PFC/RR-82-15 DOE/ET-51013-45 THERMAL & NON-THERMAL SUBMILLIMETRE EMISSION FROM ALCATOR TOKAMAK by S. EMIL KISSEL Department of Physics Plasma Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139 June 1982 THERMAL & NON-THERMAL SUBMILLIMETRE EMISSION FROM ALCATOR TOKAMAK by S. EMIL KISSEL S.B., M.I.T. (1977) SUBMITTED TO THE DEPARTMENT OF PHYSICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1982 Signature of Author Department o0 PnysIcs 31 April 1982 Certified by Prof. Ronald R. Parker Thesis Supervisor Accepted by Prot. beorge F. Koster Chairman, Department Committee Page 2 THERMAL & NON-THERMAL SUBMILLIMETRE EMISSION FROM ALCATOR TOKAMAK by S. EMIL KISSEL Submitted to the Department of Physics on 31 April 1982 in partial fulfillment of the requirements for the Degree of Doctor of Science in Physics ABSTRACT An experimental investigation was undertaken of submillimetre radiation from The Alcator Tokamak. Thermal emission at the cyclotron frequencies determine spatial electron temperature distributions. temperature electrical profiles corroborated conductivity, discharge behaviour. and classical provided was to Measurements of models graphic used of plasma displays of unusual Temperature fluctuations, such as sawteeth, were also investigated. Non-thermal frequency, radiation was observed the and occurred in two distinct fashions; feature, and a fluctuating emission of fluctuating plasmas. at emission very narrow electron plasma a steady broadband bandwidth. The has not been previously documented from Tokamak Pertinent theoretical considerations are outlined. Thesis Supervisor: Prof. R.R. Parker Page 3 TABLE OF CONTENT SECTION NULL SEC SEC SEC SEC INTRODUCTION 0.1 0.2 0.3 0.4 1.1 1.2 1.3 1.4 1.5 1.6 SEC SEC SEC SEC SEC 2.1 2.2 2.3 2.4 2.5 SECTION THREE SEC 3.1 SEC 3.2 SEC 3.3 SEC 4.1 SEC 4.2 SEC 4.3 SEC SEC SEC SEC NON-GAUSSIAN PROFILES 5.1 5.2 5.3 5.4 .......................... ........................... Current Rise ..................... Impurity Radiation ................... References & Figures ........................ 15 17 19 23 25 27 35 . .... . 35 36 40 42 43 51 51 53 56 60 60 ............ Reflections & Optical Depth ......... 62 Profile Shape ... * ..... ......... 63 .................. References & Figures TEMPERATURE FLUCTUATIONS SECTION FIVE 15 .................. Gaussian Profile ..................... Comparison To Classical Resistivity Profile Width ......................... Profile Centre .......................... ................... References & Figures THIRD HARMONIC EMISSION SECTION FOUR ................... ............... Single Particle Emission .... ...... ....... Plasma Effects Optical Depth ............................ ............ Reflections & Polarisation Temperature Profile ...................... ................... References & Figures ELECTRON TEMPERATURE PROFILES SECTION TWO 7 7 .................... The Alcator Tokamak ................. 8 Organisation Of Thesis Introduction To Instrument ............. 10 12 List Of Symbols ....................... REVIEW OF CYCLOTRON EMISSION SECTION ONE SEC SEC SEC SEC SEC SEC ................................... ....................... 68 ............. Fabry-Perot Interferometer ................... Sawtooth Instability ......... Sympathetic Plasma Oscillation References & Figures ..................... 68 69 73 75 Page 4 6.1 6.2 6.3 6.4 SEC 6.5 SEC SEC SEC SEC SECTION SEVEN SEC SEC SEC SEC 7.1 7.2 7.3 7.4 SECTION EIGHT SEC SEC SEC SEC SEC 8.1 8.2 8.3 8.4 8.5 9.1 9.2 9.3 9.4 9.5 9.6 9.7 SEC 9.8 SEC SEC SEC SEC SEC SEC SEC Cerenkov Emission ................. 84 ........... ... ................. ... Ray Trapping References & Figures ....... Temporal Behaviour 86 89 96 FLUCTUATING PLASMA FREQUENCY EMISSION ..................... 97 .................... 99 Correlated Behaviour ...... 100 Comparison With Other Experiments References & Figures 101 ................. .. 110 SPECTRAL MEASUREMENTS OF FLUCTUATING EMISSION 110 ............... Michelson Interferometer ................. Fabry-Perot System .. 112 .114 ........... 116 Shifts In Emitted Frequency ................... 118 References & Figures High-Resolution Spectra .......... ........ ..... .... 126 126 Stimulated Emission ...................... ...... 128 Related Experimental Observations 132 * .... ................. Runaway Electrons .134 Plasma Wave Modes .............. *..... . 136 ..... Other Waves ................. *........137 Saturation Of Plasma Waves 139 ............. Linear Radiation Processes 141 ................... References & Figures FLUCTUATING EMISSION: NON-LINEAR EFFECTS SECTION TEN 81 81 .................... Spectral Character ......... 83 Review Of Previous Experiments FLUCTUATING EMISSION: PLASMA WAVES SECTION NINE SEC SEC SEC SEC SEC SEC ............. DOMINANT PLASMA FREQUENCY EMISSION SECTION SIX *..... 144 10.1 10.2 10.3 10.4 10.5 10.6 ................. .144 Resonance Conditions .......... .......... 147 Dispersion Relation .................... 149 Non-linear Current .. 9..... ......... 151 Absorption .153 ............... Relation To Experiment .. ..... . ........ References & Figures 1 2 3 4 5 6 7 8 ............ ........... .. 159 Optical Path ............. 162 Reflections & Polarisation ........... 163 The Michelson Interferometer ......... 165 Fourier Transform Spectroscopy 170 ................................ Detector 9.......171 ....... .... Calibration ....174 Atmospheric Water Vapour Absorption ................... 175 References & Figures 99155 APPENDICES APX APX APX APX APX APX APX APX T 0 T H E H A P P Y F E W Page 5 ACKNOWLEDGEMENT The Alcator experiment is a collective effort requiring the labours of numerous individuals. I would like to acknowledge the contributions by the support staff, technicians and scientists which allowed this work to be performed. I am especially grateful for occasionally necessary the prodding guidance, provided by encouragement, and Dr. Ian Hutchinson. While I might question what presented is no doubt of the great personal gain which I him, there pernicious have received from our collaboration. as product our proximity Of greatest importance to has me, a student, has been the inspiration Ian has provided while serving as my role model of the true scientist and scholar. I also 'seagulls' thank with the whom eclectic I have community lived of and shared. graduate Their indomitable spirit surely makes them birds that have not lost faith virtue of flying. student in the high Page 6 Dedicated in memory of Therese Crossley with rue my heart is laden for golden friends I once had for many a rose lipped maiden and many a light foot lad by brooks to broad for leaping the light foot boys are laid the rose lipped girls are sleeping in fields where roses fade Page 7 SEC 0.0 This thesis radiation during INTRODUCTION made the presents in conjunction period requirements 1978 with through of submillimetre plasma the Alcator tokamak experiment 1981. As by stipulated the for the award of Advanced Degrees, the included material represents novel results research measurements documented of original herein is investigation. Most of related directly to the operation of Tokamak devices and should be of interest to persons involved in field, particularly the area of of plasma physics. this submillimetre plasma diagnostics. However, some of the items recorded are pertinent to the more field the general For this reason the reader is assumed to be knowledgeable in the area of plasma physics, but not specifically au fait in the arena of Tokamak research. SEC 0.1 THE ALCATOR TOKAMAK The Alcator C tokamak is shown schematically plasma is geometry. produced in fig.(1). The and confined within a magnetic field of toroidal A large toroidal field is externally produced by magnet coils, while a relatively smaller poloidal magnetic field is generated by an induced toroidal current within the plasma. is necessary both to current, the Because plasma of generated the induced This discharge time of several hundred much than steady state discharge. is characteristic said to occur nature of the can only be maintained for a finite time. longer current provide a confining force, and as a source of energy to heat the plasma. plasma The plasma milliseconds. is collision or transport times, so a during the middle portion Typical parameters for steady state operation are: of a Page 8 6 x10 14 Plasma Density: .5 x101 4 Plasma Current: 200 - 600 Electron Temperature: 1000 - 2000 Electron Volts Ion Temperature: 800 - 1500 Electron Volts Toroidal Field: 50 Majority Species: Hydrogen, Deuterium, Helium Of course the temperature, beginning and end of - - cm-3 kAmperes 100 KGauss current, and a discharge, density so are nill at the a rise and fall phase will precede and follow the steady state portion. The time evolution of a representative discharge is shown in fig.(2). The hot plasma is confined entirely chamber so that measurements of within plasma a toroidal properties, conditions at the discharge centre, must be inferred technique. For this reason by information to be so some benign since by this can be gathered without perturbing the discharge. Observations of submillimetre proven particularly the study of electromagnetic radiation emitted by the plasma is well suited as a diagnostic method vacuum useful in wave this radiation, in respect, that measurement systems of this kind have become particular, integral in have recent years components of most tokamak experiments. SEC 0.2 most ORGANISATION OF THESIS Cyclotron radiation is the principle mechanism responsible submillimetre A review of this emission from tokamak plasmas. radiation process is outlined in sec (1). emission is the capability for An implication of cyclotron of determining electron temperature, in Page 9 particular the spatial dependence or temperature second technique harmonic radiation. This profile, has become recently, so there is nothing new in the presentation profiles per se. from of the standard temperature The purpose of the first half of the thesis is to apply the measured temperature profiles towards a better understanding of the physics involved in the discharge. In sec (2), (3) it is consistent with the confidence the measured profiles The importance of this is the justification classical toward measurements. that are a classical resistance model over the full range of operating conditions. applying shown the Radiation model to of the plasma, and the generation of accuracy of the temperature profile at the third harmonic has been the topic of recent discussion, chiefly as a means of determining electron density. Sec (4) present measurements demonstrate that although the density does profiles, it of method self-consistent calibration of the important tokamak practical research. fluctuations the third harmonic emission which is inadequate represent cyclotron for a useful acquiring technique diagnostic. This for has consequences for anyone involved in this area of Sec (5) presents measurements of commonly observed during tokamak operation. temperature While these results demonstrate the wide ranged applicability of the submillimetre diagnostic, they also represent a worthwhile addition to the data base regarding phenomena, such as sawteeth, which are interest. of current topical Page 10 The remaining half of the thesis deals with Some kind of radiation of this sort has been observed by most and previous experiments. good for between theory and some types of non-thermal emission. the emission discussed in sec (6). however does expand The upon material the experiment presented existing measurements is This is true of data, distinguish two similar types of non-thermal emission. present current Theoretical explanations have been advanced to the point where the agreement section a origin, in particular emission near the plasma frequency. non-thermal quite of radiation in this and helps Sec (7), (8) of a second type of emission which has not been previously documented. The measurements in these sections clearly differentiate this new emission from the process discussed in sec (6). Sec (9), results (10) review currently the pertinent theoretical and experimental available concerning this novel type of radiation. The intent in these sections is not so much to processes attention on the sorts of involved, as it is to focus identify the mechanisms which can be reasonably expected to apply, and to actual indicate those which cannot. SEC 0.3 INTRODUCTION TO INSTRUMENT Most of the submillimetre spectral measurements presented within this thesis were made with a Michelson interferometer. discussion of the operation of this instrument appendix. is A detailed presented in the However, a brief introduction to the interferometer system is appropriate prior to a presentation of measured spectra. Page 11 Given some spectral intensity 1(w) of frequency w, the output of an ideal interferometer is: EQ (1) J(x) 1(w) cos(wx/c) du f = 0 where x is the optical path difference of the system. In operation the optical path is varied so that J(x) is measured over a range of x. The spectral intensity is related to the interferometer output by the inverse transformation: 1(w) = EQ (2) cos(Wx/c){J(x)-j} dx -f 0 where T is the average value of J(x). I(w) can be constructed provided The exact that J(x) spectral is semi-infinite range of x. In practice this is of Instead intensity measured course x is periodically varied over a limited range. impossible. The spectral intensities deduced from eq (2) are thus based on incomplete information, and therefore will spectra. Since it sets of have limited frequency resolution. The period with which x is varied sets measured over a the time resolution of the is necessary to assume a constant I(w) during one measurement cycle, the spectra will represent an average over this cycle time. Two important aspects of the interferometric measurements can be summarised by the application of some output, not frequencies. from following: form of The produced spectra result from the eq (2) to the entire interferometer individual measurements of intensities at separate Although there is a direct relation between the path Page 12 difference x and the time during which the measurement of J(x) is made, no such relation connects the intensity at a specific with a specific time. frequency The intensity at a specific frequency results from information gathered over the entire time during which x is varied. SEC 0.4 LIST OF SYMBOLS Tokamak research has field. Producing produced a lexicography thankless chore, so the usage of avoided. a vocabulary peculiar to the of Tokamak terminology would be a non-standard terminology has been The commonly used symbols are listed below as a convenience. It should be noted that most of these quantities are functions space or time or both. a...... Temperature profile width Y.. Relativistic factor B...... Magnetic field (Toroidal) t..... Optical depth n...... Plasma current I(w)... Spectral intensity Plasma frequency np... Gyrofrequency n c----- J(r)... Plasma current density Resistivity Plasma density qn...... Refractive index Subscripts, Superscripts Rn...... Safety factor Major radius of torus 0. . . . . Ordinary rlim--,. Minor radius, limiter radius x T ...... Temperature 0.... . Central value V ...... Toroidal loop voltage e...... Electron v ...... Particle velocity i...... Ion Zeff..-- Effective ion charge number ...... Extraordinary of Page 13 THE ALCATOR TOKAMAK TOROIDAL FIELD MAGNET TOROIDAL HORIZONTAL ACCESS PORT VACUUM CHAMBER Scale 10 cm Fig. (1) T EXAMPLE TOKAMAK DISCHARGE PLASMA CURRENT (kamp) - 400! 200 2.0 LINE AVERAGE DENSITY (1014 cm-) - 1.0 ____________________ - I I I ELECTRON TEMPERATURE (keV) 1.0 TIME (mwed) --- 100 200 300 400 Fig. (2) Typical time evolution of Alcator plasma discharge. Current is determined by measurement of magnetic flux, density by laser interferometry, temperature by central second gyroharmonic radiation. Fluctuations in temperature are caused by sawtooth instability. Page 15 SEC 1.0 REVIEW OF CYCLOTRON EMISSION Electron cyclotron acceleration of the emission charged results particle in from the centripetal a static magnetic field. 'Synchrotron radiation' is a term connoting relativistic energies is and misleading when applied to a plasma with non-relativistic particle energies. Cyclotron emission by the plasma can superposition be regarded as the of numerous independently radiating electrons, although the effects of dispersion, absorption and other aspects of the Tokamak environment must be considered. The submillimetre discharges results emission measured from typical entirely from cyclotron emission. Alcator Because of the larger magnetic fields encountered in Alcator, the emission occurs higher frequencies at but is otherwise similar to measured spectra from other Tokamaks. SEC 1.1 SINGLE PARTICLE EMISSION The exercise of computing the emission by a single electron in a magnetic field can be found in numerous textbooksl, 2 , and consists of determining the radiation field arising from the current of a gyrating electron. distribution For the geometry of fig.(1) the emitted power from a single particle is: W d2 ,v/c'( dild w EQ (1) .27rc n=1 = x cote (cose - vI/c) Jn(*) 2 e2w2 ICJ z (cose - vI/c) Jn(*) nv1 /c [1-(v /c)cose]~1 sinE w =. ne [1-(v 1/c)cose- 1 n MC where the direction of the vector quantity being squared indicates the Page 16 polarisation of the electric field. radiation at harmonics of the Two noteworthy features are the gyrofrequency, and the disparity of emission intensity into the two distinct polarisations. The frequencies of radiation are determined by the fourier harmonics of the periodic current distribution of the electron motion, giving rise to integrals of the form: EQ (2) which (sinwcet) exp(i .L sinwcet - wt) dt C f will be non-vanishing for integer multiples of the gyrofrequency, thus accounting for the radiation at the gyroharmonics. It is this type of integral which produces eq (1). the Bessel functions in The relative intensity of emission at subsequent harmonics is approximately: EQ (3) = [2n(*)I2n1() In-1 occurs at the fundamental, the local c most of the emitted power or first gyrofrequency. The observed radiation upon - 4n so that for a non-relativistic electron depend + = frequencies, magnetic as field, indicated and projected velocity of the emitting electron. For upon in the eq (1), energy and a single particle these quantities are constants, so emission will occur at harmonics of a narrow broadening. frequency, The the width relativistic being factor determined y can by radiational decrease the emission frequency, but only for uncharacteristically energetic particles. Page 17 The two modes extraordinary, and of the called and ordinary refer to the polarisation of the electric vector. Differing amounts of example, are propagation radiation ratio of are emission emitted into into each mode. two modes of the For nearly perpendicular propagation at any harmonic frequency is approximately: EQ (4) n(z) 2 ).2 v [nU' In(x) v, [j'() so that a non-relativistic electron r v-1 riv 2 is expected to emit cyclotron radiation primarily into the extraordinary mode. SEC 1.2 PLASMA EFFECTS Electrons will radiate at determined by harmonics of the the local value of magnetic field. gyrofrequency as The toroidal field as a function of position is: BR B(r) = --0 EQ (5) R+r where r is the radial position along the midplane geometric axis. broadening eq (1). For broadening produced example, at by the the This spread is Doppler the second much larger or relativistic terms of gyroharmonic the various processes acting upon a typical Alcator plasma (BT = 60 Kg ne = 1014 cm- 3 , Te = 1 KeV) will have the following collisional - 5x10- 8 , magnetic - 5x10- 1 . For relativistic from A large spread in radiated frequencies will result from the spatial variation of the field. than measured relativistic - 6x10- 4, angles e within 10 effect: Aw/ 2 wce = Doppler - 5x10- 2 cose, of perpendicular the broadening term exceeds the Doppler width, however, both Page 18 are small compared variation. Thus to the width by produced the magnetic field the frequency extent of radiation at any particular gyroharmonic will be determined by the variation of the toroidal Because of limitations imposed by construction of the toroidal magnetic field, this being the larger effect. magnet, optical access to the plasma is restricted to directions perpendicular to the magnetic field, hence the observed radiation must be travelling in this direction. The two polarisation states will describe propagation perpendicular to the toroidal magnetic field. The ordinary mode is polarised with electric vector oriented along the field direction, while the extraordinary polarisation has perpendicular to the field. its vector The cold-plasma dispersion relation for these two modes of perpendicular propagation is shown in fig.(2). ordinary mode is plasma frequency, non-propagation accessible while between the the to The radiation with frequency above the extraordinary mode upper-hybrid and has a region right-hand of cutoff frequencies. Due to the spatial variation of the plasma density and magnetic field, the dispersion relations for the two modes will have positional dependence. each In particular the cutoff mode. will change for and resonance each radial position. frequencies This is shown in fig.(3), where the critical frequencies are plotted as a function radius for a typical plasma condition. first harmonic propagating outward will encounter the this reason no extraordinary polarisation first harmonic This of Extraordinary radiation at the upper-hybrid layer before the plasma edge, and will be absorbed or reflected. observed. of emission For is situation would not be expected if, for example, the i Page 19 detection system was located at the inner by precluded the design and plasma edge, a situation of the Alcator tokamak. construction Ordinary mode first harmonic emission can be observed, provided that wpe/Oce < 1. When harmonic cyclotron emission is considered in the context of a magnetoactive plasma a necessary radiation absorption by the plasma. this absorption will consideration becomes that of For a sufficiently dense plasma alter the emission produced by a collection of individual electrons to the extent that a single particle treatment is inadequate. SEC 1.3 OPTICAL DEPTH In a plasma, radiation at any frequency must travel from a point of emission to a point of observation along a path which may allow for absorption and reemission of radiation at equation of the same frequency. The radiation transfer describing this process of absorption and emission along the ray path is3: d dI(w) = ds I ds 1 -- EQ (6) . dI(w) -w - L d92 where j. is the volume emissivity and a. is the spectral governing radiation at frequency w. Both of these quantities are integrals of the total electron distribution function, the combined absorptivity behaviour of individual plasma electrons. becomes: =dI(w) 1- exp(- faw ds) raypath I Q ()d and represent The emission Page 20 distribution For a Maxwellian electron law Kirchhoff's relates the emissivity and absorptivity 4 : EQ (8) -. aw bb(w) = -1 N3 Ew 3 kT _] 87*3C2 with the result that: EQ (9) bb [ds 8l raypath where Ibb ( the temperature ) remains constant across the resonance. Since the radiation quantum energies are small compared to the thermal plasma energy, hiw(kT, the blackbody formula can be expressed in the Rayleigh-Jeans limit, Ibb The optical depth absorption affects = is T, radiation and the plasma ignored For radiators. w2kT/8w 3 2 T>>1 properties to which of the transfer. If Tr1, absorption can be treated as a collection of independent the radiation will approach an equilibrium with the emitting electrons, so that its intensity thermal degree a measure of the plasma. will If the characterise optical depth is much greater than unity for a particular frequency, the emission at that in thermal intensity approaches the blackbody level, provided that the frequency electron distribution is near equilibrium. is the equilibrium, it cannot Furthermore, if the plasma radiate more than the local blackbody level, IbbDetermination absorptivity along of the T by eq (9) requires raypath for integration a particular absorption is a local process provided that the scale resonance between of the frequency. The length the of wave and electrons is small compared to variations Page 21 in electron density, or temperature, or magnetic field, and of course, that no overlap of harmonics occur. The scale length is determined by the magnetic field variation together with the appropriate resonance width: On L= EQ (O) R (2n) = where the relativistic width is used for propagation. of of as the above scale length perpendicular multiplied by the local The absorptivity can be found through the imaginary refractive index 5 , or from the emissivity. of The optical depth is thus a local quantity, and can be absorptivity a.. part case It is seen that L is much smaller than typical gradients in the plasma. thought the Kirchhoff's law and the The optical depth for the nth harmonic is6: 2 kT(r) n-i R pe(r) n n 2 c wce(r) (n-1)! 2mc2 (r) T(r) 0 = 0.15 2 kT(r) kr2 T (r) 2 for the low density, wp<<wc, case in In the a wp Caic limit eq (11) is the mildly relativistic limit. fair approximation for harmonics n > 1, but is not applicable to the first harmonic. Page 22 Absorption at the first gyroharmonic requires special treatment, for the extraordinary polarisation. particularly is that while extraordinary the ordinary mode has mode an is purely electrostatic The reason for this electromagnetic, component elliptically polarised perpendicular to the magnetic field. It that component with left-hand rising density polarisation, associated the electrostatic instead of the with the extraordinary mode. the electron gyrations and so reduces the has been right-hand shown7 produces a case normally This decouples the wave from the absorption. For Alcator the theory 8 predicts an optically thick ordinary, and thin conditions extraordinary completeness mode it at is the first mentioned harmonic. For the measurements 9 that sake of in other devices indicate an optically thick extraordinary mode, this being attributed to a mode conversion of electrostatic (Bernstein) waves' 0 . harmonic the and 3 rd extraordinary polarisation, for perpendicular propagation in properties" The contribution of plasma dielectric the regime wp<Wc. to 2 nd the The concern in this thesis is primarily with absorption at higher harmonics results from relativistic 2 (vth/c) 2 and finite density (wp/Wc) effects. It has been argued1 that a relativistic treatment is necessary for the 2 nd harmonic, even at low (1 KeV) temperature, but not for higher harmonics. The finite density effect for E = w/2 has been calculated in the non-relativistic limit for harmonic n13: E aE X =2,X(E.C.) EQ (12) S= 11 2 2 n2 - 1 n-3/2 On2 1 - (1 + - O)n1 2 n2 Page 23 x where aw (E.C.) refers to eq (11), the Englemann, Curatolo result. For a situation of interest, wp/wc = .8, eq (12) yields the following: EQ (13) n=2: a = 1.13 . x(E.C.) The absorption is enhance slightly at the slightly limit. the at the 3 rd, over fact that T>>1. that T be large. .93 a x(E.C. harmonic, and decreased 2 nd harmonic does not change It is noted here that nowhere in what follows is an explicit value of only 2 nd = the values predicted in the low density The small correction to a at the essential a n=3: T for the second harmonic required, This requirement is satisfied when the finite density are included. Since the blackbody level scales with the square of the radiated frequency, which in turn is a linear function of the magnetic field, high field devices such as cyclotron radiation. Alcator can generate higher levels of However, the radiated power from Alcator due to cyclotron emission is inconsequential compared to other power loss mechanisms, and to the ohmic input power. SEC 1.4 REFLECTIONS & POLARISATION Fig.(4) shows representative submillimetre spectra of the two polarisations from a tokamak discharge, as measured with the Michelson interferometer corresponding system. to the Emission first three harmonic feature being determined variation across the plasma. is recorded at the frequencies gyroharmonics, the width of each primarily by the magnetic field As expected, emission intensities at the optically thick frequencies are limited to the blackbody level, while Page 24 at intensities higher frequencies below this level. are harmonic extraordinary radiation is detected, in accordance predicted absorption examination shows that are polarisations description. the at the upper-hybrid layer. the with However, cursory emission of intensities No first into the two not in the ratios predicted by the single particle This is particularly true at the third harmonic where the optical depth is less than unity for both modes, so that the ratio of intensity should be nearly approximated by eq (4). The presently accepted explanation for observed ordinary mode emission at the second and third gyroharmonics requires the reflection of extraordinary mode radiation from the vacuum reflected wave chamber wherein undergoes a change in polarisation, thus appearing to the observer as if emitted with the ordinary polarisation. irregular internal unreasonable. shown the Given the construction of tokamaks such a hypothesis is not The relative polarisation of the second harmonic, 5:2, in fig.(4) is the largest value obtainable from Alcator, and is consistent with polarisation measurements from other model 15 ,16 . scrambling tokamaks 14 According total intensity of every reflected ray is some to and with the this model, the fraction r of the incident ray, where r < 1, and some portion p of the reflected ray has changed polarisation state. The polarisation ratio then becomes: EQ (14) 10 P Ix 1+p-r Furthermore it follows that the extraordinary mode intensity is: -T EQ (15) p = r-p 1- x 1 - peT Page 25 For the stated polarisation ratio, and p = .5 as measured from 3 rd harmonic emission [sec (4.1)] eqs (14),(15) are satisfied with p = .24 and r = .64. r = .85. For Alcator A17 this same treatment produced p = .15 and The errors in the measurements are such that the two sets of values are considered approximately similar internal construction, equal. so The that two machines have comparable results are disallow ignoring the frequencies for optically thick frequencies. The expected. The presence of doppler spread reason being absorbed of that by, reflections any reflected does not radiation must recross, and be the optically thick layer, and hence cannot be observed from the outside. For this reason the doppler broadening effect is ignorabl e. SEC 1.5 TEMPERATURE PROFILE For the optically thick second harmonic extraordinary mode, spectral width is governed by the magnetic field spatial variation, with intensity at each frequency blackbody level. proportional the corresponding frequency can be The blackbody level at a associated with the temperature at the equivalent radial location within the plasma, eq (9). to A conversion of frequency to spatial position can be made by use of the magnetic field, eq (5)18. particular this being done be used. treatment. thick From fig.(3) it is seen that the first harmonic ordinary and second harmonic extraordinary this with In this fashion spatial electron temperature profiles can be mapped 19 . It is only required that radiation from an optically frequency the mode are candidates for The second harmonic is preferred since it will have Page 26 the higher thus intensity, Furthermore, emission at sufficient plasma densities. plasma density during a better the first Fig.(5) signal to harmonic can shows a discharge the causes encroach upon the first harmonic, with the noise be case ratio. cutoff where for rising the plasma frequency to resulting attenuation of first harmonic emission. Electrons of spectrum through relativistic energy can broaden the mass term and the doppler shift. the frequency These effects will spread the region of emission beyond the frequency domain set the magnetic field variation. The resulting spectra will eventually become a continuum, with no distinguishable harmonic structure 20 . example of plasma An broadband spectrum is shown in fig.(6) for extraordinary mode emission. band by The low frequency cutoff results from the extraordinary frequency continuous is spectrum mode. discussed The in the non-propagation additional feature near the sec(8). in the vicinity of the A depression 2 nd of the harmonic, apparent in fig.(6), results from absorption by the thermal electrons. Electrons with y > 1 will emit radiation at the as thermal electrons in a lower magnetic field. gradient this thermal layer emission will always and the observation point. be chamber in order perpendicular to the field. entire plasma cross optically thick layers. to be between the point of Doppler broadened radiation from observed being back since After reflection section, frequency Because of the field relativistic electrons must be scattered from the vacuum same subject it wall of the it is not emitted must to traverse the absorption by any Cyclotron radiation by relativistic electrons will be absorbed at the thermal layer, and the observed radiation will Page 27 be characterised by the temperature of this layer. the 2 nd broadband corresponding cyclotron suggested that modify the that to the thermal to a temperature emission by level of with the 850 eV. relativistic measured It has been electrons will intensity of the optically thick harmonic, and so perturb the profile. seen fig.(6) emission in the frequency range of the optically thick harmonic is absorbed radiation in Thus However, from the previous argument and example, such it is emission will be absorbed to the appropriate thermal level. SEC 1.6 REFERENCES & FIGURES 1.) Classical Electrodynamics J. D. Jackson Wiley & Sons, New York (1962) 2.) W.K.H. Panofsky, M. Phillips Classical Electricity & Magnetism Addison Wesley, Reading Mass. (1962) 3.) G. Bekefi Radiation Processes in Plasmas Wiley & Sons, New York (1966) 4.) The integra 1/n2 i.e. ki similar term angle of the medium. 5.) M. Bornatici, F. Engelmann, G.G. Lister Phys. Fluids 22 1664 (1979) 6.) F. Engelmann, M. Curatolo Nuclear Fusion 13 497 (1973) 7.) M. Bornatici, F. Engelmann Comments Plasma Phys. Cont. Fusion 4 139 ovej k-sace will introduce a factor of = w /n2 c . This will be canceled by a which accounts for the.change in solid ray as it passes through the refractive (1979) 8.) Y. Dnestrovskii, D.P. Kostomarov, N.V. Skrydlov Zh. Tekh. Fiz. 33 922 (1962) 9.) A.E. Costley, T.F.R. Group Phys. Rev. Lett. 38 1477 (1977) 10.) J. Hosea, V. Arunasalam, R. Cano Phys. Rev. Lett. 39 408 (1977) Page 28 11.) 12.) 13.) 14.) C.M. Celata, D.A. Boyd Nuclear Fusion 19 423 I.P. Shkarofsky Phys. Fluids 9 561 K. Audenaerde Plasma Physics F. Stauffer (1979) (1966) 19 299 (1977) Private Communication 15.) A.E.Costley, R.J.Hastie, J.W.M. Paul, J. Chamberlain Phys. Rev. Letters 33 758 (1974) 16.) I.H. Hutchinson Plasma Physics 21 1043 (1979) 17.) I.H. Hutchinson, D.S. Komm Nuclear Fusion 17 1077 (1977) 18.) It is assumed that no overlap of harmonic frequencies occurs, so that any point in space can be resonant with only one harmonic at one particular frequency. 19.) R. Cano et.al. Nuclear Fusion 19 1415 (1979) 20.) C.M. Celata, D.A. Boyd Nuclear Fusion 17.735 (1977) Page 29 GEOMETRY OF CYCLOTRON EMISSION A X OBSERVER B Vii B 0 A z A y A4 V Sinl t] - A[FVL tj y COS 1 -e B ELECTRON GYROFREQUENCY o A t] + V,,t Z 'Ymc Fig.(1) Radiation is observed perpendicular to the magnetic field; that is, along the x axis. The radiation from a single particle will be symmetric with respect to rotation about the z axis. y is the relativistic factor. y = 1 for thermal plasmas. Page 30 DISPERSION RELATION FOR PERPENDICULAR PROPAGATION 3.0 k2c2 I I I I 2.0 / / EXTRAORDINARY POLARISATION k2 c 2 2 2 r2 (W2 U) 2_ Wh 2 0 1 22 ORDINARY POLARISATION k2 2 k2 / 1.0 1 - 2 - )p U-2--------------------------- / o/ / I I 0.0 / / / / / I / / FREQUENCY --ph P. rC 1.0 2.0 Fig. (2) Cold plasma dispersion for wpe/we frequency branches are not shown. 3.0 .57 Low Page 31 CRITICAL FREQUENCIES vs. RADIAL POSITION 1.5 C h - 1.0 Wp. 0,5 -10 0 RADIUS (cm) .10 Fi g. (3) For wpe/wce = .57 .Shaded areas are regions of non-propagation for the extraordinary mode. Ordinary mode does not propagate below the wpe curve. Radiation moves from left to right, where it is observed on the outside (low field) side of the Tokamak. ro~ POLARISED CYCLOTRON EMISSION I(W) ARBITRARY UNITS BLACKBODY LEVEL EXTRAORDINARY POLARISATION co 3 Cz O ORDINARY POLARISATION 200 400 600 FREQUENCY 800 GHz Ordinary and extraordinary mode emission Hydrogen plasma in 60 kGauss magnetic field. spectra are plotted on same scale. from Both 3~ CUTOFF OF FIRST HARMONIC Page 33 ORDINARY MODE EMISSION I(CO) WC = 1.0 TIME = 170 msec c =.8 C TIME = 100 msec - =- .6 W6TIME 200 =30 msec 400 600 FREQUENCY 800 GHz The situation Ope 'Oce is rarely encountered in the Alcator C because plasma disrupts. This case for 60 kGauss. All Fage BROADBAND CYCLOTRON EMISSION 1(W) BLACKBODY LEVEL FOR T Wp 200 = 850 eV 2W C 400 600 FREQUENCY GHz The shaded region is Fig.(6) emission level. The feature at 800 wpe above is the thermal discussed in sec (6). The dip at 130 GHz corresponds to absorption at the hybrid layer, this being an extraordinary mode spectrum for 60 kGauss field. Page 35 SEC 2.0 ELECTRON TEMPERATURE PROFILES The electron temperature profiles mode inferred from extraordinary emission at twice the electron cyclotron frequency have nearly a Gaussian shape under normal circumstances. thus measured is found classical resistivity spectroscopic Zeff. Relative as equal the determined central predicted by the temperature value based upon measured voltage and The profile width is also in agreement with this model, and along with qO = .85. to The the central changes of temperature the indicate values of position of the profile centre agree with changes in the outer flux surface position measured during plasma of the positioning experiments, while the absolute profile centre provides an accurate measure of the value toroidal magnetic field. SEC 2.1 The GAUSSIAN PROFILE optical depth of the second harmonic extraordinary polarisation from the Alcator Tokamak is1 : 23.8 n(101 4 cm- 3) T(KeV) EQ (1) B(Tesia) Under normal operating conditions of density applies over and temperature, the square of the frequency of accuracy of the eq (1.5) which temperature After dividing according to eq (1.8), the second harmonic spectral feature is converted to application > 1 most of the plasma cross section, so that emission from any position will be at or near the blackbody level. by T a spatial temperature converts frequency to position. profiles derived in this fashion by The is Page 36 discussed in the appendix. The measured temperature profile can be fit to a Gaussian: EQ (2) T(r) = To exp [ where TO,rOa refer to the central temperature, The choice of a Gaussian is arbitrary; expressions could include easy be used. integration Favourable position and width. a number of other analytic features of the Gaussian and the direct correspondence of the width parameter, a, with the perceived profile width. Its major drawback is its infinite extent, which fortunately introduces only small errors. A temperature fig.(1). The profile three and corresponding shown The weighting favours in the of peaked temperature and ignores the edges where the emitting layers are not expected to be optically thick. most are free parameters of the Gaussian are adjusted to minimise a weighted mean square deviation. region fit A common feature of Alcator profiles is the slight increase towards the inside (high frequency) edge which results from an overlap of the second and harmonics. third For this reason too the edge points are not included in a fit. SEC 2.2 COMPARISON TO CLASSICAL RESISTIVITY Classical resistivity results from collisions considered. additional between two Anomalous processes charged resistivity a treatment where coulomb particles are the only interactions then is any deviation due to such as non-collisional phenomena or collective effects in the plasma, eg. scattering from waves. The classical Page 37 resistivity calculated by Spitzer 2 for a pure hydrogen plasma will be the basis for comparison, with any deviations contained in / ratio With this definition the plasma resistivity is: n = 5.22 x 10-3 lnA T-3 /2 (eV)[n/n*] EQ (3) (ohm-cm) For an ohmic-heated plasma the local resistivity by the the ratio of is determined the local electric field and current density. central current density is related to the toroidal magnetic field The by qO: 2 B j 0 EQ (4) The presence of sawtooth determines the = j_____ u0q= R oscillations MKS (MKS) indicates qO < 1 which central current density 3 . The central electric field is established by assuming the measured edge loop voltage is across the assumptions the be ignored. With resistive voltage and central temperature are related: EQ (5) constant plasma cross section, and by sampling data during periods of constant current so that inductive effects can these then V T(eV) 3 /2 = 522 lnA B-n-(MKS) q0 r Page 38 The relation resistive voltage range the of between the measured parameters. and The data spans the available is shown in fig.(2). two temperature central Curves eq (5) representing are For temperatures less than 2 KeV the data are consistent with n/n* = 1.5. However, above this temperature a marked discrepancy indicated. In fig.(3) the ratio of measured to predicted temperature is occurs. plotted versus density for the same discharges shown in fig.(2). The deviation from predictions based upon eq (5) occurs at lower densities as well as higher temperatures. production of Because these conditions favour the runaway electrons, and because of known distortions of the electron distribution which may occur under conditions 4 , these figs.(2) and (3) suggest the involvement of some related process. For example, the presence of the cyclotron creating emission non-thermal the electrons false could impression enhance of temperature, or it might be argued that a large anomalous appears at lower a higher resistivity densities as a result of runaway driven collective modes 5 . However, the observed deviations can be explained by the enhanced impurity levels encountered during low density operation. The presence of impurities observed resistivity over the in the plasma predicted will value which Zeff = 1. This increase can be absorbed into the factor increase the is based on n/n*, which is approximated by the analytic expression 6: n EQ (6) .914 x Zeef S 1.07 +=e The Zeff is measured with continuum visible 7 and + 1.077 + Zeffaf a prediction of .58 x Zef bremsstrahlung emission in n/n* is made according to eq (6). the A Page 39 temperature measured value of n/n* is obtained from the electron and A constant coulomb logarithm is used for all data, since the eq (3). variations which would be produced by changing density and temperature are than smaller the inherent inaccuracy intrinsic the in approximations used to generate the logarithm to begin with. In fig.(4) the two values of n/n* are plotted versus one another for the same reasonable data shown considering The parameters. in the figs.(2) inherent measured central and (3). errors The agreement is in the temperature and component voltage consistent with the classical predictions provided that the effect impurities is consisdered. It is not necessary to are of include any anomalous effect in order to explain the relation between the measured electron temperatures and voltage. It would seem then that knowledge of Zeff and the would in render well measurements, introduced Zeff The loop by ohmic-heated and loop voltage plasmas. Unfortunately voltage, are measurement is not these easily made with susceptible to discounted if errors measurements are made during periods of relative steady state operation. complication results from the measurement being rather two changing currents in the plasma and in external coils. These inductive effects can be partially edge, voltage superfluous independent measure of electron temperature behaved accuracy. loop than the centre. 'The Zeff made at A further the plasma determined by continuum bremsstrahlung will depend upon an absolute intensity calibration furthermore is a chord average measurement, of Zeff will require an inversion These factors contribute of and so that the central value several chordal measurements. to an overall systematic uncertainty which Page 40 does not effect the relative precision of to measurements corresponding subsequent discharges, for example, but may influence the absolute accuracy of both by the same amount. SEC 2.3 PROFILE WIDTH If the voltage is constant across the plasma cross section the then assumption of classical resistivity combined with the temperature profile determines the current density: EQ (7) J(r) = JO exp [r - r 2 Integration of this expression yields a quantity which should equal the total plasma current: I = j0 EQ (8) a2 03 If the central current density is fixed by sawtooth oscillations the profile width can be written: EQ (9) r2 a2 = 3 qO 2 qlim lim where rlim is the limiter radius and the value of q at the limiter is determined by the plasma current and the toroidal magnetic field: EQ (10) qlim = 2w B r{im iR (MKS) u01 R Because the Gaussian profile does not become zero near the limiter radius as a real profile must, this treatment will be in error by : Page 41 E Q (11)E(1)exp -32ir im2% 2 at asa2 Fig.(5) is a plot of the profile calculated from eq (10). of the limiter This same value was used in conjunction with the previous discussion of temperature scaling with classical resistivity. that the qO inferred here is not a measure density. of central It should be noted the central current It is a number which works best within the framework of the applied models. current q A least square fit of a power law to agrees with eq (9) and indicates qo = .85. data 9O versus The data are an average of a large number of discharges, all having sawteeth. the width To date no distribution has direct been measurement made, of qO, or of the so that a comparison with the inferred qO is impossible. A good bit of scatter is present in the data because of averaging is not apparent. of fig.(5) which If the data are divided into extremes of temperature most of this scatter vanishes. This is shown in fig.(6), where the data of fig.(5) is replotted to include only the temperature extremes. if the data are this difference sawtooth The obvious separation of data does resulted the discharge, which difference results is from an variation not from the an unequal would case. actual be It sawteeth, sec (5), constant, so it seems unlikely that current density on axis. averaging noticed is effect manifestation of the fourier transform that of occur sorted by density, current, or magnetic field. periods Measurements not several during a single unclear whether the in the plasma or is the generates indicate some over If the profiles. that qO is relatively discharges have a higher The difference may be explained by allowing Page 42 an accumulation of impurities on axis, thus producing a local increase in Zeff. The increased resistivity on axis would then produce a higher temperature for the same current density. This explanation can neither be confirmed or disproved with the existing data. SEC 2.4 PROFILE CENTRE The central position inferred from the Gaussian fit depends upon the toroidal magnetic field value used. This value is not measured directly, but is arrived at indirectly through an measurement of the magnet current. from this technique. field such Other inherently Substantial errors have resulted possible influences upon the magnetic as the ripple produced by the toroidal asymmetry produced by diagnostic ports and current penetration times in the been inexact magnet have ignored since they are unmeasured quantities and are expected to be small. There is nothing in the data which suggests that these effects should be included. The cyclotron emission spectrum provides direct measure of magnetic field. for a series of discharges, reflect actual plasma measurements made techniques with reflect changes motion. the only independent Given that the field is constant relative magnetic the When pickup in profile compared loops 8, with flux position fig.(7), same relative change in position. magnetic loops are sensitive to the outer positions surfaces, the two Since the while the profile fit is governed by the plasma centre, the agreement in fig.(7) indicates motion of the entire plasma cross various applied vertical control agreement of the relative change fields. in section Fig.(7) position, and in response indicates should not to the be Page 43 interpretted as measuring an absolute shift of the centre with respect to the outer flux surface. outward, Although such a shift, approximately 1 cm is expected due to toroidal effects, the observed 'shift' is just as likely due to a systematic overestimation of the magnetic field. If, based upon evidence from the pickup loops or the soft array, the temperature profile magnetic field is determined harmonic has profile by a maximum. is assumed to be centered, then the the frequency at which REFERENCES & FIGURES 1.) Engelmann & Curatolo 2.) L. Spitzer Physics of Fully Ionized Gases Wiley & Sons, New York (1962) 3.) S. vonGoeler, W. Stodiek, N. Sauthoff Phys. Rev. Letters 33 1201 (1974) 4.) J. Rice 5.) 0. Buneman Phys. Rev. 115 503 (1959) 6.) W.M. Stacey Jr. Fusion Plasma Analysis Wiley & Sons, New York (1981) p. 264 7.) E.S. Marmar To be published 8.) MIT (1979) Rev. Sci. Instrument P. Pribyl S.M. Thesis also PFC/RR-81-21 2 nd a recalibration of the current will bring the two techniques into agreement. Ph.D. Thesis the If this value disagrees with the field determined by the magnet current measurements, SEC 2.5 x-ray MIT (1981) Page 44 Te(KeV) GAUSSIAN TEMPERATURE PROFILE FIT *. 1.0 'I -- DATA GAUSSIAN FIT ,5 WIDTH; a I T0 970eV = I i 10 9,1cm ro=-1.5cm CENTRE; CENTRAL TEMPERATURE; = 0 -10 RADIUS (cm) Fi g. (1) Electron temperature profile with fit. From steady state portion of Hydrogen discharge: 60 kGauss, density = 2.5x1014 cm 3 line average. Page 4b Te ELECTRON TEMPERATURE KEV vs RESISTIVE VOLTAGE 3.0 0 0 0 0 8 0 0 0 0 0 0 00 0 0 00 2.0 -0 0 00 7 00 =2 0 0 00 0~000 000 1.0 77 Vres I I 1.5 2.0 I 2,5 Fig.(2) Resistive voltage incorporates inductive corrections. These data from steady state portions of sawtoothing hydrogen discharges in 80 kGauss magnetic field. -1 VOLTS 1* I 3.0 Page 46 RELATIVE ELECTRON TEMPERATURE vs LINE AVERAGE DENSITY Te T* 0 00 E T* S0 - 2 380 V'q0 0 3.0 0 0 0 0 0 0 0 2.0 00 0 0 0 0 0 0 0 0 00 00 0P 00 0 1.0 -14 ne =10 A 1.0 f 2.0 cm 3.0 Fig.(3) These data correspond to the same discharges as th previous figure. The predicted temperature Te supposes n/n* = 1. -3 Page 47 7* (Te , PREDICTED RESISTIVITY vs MEASURED RESISTIVITY V) / 7 // Te 7* 7* 6 [T3 / / 0 0 0 5 0 0 LEAST SQUARE FIT SLOPE = .95 4 / / 0 3 00 0/ 00 2 0 00 0 /10 1 0 .914 Zeff 1,077 + Zeff 7 W + .58 Zeff / / 7*(Zef) I I t I I I 1 2 3 4 5 6 Fig.(4) These data correspond to the previous two figures where measured Zeff are available. The primary contribution to Zeff comes from partially ionised Molybdenum. 7 Page 48 a cm TEMPERATURE PROFILE WIDTH vs LIMITER q 12 LEAST SQUARES FIT a = 18.03 ql-,50 a 0 == q 8 =0qq = 1,85 6 2 q LIM I 1 2 3 4 5 (5) Data from discharges Fig in Hydrogen and Deuterium; magnetic fields ranging from 50 to 100 kGauss. The error bars represent a standard deviation over an averaged set of data. 6 I rEM R R a CM -ts TEMPERATURE PROFILE WIDTH vs LIMITER q 12 0 0 I0-1 10 0 0 0 *0, 0 . O 1.0 0 a. 0o= 0.8 8 6 * HIGHEST TEMPERATURES 4 o LOWEST 2 TEMPERATURES q I i I 1 2 3 I 4 5 Fig.(6) These data represent the 15% most extreme temperatures from the previous figure. This separation does not occur if data is selected according to current, density, or working gas LIM Page 50 PROFILE CENTRE vs. FLUX CENTRE CYCLOTRON ro (cm) 000 1.5 0 0 UNITY SLOPE 0 0 0 0 1.0 0 0 0.5 0 0 0 0 0 0 0 0 0 00 0 0 010 0 000 -0.5 0 0 0 FLUX POSITION (cm) I -1.5 -1.0 I -0.5 I 0.0 0.5 Fi g. (7) The offset in position between the profile centre and edge flux surface measurement most likely results from errors in magnetic field calibration, and should not be interpreted as resulting from a Shafranov shift. For these data the same field calibration was employed for discharges from a single day at constant field, so only a constant systematic error results. 1.0 Page 51 NON-GAUSSIAN PROFILES SEC 3.0 During the steady state portion of plasma from different very Gaussian are discharges, occasionally radiation portion -of plasma observed from profile most extreme cases the profiles are hollow with are deviations diffusion time of the ohmic flux. finite large During the current rise elements. impurity discharges, the to attributed by These observed. deviations result from pathologic current distribution or localised profiles a depressed In the central While these unusual profiles may be transient, they must temperature. persist for the duration of the Michelson scan period in order to be observed. SEC 3.1 CURRENT RISE The plasma current is Because the hot plasma driven by an induced electric field. is a good electric conductor, magnetic flux will penetrate inward over a diffusion time: EQ (1) T = 2 a L2 = .55 T3 /2 L2 (msec, KeV,cm) where a is the plasma conductivity and L is During time steady state this diffusion the is diffusion about Te = 1.0 keV and L = 9 cm, while the ohmic flux changes scale of 400 msec. be nearly constant across the assumption of constant voltage made in steady state. 45 msec, over for a time Because diffusion through the plasma is quicker than the change in ohmic flux at the edge, the will length. plasma induced cross sec (1) is loop voltage section. justified The during Page 52 During current rise (fall) the ohmic flux is being changed times comparable to the magnetic diffusion time, consequently flux, and hence current, will be added portion outer (removed) preferentially initial ionisation period the diffusion steady state. Once ionisation is time rises as the temperature increases. cannot penetrate the plasma at the rate of the initial the the During diffusion time is close to zero because the plasma is a poor conductor. the from The corresponding current distribution of the plasma. will be broader (narrower) than predicted for the over complete Because flux current rise, additional current will flow along the outside of the plasma. some instances a hollow current profile will be produced In by a rapid current profiles current rise. The relationship between the temperature expressed in eq (2.7) implicitly across the plasma cross section. current rise This condition does not apply during so a hollow current profile does not a similar temperature profile. with the indicates density. voltage a comparable current, since the increased conductivity at the plasma edge resulting from the higher temperature increased necessarily However, a hollow temperature profile during current rise almost certainly situation a constant loop voltage when the rapidly changing flux creates a higher voltage at the plasma edge; require requires and can only result in an Thus a measured hollow temperature together increased profile rise indicates the coexistence of a skin current. with any edge current during current Page 53 An example of this skin effect is shown in evolution rise. of the is not continuous, during current plasma breakdown. feature will rise peaked but disruptive rearrangement of the shape where the hollow profile can be followed through the current The transition from hollow to discharge fig.(1), is during such accomplished by a sudden distribution. current is profile determined The profile during and shortly after If the earliest measurable profile is hollow, persist during current a rise. this If the first measured profile is peaked, subsequent profiles will also be peaked. The persistence of the beyond the magnetic hollow current (temperature) profiles diffusion time results from the localisation of ohmic heating power to regions of highest current (temperature). disruptive conversion from ensues from the instability of hollow to hollow peaked The profiles undoubtedly profiles to various tearing modesi. SEC 3.2 Localised IMPURITY RADIATION energy loss by impurity significantly alter the temperature profile. line radiation can The energy loss rate per free electron per impurity ion is represented by the quantity so Lz that the total radiated power is: EQ (2) Prad = ne nz Lz(T) where ne,nz are the electron and element Lz particular impurity impurity. ion will be densities. a sensitive For a function of electron temperature, so that different temperature regions will have Page 54 very Unlike field diffusion the effect of impurity line radiation density. can radiation losses, even if they have the same impurity different alter profile shapes during steady state. Molybdenum is a common impurity in Alcator and an effect upon the temperature profile can attributed to this element. be A similar effect has been observed from PLT resulting from tungsten radiation 2 . Calculations 3 show that molybdenum radiated power is maximum for coronal temperatures slightly under 1 KeV and decreases sharply for temperatures above or below 1 KeV. when Alcator molybdenum of [ne < 1014 cm-31 densities are typically large, two distinct profile shapes are observed; value At low densities either a peaked profile with central several KeV, or a hollow profile with a central temperature of 1 KeV or less. Examples of such profiles are shown in fig.(2). In the case of the hollow profile, bolometrically measured radiated power is greatly increased over the peaked case, and accounts for 70% of the ohmic input power. The hollow profile discharge with the central temperature central soft oscillations. early in x-ray oscillations. is reduced throughout exceeding and 1 KeV. shows no the The sawtooth In the other case the central temperature exceeds 1 KeV the central soft emission never endures discharge, x-ray and rises to values of 2.5 - 3.0 KeV. emission is large and shows normal The sawtooth For both cases the ohmic input power is about the same, although the spatial distribution of the ohmic power may be different. If this input power can be converted only to plasma energy or to line radiation, for each case a balance of power requires: EQ (3) POH + Plasma + Radiation = neTe + nPT T + Prad Page t! where the confinement time line radiation, Furthermore, and is that expected for the condition of T is assumed to be if the impurity density is assumed equal for both constant no cases. for both cases, the difference in measured plasma energy for the two situations depends upon the difference in radiation rates ALz as follows: {ATe + ATi) = nZT ALz EQ (4) Based upon the calculations for molybdenum and the measured temperature differentials: ATe = 2.2 KeV ; AT 1 = .7 KeV + ALz = 1.75 x 10-25 watts-cm 3 EQ (5) assuming a confinement time of 10 msec the inferred is nz = 2.5 1011 cm- 3 with density and resulting measurements of a Zeff Zeff molybdenum = 5. impurity density These values for impurity are consistent line radiation with and spectroscopic spectroscopically measured Zeff. Because the radiated power increases with temperature for values below 0.7 KeV any increase increase in radiation loss. radiation term in If determined the will be balanced by an temperature exceeds I KeV the decreases with temperature so that the temperature is free to rise to higher values. is temperature quite Which of the two profiles will develop early in the discharge as no transition between profile types has been observed. In fig.(3). the profile can be seen throughout the discharge. hollow temperature Page 56 It is not clear that the hollow temperature profiles of are associated sawteeth under current with these density so hollow current conditions profiles. only requires fig.(3) The lack of visible a reduced central that q0 < 0. That no disruptions occur suggests that the profiles are not hollow, or if so, are somehow stable against disruption. Figs.(2) and (3) refer to a particular reproducible phenomenon resulting from impurity radiation, and do not represent the plasma response to the presence of impurities in Situations have been observed where a sudden influx of high results in heating, cooling, non-reproducible events which may disruption occur etc. cases. Z matter These, however, are randomly condition. SEC 3.3 all REFERENCES & FIGURES 1.) B. Carreras, H.R. Hicks, B.V. Waddell Oak Ridge Report ORNL/TM-6570 (1979) also: Nuclear Fusion 19 1423 (1979) 2.) Plasma Physics and Controlled Nuclear Fusion I.A.E.A. Vol.III 10 (1978) 3.) D.E. Post, R.V. Jensen At. Data Nucl. Tables 20 597 (1977) at any operating Page 57 EXAMPLE OF SKIN EFFECT DURING CURRENT RISE Te (KeV) 1.5 1.0 0.5 I 10 i i 0 i -10 I RADIUS (cm) Fig.(1) Temperature profiles measured during current rise 60 kGauss field; 400 kAmperes. 0 to from 3 density = 1.6x1014 cm- line average. Page 58 Te (KeV) RADIATION DOMINATED TEMPERATURE PROFILES 3.0 2.0 1.0 10 0 -10 RADIUS (cm) Fig. (2) Profiles from steady state portion of Hydrogen discharges at 80 kGauss. results from Molybdenum radiation. consecutive Hollowness In both the plasma current = 350 kAmperes; density = 8.5x1013 cm- 3 line average. cases Page 59 EVOLUTION OF RADIATION DOMINATED TEMPERATURE PROFILE Te (KeV) 1,5 1.0 10 0 RADIUS (cm) -10 Discharge in Hydro en at 0 kGauss. Density never exceeds 1.0x101 4 cm- 3 line average. No sawtooth activity is observed for the entire discharge. Hollowness results from Molybdenum radiation. Page 60 THIRD HARMONIC EMISSION SEC 4.0 Radiation at the third gyroharmonic rivals the in harmonic intensity, and is the highest order of cyclotron emission detected by from thermal Alcator plasmas the frequencies these the Michelson is plasma not emission interferometer. At to the optically extraordinary mode, but in a gray region where the second T thick < 1. Because of this will not be at the blackbody level and will depend upon electron density as well as temperature. This in contrast to the case of the second harmonic where values of T > 1 allowed the assumption of blackbody emission. the third provide analysis While the treatment of sec (2) when harmonic information of the can not regarding provide the applied to temperature profiles, it does plasma density. Conceptually, second and third harmonic emissions together allows for an absolute calibration of the diagnostic system, freeing it from dependence on other sources of temperature measurement. SEC 4.1 REFLECTIONS & OPTICAL DEPTH The optical depth for the extraordinary polarisation of the is not third harmonic isl: EQ (1) T3 = .236 neTe2 B (1014 cm-3, KeV, Tesla) The expected intensity from a thermal plasma is: EQ (2) 13 = Ibb {1 - e-T3} + I e-T3 where Ibb is the black body level. Because the optical depth Page 61 large the I term must be included to account for additional radiation reflected into the viewing angle from the vacuum chamber walls. fraction p of each ray is reflected If a when encountering the wall, summation over all reflections results in the following expression 2: EQ (3) 13 = Ibb 1 -e- 1 - -T pe The effective reflection coefficient p must account for radiation loss to the vacuum chamber, port extensions and windows. with polarisation scrambling. not absorbed by the Because the plasma, fig.(1) blackbody eq (1). during level the is third loss to plot harmonic or intensity relative to the plotted versus the calculated optical depth from run. Any systematic errors of Inherent errors discharges due to alignment or The is reasonable when the strong dependence of and temperature is considered. plasma walls (See sec (1.3)) calibration may be assumed constant for these data. this the These data come from the steady state portion a single is will reduce the effective reflectivity by increasing the likelihood of In polarisation ordinary scrambling rejection by the interferometer. It must also deal in T scatter in upon density these measured parameters limit the accuracy of computed t to about 20%. The data can be described by eq (1) if an effective reflectivity of 0.5 is chosen. When operating with various toroidal fields or different optical alignments, the value of p inferred from plots such as fig.(1) changes. Systematic errors in calibration making it a measure of more than reflectivity. will be absorbed by p Page 62 In fig.(2) the measured electron temperature is product the plotted versus the line average density and the ratio of second and of third harmonic intensities. In this format the potential of the third harmonic as a diagnostic is illustrated. usefulness Provided that a curve such as fig.(2) has been previously generated it can be used infer the density from the measured temperature, or temperature from the known density. the to second In the latter case a temperature harmonic can be achieved. calibration of In practice this technique provides a check of the calibration currently being used, but does not free the system from independent temperature measurements. alignment requires a new curve like fig.(2). be realised computed. where the Every new Ideally, a situation can systematic error remains constant, or can be A single curve such as fig.(2) would then be a useful tool for measuring temperature. SEC 4.2 PROFILE SHAPE Given that its intensity is proportional to the the third optical depth, harmonic spatial profile becomes a function of the density and temperature profiles: 13 (r) - ne(r) exp -3 EQ (4) where the Gaussian temperature profile example third harmonic profile, is plotted 2 invoked. case an The shape, as in of the second harmonic, is approximately Gaussian as shown. The narrowness of the dependence is in the fashion of sec (3) where the frequency squared coefficient is removed. the Fig.(3) profile in fig.(3) upon the temperature profile. results from the cubic Because of this the density ragc u.) profile contributes profile. In nothing to the shape of the of harmonic fig.(4) the third harmonic width is plotted against the corresponding second harmonic width for a number of ratio third widths is nearly /3 , discharges. which reflects the fact that the intensity is proportional to the cube of the temperature. only confirms to empirically between This plot that the density profile is broad, and can not be used describe a parabolic the shape. For example, the difference and square root parabolic density profile would not be discernible in the third harmonic width. SEC 4.3 The REFERENCES & FIGURES 1.) Engelmann & Curatolo Any finite density effects will be incorporated into the factor p so that only the nT2 dependence of the optical depth is important. 2.) D.A. Boyd Int. J. Infrared & MM. Waves 1 45 (1980) This reference contains a general description of the process by which density profiles can be extracted from the third harmonic emission. Page 64 RELATIVE INTENSITY vs OPTICAL DEPTH FOR THIRD HARMONIC 13 'bb 0? 07 o 0 07 0 13 - e 7P=O. 5 1-pe 'bb 0 0" 0 0 0 0 o 7 / P0 000 0 0 0 O 0 od 0 0 / 0 / I .10 0 0 / 0 0 0 0 0 0 I .15 I .20 Fig.(1) from Hydrogen intensities harmonic Third The optical depth is discharges at 80 kGauss. and temperatures measured from calculated densities. I T 3 .25 Page 65 ELECTRON TEMPERATURE vs. PRODUCT OF DENSITY & HARMONIC RATIO 0 0 00 Te(KeV) 00 0* 0 1.') %0 00 0 0 0 00 00 00 1.0 000 0 00 00 0 0 0 .8 .6 .2 fe (fringes) 12 13 I 3 I 6 9 Fig.(2) Data from Hydrogen discharges at 80 kGauss. The 'fringe' is the ex erimental unit of density, 1 fringe = 5.6x10 13 cm- line average. 12 Page 66 GAUSSIAN THIRD HARMONIC PROFILE FIT GAUSSIAN FIT CONSTRUCTED DATA WIDTH; a = 4.5 cm CENTRE; ro=-1.5 cm +A 10 0 -10 RADIUS (cm) Fi g. (3) Third harmonic profile from the steady state at 80 kGauss. discharge of, portion Density = 2.6x,014 cm-3 line average. The steep increasing signal on the outer edge is due to overlap of the second harmonic. Page 67 a 2 CM SECOND HARMONIC WIDTH vs THIRD HARMONIC WIDTH 0 12 08 ... 2 (r) e ( /0 0 10 000 0 00 8 6 SLOPE = 1.67 I3 (r) e- 03 2 a3 I I I 1 2 3 I I I I 5 6 7 Fig. (4) Data from sawtoothing discharges at 80 kGauss. Widths are determined by Gaussian fits to corresponding harmonic emissions. CM 7 Page 68 TEMPERATURE FLUCTUATIONS SEC 5.0 The Michelson interferometer requires an oscillation generate a single interferogram. The resulting period spectrum is not measured at one time but is representative of this entire period. is to Nor the spectrum swept in time, with each frequency corresponding to a different time during the sweep. If the emission intensity changes during -the scan period, this change appears in the interferogram, and consequently in the spectrum through the fourier transform. The problem with temporal fluctuations is that there is no way the fourier transform process can separate features in the interferogram resulting from temporal variation from features which are produced by the action of the interferometer. will For example, a single narrow spike of emission produce a sinusoidal perturbation on the spectrum. is sampled at emission irregular intensity time will not intervals, produce periodic periodic Because data fluctuations in fluctuations in the interferogram, hence no sharp spectral features will result. The study of emission monochromator, such fluctuation is best performed with a as the Fabry-Perot interferometer which provides adequate time resolution, while relying upon discharge reproducibility to measure separate frequencies one at a time. Two common fluctuations in cyclotron emission are due to the sawtooth instability and to plasma motion. SEC 5.1 FABRY-PEROT INTERFEROMETER Page 69 The Fabry-Perot interferometer is an etalon of two parallel The separation between meshes is adjustable. reflective metal meshes. Radiation will be transmitted through the device if its wavelength an half integer times the mesh separation. Thus the device will transmit a fundamental frequency and higher orders of this So, for example, when fundamental frequency, transmitted using the separate viewing radiation in the second order. Fabry-Perot. polarising grid The the at the frequency. gyroharmonic fourth harmonic as will the be This effect must be considered when device must second is be is non-polarising so that a used if only one polarisation is desi red. The resolution, or finesse, in the first order of transmission is determined by the wavelength viewed and by the mesh spacingi: F = w{2g ln(g/2wa)}- 2 EQ (1) X2 where g is the mesh spacing (lines per inch)-and While a large finesse allows high amount of available incident signal. more desirable so that a the wire width. resolution it reflects a large In practice a modest finesse a reasonable is signal to noise ratio can be maintained. SEC 5.2 The SAWTOOTH INSTABILITY sawtooth perturbation instability has been modelled 2 by a growing of the q = 1 surface resulting-in an internal disruption which flattens the central temperature and densities to the values the q = 1 surface. at The redistribution of energy extends beyond this Page 70 surface, so that the density and temperature will increase for disruption regions just outside the q = 1 surface. surface the sawteeth are 'normal', outside, 'inverted'. surface is identified with the change from normal to inverted. point the after Inside this q = The 1 of inversion where sawteeth This is the model which will be used as a basis for comparison of measured quantities. When observing sawteeth in polarisation of the the optically thick extraordinary second electron gyrofrequency, the amplitude of the sawtooth is a direct measure of the temperature variation. of this change relative to the A plot time average local temperature is plotted as a function of position in fig.(1). Rather than absolute magnitude a ratio of temperature change to average temperature is used to characterise the sawtooth. This eliminates the need for precise calibration and corrects for any shot to shot variation. The model used to describe sawtooth temperature fluctuations outlined in fig.(2). Before the disruption the profile is assumed Gaussian, though not necessarily the same as the 'average' profile sec (2). and For narrower, is of example, the pre-disruption profile may be more peaked reflecting a slightly smaller value of qo. The post-disruption profile is taken to be flat in the central region such that q = 1 throughout this area. Using this model, the predicted fractional sawtooth amplitude as a function of position is: EQ (2) 2 TI)= 1 - exp r a2 I T(r) where x is the 'inversion point' of the 2 sawtooth. This position depends upon -the temperature profile width a, and the q value on axis, Page 71 qo. The inversion point and related, but not equal. q = 1 surface pre-disruption the For a peaked current are profile the pre-disruption q = 1 radius will always occur slightly outside of the inversion point, according to this model. The radial extent of the sawtooth perturbation is plasma is approximated by rmax density, and restricted by values of qo and a, as inferred from requiring measured = conservation of /2 x. For reasonable temperature profiles, the inversion point occurs at radii varying from x = 2 cm to x = 5 cm. The corresponding central sawtooth amplitude is expected to vary from 7% to 20% according to the model. A further measuring consideration the spatial resolution The Fabry-Perot is used the same optical path described in the appendix. along the viewing axis is governed by the being of the instrument, which is not negligible compared to the size of the feature being measured. with is observed, optics employed. while finesse at in conjunction The resolution the frequency perpendicular resolution is determined by the The measured fluctuation will be a weighted spatial average of the actual fluctuation amplitudes: EQ (3) AT(r) measured T(r) = where g(r,r') is a normalised J g(r,r') 6T(r') actual dr' T(r') response resolution limits of the apparatus. function g(r,r') =+ R[1 +.2F T the The appropriate choice of g(r,r') for the Fabry-Perot interferometer is: EQ (4) incorporating 2 2]1/2 sin2 R+r1 R+r Page 72 where F is the finesse. With an expected finesse of 18 for the Fabry-Perot at 420 GHz (B = 75 KG), the spot size (<2 cm diameter) and diffraction width (.3 cm) are not limiting factors in the instrumental resolution. As shown in fig.(1) the measured central sawtooth amplitudes are to insensitive changes qlim, in variations in plasma current or point does in whether these changes result from field. magnetic The with decreasing qlim as expected, but the profile change width also changes, leaving the ratio x2 /a2 constant. EQ (5) EQ(5 inversion AT2 x- q= qo Since: a. T (r=0) = 1-exp2 2 a the constancy of central sawtooth amplitude indicates an invariance of q0 Within the context of the model of fig.(2) the central amplitude is determined relate a with qo and qlim. a. It is by x and a, while the arguments of sec (2) Finally, eq (5) identifies x with the uncertainties of in it is possible to nearly satisfy them all. in fig.(1) can be made consistent with the model if value qO and unlikely that a single value of qo will satisfy all these relations, however, by invoking quantities sawtooth qO = .80 is chosen. a since between sawteeth the observed suggests measured The data shown pre-disruption The variation of qo between .8 and 1.0 during a sawtooth compares favourably with the 'average' sec (2), the linear a 3/2 temperature power current qo = .85 change density, of with time or qo, rage I3 variation. Fig.(3) shows the sawtooth amplitude for otherwise under identical as conditions for those helium in discharges The fig.(1). inversion point is the same in the two figures, but the amplitude in helium is doubled what it is in hydrogen or deuterium. While doubling the amplitude, the sawtooth frequency is halved, indicating the that heating of the electrons occurs at approximately the same rate in both The amplitude difference can not be related cases. to an ion mass difference since no dissimilarity was detected between hydrogen and deuterium. There is insufficient regarding data to make a similar claim If the same sawtooth model is to apply to helium as Zeff. well as hydrogen it is necessary to require a smaller qo (i.e. higher current density) in helium. SYMPATHETIC PLASMA OSCILLATION SEC 5.3 Another source motion pre-disruption produced by of emission fluctuation oscillations of results from plasma the controlling fields. fields are powered by a solid state rectifier which imparts ripple. field. When polarisation a 360 Hz to noise pickup, the ripple produces a genuine In addition effect upon the These plasma position, presumably through the vertical viewing a portion of the second harmonic extraordinary where the emission profile has a non-zero spatial derivative, plasma motion will produce the following fluctuation: EQ (6) -2 r Ar iAT(r) T(r) where Ar is the amplitude a2 of plasma motion. Fig.(4 shows these Page 74 fluctuations for radial positions of -10 cm from two concecutive discharges, and compares them to the loop voltage two signals measurements 3 . centroid The this loop proportional to the positive time derivative ripple, so The out of phase with respect to each other as expected are and both indicate a plasma motion of t3 mm, x-ray oscillations. fig.(4) indicates field by about T/2 rads. that in agreement with voltage oscillation of vertical the is field plasma motion lags the vert.cal This lag is attributed to a 4 msec flux penetration time through the edge plasma 4 . No motion is observed when viewing the plasma Sudden changes in centre this because the oscillation profile amplitude alterations of the temperature gradient. result zero. from rapid Fig.(5) shows this effect in During the marfe the fluctuation level is 1% . the average When the marfe plasma motion ceases temperature increases 20% and the fluctuation level returns to 3% , as in the previous examples. of is with a 'marfe'5 while viewing the -10 cm radial position. conjunction abruptly, gradient is constant, the If the discontinuity fluctuation amplitude represents a drastic change in amplitude in measured profile width; from a = 15 cm to a = 10 cm. These oscillations are an example where an apparent fluctuation temperature in fact results from changes in position alone. It would be incorrect to assume that the fluctuations of fig.(4) resulted, example, from fluctuation. for ohmic heat.ing variation produced by the for voltage Furthermore such fluctuations will complicate any search small scale temperature profile variance, such as those suggested to occur in conjunction with the onset of certain MHD activity6 . Page /b SEC 5.4 REFERENCES & FIGURES 1.) K.F. Renk, L. Genzel Applied Optics 1 643 (1962) 2.) G.L. Jahns et.al. Nuclear Fusion 18 609 (1978) 3.) R.S. Granetz Private Communication 4.) P. Pribyl Private Communication 5.) The marfe is an episode of enhanced Ha emission that often occurs during high density operation. It is localised to the inside plasma edge, and is accompanied by sudden increases of plasma density at the inner edge. The cause is unknown. The name 'marfe' results from the modesty of its two principle investigators; E.S. Marmar & S.M. Wolfe. 6.) R.S. Granetz Submitted to Phys. Rev. Letters (1982) Page 76 RELATIVE SAWTOOTH AMPLITUDE vs RADIAL POSITION 0 --- ATe % - ql~~m - 3.6 .0im Te 10 Q...- lm ' /0 / I, (I / / 5 '0 I I I \ 0 I -5.0 I I I I I +5.0 I - i I I~j 0 a CM -* 00 07 0 -5 0 H-i Fig.(1) Data from discharges in Hydrogen at 80 kGauss. Densities = 3.0-4.0x1014 cm- 3 line average. No difference in amplitude results from changes in density, current, or magnetic field. Page 77 SAWTOOTH MODEL Te T NORMAL SAWTEETH INVERTED SAWTEETH -- .5 Fx x PRE-DISRUPTION PROFILE-+ 10 0 RADIUS (cm) Fig.(2) -10 A positive ATe/Te iimpeies a reduction in temperature - normal sawteeth. Negative ATe/Te implies an increase in temperature - inverted sawteeth. Page 78 RELATIVE SAWTOOTH AMPLITUDE vs RADIAL POSITION IN HELIUM 10 15 0 0 Te 0 10 FOR HYDROGEN, DEUTERIUM WITH SAME PLASMA PARAMETERS / / / / -5 0 I I I I I I +5.0 e...% -5.0 i I / / / / -~~0 0 5 Fig.(3) Data from discharges at 80 kGauss. Change between Hy.drogen and Helium was executed during single run (day) with immediate doubling of sawtooth amplitude in Helium observed. CM I Page 79 CORRELATION OF CYCLOTRON EMISSION WITH LOOP VOLTAGE OSCILLATIONS I I 2 e 1 6.0 7% LOOP VOLTAGE INSIDE OUTSIDE r = -10 cm r = +10 cm 24 e 61% 6.0 -F LOOP VOLTAGE 4.0 8,0 I I 12.0 16.0 Fig.(4) Data from consecutivedischarges in Hydrogen at 80 kGauss. Density = 3.3x1014 cm- 3 line average. msec Page 80 CORRELATION OF CYCLOTRON EMISSION WITH LOOP VOLTAGE DURING 'MARFE' I I LOOP VOLTAGE DURING 'MARFE' AFTER 'MARFE' 6.0 % 2q e 2.0 % I 10 I 20 I 30 Fig.(5) Data from current rise portion of Hydrogen discharge at 80 kGauss. Density = 2.5x101 4 cm-3 line 'average; current = 250 kAmperes at time of marfe. I 40 msec Page 81 SEC 6.0 DOMINANT PLASMA FREQUENCY EMISSION 'Dominant wpe emission' refers to a characteristic radiation near the plasma frequency. electron spontaneous It is usually a few times the thermal level in intensity and has spectral extent from central plasma to the edge cyclotron frequency. Dominant emission has been observed from the Alcator Tokamak as well as other is most noticeable when wpe/wce the electron distribution nearly devices, highest .6 and a non-thermal component of present. The emission intensity Because the trend in Tokamak operation is to is density possible for a given magnetic field; achieve i.e., to maximise a, the condition wpe/wce = .6 is not a usual situation. this and constant in time and can easily be measured with the Michelson interferometer. the is the reason For the appearance of dominant wpe emission is not expected, except perhaps early in a discharge while the density is rising to its steady state value. The most successful explanation of this radiation involves the Cerenkov result that the resonance emission with intensity runaway electrons, with the is proportional to the current carried by these electrons. SEC 6.1 SPECTRAL CHARACTER Example spectra of dominant wpe emission are shown In in fig.(1). each case an intense feature at lower frequency rivals emission at the second gyroharmonic in intensity. Otherwise the spectra are completely normal, with second and third harmonic intensities equal to the values expected from a usual discharge. in fact occurs That the emission feature at the central plasma frequency is shown in fig.(2), where the frequency of maximum emitted intensity is plotted as a Page 82 function of line averaged fig.(1). If a peak to averaged density ratio of 1.3 is measured frequencies are density in inferred from is the of representative measurements1 spectroscopic assumed, agreement with the expected excellent This ratio of 1.3 central plasma frequencies. values for many cases such as those of and from fits to multi-chord interferometric measurenents 2 of the plasma density. Although apparently narrow in gyroharmonic, second resolved by the plasma the Michelson. The comparison frequency spectral to emission the is just barely feature width at is equal to separation of the central plasma and edge cyclotron frequencies. the This is shown in fig.(3) where the half-width of the feature is plotted as frequency is a function of As frequency. the edge cyclotron approached the feature width becomes progressively narrower. same time the emission amplitude progressively decreases. the intensity relative to frequency, versus and the thermal this plasma frequency exceeds the blackbody level decrease can be seen. edge cyclotron value At the In fig.(4) is plotted When the central no emission is observed. Dominant emission usually occurs early in discharges which would otherwise be considered normal with regard to macroscopic parameters such as current, dominant cyclotron emission density, are radiation, electrons, sec (1.4). and generally indicating However, temperature. preceded the by presence of periods of broadband of relativistic The emission intensity usually decays away over a period of about 100 msec, but if the plasma density is low, episodes sufficiently so that wpe/wce is less than unity, the emission may persist for the duration of the discharge with nearly constant intensity. Page 83 When present, the emission is For higher most intense for Ope/wce = .6. frequencies the intensity decreases, as shown in fig.(4), while for lower frequencies the emission , if observed at all, is much Fig.(5) is an example of dominant emission for the case less intense. exhibits wpe/wce = .4. While the intensity is small, the feature now a double which structure humped was not apparent in the previous examples. The dominant emission feature occurs in both polarisations no difference noticeable extraordinary modes. than the in spectral shape between with ordinary and The extraordinary mode emission is more intense ordinary, but the actual degree of polarisation is not well measured. REVIEW OF PREVIOUS EXPERIMENTS SEC 6.2 Measurements on TFR 3 show early in certain types of of 100 msec. that the dominant emission occurs discharges, and decays with a time constant These discharges are characterised by low densities and high temperatures with wpe/wce = .6. The emission occurs at about the central plasma frequency and extends to the edge cyclotron In this the results are identical to those presented in the respect The intensity of emission, while greater previous section. thermal level, is not than the as intense as the examples shown in fig.(1). This situation also reflects the dominant emission seen T-II Tokamak 4 . frequency. on the JIPP Page 84 The Alcator A 5 results emission again intensity are similar being less to than those from double humped clearly An show feature which is only just .noticeable in fig.(5). This difference probably reflects between the two machines. the disparity in aspect ratios Alcator A has an aspect ratio R/rlim = 5.4, while for Alcator C R/rlim = 3.9. the the the Alcator C case. important difference is that the Alcator A measurements the TFR, For wpe/wce = .6 the separation of central plasma and edge cyclotron frequencies will be 20% greater in Alcator A. It is reasonable to assume that this larger spectral region increases the likelihood of resolving any emission structure. The Alcator A results also conclusively show the emission to polarised, with more radiation coming in the extraordinary mode. be The degree of polarisation, about 3:2, is consistent with the polarisation scrambling model used to explain observed ordinary mode emission at the second gyroharmonic. SEC 6.3 CERENKOV EMISSION Cerenkov emission occurs when the particle velocity can exceed the local electromagnetic phase velocity, that is, when the refractive index is situation the greater can vicinity optically thin than only of For electron plasma relative plasma this frequency. The plasma is this region, so the emission can be considered by summing over all runaway electrons. the a magnetoactive be realised by the extraordinary polarisation in the in unity. constancy of the Because of emissivity velocities, under certain conditions the emission proportional to the number of runaway electrons. this for summation, large intensity and electron will be Page 85 The plasma is described in the cold plasma approximation by by both thermal properties of This requires ignoring effects dispersion relation. Appleton-Hartree and electrons relativistic upon the dielectric Effects caused by runaways can the background plasma. be ignored since the density of these electrons is assumed than that of the thermal plasma. the phase velocity of much In the frequency region radiation the will always less wpe< W <wce exceed thermal the electron velocity, so warm plasma effects may be ignored. The emitted wave is required to be in phase with the emitting electron: ... = EQ (1) v cose k where v is the particle velocity, which is direction of the ambient magnetic field. assumed to lie with the electron velocity. fig.(6). The is always less than unity for the ordinary mode. condition can never be satisfied, so expected. angle of The.dispersion relation for a particular choice of frequency is shown in index the Combined with the dispersion relation this coherence condition relates the frequency and emission in no emission in refractive The coherence this mode is For the extraordinary mode the refractive index is greater than unity everywhere, particularly for small angles with respect to the magnetic field. It has been shown 6 ,7 for reasonable estimates of the density and energy (nr/ne = 10-5, E = 200 keV) the emissivity runaway will be strongly peaked near equal that to or the plasma frequency, with intensity greater than the expected second harmonic emission from Page 86 the 1 keV background plasma. This result depends upon the exact used to describe the runaway population, but is shown to distribution be insensitive to the perpendicular component of the runaway velocity. RAY TRAPPING SEC 6.4 Although a sufficient emissivity exists, the radiation must exit the plasma in order to account for the observations. While the phase velocity of Cerenkov radiation will typically be within 15 degrees the of magnetic field, the group velocity will be at 70 degrees or more, that is, closer to the perpendicular direction. This can be seen in fig.(6), where the direction of the group velocity is indicated by the normal to the dispersion emitted radiation is curve. close the group velocity of the to perpendicular does not mean that the radiation can be directly observed refractive That from the outside. Because the index for the extraordinary mode decreases moving from the inside to the plasma edge, radiation may be reflected back toward the centre. The plasma is homogenous in the direction of the magnetic field, so that the refractive index component in this direction is constant along a ray path. In cylindrical coordinates (infinite aspect ratio): n(rl) cosG(rl) EQ (2) n(r) = = n(r2 ) cose(r 2 ) = constant refractive index at r At the point of emission this constant is determined by the condition: EQ (3) {n coselemission point I [v- ) coherence Page 87 At the plasma edge the refractive total index reflection . occurs. internal approaches unity, so that The situation is less severe in toroidal coordinates, where the requirement for radiation to escape is 8 : rlim 4 1 + {n coseiemission point EQ (4) This condition is less restrictive on Alcator C than on Alcator A and TFR, which may in part explain the larger intensities observed from Alcator C. For example, spectra similar to fig.(1) have been measured DITE9 , which has R/rlim = 4.5. from If observed Cerenkov emission in the ordinary polarisation results from multiple reflections within the vacuum chamber, the escape condition should be applied at the chamber wall, if that is the source of 'emission.' Since the refractive is unity index at this point, from eq (4) it is seen that the radiation is capable of escaping. The most recent treatment 10 of Cerenkov the effect of multiple reflections emission over a 2w solid angle, angles of emission. This that work by by is limit to angles over all successful fig.(5). The in spectral of emission. Additionally a the considered runaway electron velocities is set by the escape condition as determined limit included a consequence of vacuum wall reflections, which necessitate some type of average over all lower integrating particularly explaining the double humped spectra such as shape has isotropising the reflected is, is emission from eq (4). The chosen lower of .8 c restricts the velocity range of emitting electrons such that only the total number of electrons in this region has any Page 88 significant effect the upon intensity and the current Using emission intensity. by carried these Hence the average are runaways related. the result from ref(10) with the reflectivities of sec (1), the runaway current is approximated by: Ix Irun(KAmp) EQ n11 n 1014 cm - 4 m3-1/2T Ix - Ibb Te(KeV) 3 average intensity of extraordinary mode Cerenkov emission averaged over a small frequency interval. This equation is meant to apply only to cases such the fig.(5) where emissivity is nearly constant over some small frequency interval, and a good spectral fit can be obtained. this as particular Application of eq (5) to example predicts a runaway current of 5 Kamps, which amounts to 2% the total current. This result is nearly identical to As the plasma frequency approaches the cyclotron frequency, two the original treatment of Alcator A data. competing effects occur. The Cerenkov emissivity increases 1 l, which enhances the emission intensity; encroaches upon cyclotron Thus the radiation resulting wpe/uce = -4. Fig.(5), Under this interval over which the emissivity is intensity edge from for example, deals a fixed nearly has meaning, and eq (5) applies. 15% the total. with the condition there is a frequency constant, and average This is not the situation in fig.(1) where eq (5) would predict a runaway current or frequency current -will depend upon the particular values of the plasma and cyclotron frequencies. situation the the spectral region of emission, which decreases the intensity by absorption. runaway and of 50 Kamps, Certainly it would be a mistake to apply eq (5) to Page 89 fig.(4) and interpret the decreasing emission consequence of a diminishing runaway current. SEC 6.5 REFERENCES & FIGURES 1.) E. Marmar Private Communication 2.) S. Wolfe Private Communication 3.) A.E. Costley et.al. Phys. Rev. Lett. 38 4.) 1477 (1977) K. Sakai et.al. Int. J. Infrared MM Waves &1 77 (1980) 5.) I.H. Hutchinson, D.S. Komm Nuclear Fusion 17 1077 (1977) 6.) H.P. Freund, L.C. Lee, C.S. Wu Phys. Rev. Lett. 40 1563 (1978) 7.) H.P. Freund et.al. Phys. Fluids 21 1502 (1978) 8.) M. Bornatici, F. Engelmann Phys. Fluids 22 1409 (1979) 9.) A.E. Costley 10.) K. Swartz, I.H. Hutchinson, Kim Molvig Phys. Fluids 24 1689 (1981) 11.) A.G. Sitenko, A.A. Kolomenskii Zh. Eksp. Teor. Fiz. 30 511 (1956) Private Communication intensity as a Page 90 'DOMINANT )pe EMISSION' EXAMPLE SPECTRA ARB. UNITS 80 KILOGAUSS ARB UNITS 60 KILOGAUSS 200 600 400 FREQUENCY 800 GHz Fig.(1) Extraordinary mode emission from times early in discharges when: = I.ixi01 4 cm-3 (60 kGauss) density central density = 1.5x1014 cm-3 (80 kGauss) central ragc .7 SPECTRAL FREQUENCY vs, LINE AVERAGE DENSITY FOR 'DOMINANT Wpe EMISSION' W/27r P11-. X Gl x X X X XXXC 150 1 x x Xx 47r X n. e2 X x x - 100 I x XX x XX 50 1.0 2.0 3.0 Fig.(2) Data from ordinary and extraordinary mode emission; magnetic fields from 60-100 kGauss. The drawn curve assumes a peak to line average density ratio of 1.3 101 cm 3 raye t SPECTRAL WIDTH vs. FREQUENCY FOR 'DOMINANT cope EMISSION' Co I I 8 Q =W wce(r=a) - co 0 6 /e p0 4F / / 40 / 90 / 0 /7W 0 21- 00 I W/wce(r=O) I I .5 .6 .7 Fi g. (3) Spectral width is the FWHM of the plasma frequency line. Because of finite instrument resolution, Q in excess of 8 is not achieved. Page 93 INTENSITY RELATIVE TO THERMAL LEVEL FOR 'DOMINANT Ope vs, FREQUENCY EMISSION' 1bb 8 6 4F| 0 2 F Wce (r=a) 0 %@ .60 0 Wce( r=O) *.'D I f I ,65 .70 ,75 Fig. (4) Data from repeatable discharges at 80 kGauss. I /Clce (r=0) * F RESOLVED 'DOMINANT Wpe EMISSION' pe wce(r=alim) pe 200 Wce 40 400 600 FREQUENCY 800 GHz Fig. (5) Extraordinary mode emission at 60 kGauss. The double humped feature of dominant emission is just being resolved. In these cases the intensity is reduced in comparison to previous examples. ~ - sr REFRACTIVE INDEX vs, ANGLE FOR FREQUENCIES NEAR Page 95 (0pe nil (= 1,0'4 (. pe EXTRAORDINARY WAVE ORDINARY WAVE n_- II nil EXTRAORDINARY WAVE ORDINARY WAVE = .60 = 1.02 wpe -1 Nj I oce = .25 Fig.(6) Only the extraordinary mode has phase velocity greater than the speed of light, particularly for direction. field the to near propagation Direction of group velocity is given by a normal Page 96 SEC 7.0 FLUCTUATING PLASMA FREQUENCY EMISSION Another type associated with emission.' It of radiation non-thermal is processes characterised feature, near the electron temporal fluctuations. The observed by from is an Alcator termed which 'fluctuating wpe extremely narrow spectral plasma frequency, which undergoes rapid the exceed may intensity combined While radiation at the cyclotron harmonics by factors of ten or more. sharing similarities theories, the with results from various experimental Alcator phenomenon and differ in some way from all of In this sense fluctuating wpe emission is a novel these. is result not previously documented. The fluctuating emission is present to some degree in nearly one half of all Alcator discharges. deuterium, and 100 KGauss. the helium plasmas It has been observed from hydrogen, in toroidal fields when electrons. conditions However, the favour the production fluctuation may reoccur termination or may persist throughout the discharge. not related with of to suggests runaway during plasma The emission is fluctuations of other diagnostic signals, such as current or density, however, occasional correlation activity 40 The emission usually occurs during the early portions of discharge usually from with MHD that such activity may be involved in the emission process. The temporal behaviour studied with a detector submillimetre radiation. of the operating fluctuating as emission a broadband is monitor The large intensity of the feature makes easily recognisable against the background level. best of it While the Michelson of interferometer as previously discussed is not suited for the study rapidly radiation, varying is a provide does amount of limited A scanning Fabry-Perot interferometer spectral information. is used to generate high-resolution spectra of the fluctuating feature. TEMPORAL BEHAVIOUR SEC 7.1 Examples of fluctuating wpe emission are shown in fig.(1), where interferometer. the operating is detector the is shortly after and ionisation persists Comparison of these examples from consecutive varying levels The shows discharges that of emission occur while macroscopic plasma parameters As shown, intensity of fluctuating emission can be associated with the flux of hard x-rays. in bursting about 50 msec. for such as temperature, density, and current remain unchanged. the on evident Superposed upon this background the fluctuating emission appears as a series of high frequency bursts. begins Michelson the with conjunction The resulting interference structure signal. analog in the presence limiter. emission of This suggests participation of non-thermal process, non-thermal electrons and emission requires the x-ray their collision with the The absence of hard x-rays alone does not guarantee lack of non-thermal electrons, plasma hard since electrons discharge however, during the first milliseconds of confinement of fast electrons will be poor, so that the presence of these particles and the subsequent hard x-ray activity can be related. The early portion of a discharge is shown in fig.(2). Fluctuating emission begins immediately after ionisation is and precedes the first intimations of x-ray flux by 1 msec. voltage is a nearly constant 50 Volts and relativistic electrons in .5 msec. is capable of completed The loop generating Page 98 A notable feature of fluctuating wpe with periodicity which the regular the bursts of emission often occur. Fig.(3) show examples of such cases. of rates 5 - 20 kHz emission Typically these of bursts. narrow spikes at a total duration of several dozen cycles. with continue with a new The fluctuations are best described as a series of spaced at regular The oscillations. sinusoidal occur fluctuations The fluctuation may cease at this time, or may series is time intervals, fluctuation rather spectrum by than by a produced frequency analyserl will incorporate the non-sinusoidal nature of fluctuations by including overtones of the repetition rate. Harmonic structure on such a spectrum will reflect the structure of the and not averaged over a small during the time of fluctuation emission. persists bursts necessarily the periodicity with which the bursts occur. measure just this periodicity, the time number emission frequency is of cycles. shown gradually This property can interval in decreases only be between the spikes To is A plot of this frequency fig.(4). As through observed seen, the the duration of when the emission for many oscillation periods and when the fluctuations are a series of periodic spikes. Often the spikes occur at there is no definite irregular fluctuation frequency.- explained by a superposition of differing justification await frequencies. discussion frequencies. of The spectral intervals several in Such periodic case which cases may be fluctuations of of this supposition must measurements of the radiation Page 99 SEC 7.2 CORRELATED BEHAVIOUR If the plasma density is sufficiently low so that wpe < wce, the fluctuating emission in fact happens at may be observed throughout the discharge, if it these times. When this occurs the temporal behaviour changes dramatically, and a correlation between emission and sawtooth disruptions becomes manifest. This correlation is fig.(5) several where the time Whenever fluctuating discharges emission is observed sawteeth is expanded. during the spikes. number from one discharge to another. highest fluctuation frequencies largest number of emission spikes. fluctuation one frequency sawtooth interpreted the sawtooth. observed to The fluctuation frequencies for than are the previous associated example. with the the the is plotted against the number of spikes during period. for as is This is seen in fig.(6) where a number of discharges. This may be meaning the total time of emission is nearly constant so that higher rates will produce more rates of of spikes during a sawtooth cycle remains a constant over sawtooth correlated emission are faster The of However, they are fewer in number and are confined many sawtooth periods during a single discharge, but vary portion Like the previous examples the emission is a series to a small time region surrounding the falling edge of The in when sawteeth are present, it always appears in phase with the sawteeth. periodic surrounding shown spikes. However the faster are also associated with higher plasma density, suggesting that the frequency is somehow dependent on this or some parameter. associated plasma- Page 100 A closer examination of the sawtooth correlations the of onset emission to corresponds that reveals of large m = 1 MHD periods oscillations which either precede or follow the sawtooth falling edge. The emission is also seen to occur during periods of m = 1 activity in This the absence of sawteeth. evident that the mode. result the is shown in fig.(7), it where radiation occurs in phase with the oscillations of These oscillations in the soft x-ray signals are thought to rotation, from q = 1 flux surface. either poloidal or toroidal, of the perturbed The correlation of fig.(7) suggests this that Fig.(8) phenomenon is also related to the emission process. shows the time history of fluctuating emission from a single discharge, displays a large variety of correlated emission behaviour. bits of emission are correlated with the small disruptions with the is and The first associated After this the emission continues at a current rise phase. fairly steady rate until m = 1 oscillations become evident on the soft x-ray trace. At this time the bursts of emission come in phase with these oscillations. Finally, once sawteeth is largest. Sawteeth and m = 1 activity times the q = 1 surface. near region to established the near the falling edge when m = 1 is emission confined become are confined to the plasma Because of the correlation with this activity the region of emission is probably also localised to the plasma centre. SEC 7.3 COMPARISON WITH OTHER EXPERIMENTS Fluctuating emission has Although behaviour sawteeth. also been observed from Alcator A. less carefully studied, the emission shows the same temporal as the Intense Alcator C bursts of results, including correlations with submillimetre emission have also been vage iut observed from DITE 2 and Microtor 3 , although it of consequence interferograms occur useless, fluctuating briefly from measurements with fluctuating signal A the the Since signal. fluctuations during the early phase of discharges, they do steady Michelson as ionisation, and ignored. of since the Fourier transform technique cannot not ruin the more important (from a confinement study point interferograms these if rendering the is emission fluctuating a rapidly deal with usually the unknown the same character as the Alcator results. have events radiation is may interferometers a nuisance, For portion. state or this reason, this regard 'noise' associated with as A submillimetre view) of experiment based upon a heterodyne detector, or monochromator, concerned with radiation at the second gyroharmonic will necessarily fail to detect frequency Furthermore, even if measured by a broadband detector, the emission. fluctuating signal may not be display plasma and storage. For noticed example, due to Alcator the format of data data is digitised for archival purposes at an effective rate of 3 kHz, making it useless for the study of the much faster fluctuating emission. SEC 7.4 REFERENCES & FIGURES 1.) I.H. Hutchinson, S.E. Kissel Phys. Fluids 23 1698 (1980) 2.) A.E. Costley 3.) R.J. Taylor Private Communication Private Communication WITH CORRELATION TO SUBSEQUENT HARD X-RAYS PLASMA CURRENT DENSITY SUBMMI .MICHELSON) ARD X-RAY, 20 MSEC/DIV. Fi g.(1) Discharges in Hydrogen EaO kGauss from same run. All pictures have the same scale. Page 102 vage iu. ONSET OF 'FLUCTUATING ope EMISSION' LOOP VOLTAGE 50 VOLT MAX. VISIBLE LIGHT BROAD-BAND SUBMILLIMETRE . HARD X-RAY 1 MSEC/DIV, Fig. (2) lonisation begins 1 msec from the left edge. Visible light results from molecular and hydrogen line emission. Decrease in visible light signal indicates complete ionisation. Page 104 PERIODIC BURSTS OF 'FLUCTUATING Wp, EMISSION' .1 msec/div Ave. fluctuation rate = 40 kHz central density = 3.0 1013 c1-3 .1 msec/div Ave. fluctuation rate = 10 kHz central density = 1.9 1013 c1-3 Fig.(3) average higher that is trend general A density. increased with occur fluctuations However, the rate tends to decrease in time for a sequence of bursts. This is noticeable in the top examole. Page 105 PERIODICITY OF 'FLUCTUATING RATE OF BURSTS 40 EMISSION' ope kHz ± 1.1 30 2 I L 20 I I 10 TIME FROM START OF EMISSION /ksec I I 200 I I 400 I I 600 Fig.(4) Data. from two sequences of periodic bursting emission from single discharge in Deuterium at I Page 106 EXAMPLES OF 'FLUCTUATING Ope EMISSION' CORRELATION WITH SAWTEETH PLASMA CURRENT PLASMA DENSITY SOFT X-RAY CENTRAL CHORD BROAD-BAND SUBMILLIMETRE 50 MSEC/DIV. 2 MSEC/DIVI 50PySEC/DIV. Fig. (5) Successive expansions of two dischargj Hydrogen at 60 kauss; 3 left ne = 7.3x10 right ne = 1.5x10 cm~ c n cm, Page 107 FREQUENCY OF BURSTS VS. NUMBER DURING SAWTOOTH CORRELATED EMISSION 80 0 / / AVERAGE FREQUENCY / kHz / 601- / / S 40 PREDICTION BASED UPON CONSTANT EMISSIN TIME t = 190 pksec / / / / / / / 20 S / / / NUMBER OF BURSTS I I 5 10 Fig. (6) Data from cases such as previous figure. I -I 15 20 Line is Page 108 CORRELATION OF 'FLUCTUATINGcope EMISSION' WITH m=1 OSCILLATIONS ASOFT i' X-RAY CENTRAL CHORD BROAD-BAND SUBMILLIMETRE .5MSEC/DIV. Fig.(7) The oscillations in the x-ray signal correspond to rotation of the m=1 mode. Bursts of wpe emission occur in phase with this rotation. Page 109 EVOLUTION OF 'FLUCTUATING Wpe EMISSION' CORRELATION WITH MHD ACTIVITY UPPER TRACE: BROAD-BAND SUBMILLIMETRE LOWER TRACE: SOFT X-RAY CENTRAL CHORD Fig.(8) Continuous picture of discharge after ionisation. 1 MSEC/DIV. starting 50 msec Page 110 SEC 8.0 SPECTRAL MEASUREMENTS OF FLUCTUATING EMISSION Spectral measurements of the fluctuating wpe emission show radiation occurs in an that extremely narrow bandwidth near the central electron plasma frequency. The spectral narrowness of the fluctuating emission distinguishes it from the broader radiation feature discussed in Sec.(6). detailed Because of the novelty account of the with the fluctuating techniques measurements will be presented in comparison of this and emission results section so of that a spectral meaningful other experimental results and discussion of various theoretical interpretations can be made. SEC 8.1 MICHELSON INTERFEROMETER As mentioned in sec (5) the Michelson interferometer is not well suited for the study of signals which vary during the 10-15 msec sweep time of the vibrating mirror. This insensitive fluctuations, to more rapid is not because but the due system to the is false interpretation of these temporal fluctuations by the Fourier transform routine as instrument. resulting In the from case the of changing sawtooth path or difference loop voltage of the induced oscillations, the amplitude of fluctuation is small enough so that the effect on the generated spectra is very small. fluctuating thermal wpe emission background catastrophic. Even the when However, because the is many times larger in intensity than the effect upon the resulting spectra is the interferogram is smoothed prior to the Fourier transform the spectrum is usually dominated features resulting from the sudden bursts. by unreal noise Page 111 Fortunately Michelson it is not interferometer concerning the interferogram necessary signal incident occurring in to order radiation. Fourier transform to recover Fig.(1) is the information an example during a period of fluctuating emission. In addition to the characteristic 10 KHz spiking there is a slower signal modulation resulting from the action of the interferometer. That this effect is not the manifestation of a peculiar time history of emission can be not demonstrated by a single example. observations bear out the contention that this indeed caused by the interferometer. However repeated modulation effect is In this example, the minima occurring at approximately 1.8, 3.5, 4.9, 6.3, 7.6 msec after the zero path difference correspond to situations where the path differences are integral multiples of the conditions, Axi=NiL, can to an that interval motion is nearly sinusoidal observed case the minima can indicated wavelength(s). Since these be satisfied for only a single wavelength, the recurring minima imply restricted emission the fluctuating emission about a single frequency. with known amplitude must As the mirror and period, be converted to a radiation frequency. frequency is about 100 GHz, be the In this which compares favourably with the expected central plasma frequency. The persistence of the modulations throughout the is a reflection of the spectral narrowness feature will no longer produce coherent difference feature: is equal AX=c/Av. interferogram the to the of the radiation. interference when the A path inverse of the frequency spread of the Thus by measuring the 'coherence spectral interferogram width This technique was employed in the length' of the of the feature can be determined. 1 9th century by Fizeaul and others Page 112 to measure widths of optical lines resulting from atomic transitions. A similar attempt to measure the coherence length of emission is by restricted the mechanical fluctuating the limitations instrument, however an upper limit of 5 GHz can be assigned of the to the bandwidth of the emission feature. SEC 8.2 In FABRY-PEROT SYSTEM order fluctuating wpe spectra than discussed doublet to feature the in investigate any spectral structure Michelson allows. As an example, for reasons sec (9) the emission might be expected to occur as the For Alcator wpi is approximately 1 to 2 Ghz, which is smaller than the resolution capability of measurements system. the it is desirable to obtain higher resolution wpe±wpi where wpi is the ion plasma frequency. conditions of are made the Michelson. with Because this device High resolution spectral a rapid scan Fabry-Perot interferometer is not a standard instrument it is necessary to describe its operation before a presentation of resulting measurements can be undertaken. The experimental arrangement of the system is shown in fig.(2). Fabry-Perot Radiation from the Tokamak follows the same optical path described in the appendix, and is wire mesh This polariser polariser is which oriented to couple equal upon a amounts of both One of these detectors is a broadband monitor having a nearly constant response from coupled incident splits the signal between two detectors. polarisations to each detector. radiation interferometer 50 to 200 GHz. The to the second detector must first pass through the Fabry-Perot etalon which transmits only specific frequencies. The Page 113 is etalon to that described in sec (5), and consists of two similar wire meshes of loudspeaker 2 . When the loudspeaker is driven the adjusting by the loudspeaker dome. these be the position of the static mesh with respect to three generates system the During operation as a position monitor for the moving mesh. from of from each of the two detectors and a third which serves one signals, can range this of The location for example from 54 to 68 GHz. varied amplitude etalon can be swept over a small frequency range, the Thus Typically the oscillation an with 110 Hz loudspeaker is driven at between separation the two meshes varies with time in a sinusoidal manner. .3mm. is a moveable stage, the other on the dome of a on mounted statically mesh One constant. mesh inch per lines 100 is it signals position frequency is related to the to necessary In order to make any sense know monitor how the transmitted also is It signal. important to know the frequency resolution of the instrument. This calibration is with made an Fig.(3) shows such a calibration. source. respect the static mesh with location changing to the 83 GHz klystron radiation By varying the location of moving mesh and noting the the 83 GHz signal with respect to the position of monitor signal, an accurate calibration of the position monitor signal is made, frequency. mesh thus allowing the conversion of this signal to transmitted The position monitor signal is nearly proportional to the a small phase shift must be included. The separation, though frequency resolution is measured by noting the line. The line width of the The measured of the 83 GHz source can be neglected so that the measured width results entirely from instrument. width finesse the finite resolution of the of 114 at 83 GHz is in agreement Page 114 with calculated values 3 . When measurements are other made at frequencies 83 GHz it assumed that the finesse is proportional to the than square of the wavelength as dictated by theory and experiment. The relative spectral shape of the emission signals obtained feature determined by detectors. It is not necessary that the two detectors have response, the only frequencies. of the from since fluctuation will be results both lost detectors when show will the the be independent occur in the of same the first ratio is fluctuating order of temporal be effected equally and the taken. emission Because to narrowband, radiation passing through the Fabry-Perot can to the two that their responses be in the same proportion at all The measured spectra will fluctuation Michelson ratios is transmission be be only. the quite assumed The signal intensities are measured with respect to any existing baseline so that non-fluctuating thermal components will be ignored. SEC 8.3 HIGH-RESOLUTION SPECTRA Examples of the three signals from the Fabry-Perot system during the occasion of fluctuating emission are period occurs 10 msec from the initiation During of shown in fig.(4). the plasma discharge. this interval the mesh motion corresponds to a frequency sweep starting at 50.5 GHz and increasing to 60.5 GHz. fluctuating signal remains nearly While the sweeps through the emission feature. the expansion of the region of emission mentioned that a good bit of broadband constant, the Fabry-Perot signal shows a maximum of transmission indicating that during this etalon This chance in time the This is shown clearly in fig.(4). It should be is involved in making these Page 115 measurements. Since the preselected, one frequency must wait region for to be scanned must be the necessary conditions of plasma density and presence of fluctuating emission in order to measure the spectra. The relative spectrum obtained from this example by dividing the Fabry-Perot signal by the broadband signal is plotted as a function of frequency, determined fig.(5). The spectrum very narrow bandwidth. primarily from from the calibrated radiation monitor, in shows an unambiguous transmission feature of The scatter of the plotted points arises uncertainties in reading the positions and amplitudes of the various bursts of emission. the position frequency However, since it is assumed that of the emission remains constant during the sweep time, the scatter might also result partially from small changes in the frequency. The residual level in the wings of the transmission feature is due to leakage of baseline level around the etalon, so the should be regarded as slightly positive to compensate for this problem. Fabry-Perot radiation The resolution corresponding to the FWHM of the is indicated on the spectrum and is such that the feature is regarded as being unresolved. The electron density at which the plasma measured frequency of 54 GHz frequency is not too far from the approximately typical these small values is considered. 1.6 measured particularly when the relative accuracy of the line at at this time. the radiation is assumed to occur at the central plasma frequency, a peak to averaged density ratio of This the is 3.6 10-13 cm- 3 . The line averaged density is about 2.8 10-13 cm- 3 and is rising rapidly If equals is required. values of 1.3, averaged density Because accurate values of Page 116 central it available, emission does be not can demonstrated the central plasma at occur However, no examples have been found frequency. frequency the that conclusively not are density where the measured The of emission is not consistent with such an assumption. measure emission frequency conceptually provides a sensitive of the plasma density since the instrumental resolution of 0.2 GHz translates to density changes of typically 1%. technique The accuracy absolute of this upon the association of the radiated frequency with hinges the central plasma frequency. Occasionally the emission feature is observed the characteristic situation where a non-fluctuating type of emission is the Fabry-Perot system. by the transmitted signal. although it resolution varieties appears at are Once again the the because frequency. measured, encountered by The shape of the spectrum is directly shown broader higher occur without Fig.(6) illustrates the behaviour. fluctuating to the Whenever signal fluctuating case, but they have the of feature same narrow unresolved, reduced instrumental these intensity is is non-fluctuating less spectral than the shape as their namesake. SEC 8.4 SHIFTS IN EMISSION FREQUENCY The assumption that the radiation frequency constant is than relatively necessary because the Fabry-Perot requires a finite time to sweep through a frequency interval. more remains The required time is never a few msec, which is small in comparison to the time scale of density changes. Because of this it is not likely that variations in plasma frequency caused by density changes will have an effect upon Vage ii/ the measured spectra. There is evidence of sudden change successive example. bursts of fluctuating in radiation emission. frequency of Fig.(7) shows such an The broadband signal shows a series of individual bursts of radiation, most of which belong to a sequence of evenly spaced spikes. In addition there are a few spikes which occur outside this The Fabry-Perot signal shows the transmission of sequence. the sequential spikes, but also shows that the additional spikes are not transmitted. The situations are indicated where two successive bursts occur, but only the one belonging to the evenly spaced sequence The frequency content of these sudden shifts transmitted. bursts must differ by at least the instrumental resolution, which is 0.3 GHz for this demonstrating is in radiated example. frequency Besides fig.(7) suggests a connection between radiation frequency and the also periodicity of temporal fluctuation. It is difficult to make where the bursts are good regularly spectral measurements of spaced and few in number. This is because the peak spectral frequency will most likely be swept during an interval are irregularly does not happen. time it will spaced In cases such as fig.(4) be so where the but much denser in time this problem However, if the radiation frequency is not through between bursts, and the few points obtained will occur in the wings of the line. bursts cases easily noticed in changing these cases. in If the situation of irregularly spaced bursts is treated as the superposition of sets of periodic bursts, each with different periodicity, each set may have assigned to it a different spectrum shown in fig.(5) radiation frequency. Thus the may be some average of many frequencies. Page 118 The problem of changing frequency makes it that every in case the However, no apparently broad which could not be is spectrum spectral difficult to be certain instantaneously very narrow. features have been observed attributed to frequency changes during the scan time. Because polarisation the two equally, from a polarisation occurring in detectors the fluctuating emission coupled such as the polarisations. show each no clear always nearly equal in each mode. preference The successive two of mode, absence of polarisation of the nor any emitted polarisation at assume REFERENCES & FIGURES H. Fizeau Ann. Chim. Phys. 66 429 (1862) 2.) D.S. Komm, R.A. Blanken, P. Brossier Applied Optics 14 460 (1975) 3.) Renk & Genzel that the detector implies the similar situation at the region of emission within the plasma. 1.) bursts Because of the large effects of diffraction and measured SEC 8.5 of The measured intensity is reflection at these wavelengths it would be imprudent to the mode Polarisation measurements of evidence of sudden changes in polarisation. radiation is unknown. to situation shown in fig.(7) can not result effect; different are Page 119 MODULATION OF 'FLUCTUATING wpe EMISSION' BY MICHELSON INTERFEROMETER 2 MSEC/DIV. Fig. (1) The indicated point is the zero path difference of the Michelson interferometer. Extremes of mirror vibration occur 3 msec prior to and 12 msec after this time. Page 120 SCHEMATIC DIAGRAM OF FABRY-PEROT SYSTEM BROADBAND DETECTOR DATA * STORAGE F DISPLAY LOUDSPEAKER DRIvE RADIATION FROM TOKAMAK DETECTOR B BEAM SPLITTER * DETECTOR: * MESH : DRIVE : * FINESSE: F XED GI RID MOV11 GRII InSb crstal at 4.20 Kelvin 100 lines per inch electroform copper 110 Hz amplitude 114 Fig. (2) = - .3 mm Page 121 CALIBRATION OF FABRY-PEROT SYSTEM WITH 83.5 GHz KLYSTRON 1 msec/division Fig.(3) position the Fabry-Perot is trace The upper transmitted the lower trace is the monitor, The appropriate horizontal klystron radiation. scale is 1 GHz/division. This implies a FWHM'of 0.73 GHz, or a finesse of 114. Page 122 'FLUCTUATING W0 pe EMISSION' EXAMPLE SIGNALS FROM FABRY-PEROT SYSTEM (b) (a ) FREQUENCY MONITOR FABRY- PEROT SIGNAL UPW-WMWW- - 0.5 ms W, , WV 'W-= EXPANSION BROAD BAND SIGNAL 0.1 ms Fig.(4) Emission from Hydrogen plasma in 60 kGauss magnetic field. Density = 2.8x1013 cm- 3 line average; Current = 120 kAmperes at time of emission. a.) The Fabry-Perot is sweeping from 50.5 to 60.5 GHz left to right. b.)an expansion about the region of maximum transmission. Page 123 HIGH RESOLUTION SPECTRUM OF 'FLUCTUATING Cope EMISSION' RESOLUT ION 0. I- 0 C,, UJ F- w * 0 1-J *0 w 0 4# * 0 0e .0 e0 55,0 54.0 FREQUENCY (GHz) Fig. (5) Spectrum obtained from previous figure. Instrumental resolution of .2 GHz is determined by the meshes used in the construction of the Fabry-Perot interferometer. Page 124 'FLUCTUATING Wpe EMISSION' SPECTRUM OF NON-FLUCTUATING COMPONENT BROADBAND SIGNAL POSITION MONITOR FABRY-PEROT SIGNAL .25 msec/division Fig.(6) Horizontal scale corresponds to 1.3 GHz/division with the feature occurring at 71 GHz. The instrumental resolution at this frequency is 0.5 GHz, so that the feature is regarded as unresolved. Page 125 SUDDEN SHIFTS IN RADIATED FREQUENCY OF 'FLUCTUATING W0 pe EMISSION' POSITION MONITOR BROADBAND SIGNAL FABRY-PEROT SIGNAL t I ,imsec/division Fig.(7) Horizontal scale corresponds to sweep in frequency from 55 to 52 GHz, left to right. At indicated times only one of two consecutive bursts is transmitted, indicating a frequency difference of .3 GHz or more. Page 126 SEC 9.0 FLUCTUATING EMISSION: PLASMA WAVES The peculiar conduct of fluctuating wpe numerous explanations, Tokamak environment. untenable when particularly when emission dealing Nonetheless, explanations can they can generate with the complex be dismissed predict behaviour contradicting the established experimental results, or when they impose unrealistic restrictions plasma parameters. as to A candidate process must require the presence of runaway electrons, and must generate intense narrowband radiation near the plasma frequency. fluctuations and It must correlations allow for the variety of temporal observed, and must be capable of operating under a continuous variety of plasma conditions. SEC 9.1 STIMULATED EMISSION An example of a process eliminated spontaneous from consideration Cerenkov resonance discussed in sec (6). is the As demonstrated by experiment and theory, the spectral width of this radiation is very much broader than measurements of fluctuating emission, sec (8). frequency domain dispersion of relation Cerenkov by the emission is determined in For radiation similar discounted. through density and magnetic field; which are constant in neither time or space reasons spontaneous cyclotron the The the quantities Tokamak plasma. by runaways is This process, while capable of generating emission far in excess of thermal levels, is fundamentally very broadband in nature. These spontaneous emission processes produce broadband because the dispersion relations governing electromagnetic waves are -not particularly restrictive, allowing a single fast resonant emission electron with, and produce radiation at many frequencies. to be Stimulated Page i/ processes can produce narrow spectra by coupling the emission back with the emitting plasma, allowing the most unstable frequency (the one with greatest growth rate) to dominate the spectra. possibility is product the stimulated emission of One such extraordinary waves by Cerenkov resonance with runaway electrons. In analogy with an optical laser, medium this would require an unstable characterised by a population inversion or negative temperature, a coupling process, and some effective cavity to provide the feedback between wave and medium. Or, it is possible for the gain to be so large that in a single pass As through the medium a dominant wave emerges, requiring no 'cavity'. discussed in sec (6) trapping of extraordinary waves can result from the total internal reflection due to the radial decrease in refractive index. However, this trapping appears to be weak in the toroidal case, by the very fact that the spontaneous emission is observed. It seems unlikely radiation is that stimulated responsible for the observed accompanying spontaneous emission is ever fluctuating emission. of emission seen extraordinary feature, in since conjunction no with Furthermore, the Cerenkov resonance requires a high electron energy, since the refractive index is close to unity, which contradicts the observation of fluctuating emission from plasmas which would appear to have marginal runaway energies. The considered explanations involve some of fluctuating emission all intermediary 'mechanism, electrostatic plasma waves in particular, which greatly restricts the range eventually wpe radiated. The utility their narrow frequency spectrum marginally non-thermal electrons. and of frequencies to be of the plasma waves results from their possible resonance with In addition, the trapping condition Page 128 on electrostatic waves is much stronger, so that and electrostatic plasma centre. waves a 'cavity' exists, can be described as modes confined near the Given the existence of runaway driven plasma waves, it must be shown how these waves give rise to electromagnetic radiation. SEC 9.2 RELATED EXPERIMENTAL OBSERVATIONS Related experimental observations of an extraterrestrial concern a class of events known as type III solar characterised by radiation at the fundamental and second the local plasma frequency. origin burstsi, harmonic of Energetic electrons (100 keV) are ejected from the sun during a solar flare and stream along field lines through the solar waves. would wind plasma, where they drive linearly unstable plasma Saturation of these waves by a quasi-linear plateau limit their formation propagation to only a few kilometers, conflicting with the observation of waves in the vicinity of earth. An alternate non-linear stabilisation mechanism has been proposed 2 which allows the plasma waves to propagate as This mechanism produces localised wave packets, fluctuations. solitons. a spread in phase velocity of plasma waves through a parametric decay which generates an enhanced acoustic or Radiation at the level of ion fundamental results from scattering of electrostatic waves from the density fluctuations of the ion waves, while radiation at the second harmonic occurs by coalescence of two plasma waves. A somewhat contradictory model 3 of solar burst radiation involves the 'collapse' of the plasma wave packets on time scales much shorter than stabilisation the growth of the mode. time of the beam mode, rather than a The collapse of the wave packet is in the Page 129 spatial sense, and is produced by a spreading Radiation at wpe and plasma waves. 2wpe in wavenumber of the occurs by the same scattering process described previously. Regardless of what model is used, the fact remains that electromagnetic radiation associated with type III solar bursts occurs at twice the intensity plasma at frequency twice the as well as the fundamental. plasma frequency is as or more intense than that at the plasma frequency, which is not the case where results, 2 wpe no radiation at is seen. for the relation for Alcator Furthermore, the solar wind plasma is not strongly magnetised, (wpe/wce = 100), so dispersion that the case for the plasma waves is not strongly dependent upon the wave vector direction with respect to the magnetic field. not The wpe < wce, when the appropriate This is branch of the dispersion relation becomes 4 l EQ (1) 2 2 h_ 1 2 _ 4w2c2cos2e "W - 1 1/2 cos2e This means that the envisioned decay to a low frequency wave can occur for resultant plasma waves propagating at small angles to the field, reducing the problem to one dimension, whereas models are two dimensional. the so that acoustic waves will be strongly damped. into ion 'collapse' A final difference arises from the fact that the Alcator plasma has comparable electron and ion to proceed only quasi-modes, and Decay would then have radiation scattering from the thermal fluctuation level. temperatures, would result by Page 130 Back on measurements earth, of some of the earliest reports type of spectrum stellarator 5 ,6 . A microwave emission above the thermal level lies in the vicinity of the electron plasma noise spectral a fluctuating plasma frequency emission were made in conjunction with turbulence studies on the Uragan bursting of shows while frequency, fluctuations the turbulent at harmonics of the ion acoustic frequency. The radiation has an instantaneous spectral width while spectrum of fluctuations shows two peaks at twice and four the times the ion frequency. and The radiation can be described by of 10% wpe+-wpi wpe+-2wpi, and is explained by the nonlinear scattering of plasma waves by density fluctuations at the ion acoustic frequency. A fundamental results is that difference the Alcator concerning and Landau damping of ion Alcator Significantly, when auxiliary Ti > Te, the heating produces equivalent temperatures in Uragan, fluctuating emission vanishes. ion Uragan plasma is nearly isothermal, Te = Ti, while for Uragan it is not, Te = 10 Ti. ion the This can be explained by the strong waves which occurs when the ion thermal velocity approaches the ion sound speed. Also, the Alcator plasma is not in the same turbulent state, with the marked resistive anomaly, as the Uragan plasma. Finally, the spectral measurements from the two machines have a dissimilar character. A more recent experiment results which stellarator. a uniform W > are similar on a linear to those reported has from generated the Uragan An electron beam is injected along the magnetic axis plasma. of these of High frequency electrostatic waves with frequency wpe are seen to grow and quickly saturate. dispersion device 7 Measurements of the waves identifies them as electron plasma waves. Page 131 Saturation of the plasma wave amplitudes corresponds to the onset of growth low density fluctuations, and the simultaneous frequency emission of electromagnetic radiation near the plasma frequency. density fluctuations and are confined The to a relatively narrow frequency band, and display the dispersion of ion acoustic waves propagating the direction of the electron beam. The frequency spectrum of the electromagnetic radiation is a few percent wide, with centre above the the direction of the incident beam (ordinary mode). results are shown to agree very slightly The radiation is polarised in plasma frequency. electron in well with The experimental a mechanism wherein radiation results from scattering of intense plasma waves from self consistently produced ion acoustic waves. In this and other linear experiments, waves which are only moderately damped. high Experimentally, beam downstream tracing convective of injection wave allows acoustic In addition, there is often a at radiation even and turbulence, of degree Te > Ti 2wpe 8 at one end of a linear device allows growth, so that distinctions and absolute instabilities 9 can be made. Tokamak, the 'beam' is injected continuously in between In the case of a space, so that the distinction between a instability that travels spatially with the beam and one that grows in time for all space can not be made. Finally, some mention in passing should class be made concerning a of runaway instabilities noted in Tokamaks, which arise from an effect1 0 . anomalous Cerenkov resonance produce periodic explain the simultaneous bursts radiation of that fluctuations plasma is While waves observed), this mechanism does (the mechanism does not it also generates in the loop voltage, temperature, current, Page 132 hard x-rays etc. The driving force of the and anisotropy, temperature the periodic sudden sudden attempt to equilibrate the electron temperatures. instab'ility is a severe fluctuates result from a parallel and perpendicular This effect is not observed in Alcator, nor is it expected, since the total energy of the runaways is negligible compared to the background plasma, which is well behaved. SEC 9.3 An RUNAWAY ELECTRONS initial experimental circumstance required in all bursts external to and the Stellarators beam plasma plasma with In the case of studies, these electrons are created which they interact, whereas in and Tokamaks creation occurs within the very plasma. these toroidal devices the source of runaway previous results, including those from Alcator, is the generation of a detached component of suprathermal electrons. solar the phenomenon electric field. associated with suprathermal the electrons current driving By equating the energy gain during free fall is In the induced in the electric field with energy loss through coulomb collisions, a critical electron runaway field is found'l: 47r e3 ne lnAc Ec - EQ (2) m vth 2 4 lnAc ne 15 Te {Volt cm~1 , 1014 cm- 3 , KeV} The highest electric fields, .12 V/cm, are encountered ionisation and shortly thereafter. during These fields do not exceed the critical runaway field (Ec = 2 V/cm) during this time, so as a whole will not plasma the plasma runaway, rather only electrons with motion in excess of a critical velocity will: Page 133 [L 1.2 Ec .c EQ (3) vth The rate of runaway production is determined by the ratio of the thermal electron velocity to this critical runaway velocity: rvthl 3/4 exp . EQ (4) K ne v(vth) --- - c 2 /2 vc 2vth vth An estimate of the number of runaways can be made by applying [vC to period shortly after breakdown, while the voltage is high and the before hard x-ray activity indicates the loss case eq (4) of fig.(2), of runaways. In the sec (7)this approximation is between 106 cm- 3 and 107 cm-3. The presence of runaway electrons alone does not a plasma make. The traditional linear formalism requires a positive slope to the runaway distribution, as in the case of the well tail' problem. This type critical runaway velocity known 'bump on of distribution can be expected when the decreases rapidly, so as to detach an isolated segment of velocity space from the bulk distribution. situation can be realised during the early Tokamak current rise Such a portion of discharges when the critical velocity changes from Uc = 16Vth to Uc = 60Vth in a period of 3-5 msec. the wave presence of neutral ionisation phase 12 , which Bumps in velocity space due to atoms have may account also been noticed during the for the observation fluctuating emission immediately following complete ionisation. not unreasonable to suppose that a beam type accompanies electron of It is distribution fluctuating emission, but since no direct measurements of the distribution have been made this can only be an assumption. It is Page 134 noteworthy that grossly runaway discharges, produced by withholding neutral gas injection, do not suggests the necessity of generate fluctuating a subtle perturbation emission. This of the electron distribution, rather than a wallop of runaway current. SEC 9.4 PLASMA WAVE MODES A beam of runaway electrons will drive linearly unstable waves which satisfy the resonance condition; plasma w - k vb = 0- runaways are restricted to having only parallel velocity, the use If of the cold plasma wave dispersion, eq (1), yields: EQ (5) wp cose = Ik| vb cosE ki ; vb- This means that a particular beam velocity will be resonant with waves having the same magnitude wavenumber. Since the duration of a single burst of radiation is much greater than time of the driving the circumferential transit electrons, it is reasonable to assume that the plasma waves can be represented by a toroidaily uniform waveform or mode. Inspection of the dispersion relation, w = wb cosE shows that as the wave propagates radially outward into regions of reduced density, cose must increase to maintain constant frequency. The condition cose = 1 limits the radial extent of the wave, so that peaked density profiles will result in the trapping of plasma waves near centre. mode plasma If the torus is treated as a cylinder with periodic boundary conditions over a length 2WR, undamped plasma waves can by the numbers be described representing the allowed perpendicular and parallel Page ijb components of the wave vector. 'i'(r,z,t) = Om,n(r) exp i{knz - wm,nt} EQ (6) kn = n/R om,n(r) + 0 for r > 0 The parallel wavenumber must be an integer multiple of the major radius. reciprocal Because of the relative large size of the major radius with respect to typical plasma wavelengths, the axial mode number will be quite large, n > 800. Changes produce small changes in the wave which is in the frequency, axial mode number will typically Aw = 70 MHz, smaller than the resolution of the measurement system. The perpendicular extent of the wave is determined by the density profile, and by the frequency of the wave with respect to the central plasma frequency. a rr dr k1 2 = om, n~r k (r)tm,n(r) + dr = 0 k2k2 p(r)2 2 EQ (7) 1/ k 2(r) = where a square root particular k2{ [I parabola - r2/rlim 2 ]1/ 2 _ w2 density k, as- determined by the 2~ profile- is number radial This correspondence frequency, activity. restriction 'offers of a The two lowest solutions are shown in fig.(1), where it is seen that the radial extent of the plasma wave is confined centre. For beam velocity, bounded radial solutions will exist for discrete values of w only. mode assumed. the emitted a possible frequency to to the explanation the central plasma for the plasma and the correlation of emission with centrally located MHD A shift in radial mode number, for example m = 2, will produce a frequency change of from m = 1 to 430 GHz under typical Page 136 This conditions. so resolution, is difference that observed the frequency may be related to radial in the emission discontinuities of transitions mode instrumental the than larger the plasma wave. OTHER WAVES SEC 9.5 Fig.(2) nearly shows parallel the Appleton-Hartree propagation. The dispersion waves that As shown, these branches branch spontaneous Cerenkov are resonant because of to with their electrostatic waves. electron mode velocities. pass through the w = wpe region. also corresponds ordinary waves occur at or just slightly below the plasma frequency, and can be resonant with many Additional for have been discussed belong to the electrostatic limit of the low frequency branch. relation the extraordinary emission. mode involved One such with the It is seen that waves in this region relativistic electrons large velocity cannot be treated as purely phase Notice that the only, 1.0 >n> 1.1. extraordinary mode and resonance with relativistic electrons extends nearly to the cyclotron frequency. This is why broadband. the spontaneous Cerenkov emission is fundamentally Ignoring stinulated Cerenkov emission means that waves on this branch are not considered. Electrons resonant with the magnetised resonate with plasma electron cyclotron waves near wce. waves made. of the also If the fluctuating emission involves driven waves belonging to the wp branch, consideration can then some higher frequency electrostatic waves should be These waves have the following angular dependence: Page 137 2h EQ (8) W2 = w 2 1 2L1 1/2 W4 _ 4w2 W2 COS20 p 2 Wc2 w5j2 sin2e ccoe As the wave propagates outward to regions of decreasing magnetic field and density, the angular term increase to maintain constant must perpendicular Eventually the wave travels frequency. to the field it is absorbed at the upper-hybrid layer, which will always lie where between the propagates resonance cyclotron the observer. are radiation process the from excluded stimulated process is expected to occur with were to frequency, it would not be observed from since the wave In contrast to the magnetised plasma waves, the waves cyclotron non-thermal If the increase in field and density will decrease e inward, and reflect the wave. electron and these occur the Thus no centre. If waves. at outside the in some cyclotron any case, the fundamental is optically thick in the ordinary mode, and is cutoff by the hybrid layer in the extraordinary mode. SATURATION OF PLASMA WAVES SEC 9.6 Given the instability of plasma waves to linear growth, concerns the saturation mechanism of these waves. question mechanisms are beam trapping 13 , and cold small the beam, initial determined by trapping of beam driven wave, while decay limitation processes Two known instabilities1 4 . decay electrons a fair of in require the the For instability potential of a is the larger levels of beam energies. According to the trapping model, a single wave emerges from initial unstable band of wavenumbers. the If a beam has density nb and velocity vb, this single wave will have the maximum growth rate: Page 138 EQ (9) Yx= 131/2 2-4/311 /3J k Wpne k P p vb e When the beam density is small, the wavenumber spread of the growin'g waves will be negligible, and trapping of beam electrons will occur in the potential wells of the After several trapping nearly times, sinusoidal the wave wave electric energy tends field. toward a meta-equilibrium: 2-4/3 nb m Vb 2 W = EQ (10) n ne In the most extreme case this energy is expected to be less than 1% of the thermal electron energy, so in no way is this a strong beam plasma case. to The meta-equilibrium will persist until other waves have a similar amplitude as the original unstable quasi-linear description becomes appropriate. broadening This mode, will grown when a produce a of the wave spectrum and the formation of a plateau in the beam distribution. The decay transfer of instability energy from limits growth plasma wave decays into wavenumber, and an ion acoustic wave. have lower of the non-linear solar another bursts, plasma The product where wave plasma the of larger wave will phase velocity, so it will be non-resonant with the beam, but may be damped spectrum to beam resonant waves to non-resonant modes. This is the process applied to the case of original due high by the ambient frequency plasma. fluctuations, stated which saturation process is occurring. Without it measuring the cannot be definitely However, the trapping I Page i39 model only requires a coherent linearly operate for small values of beam energy. an upper limit wave, and will unstable Eq (10) may be regarded as to the expected wave energy, since any decay process will only reduce this value by transferring energy to other modes. SEC 9.7 LINEAR RADIATION PROCESSES A plasma wave by itself will not radiate. test electron will undergo an oscillation Although which produce radiation, the particle under consideration medium of changing dielectric. a thermal in vacuum would is moving in Because of this, a dipole moment is generated which cancels the effect of the electron motion, so that radiation results. a no Alternately it can be said that since the density and velocity perturbations of the plasma wave are out of phase, no net oscillatory current results. If the test particle has a relativistic mass, radiation can result from its motion through the plasma However, the plasma waves are being driven by non-thermal electrons and have phase velocities close to the speed of light, relativistic particles through the wave. broadband will be moving in spectrum, so that most phase with, rather than Furthermore, this relativistic radiation wave1 5. effect produces a which is contrary to the experimental measurement. Another mechanism providing a direct conversion of plasma waves to electromagnetic radiation is linear mode conversion in the presence of an inhomogeneity, such as the steady state has been density gradient. It shown by theory and experiment1 6 that such a process occurs in an unmagnetised plasma, however, the treatment cannot to the strongly magnetised case. be extended The reason for this is the inability Page 140 to match both frequency and wavenumber of the two modes, by the linear and the resonance condition: 2 k W2 - = = The conversion must occur near propagating the to - k It is seen that for to C2 w = wpe field, into that waves electromagnetic is, into left circularly W2 2 2 = c possible 2 The appropriate cold plasma dispersion is: polarised waves. EQ (12) c2 -7 v v parallel required The plasma wave can be described by the cold theory. plasma dispersion, eq (1), EQ (11) as 1 I a typical <T2 W(W + WC) I situation, wce/wpe = 4, it satisfy both conditions on the wavenumber. is not The two mode branches remain separate, and no linear conversion may occur. It has been assumed that the density gradient is totally in perpendicular direction, that the is , Vn - B = 0. This is expected to hold in steady state since the rapid parallel diffusion will produce a toroidally symmetric density distribution. The treatment also assumes that the scale length of the much not greater true, continuously it mode inhomogeneity than the wavelengths of the modes involved. is across becomes appropriate. no longer the possible to treat conversion is If this is the wave modes inhomogeneity, and a scattering description Page 141 REFERENCES & FIGURES 1.) E.N. Kruchina, R.Z. Sagdeev, V.D. Shapiro Pis'ma Zh. Eksp. Teor. Fiz. 32 443 (1980) 2.) H.P. Freund, K. Papadopoulos Phys. Fluids 23 732 (1980) 3.) M.V. Goldman, G.F. Reiter, D.R. Nicholson Phys. Fluids 23 388 (1980) 4. This approximation to the actual cold plasma dispersion is accurate to better than 5% for all angles E under the Alcator plasma conditions. It is better than 1% accurate for angle 0 < 300. 5.) N.Perepelkin, V.Suprunenko, M.Vasiliev, A.Dikii Paper 19 III Int. Sym. Toroidal Plasma Confinement Garching, FDR (1973) 6.) A.Longinov, N. Perepelkin, V.Suprunenko Fiz. Plazmy 2 626 (1976) 7.) D.A. Whelan, R.L. Stenzel Phys. Rev. Lett. 47 95 (1981) 8.) Y.Kalinin, D.Lin, V.Ryutov, V.Skoryupin Zh. Eksp. Teor. Fiz. 55 115 (1968) 9.) M. Seidl, W. Carr, D. Boyd, R. Jones Phys. Fluids 19 78 (1976) 10.) L. Muschietti, K. Appert, J. Vaclavik Phys. Fluids 24 151 (1981) 11.) H. Knoepfel, D.A. Spong Nuclear Fusion 19 785 (1979) Review paper on runaway electrons in Tokamaks 12.) G.A.W.Lins, H.-J.Kunze Phys. Fluids 24 839 (1981) 13.) T.M. O'Neil, J.H. Winfrey, J.H. Malmberg Phys. Fluids 14 1204 (1971) 14.) A.L. Throop, R.R. Parker Phys. Fluids 22 491 (1979) 15.) A. Gailitis, V.N. Tsytovich Zh. Eksp. Teor. Fiz. 46 1726 (1981) 16.) R.W.Means, L.Muschietti, M.Q.Tran, J.Vaclavik Phys. Fluids 24 2197 (1981) Page 142 TWO LOWEST MODE RADIAL PLASMA WAVEFUNCTIONS 1.0 4(r) a8 6 ,4 .2 Radius (cm) 0. :1. 2 3 4 -.2 -,4 -.6 Fig. (1) Numerically evaluated solutions for radial langmuir wavefunctions, based upon a square root parabola steady state density profile. Both solutions are for the case kp =13cm- . The two modes are separated by 430 MHz for a central plasma frequency of 60 GHz. 5 Page 143 DISPERSION RELATION FOR PARALLEL (0=10) PROPAGATION kc 3 Electron Cyclotron Waves WP coso waves 2 Possible Cerenkov resonance o 1 0 -I 4 =3 p I I I 1 2 3 Fig. (2) Appleton-Hartree dispersion for cold plasma. The two nearly electrostatic waves which may be resonant with non-thermal electrons occur P Page 144 SEC 10.0 FLUCTUATING EMISSION: NON-LINEAR EFFECTS The resonant three wave interaction is a non-linear coupling which modes allows oscillating waves. emission is the Pertinent the of transfer interaction to energy a discussion involving The former wave is of three the fluctuating frequency as two of the expected to exist by the linear coupling to runaway electrons, and is described in terms of axial radial mode numbers per the previous section. wave is known to exist, since it is directly having exited the plasma. feature. of and by the and The electromagnetic measured, albeit after The identity of the third wave is unknown, however, its frequency and conditions, wp magnetised plasma waves and electromagnetic radiation near the plasma constituents. between of wavenumber measured are spectral restricted width by resonance of the radiation Because of the numerous experimental and theoretical studies three wave interactions involving acoustic fluctuations, this mode is of particular interest as a possible third wave. SEC 10.1 RESONANCE CONDITIONS The process under consideration can be schematically represented by the following expression: [ (U, kl J+ [ ws, ks ] EQ (1) [ w1 , kl ] where subscripts the electromagnetic -- i l,t (s, ks refer wt, t kt ] ] + E wt, to the kt J plasma (langmuir) (transverse) waves, and s to the scattering wave. restriction has yet been placed on the scattering wave, which in can and No fact be a particular fourier component of a steady state perturbation; Page 14!3 i.e ws = 0. that the The resonance condition is. the fairly obvious requirement sum of the constituent waves and the product wave must maintain a nearly constant phase in both time and space: Wt kl + ks 5 kt W1 5 WS + Wt ki 5 ks + kt l + Ws EQ (2) Because the measured frequency of equal the emitted radiation is nearly to that assumed for the plasma wave, the third wave must have a relatively example, small the frequency scattering in comparison. process involving This two eliminates, for plasma waves. The resonance condition thus restricts the scattering wave: EQ (3) WtWl While the close proximity of the transverse frequencies limits the is that the allows for particular frequency. depends strongly upon the allowed wavenumbers. within a variety the plasma, where The the The refractive index near the plasma the strong of transverse wavenumbers for any frequency wave polarisation, and upon the angle of emission with respect to the magnetic field [see fig.(6) As wave resonance condition applies to the waves at the location of scattering, i.e. dispersion langmuir magnitude of ws, this consideration alone is far less restrictive when applied to reason and of sec(6)]. these quantities cannot be measured, the appropriate wavenumber of the emitted transverse wave is uncertain. density is not known well enough to The determine transverse wave frequency lies above or below the central if plasma the measured associated plasma Page 146 frequency. However, the separation of the lowest order langmuir mode and the central plasma frequency is 430 MHz for a typical Thus, if the situation. scattering wave has a frequency above this difference, the condition wt > wp can be satisfied everywhere in the plasma. a situation ws = 1 GHz. both could result scattering by acoustic waves where The case wt/wp > 1 allows the transverse wave polarisations for all emission angles. alternate situation, wt < centre. from This can Such access to Since ws is small, the p, can only be satisfied near the plasma be seen in fig.(1) of sec(9) where the inflection point (change in sign of curvature) of the radial langmuir mode occurs at the point where wl = wp. Thus the condition wt < wp can only be satisfied for a region within a few cm. appropriate to characteristic maintain localise dimension less emission than centre. process to It is not a volume a few wavelengths and of still of the wave as being everywhere a mode of a continuously varying medium. Even if the transverse wave frequency is slightly a description the of the less than the central plasma frequency, in the plasma volume outside of the region of emission the dielectric is still described by the condition wt > wpThe inability to experimentally discern a preferred polarisation for the transverse the emitted wave. effects of polarisation. frequencies wall wave does not indicate a lack of polarisation of As was the case for the reflections diffraction This effect is expected to when the wavelengths dimensions of the flange and consideration and of the are be of gyroharmonic, will more comparable bellow internal polarisation second the scramble severe to the lower the physical assemblies. non-linear at the Although oscillating Page 147 current favours justification radiation into the ordinary mode, there is no for disallowing the extraordinary mode, particularly if the waves are emitted with small angle to the magnetic field. SEC 10.2 DISPERSION RELATION Fig.(1) shows the wavenumbers as a function of angle for the two polarisations above the of transverse plasma dispersion waves frequency. relation is the emitted The at a frequency slightly Appleton-Hartree cold source of these curves. In addition, a region of k-space is indicated where the langmuir waves are as plasma expected, determined by their required resonance with non-thermal electrons. These electrostatic modes are also solutions to the equation refractive index. for the special case of a large Appleton-Hartree satisfy the resonance condition, the scattering wave must connect wavevectors of the langmuir and transverse possible scattering waves are indicated. fairly restricted in direction, so waves. Fig.(1) that shows To the Several such that ks is the scattering wave must propagate mostly along the magnetic field, and in a direction opposite to the plasma waves. This means that a purely perpendicular wave cannot satisfy the resonance condition. unrestricted. For values The magnitude of ks is of ks ranging from near 0 to greater than 2.5 wp/c a possible scattering solution allowed wavenumber apply scattering quite exists. only to The emission extraordinary mode, ks must satisfy .68 wp/c<ks<2.2 wp/c in produce radiation in the ordinary mode. extrema into order of the to Page 148 The escape condition can emission Transverse be also discussed applied emitted waves to close in conjunction to field the Cerenkov transverse waves. possible the with direction in the extraordinary polarisation will be internally reflected and trapped in the plasma centre. occurs angles for shown For the conditions less than .14 rad. propagates along the magnetic field, in trapping fig.(1), Thus, if the scattering wave it is possible for both langmuir and transverse waves to be trapped in the same volume. the W.hile this trapping may provide for strong interaction between the waves, it forbids of emitted transverse waves unless some disturbing detection allows perturbation (MHD activity?) course, if the escape of radiation is emitted in the ordinary polarisation no the such trapping may occur, regardless of the emission angle. the Of radiation. Because of small perpendicular scale of the langmuir modes, it can be argued that coupling is more efficient for transverse waves nearly which propagate parallel to the magnetic field direction, this because in such a case the interaction volume of the waves is maximised. Fluctuations which electrostatic frequencies ion near considerably the by frequencies and plasma and and include waves, drift with waves at Measurements of low frequency density perpendicular - fields magnetosonic frequency, scattering w = 300 magnetic 400 kHz. indicate a broad spectrum wavenumbers; These of typically fluctuations are during most discharges, showing no correlation with plasma frequency emission. plasma ion across waves frequency. laser ki = 0 - 25 cm-1, continuous cyclotron lower fluctuationsl propagate While the fluctuation level is highest at the edge, a tentatively identifies secondary maximum occurs at the I Page 149 It centre, with typical magnitude of n/n = 3% . is that suspected these fluctuations result from drift waves, though this identification is very uncertain 2 . Regardless of their identity can these fluctuations only satisfy the resonance condition if the parallel component of the wavevector is sufficiently large. been placed on this parallel An upper bound component, that 6 cm-1 of is, the wavevectors are preferentially perpendicular, and cannot resonance condition. typical satisfy the However, the range of measured values of k1 and kp allow for possible mentioned has solutions. These possible drift waves are since they constitute the only measured fluctuations in the Alcator Tokamak. It seems that they are unlikely candidates for the non-linear emission process. NON-LINEAR CURRENT SEC 10.3 When all three waves are plasmons the resonance condition is equivalent to a statement of conservation of energy and momentum 3 . In the case considered here, the necessity follows directly source term distribution, three wave oscillating for from the of the resonance condition mechanics of the radiation process. electromagnetic waves is an oscillating current which cannot be produced by the plasma wave alone. interaction allows the generation of a The The non-linear current, which results from the mixing of the density and velocity perturbations from the coupled waves: EQ (4)~~ Enl(r,t) = where only the electron -e nl(r,t) vs(r,t) fluctuations need e vl(r,t) ns(r,t) - be considered. If the scattering wave has a small phase velocity in comparison to the plasma Page 150 wave (such as an acoustic wave), or if perturbation, the it is a steady state density first term in eq (4) is negligible compared to the second, so that the current density becomes: -e w n1 ns i0w1 + ws] t} exp {i[k1 + ks]r EQ ( Jnl(r,t) = JO exp i[kj r - wj t) The radiation field resulting from the non-linear current distribution is found by solving for the vector potential: EQ (6) A(r,t) = iC-r - w c -exp cr t] exp i[k ik - c n r' d dr' The oscillating coefficient determines the frequency and wavenumber of the radiated wave, i.e.; wt = wj = wl + ws and kt = wj/c, and the integral insures that kt = kj = kl + ks These . are precisely the resonance conditions of eq (2). The degree to which the resonance condition must be satisfied is determined by the extent of the integral eq (6), which in turn is controlled by the degree of coherence of waves. wave For packet Akj = 2w/L, The measured a&t/wt < 1%, example, of plasma extent L, a spread scattering in wavenumber, thus radiated frequency, Aot = wt* Akj/kt, results. upper limit of the radiated frequency bandwidth, indicates a minimum wave packet extent, L = 50 cm, which is comparable to the toroidal circumference, so representation and if the current density is treated as a spatial parallel and the becomes unacceptable. mode description of sec (9) is that a wave packet For this reason the plasma wave favoured. The narrowness of the bandwidth of radiated waves also restricts the frequency extent of the Page 151 scattering wave by a similar argument. wave is known to have However, since the scattering a low frequency anyway, applying a 400 MHz bandwidth for example, does not require an especially coherent wave. SEC 10.4 ABSORPTION When the size of the interaction region becomes large enough that the radiated wave amplitude is comparable to that of either the plasma or scattering wave, it is necessary processes, so whereby energy is returned to the plasma modes. removed This to consider the inverse from the transverse wave and situation is analogous to the treatment of emission and absorption of cyclotron radiation, in that a transport equation results. When wave-particle interactions can be ignored, the transverse mode can be described by4: d d + Vt (group) L EQ (7) = Nt Atls where Na is the number density of mode energy density by Wa = waNa. It NlNs - NtNl - NtNs a, related is assumed to in the spectral eq (7) that the resonance condition is satisfied by the three waves, so that frequency and wavenumber indices are ignored. the evolution of the other waves. upon the identity of the Two similar equations describe The interaction kernel Atls depends scattering wave, relationship between the non-linear current and the and of the plasma and scattering waves. expresses energy the densities Page 152 By eq (7) it is evident that a quasi-equilibrium solution can If the density of scattering waves is small, so that N1 >> Ns, exist. the last term in eq (7) is ignorable, so that for steady state: EQ (8) Nt = Ns + Wt = This has been considered in the Ws context of explaining dominant wp emission by the scattering of plasma waves from a thermal level of ion acoustic fluctuations 5, the intensity being determined by the of equipartition the acoustic The result is an theorem. intensity of emitted radiation which is independent of plasma wave fluctuations the level of energy, and is described by the effective temperature of the acoustic fluctuations: EQ (9) it = Ibb(Te) ; Te w Ti] 1 /2 ET For the specific case of ion acoustic waves, an intensity than 40 times the thermal level results, which representative of typical fluctuating wp intensities. the damping (second) of greater is fairly Provided that term in eq (7) is large enough, the results of eq (9) will be true regardless of the type of scattering wave used. Accordingly, other branches of the ion spectrum will produce a similar result. also If the damping term is small, then presumably will be equal to the or effective greater temperature, and the frequency of the scattering the emission term will be small, so that a negligible amount of radiation results. radiated intensity will be determined by which the scattering than wave. The temperature, the plasma From eq (3) wave is known to have a low frequency compared to wp, Page 153 so the predicted emission intensity will thermal Drift level. be much than the waves, because of their lower frequency, will The generate an even greater effective temperature. waves greater case of drift in fact cannot be treated by eq (8) because the drift waves are saturated modes. The saturation mechanism, whatever it may be, must be included in eq (7),- so that the process will no longer involve only three waves. RELATION TO EXPERIMENT SEC 10.5 A consequence intensity. Quite of eq (9) is often the a constant observed described as having a nearly constant expected fluctuating level. This emission emission can be is particularly true with regard to radiation which accompanies MHD activity, or which occurs in examples periodic of this. bursts; figs. (3), (The discerning (5), (7) of experimentalist sec (7) are refrains from citing those cases where constant intensity results from saturation of detector and/or electronics.) According to the scattering model, the emission fluctuations result from like oscillations in the current. This requires either thermal level, hard x-ray an then only fluctuations of the plasma waves can produce the observed emission correlated the density of plasma or scattering If the scattering waves are populated to waves to change rapidly. effective non-linear emission, behaviour. the Because oscillation in there plasma is no wave amplitude probably does not result from the wholesale loss of resonant runaway particles, but rather some subtle alteration of their velocity distribution which destroys the positive growth of the wave. Page 154 The correlated behaviour activity m = 1 with emission fluctuating result either from a direct involvement of the MHD mode can in the radiation process, i.e. indirectly, of for example by the m = 1 is the scattering wave, or trapping of the transverse and langmuir waves in the local density maxima. If the m = 1imode is regarded as perturbation of a few percent, a nearly with stationary density infinite parallel extent and perpendicular scale length of order .5 cm, reference to fig.(1) that a ksJ. = 3 cmnC shows could satisfy the wave matching condition only if a similar or larger ksl were present. However, an extent implies a 0 parallel wavevector component. infinite parallel The MHD mode cannot satisfy the resonance condition because it is a purely perpendicular perturbation. If the scattering wave is independent radiation can correlation untrapping be with of imagined state m = 1 activity, as occurring continuously. m = 1 rotation may result from The observed trapping density perturbation. could and If the density profile is peaked enough on axis, the radiation may escape only when a large rotating mode flattens the profile. then the the emitted transverse wave in the region of increased refractive index associated with the steady of explain the This result that during steady state emission is only observed in phase with large m = 1 oscillations which precede follow sawtooth disruptions. or Unfortunately, the absolute phase of the rotating mode relative to the emission is unknown. vage it0 SEC 10.6 1.) REFERENCES & FIGURES R.L. Watterson et.al. Bull. Am. Phys. Soc. 26 885 (1981) 2.) C.M. Surko, R.E. Slusher Phys. Fluids 23 2425 (1980) 3.) R.C. Davidson Methods in Non-Linear Plasma Theory Academic Press, New York (1972) p.117 4.) V.N. Tsytovich Nonlinear Effects in Plasmas Plenum, New York (1970) p.9 0 5.) I.H. Hutchinson, Kim Molvig, S.Y. Yuen Phys. Rev. Lett. 40 1091 (1978) Page 156 SCATTERING GEOMETRY OF WAVEVECTORS LANGMUIR WAVES Kj POSSIBLE SCATTERING WAVES KL[WP} &- U _____________________ 1.0 TRANSVERSE WAVES EXTRAORDINARY ORDINARY -1,0 _ Wc t 3CO Fig.(1) Because of only approximate knowledge of the plasma frequency (central density) the choice of Wt/wp is somewhat arbitrary. Dashed line indicates trapped transverse waves. Langmuir waves are assumed to be resonant with runaway electrons; ve Z c, v, >> v, - 1.02 Fage i:>/ CONCLUSION SEC 11 The diagnostic utility of of submillimetre plasma emission measurements as a condition should be manifest from the previous sections. As a technique for determination of evolution of temporal and spatial electron temperature the measurement of emission at the optically thick- second gyroharmonic is demonstrated to be particularly well suited. The ability to describe the temperature profile by the simple Gaussian expression should assist those persons modelling transport properties in interested in Tokamak plasmas, or to anyone in need of any analytic expression for the spatial temperature variation. That the temperature and profile width are consistent with a classical resistivity model indicates a relative small and possible neo-classical effects. influence by anomalous In addition, the repeatability of the scaling of profile width with limiter q allows the prediction of results in new parameter ranges, such as high current and field. Radiation at the third gyroharmonic has been shown to be as a method for self consistent calibration of the entire system. This is not a trivial matter, as calibration remains difficult for problem. As a possible suggested that a meticulous attempt to undertaken. useful topic calibrate a serious future the and work, it is diagnostic be This would probably require the development of an intense broadband reference source, and possibly the resolution of the wall reflection, polarisation scrambling problem. Sawtooth measurements have been made which do ambiguities emission. inherent not involve the in chord-average techniques, such as soft x-ray These results together with profile scaling suggest a range Page 158 of central current density such that qO fluctuates between .8 and 1.0 during the peculiar sawtooth, with time average value qo = .85. of The behaviour of sawteeth in helium plasma should be of interest to persons attempting an explanation of the sawtooth phenomena. The observation of non-thermal used as an indicator of the emission of the from equilibrium extremely Furthermore, the emission events. others is be uncertain a subtle In fact it would appear that only electron distribution fluctuation emission occurs in such a narrow for can process, at least in the case of fluctuation wpe radiation, is uncertain. perturbation emission a non-equilibrium electron distribution. However, the degree of the deviation since submillimetre accurate interesting density is required. bandwidth, measurement correlation behaviour Since the a technique suggests of itself. fluctuating with m = 1 modes may provide new information regarding these It is hoped that in will be stimulated reporting to this duplicate novel radiation event the measurements on other devices, and perhaps to offer explanations regarding the origin of the process. Page ib9 OPTICAL PATH APX 1 construction Optical access to the Alcator plasma is limited by to six diagnostic Each port offers vertical and horizontal ports. views of the plasma through narrow gaps in the toroidal called keyways. field The horizontal view is preferable for the measurement of cyclotron radiation since it allows sampling of the full generated spectrum magnet by the magnetic field variation. frequency Because of the narrowness of the keyway, only radiation emitted within the When viewed from the perpendicular can be observed directly. outside this keyway is the limiting constriction that observable plasma area. This .10 rads determines of front surface mirrors as shown in fig.(2). mirror images the detector keyway. the situation is illustrated in fig.(1). Having passed from the torus, radiation is brought to the detector a series of aperture onto the inside by A spherical edge of the Since the wavelengths involved are not ignorable compared to the physical dimension of the optics employed, every mirror, window, and aperture is a source of diffraction. The effective aperture diameter of the Michelson is 5 cm. Plane radiation incident on this area will be focussed into the detector. be ignored, In the very short wavelength limit when diffraction can this 5 cm diameter would approximately equal the area of the plasma viewed, since between the plasma interferometer. second and the the spherical entrance For a-wavelength gyrofrequency of mirror is aperture .67 cm, located of the corresponding halfway Michelson to the at 80 KGauss, diffraction effects will increase this spot size slightly. incident interferometer Tracing backwards from the detector, rays upon the spherical mirror will be nearly plane waves, so the Page 160 resulting diffraction will be the usual circular aperture Airy pattern. dP . a = 1/2 ka sine 2 Jj(a) dn a k = 95 cm-1 EQ (1) a = mirror diameter- 15 cm Jj= first order Bessel function The image upon the 10 cm vacuum window determined by will have a maximum extent the imaged aperture size plus the diffraction from the spherical mirror. At 1.5 m from the mirror the first zero of the Airy pattern will be determined by: Ar= Lemin = .4 cm L = 1.5 m EQ (2) emin = 2.7 x 10-3 rad (first zero of Bessel function) The spot size is 7.5 cm, so no diffraction effect from the expected. The window is keyway is 4 cm wide by 30 cm tall, so it will produce the characteristic diffraction pattern of a vertical slit: dP sina d12 a 2 a = 2 ka sine EQ (3) k = 95 cm-1 a = slit width = 4 cm This will effect only the horizontal direction. a horizontal diffraction By analogy to eq (2) size is found at the plasma centre a 20 cm distance from the 'slit' exit. Ar = Lemin = .16 cm EQ (4) emin = sin- (7r/ka) = 8 x 10-3 L = 20 cm Page ibI This distance is added in the dimensional spot size separate horizontal and at horizontal the direction only. The two plasma centre is found by adding the vertical diffraction effects the to 5 cm aperture image: Ar = Aemin(mirror)x3m = .81 cm Vertical: EQ (5) Horizontal: Ar e Aemin(mirror)x3m + Aemin(slit)x2Qcm =.97cm The spot size is an oval with first minimum defined by 6.6 cm vertical and 6.9 cm horizontal diameters. To determine the temperature profile resolution that size mandates it is toroidally symmetric. using the assumed that determined magnetic This ripple). is unmeasured and assumed small). effect by the is will straight doppler ignorable volumes. spot any (or by resolution The effect upon the measured central temperature can be ignored since the correct (and When viewing a spatial produce a 5% overestimate of the profile width. will width The axial (in-out) extent is fixed 20 GHz Michelson frequency resolution, &r = 3 cm. process forward; In the horizontal totally temperature profile of 20 cm diameter, the finite spot is vertically and criterion, Ar = 3.3 cm. direction the resolution is the profile The vertical resolution Rayleigh-Jeans field the this calibration error resulting from averages over finite In the case of the Fabry-Perot, sec (5), both the aperture size and the frequency resolution were much better, resulting in a more accurate measurement. Page 162 REFLECTIONS & POLARISATION APX 2 None of optics the employed can from result may will polarisations mirror remain distinct reflections, in this but orthogonal process. Reflections within the keyway will retain their polarisation identity extraordinary (ordinary) mode polarisation vector. mixing mentioned effect the The polarisation in sec (1) must occur elsewhere if it is to account In an effort to demonstrate for the observed ordinary mode radiation. this since surfaces are planes parallel (perpendicular) to the metal reflective the Rotation of the electric polarisation state of the emitted radiation. vector alter significantly a reflective mirror was affixed to the back wall of the vacuum chamber [see fig.(1)]. This mirror was designed to vacuum so that no randomly reflected radiation window onto could be observed. by this state. itself, Reflections within the keyway are addition, but The effective improvement in of presumably this was reflective had no prohibited will not effect the polarisation dump was only a modest A further refinement involved of a glass liner on the inside of the port extension. intended to absorb polarisation effect the 10% the degree of polarisation,.with a significant amount of ordinary mode still present. placement beam not image upon the any incident radiation, thus mixing in the port extension. measured degree of the This preventing This absorber polarisation. In retrospect, neither experiment provided a severe enough test to either confirm or disprove the polarisation scrambling model. The beam dump was not large enough, and unfortunately could not be aligned properly, while the glass covered only a small surface. portion of the exposed metal I Page 163 THE MICHELSON INTERFEROMETER APX 3 The plasma emits a broad spectrum the intensity varying in time of during radiation, submillimetre a tokamak discharge. function of the Michelson interferometer system is to measure as of spectrum this possible. include the For the as Alcator first three Tokamak beyond this since intense radiation The encountered. the cyclotron mechanics of measured harmonics, higher at much the best temporal resolution with necessary, The spectrum should but need not extend harmonics is rarely the interferometer inversely relate time and frequency resolution, so that improvement of one comes at the expense of the other. important. Unless fluctuation times, Of the two, the spectral resolution is the more beyond improvement such as sawteeth, reason for the pursuit of higher temporal emission characteristic can be achieved, there is no resolution. However, the spectral resolution should allow determination of temperature profiles with reasonable spatial resolution since these profiles are the raison d'etre of the interferometer. In simplest form the Michelson splitter with two reflective beam paths. interferometerl is Radiation from the two paths recombines constructively or destructively depending upon difference of the two interferogram is generated. recover the radiation The interferogram spectrum. In the can case be transmits or reflects of couple equal intensities a wire according to polarisation. into to the polarising wire mesh selects one polarisation and the beam splitter to length analysed Michelson interferometer, Fig.(3), the beam splitter is which the varying this path difference an By paths. a beam the two beam paths. mesh An initial is arranged If the path Page 164 difference is polarisation, mesh. zero, the beams recombine to form the and are rejected from the interferometer by the initial If the path difference is one half wavelength, beam is rotated reflected by conventional originiA the recombined w/2 with respect to the original polarisation and is the initial device, mesh into the detector. Unlike the the polarising interferometer detects a minimum at the zero path difference. The length vibrator of attached one to path is a reflecting varied by mirror. motor is driven at 35 Hz to produce a swing of path length can Two electromechanical Typically the vibratir +-5 mm. The second be varied manually and is adjusted so that the zero path difference state occurs near the extreme excursion. an complete interferograms of the result moving mirror from one vibration period. The position of the moving mirror is measured HeNe laser, so by auxiliary that the path difference is known at all times. data is sampled at equal intervals of path difference, of an which The because the sinusoidal mirror drive does not correspond to equal intervals of time. The sampling interval is variable, but is usually 126.5 um Data sampling is controlled by a Z-80 micro-processor. Input to of path difference the Z-80 includes the analog interferometer signal, a mirror drive reference signal, and the output of the auxiliary HeNe interferometer. The number of complete vibration periods to be sampled is preselected; typically about 12, which corresponds to plasma discharge. The entire sampling the full process duration of the is initiated by a Page 165 trigger from the sequencer. Alcator After sampling the data is transferred to a PDP 11-55 computer for treatment. APX 4 FOURIER TRANSFORM INTERFEROMETRY The output of the upon the idealised Michelson interferometer depends intensity I(w) and the path difference of the two spectral arms x: J(x) EQ (1) I(w) cos(wx/c) dw 1 = J- 0 J is the average level of the interferogram J(x) and is equal to one half the total intensity of all frequencies. Experimentally U the non-fluctuating baseline of the interferogram. is I(w) is related to 0(x) by the inverse transformation 2: I(W) EQ (2) f 4 = cos(wx/c){J(x)-J} dx 0 Provided that J(x) is measured for a semi-infinite entire spectrum can be recovered. measured for a finite range of path In range practice difference, of x, the J(x) can only be hence limiting the accuracy of the deduced spectra. A standard technique for treating extent is the process of apodisation. interferograms of According to this procedure the inverse transformation is: L EQ (3) fA(xL) cos(wx/c){J(x)-J} dx 1(w) = car 0 finite Page 166 the true spectrum with an instrumental line shape which finite correspond to different function is fairly functions apodisation A shapes. line standard, the reflects Different apodisation capabilities of the system. resolution the called is A(x,L) Apodisation may be considered a convolution of function. apodisation and difference path where L is the maximum and triangular been has employed throughout this thesis: L EQ (4) where L may be chosen equal According difference. to or to less eq (3), the the than spectrum maximum produced triangularly apodised interferogram resulting from a single path by a frequency source, IO(wO) is: I(w0 ) = JO L sinc 2 [(W-W 0 )L/c] 2wc EQ (5) By way of comparison, the unapodised spectrum is a sinc sinc 2 function. condition that Applying two the frequencies Rayleigh of criterion Aw separation to be rather than eq (5), the resolved is expressed by: Aw = EQ (6) Thus the resolution is inversely difference, displacement. difference) which in turn 2wc L proportional to the maximum path is the amplitude of the vibrating mirror For example, a 1 cm total mirror travel provides a conceptual 15 GHz resolution. (2 cm path Since data from Page i)/ the entire mirror oscillation may not be usually be less. For used, the will resolution data presented in this thesis, the frequency resolution is typically 20 GHz. The interferogram is actually a series of sampled points, rather a than continuous curve. Hence the integral over x in eq (3) is in reality a summation: N-1 EQ (7) 1(w) = d cos(wjAx/c) A(jAx,L) [J(jAx)-T] j=0 where N is the total number of interferogram points can sampled. Eq (7) be used to generate spectral intensities I(w) for all frequencies w, so it must be borne in mind that the resolution is still determined by eq (6). As the frequency in eq (7) is increased, at some point I(w) = I(w-2wc/Ax), and the spectrum will repeats itself. frequency that may be recovered is determined by The maximum the sampling interval: =c Wmax =.... EQ (8) Ax Thus the standard 126.5 um separation implies a maximum measurable frequency of 1186 GHz. Although eq (7) as it stands is a perfectly reasonable means of generating spectra, derivative expression employed. Called the fast fourier transform, this routine introduces it is' rarely based upon used the in the favoured process in form. Cooley-Tukey a significant savings in computation time, and become this for Instead a algorithm this reason fourier transform spectroscopy. is has A Page 168 description of the Cooley-Tukey algorithm can be found elsewbere 3 , and for the sake A requirement of the of brevity will not be included. fast fourier transform routine is that the number of sampled points to be transformed be an integer power of 2, i.e.: N = 2 n. To satisfy this requirement a zero-filling technique is used generate additional interferogram points. seen The additional points all generated have zero value, and do not effect the to This spectra. is in eq (7) where the summation can be extended to larger integers without effect provided that the cbrresponding J(x)-~J 'are zero. This is precisely what zero-filling accomplishes. A complication interferogram arising from the nature of the is a phase shift that results from not sampling a point precisely at the zero path difference. sampled discrete For example, if the nearest point is measured at some displacement 6 from the actual zero point, eq (7) should be recast as: N-i I(W) = A(j,L) cos[(jAx-6)w/c] [J(j)-~T] -4 j=0 N-I EQ (9) .Q-4 cos(w6/c) cT A(j,L) cos(wjAx/c) 2 {J(j)-~J) j=0 N-i + 4 sin(w6/c) A(j,L) sin(wjAx/c) [J(j)-~] j=0 Thus if 6 is known, the spectra can be recovered provided that the I Page 169 and cosin transforms sin are Fortunately, this is both computed. manifest in the fast fourier transform. The raw digitised data from a typical discharge is shown in fig.(4), where the entire analysis process is schematically followed. individual The data is segmented into blocks corresponding to (1) oscillation each of direction backwards Each block contains two interferograms, one for periods. with mirror respect motion. second The interferogram to the first, and must be reversed. is Now all interferograms may be treated equally. (2) The I level is level is subtracted found interferogram. if the by fitting a from each to line interferogram. the flat part This slope The line may have a slight positive or negative emission intensity is increasing or decreasing. the of The data is scaled accordingly, under the assumption that all frequency components change in the same way. Subtraction of the base level is necessary to avoid generation of large zero frequency spectral components when fourier transform is performed. the Because of the inverting effect of the polarising meshes, the interferogram is next reflected about the zero path new zero level. (3) Because the sampling of data occurs before the difference is encountered, and since it is unlikely that a -sample will be taken exactly at this time, the interferogram must be three largest data points, location approximated. and the center The interferogram is next apodised. function is used. this point in the A parabolic fit is made to the identified with the zero path difference. (4) of of this parabola is 6 is determined. A triangular apodisation Because the employed fast fourier transform routine Page 170 requires 2n data points, the zero filling technique is used, typically to create 29 points. Zero filling interferogram to larger values of path involves the extension of the difference by adding data points of zero magnitude. (5) Since the fourier transform starting from the zero requires to the actual spaced zero path data point. point The and small the which is the of other allowed because of the periodicity of interferograms. Because the first data point is not the actual interferogram point section interferogram occurring before this point is reattached at end, points path difference point, the interferogram is shifted by the spacing between the first closest equally is zero not symmetric about this point. path point the Both cosin and sin components should result from the fourier transform. (6) The fast fourier transform is now computed, and cosine components are the combined as indicated in eq (9). sin and Corrections due to calibration of the detector response, or absorption in windows or other effects may be included. The spectra are now available for further analysis. or third harmonic may be extracted components may be investigated. and studied, The second or non-thermal For purposes of data storage, the raw interferometer signal is used, thus allowing for retrospective changes in calibration etc. The time required to reprocess the data is compared to the benefits gained in this caution. APX 5 DETECTOR small Page 171 Regardless of which, if any, interferometer system was used, the This consisted of an InSb crystal mounted same detector was employed. The crystal was in a liquid helium cooled cryostat. QMC Ltd 4 . of the United Kingdom. manufactured When biased with 100 vamps current, the detector has a sensitivity of about 1 Volt per mWatt radiation. by of incident For typical emission power levels of a few vWatts incident level on the detector, a 1000 gain amplifier produced a signal of a Calibration with a modulated klystron source show that the few volts. temporal frequency response of the detector is adequate to at least 100 kHz, so that in general, finite resolution limitations will be set by the data gathering system rather than by the detector itself. APX 6 CALIBRATION The problem of calibration is two-fold. spectral features such as cyclotron When harmonics, measuring broad which may vary in frequency due to changing magnetic field, the relative sensitivity of the detection system to the complete range of radiation frequencies is required. is to be Furthermore, if the emission at some equated with a physical particular parameter, such as the second harmonic intensity with electron temperature, an absolute the system sensitivity is needed. the easier, since it requires relative spectral shape. frequency measure of Of these two, the first is by far only a radiation source of known An absolute calibration requires a source of known intensity, as well as a precise determination of the angles subtended by the source and by the plasma being measured. solid Page 172 Relative calibration is performed using a mercury arc lamp which a approximates blackbody of thousand a few degrees temperature. Additional sources such as liquid nitrogen cold loads have used with similar result. also The. problem with this technique is that no available broadband radiation source begins to approach the of plasma. the To overcome this intensity inherent source weakness during calibration, many interferograms are averaged by the Z-80. of resulting spectrum is shown in fig.(5). the An example Bearing in mind that it this spectrum represents the emission from a blackbody source, 400 GHz. is the system response begins to decrease substantially above that seen been This rolloff results mainly from the detector. response, at high frequencies the interferometer wire meshes will begin however to fail as the radiation wavenumber approaches the mesh constant. problems at low frequency evident in fig.(5) result from the Fourier transform process. due to The noticeable features near 560 and 760 absorption by molecular water vapour present interferometer chamber, a topic of later discussion. field, and the are in the magnetic emission scaling square of the frequency, and by the increase of temperature with magnetic field [sec emission As the GHz consequent cyclotron frequencies, is raised, the decrease in the detector response is offset by the blackbody with The (1)]. For example, the second harmonic will increase as w7 /2 , so that the detected signal intensity will not degrade until about 700 GHz, corresponding to a field of 120 KGauss is reached. Also shown in fig.(5) is a calibration spectrum QMC detector before its modification. made with the The spectral fluctuations at 400 GHz result from channelling in a 1 mm thick quartz disc on which Page iij the InSb crystal was mounted. Modification involved wedging this disc so the faces were no longer parallel, thus preventing channelling. can be For the this simple expedient eliminated the problem. seen, As same reason the window on the detector cryostat is also wedged. By dividing the spectrum of fig.(5) then. used by the spectra from computer cyclotron the Not plasma. the assumed to routines correct this by considered blackbody This curve is curve is obtained. sensitivity a relative spectrum, by measured technique is absorption by the .25" fused quartz vacuum window, an important factor at higher frequencies. This effect need dealing with a small frequency range, such not as be considered when the second or third harmonic alone, because the attenuation coefficient of fused quartz is a weak function of frequency. of sec However it has been included in fig.(1) (4) where emission at the two harmonics is compared, the net effect being a 15% increase in the relative harmonic window emission. will become Above 800 GHz particularly intensity absorption severe, and of the third by the fused quartz a crystal quartz replacement window should be used. It has been found that the system response is different for two polarisations. This is due in part to the differing condition of the meshes used to select the incident polarisation. fig.(2) of the Consequently in sec (1) a separate calibration was used for each mode to account for this difference. Absolute calibration is performed by scaling the optically thick second harmonic extraordinary mode measure of the electron temperature. emission with an The temperature as independent measured by Page 174 soft x-ray pulse height analysis is the usual basis for this scaling, although the results of sec (4) have been used succesfully in a pinch. The calibration factor arrived at in this fashion remains constant with plasma parameters such as density and current, so that once agreement same electron the temperature. the two techniques always generate the in The calibration factor does change when modifications of optical system due to deliberate or accidental realignment occur. However, if the system is left undisturbed, the calibration will remain in effect indefinitely. APX 7 ATMOSPHERIC WATER VAPOUR ABSORPTION The absorption lines seen in from rotational transitions transitions at 557 and 752 GHz intense lines at 380 the of are calibration molecular spectrum water particularly to be ignored. cyclotron harmonics problem exists. evident, but and 450 GHz can also be detected. less Five other sufficiently As long as the magnetic field is such that the avoid At The vapour. lines occur in the interval from 100 to 900 GHz, but are weak result these certain absorption fields, lines, no significant particularly above 90 KGauss, atmospheric water vapour alters the dramatically. to alleviate this problem, the entire optical system plexiglass ruining attempt including and the this Fig.(6) temperature is spectra interferometer volume was A noticeable' improvement remained. the an chamber, nitrogen gas. still In measured an profile, example the was and profiles enclosed in a continuously purged with resulted, but distortions of such an effect. absorption feature While does demonstrate the spatial resolution of the system, since the line width of the feature is due entirely to the finite instrumental resolution. Page 175 It is useless to hope for a dry day, or to move the detector closer to the vacuum chamber to avoid this problem. absorbent, The 557 GHz is so that under typical conditions a 1 cm air path will produce nearly a 5% reduction in signal. APX 8 strongly REFERENCES & FIGURES 1.) Principles of Optics M. Born, E. Wolf Pergamon Press New York (1964) 2.) Interferometry W. Steel Cambridge University Press, Cambridge (1967) Contains pertinent discussions of transformations apodisation, phase shifts, zero-filling ......... 3.) R. Bell Introductory Fourier Transform Spectroscopy Academic Press (1972) Contains a graphic demonstration of the Cooley-Tukey algorithm for a case of N=8. 4.) QMC Instruments Ltd. 229 Mile End Road London El 4AA Trivia question: What does QMC stand for? 11 Page 176 OPTICAL PATH TOKAMAK INTERIOR Quartz Vacuum Windnw Glass Absorb er Port Extension Scale 10 cm Keyway Bellows Plasma Chamber Concave Mirror Vertical through view the Fig. (1) of torus section slit mid-plane. Internal structure is entirely stainless steel except for absorber and window. Page 177 OPTICAL PATH / 2 PLEXIGLASS ENCLOSURE TOKAMAK VACUUM WINDOW 3 M RADIUS SPHERICAL MIRROR FLAT MIRRORS F- SCALE IM -H Fig.(2) All mirrors are front surface aluminium coated reflectors. The enclosure is continuously purged with nitrogen gas, when necessary, to minimise radiation absorption by water vapour. INTER :EROMETER ENTRANCE APERTURE Pave 178 THE MICHELSON INTERFEROMETER SCALE 10cm POLARISER STATIONARY MIRROR X1 CIDENT RADIATION X2- FOCUSSING LENS BEAM SPLITTER OSCILLATING MIRROR Path Difference of Two Interferometer legs = 2(X1- X2 ) DETECTOR Fig. (3) Both beam splitter and polariser are wire meshes. Mirrors are flat front-surface aluminised. The initial polariser is selected according to the mode of radiation (Ordinary, Extraordinary) to be viewed. The beam splitter couples equal amounts of either polarisation into the two legs. If the path difference is zero, a minimum signal is detected. By varying the path difference via the oscillating mirror, an interferogram is generated. Page 179 DATA PROCESSING .J I & Invert & temn Phs Sif ion I .Zero-Fill To Here Spectrum Fast Fourier Transform Fig. (4) Reorder Data From Sampled Point Nearest Zero Phase Page 180 CALIBRATION SPECTRA BLACK BODY (1) BLACK BODY (2) SYSTEM RESPONSE FREQUENCY 500 1000 GHz Fig. (5) First two spectra are produced by black body (a w2 ) mercury lamp. Periodic structure in first spectra results from channelling in detector, which has been corrected; second spectra. Relative system respon e is determined by dividing black body spectrum by w . Vertical scales are arbitrary linear units. Absolute calibration results from scaling 2wc emission to an independent temperature measurement. Page 181 MOLECULAR WATER VAPOUR ABSORPTION T(arb units) DIFFERENCE BETWEEN MEASURED AND ASSUMED GAUSSIAN PROFILE RESOLUTION 500 550 20 GHz 600 GHz RADIUS (cm) 10 0 -10 Fig. (6) Example of 557 GHz absorption feature due to molecular water vapour in the air path between Tokamak and detectoe. The line width is determined by the instrumental resolution, and is consistent with the 20 Ghz resolution determined by the apodisation technique employed in the fourier transform process. BIOGRAPHY B Y WALLACE STEVENS One must have a mind of winter To regard the frost and the boughs Of the pine-trees crusted with snow; And have been cold a long time To behold the junipers shagged with ice, The spruces rough in the distant glitte- Of the January sun; and not to think Of any misery in the sound of the wind, In the sound of a few leaves, Which is the sound of the land Full of the same wind That is blowing in the same bare place For the listener, who listens in the snow, And, nothing himself, beholds Nothing that is not there and the nothing that is. THE SNOW Fig. (0) MAN Distribution B. Coppi MIT 26-201 R.C. Davidson MIT NW16-205 PFC Library MIT NW16-255 U.S. Dept. of Energy 20545 Washington, D.C. ATTN: DOE Library General Atomic Company P.O. Box 81608 92138 San Diego, CA ATTN: Library Princeton Plasma Physics Laboratory Princeton University P.O. Box 451 08540 Princeton, NJ ATTN: Library Associazione EURATOM - CNEN Fusione C.P. 65-00044 Frascati (ROME) ITALY ATTN: The Librarian (Miss C. Depalo) A sociation Euratom - CEA Dtpartement de Physique du Plasma et de la Fusion Control4e 92260 Fontenay-aux-Roses FRANCE ATTN: Bibliotique Prof. Bruce Kusse Cornell University Laboratory for Plasma Studies 14850 Ithaca, NY