce the noise while in ELM a veraging, w e a verage the log-sp ectrum to determine the sp ectrum of the noise. Both the ELM duration and the length of time b etw een ELMs v aries from ELM to ELM. (See T able 1.) W e find that the ELM duration is 1.2 ± 0 . 3 msec and that the quiescent time b et w een ELMs is 6 . 1 ± 1 . 5 msec. After the 10th ELM, a large amplitude fluctuation burst o ccurs at t = 79 . 15 cen tered at 450 kHz. The length of time b etw een the 10th and 11th ELM is considerably larger, 9.45 msec v ersus 5.8 msec. W e b elieve that this ion fluctuation burst indic ates the r ele ase of some of the fr e e ener gy which c auses the ELM instability, and ther eby delays the onset of the 11th ELM. Excluding the time b et w een the 10th and 11th ELM gives the typical time length b et w een the end of the ( k − 1)th ELM and the b eginning of the k th ELM to b e 5.8 ± 1.1 msec. Since the 14 quiescent perio ds v ary in length, we need to standardize the in terv als to all b e of the same length prior to av eraging the sp ectral estimates. With this standardization, the sp ectral ev olution is clearly exp onen tial. Figure 9 displa ys the mean v alue of log 10 [ ˆ ¯ S ]( ¯ f , t ) after all quiescen t p eriods are standardized to a length of 5.8 msec. The fluctuations in the electron drift direction increase rapidly in the first 0.5 msec after the ELM subsides. During the first half millisecond, the sp ectrum b et w een 1500 & 1800 kHz gro ws b y 50 p ercen t and the sp ectrum betw een 1200 & 1500 kHz gro ws by 80 p ercent. During this time, the “ion” fluctuations actually decrease sligh tly . T o illustrate this sp ectral ev olution, Fig. 10 plots the sp ectrum for three time slices, 0.1, 0.3, and 0.5 msec after the second ELM. The sp ectra w ere calculated with 8 tap ers on 1000 data p oin t segments, and hav e a frequency resolution of 20 kHz and a time resolution of 0.1 msec. The 1 MHz p eak decreases in amplitude and in width. This narro wing of the 1 MHz line results in a reduction of the “ion” spectrum. The “electron” sp ectrum b etw een 1600 and 1800 kHz is gro wing noticeably . In detector #2, the secondary maximum do es not grow appreciably after the end of the ELM. Figure 11 plots the corresponding sp ectral evolution for detector #2. The “ion” 12 sp ectrum hardly grows during the initial 0.8 msec after the ELM. The gro wth rates of the “electron” sp ectrum for detector #2 are roughly half of those of detector #3. But the initial phase of rapid growth lasts for nearly a full millisecond in detector #2 while the the rapid growth p erio d in detector #3 is only half a millisecond long. Th us, the total growth of the “electron” sp ectrum in the tw o detectors is nearly the same. In the first millisecond, the “electron” sp ectrum gro ws by 50 % for f 0 in 1500 – 1800 kHz and by 100 % for f 0 in 1200 – 1500 kHz. (W e contin ue to use f 0 ≡ 1MHz – f for detector #2.) This suggests that the saturated amplitude is a weak function of the initial growth rate. Estimating these initial gro wth rates, γ , with S ( t ) ∼ exp(2 γ t ), we find that for detector #3, γ = 0 . 5 msec − 1 for f in 1200 – 1500 kHz, and γ = 0 . 37 msec − 1 for f in 1500 – 1800 kHz; for detector #2, γ = 0 . 28 msec − 1 for f 0 in 1200 – 1500 kHz, and γ = 0 . 24 msec − 1 for f 0 in 1500 – 1800 kHz. After this initial phase, the “electron” gro wth rates slo w by a factor of three to five in detector #2 and by even less in detector #3. During the 5.3 millisecond time p erio d b eginning 0.5 milliseconds after the end of the ELM, the level of the electron drift fluctuations grows slowly . In contrast, the fluctuation level in the ion drift dir e ction incr e ases by a factor of five in the 500-800 kHz r ange and by a factor of eight in the 200-500 kHz r ange. At the time of the onset of the ELM, the electron drift spectrum in 1200 & 1500 kHz range is roughly 50 % larger than the corresp onding ion drift fluctuations. F or detector # 2, the increase in the “ion” sp ectrum is less: a factor of 1.6 × in the 500-800 kHz range and by nearly a factor of 2.2 × in the 200-500 kHz range. The ratio of the total sp ectral energy in the ion drift direction to that of the energy in the electron drift direction is muc h less for detector #2 than for detector #3. This occurs b ecause the electron is larger in detector #2 and the ion energy growth b et w een ELM is less. Beginning 0.5–1.0 millisecond after the ELM, the evolution of the “ion” sp ectrum en ters in to an exp onen tial gro wth phase. During this later phase, we find γ = 0 . 2 msec − 1 for f in 200 – 500 kHz, and γ = 0 . 15 msec − 1 for f in 500 – 800 kHz for detector #3. F or detector #2, γ = 0 . 1 msec − 1 for f 0 in 200 – 500 kHz; for f 0 in 500 – 800 kHz, γ = 0 . 17 msec − 1 for the time 0.3 to 1.3 msec after the ELM and γ = 0 . 01 msec − 1 for later times. In summary , the “ion” growth rates are smaller by a factor of 2.5 – 3.0 times the initial “electron” growth rates. W e stress that these growth rates v ary appreciably from one discharge to another. When the “ion ” sp e ctrum gr ows mor e slow ly, the onset of the ELM is delaye d . This suggests that the ELM is triggered when the “ion” sp ectrum near the edge reaches a critical level. By a veraging l og 10 [ ¯ S ( ¯ f , t )] o v er 0.44 msec, w e find that the critical lev els 13 are log 10 [ ¯ S (350kHz , t )] = 2 . 40 ± 0 . 14 and log 10 [ ¯ S (650kHz , t )] = 3 . 45 ± 0 . 14. (See T able 2.) This corresp onds to relative v ariations of 40% in the critical level. This v ariation seems large, but ¯ S ( ¯ f , t ) increases by a factor of five to ten. Thus, the v ariance in the critical lev el is small relativ e to the total gro wth in the “ion” spectrum. The onset of the ELM need not b e caused b y the increased fluctuation level. Instead, b oth phenomena may b e caused by the same destabilizing mec hanism. Standardizing the 14 in terv al lengths is successful b ecause the final level of the ion fluctuations v aries less than either the length of the quiescen t interv al or the exp onen tial growth rate of the “ion” sp ectrum. If we b eliev e that the interesting sp ectral evolution o ccurs immediately after the end of the ELMs, w e should align the k th quiescen t p erio d at the end of ( k − 1)th ELM. If we b eliev e that the interesting ev olution o ccurs immediately prior to the start of the ELMs, we should align the k th quiescen t p erio d at the b eginning of the k th ELM. W e tried b oth of these normaliza- tions; ho w ever, no clear pattern emerged and the ELM to ELM v ariance w as larger than when in terv al lengths were standardized. W e estimate the v ariance of l og 10 [ ˆ ¯ S ] using the empirical v ariance estimate from the 14 indep endent subsequences. W e find that the empirical standard deviation, σ ( ¯ f , t ), of l og 10 [ ˆ ¯ S ( ¯ f , t )] v aries little in time. F or ¯ f = 1350 and 1650 kHz, σ ( ¯ f , t ) ∼ 0 . 1, which corresp onds to a relativ e v ariation of 25 %. F or ¯ f = 650 kHz, σ ( ¯ f , t ) ∼ 0 . 15, whic h corresp onds to a relativ e v ariation of 40 % and for ¯ f = 350 kHz, σ ( ¯ f , t ) ∼ 0 . 19, which corresp onds to a relative v ariation of 55 %. These v alues of σ ( ¯ f , t ) are for detector #3. This v ariation is small relative to the total increase in the ion fluctuations o v er the 6 milliseconds. F urthermore, the v ariance in the mean is 1 / 14 of σ 2 ( ¯ f , t ). F or detector #2, σ ( ¯ f 0 , t ) ∼ 0 . 1 for ¯ f 0 = 350 kHz and 650 kHz. Thus the “ion” sp ectrum of detector #3 is more v ariable than that of detector #2. This enhanced v ariability is partially due to precursor activity , which w e analyze in Sec. VI. The empirical v ariance is m uch larger than the theoretical estimate based on Gaussian statistics because it includes the ELM to ELM v ariation due to random effects includ- ing precursor times. These effects are random in the sense that the v ary randomly from ELM to ELM. In previous works, we hav e included the effect of tok amak to tok amak v ariation on energy confinemen t 28 and the effect of discharge to disc harge v ariation on the temp erature profile shap e 29 . VI. Time-freqency iden tification of ELM Precursors In examining the autoregressive residual fit errors, we identified a n um b er of very 14 short nonstationary bursts in the one and a half millisecond interv al prior to each ELM. The t ypical duration of the precursor burst is roughly 10 µ sec long. The frequency resolution is only 75 kHz due to short length (N = 100) of the segment. Due to its short life, the concept of a sp ectrum for the precursor is somewhat tenuous. Figure 12 plots the sp ectrum of one of the longest (11 µ sec) and most prominent precursors from detector # 3. The precursor is centered at 500 kHz, and o ccurs 0.45 msec before the second ELM. The dashed line in Fig. 12 is the corresp onding estimate on a data segment tak en 40 µ sec after the precursor corresp onding to the ambien t sp ectrum. In Figure 12, w e hav e used three tap ers with a frequency half bandwidth of 75 kHz and then kernel smo othed ov er an additional 50 kHz half-width. In comparing the tw o s pectra, w e see that the sp ectrum in the 400-600 kHz range is not only m uch larger than its normal size, but also larger than the fluctuations in the 1450-1750 kHz range, and this is not typical. The ambien t fluctuations in the 1450-1750 kHz range corresp ond to electron drift direction while the precursor frequency of 500 kHz corresp onds to ion drift. T o b etter quantify the occurrence rate, the duration and strength of the precursor, w e ha v e computed the time-frequency distribution for the 4 milliseconds prior to eac h ELM. T o hav e a very short time resolution, w e use only t wo tap ers and smo oth o v er 100 kHz. W e compute the mean, ¯ S ( f ), and standard deviation, σ ( f ), of the time-frequency distribution av eraged ov er time for eac h frequency . W e standardize the time-frequency plot b y subtracting off the mean and dividing by the standard deviation, so that the resulting function is S z ( f , t ) − ¯ S z ( f ) σ ( f ) . W e identified t w o types of short bursts: bursts which are centered in the 250-650 kHz range, and bursts which ha v e a substantial part of their energy conten t in the 1 MHz range. After insp ecting the fourteen time-frequency plots, we concluded that the bursts cen tered at 1 MHz did not hav e any noticeable correlation with the onset of the ELMs while the low frequency bursts o ccur almost exclusiv ely in the one and a half milliseconds prior to the ELM. F urthermore, the bursts in the 1 MHz range o ccur at a lo wer amplitude (10 σ ( f )) than those in the 250-650 kHz range (30 σ ( f )). T o enhance the low frequency fluctuations and to filter out the ambien t sp ectrum cen tered at 1 MHz, we prewhiten the time series by applying a tenth order autoregres- siv e filter prior to computing the time-frequency distribution. Prewhitening reduces the amplitude of the bursts in the 1 MHz range. F or graphical effect, w e hav e set all v alues of the standardized time-frequency distribution whic h are less than three to exactly zero. 15 Figure 13 displa ys a three-dimensional plot of the prewhitened transformed time- frequency distribution for the four milliseconds immediately prior to the second ELM. The precursor burst at t = 12.734 msec is clearly visible and is cen tered at 500 kHz. Figure 14 presen ts a similar time-frequency plot for the 4 millisec time in terv al just prior to first ELM. The large amplitude, 10 µ sec precursors burst are visible at t = 5.186 and 5.56 msec. Only the first burst is visible on the AR residual plot of Fig. 3b. The precursors bursts in the first t wo ELMs are lo calized around 500 kHz. How ev er, the frequency range of the precursors for later ELMs v aries b et w een 250 kHz and 650 kHz. Figure 15a plots the integrated energy in the frequency band, [300 − 700] kHz, v ersus time. The precursor bursts at t = 5.186 and 5.56 msec and at t = 12.734 msec are clearly visible and ab o ve the noise. The ELM oscillations b egin at t = 6.07 and 13.19 msec. Figure 15b gives the corresp onding time history in the [900 − 1100] kHz band. Many fluctuations occur in this frequency band, and there app ears to b e no corresp ondence b etw een these fluctuations and the imminen t onset of ELM activity . Figure 15c gives the time history in the [1300 − 1700] kHz band, corresp onding to electron drift direction. No evidence of precursor activity is indicated. The sp ectral in tensit y in this frequency range is virtually constant b et w een ELMs. Thus, Fig. 15 sho ws that the precursor bursts are lo calized in the 300 − 700 kHz band. W e set a threshold of 28 σ ( f ) on the standardized prewhitened time-frequency estimates. W e then record all of the ev en ts whic h exceed this magnitude during the four milliseconds immediately prior to each ELM. The threshold v alue of 28 σ ( f ) was c hosen to give the b est identification rate for the ELMs. W e find that the amplitudes of the instan taneous bursts ha v e an appro ximately t wo-h ump ed distribution, i.e. v ery few bursts o ccur with amplitudes of 18–30 σ ( f ). Thus, we can change the threshold from 15 σ ( f ) to 30 σ ( f ) and only sligh tly reduce the success rate of predicting ELMs. T able 3 summarizes our findings on the ELM precursor bursts. Column 2 giv es the time until the next ELM. Column 3 gives the maxim um burst amplitude in σ ( f ) and Column 4 gives the characteristic frequency of the burst. Column 5 gives the duration in µ sec. The sev enth ELM has a num b er of bursts in the 250-650 kHz, but they ha ve relatively small amplitude and therefore do not exceed the threshold. The elev en th ELM has a high amplitude burst at 1350 kHz instead of in the 300-600 kHz range. The large amplitude, low frequency burst after the 10th ELM is not listed in T able 3 b ecause it o ccurs more than 4 milliseconds b efore the next ELM. The shorter duration (2-11 µ sec) of these precursors is probably due to a combi- nation of ph ysics differences, the fast sampling rate on the TFTR diagnostic, and our high resolution estimation pro cedure. The short dur ation of the pr e cursor events r e- 16 duc es their physic al signific anc e as a c ause of the ELMs and pr ob ably me ans that they ar e mor e of a symptom of a change in plasma c onditions as the stability b oundary is appr o ache d. W e also caution that our analysis only applies to this particular TFTR disc harge and that other disc harges may b e differen t. These short precursor bursts o ccur only in detector #3, and we hav e found no corresp onding phenomena in detector #2. One explanation is that the precursors are o ccuring at the plasma edge and that the scattering volume for detector #2 is to o far in the plasma in terior to sense the precursors. Alternatively , the absence of precursors in detector #2 could b e due to the difference in measured k v alues of the t w o detectors. W e hav e examined several other disc harges, and the phenomena of high frequency bursts in the ion drift direction app ears to b e robust. In several of the other discharges, the precursors w ere weak er and less reliable. VI I. Summary In this article, we describ ed sev eral techniques whic h we hav e found useful in examining the nonstationary bursts: autoregressiv e filters to isolate nonstationary phenomena, prewhitening to reduce bias, and the singular v alue decomp osition to isolate the fundamen tal components of the signal, including the sp ectrum during the ELMs. W e hav e used a high resolution smo othed m ultitap er estimate of the ev olutionary sp ectrum to isolate short-lived high frequency ELM precursor bursts. Our evolutionary sp ectra estimate uses many few er degrees of freedom than in our previous w ork for t w o reasons: first, the bursts are so transient that segmen t lengths of more than 100 p oin ts blur the time resolution; second, the even ts app ear to b e coheren t (i.e. consist only of a single mo de), and th us are easier to estimate than an ensem ble of transien t mo des. Our main experimental findings are: 1) the ELM sp ectrum is more symmetric and broader than the stationary sp ectrum; 2) the existence of high frequency ELM “ion” precursor bursts; 3) the “electron” sp ectrum returns to its am bien t v alue within a half a millisecond after the ELM while the “ion” sp ectrum grow b y a factor of eight b et w een ELMs; 4) the onset of the ELMs app ears correlated with the level of ion fluctuations (for detector #3); 5) growth rates of 0.1–0.5 msec − 1 are estimated after the end of an ELM. Results 3-5 are not alwa ys present in other disc harges. In some disc harges, the “electron” sp ectrum grows b et w een ELMs and the “ion” sp ectrum is saturated. 17 Figure 9 sho ws that the “ion” sp ectrum grows exp onen tially b etw een ELMs. When the ion fluctuation sp ectrum increases more slo wly , the onset of the ELM is delay ed. It app ears that when the “ion” sp ectrum reaches a critical lev el, the ELM is triggered. Ho w ever, this “critical level” v aries by 40 %. The onset of the ELM need not b e caused b y the increased fluctuation lev el. Instead, b oth phenomena ma y b e caused b y the same destabilizing mechanism. Some other discharges ha v e a fixed lev el for the ion sp ectrum, and hav e no indication of an ion character to the ELM onset. The electron drift sp ectrum also app ears to grow exp onen tially to a critical level and then enters a slo w gro wth phase. The initial “electron” gro wth rates are faster in detector #3 than in detector #2, but the length of time of rapid growth is longer in detector #2; th us the total increase in the electron sp ectrum is the nearly the same in b oth detectors. Be c ause the sp e ctrum gr ows exp onential ly b etwe en ELMs, we interpr et this as the line ar phase of the plasma instabilities and the 0.1-0.5 mse c − 1 gr owth r ates to b e the line ar instability gr owth r ates . Our in terpretation is based on the assumption that nonlinear saturation is an algebraic function of time. Similarly , if the turbulence level were ev olving slowly due to c hanging plasma conditions, suc h as changes in the electric field 20 − 21 , the sp ectrum would almost certainly b e evolving more slowly than exp onen tially . The precursor bursts, which w e identify , ha v e muc h shorter lifetime than previ- ously rep orted measurements 2 , 4 , and are intermitten t. Their high frequency , ion drift nature further suggests than the ELM onset is related to increased ion turbulence. In termitten t fluctuation bursts occur at other frequencies, esp ecially near 1 MHz. Ho w ever, these bursts o ccur at a low er level (10 σ ( f ) instead of 30 σ ( f )), and are not correlated with the onset of an ELM burst. W e note that the precursors on this TFTR discharge occur with sufficient regularit y that the o ccurrence of the next ELM can b e predicted with ab out 80% accuracy . 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Lac kner, Nuclear F usion 31 , 1595, (1991) 21 T able 1: ELM and quiesc ent p erio d dur ation ELM # Quiescen t time Duration of ELM T otal p erio d (msec) (msec) (msec) 1 4.45 1.92 6.37 2 5.2 0.8 6 3 4.08 1.4 5.48 4 5.38 0.8 6.18 5 7.71 1.0 8.71 6 4.46 1.55 6.01 7 7.21 1.1 8.31 8 5.73 1.39 7.12 9 6.6 2.12 8.72 10 6.3 1.26 7.56 11 9.44 1.11 10.55 12 5.64 0.6 6.24 13 7.19 1.14 8.33 14 5.75 0.85 6.6 Mean 6.08 (5.82) 1.22 7.3 (7.05) Std. dev. 1.46 (1.14) 0.43 1.45 (1.15) The mean and standard deviation in paren theses are computed b y excluding the in terv al b et w een the ten th and elev enth ELMs. 22 T able 2: Critic al ion lo g-sp e ctrum levels ELM # 200–500 kHz 500–800 kHz 1 2.19 3.54 2 2.51 3.54 3 2.50 3.68 4 2.59 3.41 5 2.37 3.32 6 2.42 3.54 7 2.30 3.30 8 2.23 3.27 9 2.16 3.22 10 2.41 3.44 11 2.50 3.47 12 2.51 3.44 13 2.55 3.72 14 2.39 3.40 Mean 2.40 3.45 Std. dev. 0.14 0.15 The critical lev el of the ion fluctuations is calculated by a v eraging l og 10 [ ˆ ¯ S ( ¯ f , t )] o v er 0.44 msec. 23 T able 3: Ion pr e cursor bursts ELM # Time until ELM Amplitude F requency Duration (millisec) ( σ ( f )) (kHz) ( µ sec) 1 0.88 35 300–800 2 1 0.51 31 650 2 2 0.45 35 500 11 3 1.14 65 300 6 3 0.80 50 400 4 3 0.56 89 600 7 4 0.37 34 400 2 5 1.04 90 200–700 6 6 0.24 36 650 8 8 1.49 42 450 10 9 1.12 52 350 3 10 1.44 19 500 3 11 0.49 29 1350 2 12 ∗ .1-.2 27 200- 550 - 13 1.83 37 500 6 13 0.54 37 750 3 14 0.13 43 200–600 3 * No iden tifiable mo de, only broad-banded activity ab o v e threshold 24 25 10 ° I I \ 10  ' �             ' - _  -       0           Frequency (kHz)                w =                                                             #                                 26 27 28  1 ,    ,/ /                       Frequency (kHz) 1                                   29      400000   5           o L   800    Frequency (kHz) 1400             S(/)                                                                         30 31 32 33 34 35 36                             1                                                                                                                                      37 38 39 40 41  value        i  5 300000       0                    S(/)                                                                        42 43 Figure Captions: Figure 1a: Ra y tracing calculation of the extraordinary mo de w av e path launc hed from the top of the v acuum v essel. The center of the b eam path is the cen tral curve while the outer curves mark the b eam half-width of 5.0 cm. Detector #3 is at the b ottom of the v acuum v essel, and the line of sight of the detector is display ed as w ell. Detector #3 measures | ~ k scat − ~ k inc | ∼ 3 . 3 cm − 1 and ~ k is parallel to the p oloidal magnetic field at r a ∼ 1 . 0 ± 0 . 1. Figure 1b: Line of sight for detector #2 and ra y tracing calculation of launc hed extraordinary mo de. Detector #2 measures fluctuations somewhat farther in the plasma interior, at r a ∼ 0 . 75 ± 0 . 1 with | ~ k scat − ~ k inc | ∼ 8 . 5 cm − 1 . Figure 2: Smo othed sp ectrum of entire 524,288 p oin t segment, estimated with 20 orthogonal tap ers with w = 0 . 1 kHz and then kernel smo othed ov er 20 kHz. The cen tral p eak at 1 MHz is partially coherent and corresp onds to the receiver in ter- mediate frequency caused b y w all and wa v eguide reflections. The lo cal broadening from 1450 kHz to 1750 kHz is due to plasma fluctuations which are rotating in the electron drift w av e direction. W e normalize the sp ectrum of detector #2 suc h that the t w o sp ectra hav e the same amplitude when the broadening of the 1 MHz line b egins. The “ion” sp ectra are iden tical while the “electron” sp ectrum of detector #2 is three times larger. Figure 3a: The 12 millisecond data segment from t = 2 msec to t = 14 msec of the micro w a v e scattering diagnostic for T.F.T.R. disc harge # 49035. The ELMs o ccur at t = 6.07 and 13.19 msec. A num ber of fluctuation bursts prior to the ELM are visible; how ev er, most of these bursts are due to c hanges in the 1 MHz frequency range. Two p recursor bursts in the 250 − 650 kHz range occur at t = 5.186 and 12.734 msec; how ev er, it is difficult to distinquish these bursts from the IF frequency bursts. Figure 3b: Filtered data after using a ten th order autoregressiv e filter. Since the sawtooth sp ectrum is broader than the ambien t sp ectrum, the ELM amplitude is enhanced relative to the bac kground lev el. The intermitten t bursts whic h are concen trated in the 1 MHz range are reduced by filtering. In contrast, the tw o precursor bursts in the 300 − 600 kHz range, at t = 5.186 and 12.734 msec, hav e b een enhanced. Figure 4a: Microw a v e scattering data for TFTR discharge # 50616. Sa wto oth 44 o ccurs at t = 3 . 86 msec and is barely visible. The v ariance app ears to b e growing linearly in time. Figure 4b: Filtered data after using a ten th order autoregressive filter. Since the sa wto oth sp ectrum is broader than the ambien t sp ectrum, the effect of the sawtooth is greatly enhanced b y filtering. The linear gro wth of the v ariance is eliminated b y filtering. This indicates that the nonstationarity is concen trated in the 1 MHz p eak. Figure 5: Autoregressive estimate sp ectrum of en tire data set, estimated b y the metho d of moments. The 1 MHz p eak is artificially broadened due to mo del misfit. Fig. 6: Mean sp ectral estimate, ¯ S ( f ) of the time-frequency distribution, S ( f , t ), estimated using 1000 p oint samples with eight tap ers ( w = 20 kHz) and then k ernel smo othed ov er 20 kHz. The mean sp ectrum is broader than the estimated sp ectrum in Fig. 2 b ecause the frequency resolution is low er. Fig. 7: First time v ector of the singular v alue decomp osition of the time-frequency distribution. Each of the 14 p eaks corresp onds to an ELM burst. The rise time of the ELMs in Fig. 7 is significantly longer than the actual rise time b ecause w e ha v e reduced the time resolution to increase the frequency resolution in Fig. 8. W e remov e the IF frequency at 1 MHz prior to computing the singular v alue decomp osition. Fig. 8: First frequency v ector of the singular v alue decomp osition of the time- frequency distribution. The sp ectrum during the ELM is broader and more symmetric than the mean sp ectrum of Fig. 6. Fig. 9: ¯ S ( ¯ f , t ) ≡ R ¯ f +150 kH z ¯ f − 150 kH z S ( f , t ) d f of detector # 3, a v eraged ov er the 14 quiescent p erio ds. The quiescen t p erio ds are standardized to a length of 5.8 msec. Since the curv es are nearly straigh t, the gro wth rates of the ion fluctuations are exp onen tial. When the lengths of the 13 in terv als are not standardized to be the same length, more complicated dep endencies result. The fluctuations in the electron drift direction increase rapidly in the first 0.5 msec after the ELM subsides. During the next 5.3 msec, the level of the electron drift fluctuations is virtually constant. In contrast, the fluctuation lev el in the ion drift direction increases b y a factor of five in the 500-800 kHz range and b y a factor of eight in the 200-500 kHz range. During this time, the gro wth rates are γ = 0 . 2 msec − 1 for f in 200 – 500 kHz, and γ = 0 . 15 msec − 1 for f in 500 – 800 kHz. When the “ion” sp ectrum gro ws more slo wly , the onset of the ELM is dela y ed. 45 Fig. 10: Sp ectrum for three time slices 0.1, 0.3, and 0.5 msec after the second ELM. The “electron” sp ectrum b et ween 1600 and 1800 kHz is growing noticeably . The 1 MHz p eak decreases in amplitude and in width. This narro wing of the 1 MHz line results in a reduction of the “ion” sp ectrum. In detector #2, the corresp onding secondary maximum do es not gro w appreciably after the end of the ELM. The sp ectra w ere calculated with 8 tap ers on 1000 data p oint segments, and hav e a frequency resolution of 20 kHz and a time resolution of 0.1 msec. Fig. 11: ¯ S ( ¯ f , t ) ≡ R ¯ f +150 kH z ¯ f − 150 kH z S ( f , t ) d f of detector # 2, a v eraged o ver the 14 qui- escen t p erio ds. In the first millisecond, the “electron” sp ectrum gro ws by 50 % for f 0 in 1500 – 1800 kHz and by 100 % for f 0 in 1200 – 1500 kHz. (W e con tin ue to use f 0 ≡ 1MHz – f for detector #2.) F or detector #2, γ = 0 . 28 msec − 1 for f in 1200 – 1500 kHz, and γ = 0 . 24 msec − 1 for f in 1500 – 1800 kHz. After this initial phase, the “electron” gro wth rates slow b y a factor of three to fiv e. The “ion” sp ectrum increases b y 1.6 × in the 500-800 kHz range and by 2.2 × in the 200-500 kHz range. In detector #2, the “electron” sp ectrum is larger than in detector #3 and the growth of the ion sp ectral energy b etw een ELM is less: γ = 0 . 1 msec − 1 for f 0 in 200 – 500 kHz; for f 0 in 500 – 800 kHz, γ = 0 . 17 msec − 1 for the time 0.3 to 1.3 msec after the ELM and γ = 0 . 01 msec − 1 for later times. Thus the “ion” sp ectrum appears to b e more imp ortan t near the plasma edge. Fig. 12: Estimated spectral densit y during a precursor 0.45 msec b efore the second ELM. The secondary peak at 500 kHz o ccurs only during the precursors. This pre- cursor is particularly long lived and resolv able. Sp ectrum is computed on a 100-p oint segmen t using 3 tap ers with a bandwidth of 75 kHz follow ed b y a k ernel smo other with a half-width of 50 kHz. Dashed line: Corresp onding estimated sp ectral density for the 100-p oin t segment 40 µ sec later. The precursor p eak has totally disappeared and the spectrum has returned to its am bien t shap e. Since the frequency resolution of Fig. 12 is less than that of Fig. 5, the sp ectrum in Fig. 12 is corresp ondingly broader (just as the sp ectrum in Fig. 6 is broader than the sp ectrum in Fig. 2). Figure 13: Time-frequency distribution of the 4 millisec time in terv al prior to the second ELM. The large amplitude, 10 µ sec precursors burst is visible at t = 12.734 msec. The data has been filtered using a tenth order autoregressiv e filter to reduce the sp ectral range. Prewhitening remov es the fluctuations which are asso ciated with nonstationary activit y of the central 1 MHz p eak. The ev olutionary spectrum is 46 computed on 50-p oin t segments with 50 % ov erlap. W e use t wo Slepian tap ers with a bandwidth of 100 kHz follow ed b y a k ernel smo other with a kernel half-width of 100 kHz . Figure 14: Time-frequency distribution of the prewhitened data for the 4 mil- lisec time in terv al just prior to first ELM. The large amplitude, 10 µ sec precursors bursts are visible at t = 5.186 and 5.56 msec. Only the first burst is visible on the AR residual plot of Fig. 3b. Figure 15a: Integrated energy in the frequency band [300 − 700] kHz versus time for the first tw o ELMs. The precursor bursts are clearly visible in the 300 − 700 kHz band. Figure 15b: Integrated energy in tw o frequency band [900 − 1100] kHz versus time. The numerous bursts in the IF frequency range o ccur with no clear pattern, and th us are ill-suited to forecast ELM activity . Figure 15c: Integrated energy in the frequency band [1300 − 1700] kHz versus time. The precursor bursts are barely visible in the electron drift w av e frequency band. 47