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Whistler mode electron cyclotron emission from energetic electrons in TMX-Upgrade

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1988 Plasma Phys. Control. Fusion 30 491

(http://iopscience.iop.org/0741-3335/30/5/002)

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Plasma Phlsics and Controlled Fusion. Vol 30, No. 5 . pp. 191 to 513, 1988 Printed in Great Britain.

0741-3335 88$3 00- 00 I O P Publishing Ltd and Pergamon Press plc

WHISTLER MODE ELECTRON CYCLOTRON EMISSION FROM ENERGETIC ELECTRONS IN TMX-UPGRADE

C. J. LASNIER, R. F. ELLIS and G. D. TSAKIRIS Laboratory for Plasma and Fusion Energy Studies, University of Maryland, College Park, M D 20742,

U.S.A.

and

R. A. JAMES and T. A. CASPER Lawrence Livermore National Laboratory, Livermore. CA. U.S.A.

(Received 8 June 1987)

Abstract-Measurements are presented of axially propagating whistler mode electron cyclotron emission (WECE) from the energetic electrons in the East end cell of TMX-Upgrade. The measurements were made in the frequency ranges 12.5-18 GHz and 2 9 4 0 GHz employing swept and fixed frequency superheterodyne receivers and a spherical dish antenna. These measurements are compared with a one dimen- sional radiation transport model which treats whistler mode emission, absorption, and propagation for an electron population which has two components: a cold thermal component and an energetic relativistic, mirror trapped component. It is shown firstly that the measured emission is in good agreement with the predictions of the WECE model. Secondly, for low to moderate cold electron densities, and for detection frequencies slightly less than the minimum electron cyclotron frequency in the magnetic well, the emission achieves black body levels, yielding an effective hot electron temperature. At higher cold electron densities and, or higher frequencies cold plasma effects limit the diagnostic capabilities of whistler emission in so far as the hot electrons are concerned.

1. I N T R O D U C T I O N PLASMA electrons gyrating in a magnetic field emit electromagnetic radiation due to their centripetal acceleration, and this electron cyclotron emission (ECE) can be used to diagnose certain quantities associated with the electron component of the plasma (BEKEFI, 1966a). The most successful diagnostic application has been to measure the radial electron temperature profile in Tokamak devices, where ECE perpendicular to the magnetic field is employed (COSTLEY et al., 1974; BORD et al., 1976; EFTHIMION et al.. 1980; TAIT et al., 1981). More recently, ECE parallel to the magnetic field in the whistler mode (WECE) has been shown to yield the electron temperature in a small low temperature mirror device (ELLIS et al., 1982; ELLIS et al., 1983).

In both the above cases, the electrons under consideration were in thermal equilibrium and of nonrelativistic energy. Increasingly in fusion research the electron distribution may be both nonthermal, multicomponent, and relativistic. The most prominent example of this are the electrons generated by electron cyclotron resonance heating (ECRH) in mirror traps, either in single mirrors (DANDL et al., 1964; MAUEL, 1984: SMATLAK et al., 1984; BOOSKE et al., 1985), tandem mirrors (SIMONEN et al., 1985; POST et al., 1985; INUTAKE et al., 1985; QUON et al., 1985), or bumpy tori (BERRY et al., 1985; FUJIWARA et al., 1985). These electrons are characterized as having two components: a cold (T, < 1 keV), more or less, thermal component and an energetic, relativistic component which is nonthermal, possessing

49 1

492 C . J. LASNIER e t al

a loss cone pitch angle distribution and possibly a non-Maxwellian energy distribution. In addition, the components are radially and axially inhomogeneous.

Now, in such a two component case, ECE originates predominantly from the energetic component; it is the diagnosis of this component that is feasible and is the focus of the work. However, the nonthermal nature of the distribution, the inhomo- geneity, and the presence of a cold refractive andior absorbing component make analysing the emission a challenging problem. In general, one must employ a radiation transfer model and include plasma dielectric effects. For high harmonic ECE perpendicular to the magnetic field these difficulties can be avoided, but for whistler emission this is not the case.

In this paper we describe measurements and analysis of whistler mode ECE from the energetic ECRH generated electrons in one end cell of TMX-Upgrade. The general purposes of the study are twofold:

(a) To investigate whether the measured emission can be explained by whistler mode ECE theory in combination with a radiation transport model.

(b) To evaluate the diagnostic capabilities of WECE, in particular for use in measuring the effective perpendicular temperature of the energetic electrons.

To preview our results, our basic conclusions are that the emission is well described by WECE theory and that under some circumstances the perpendicular temperature can be measured, but that cold plasma effects can prevent this.

The paper is organized as follows. Section 2 describes the theoretical modelling, including predictions for TMX-U. Section 3 describes the experimental set-up and Section 4 presents the results and comparison with theory. The summary and conclusions are presented in Section 5.

2. T H E O R E T I C A L M O D E L I N G Interpretation of the experimental results is based upon a radiation transport

model for the emission, absorption, and propagation of whistler mode ECE, which is discussed in some detail in TSAKIRIS and ELLIS (1982). At that time our focus was on diagnosing the cold (or warm) electrons and the results were nonrelativistic. We have since changed our focus to the energetic electrons and in the subsequent papers extended our emission and absorption coefficient calculations to the fully relativistic regime. In this section we will only briefly review relevant aspects of the previous work so as to concentrate on the results important for the current work.

2.1. Fundamental concepts We are concerned here with synchrotron emission and absorption, and it is useful

to state some fundamental ideas. Firstly, we note that a single electron with velocity v, with c I l (ui) parallel (perpendicular) to B emits and absorbs synchrotron radiation only at frequencies given by the resonance condition:

where m is the harmonic number, y = (1 - u2jc2)-), c = (U; + U:)’, p = ujc, and cp is the angle between the direction of propagation and p, w, = eB/m,, m, = rest mass of the electron (BEKEFI, 1 9 6 6 ~ ) . Note that there is a relativistic downshift from

Whistler mode electron cyclotron emission from energetic electrons in TMX-Upgrade 493

o = mw, and a Doppler shift due to p For the whistler mode of propagation (right- hand polarized wave propagating parallel to B) coscp = 1 and emission and absorption occurs only for m = 1. This latter condition arises because the electron gyration, and thus acceleration, occurs in a plane perpendicular to the wave vector and is purely circular in this plane; thus no harmonics of the cyclotron frequency are generated. If a component of the circular motion occurs along the wave vector then all harmonics of the cyclotron frequency are generated.

For emission and absorption from a low density collection of electrons (tenuous plasma limit) with a distribution of velocities one must sum up the contributions from individual electrons. The first result of this is simply that the emission and absorption are proportional to the density of the electron distribution. In addition, however, the range of velocities in the distribution leads to a shifting and broadening of the emission and absorption lines. In particular, relativistic effects lead to a downshift of the spectrum from o / w , = 1 to w/w, = ( l l y ) and a relativistic broadening Awiw, - kBT:",c2. Doppler effects lead to broadening both above and below w/o, = 1 with Awlo, - ( p In these equations T is simply an effective temperature and ( l / y ) and (&) represent averages over the distribution function.

If the electron density is sufficiently high, then kinetic theory must be employed to calculate the emission and absorption (and propagation). The general concepts of relativistic downshift and Doppler broadening still apply, and for the whistler mode both absorption and emission continue to increase as the density increases (ELLIS et ul., 1982; ELLIS et al., 1983).

2.2. Modeled configuration, distributions, and profiles

2.2.1. Geometry. This work is concerned with axially propagating whistler emission in a mirror end cell geometry. The antenna which detects the emission (Section 3.2) is located outside the end cell, close to the machine axis, and the radial extent of the antenna pattern subtends the entire hot electron cloud. For such an arrangement it is the axial dimension z , i.e. the direction of propagation, which is important and which enters the radiation transport equation. Thus, we treat the problem one-dimensionally, with variables being functions of z . The radial variation enters only in the sense that the apparatus measures radially averaged quantities.

2.2.2. Electron components and distributions. We assume the electrons to consist of two components. The first is a Maxwellian component of density n, and temperature T,; this component represents electrons which are generated in the end cell via ionization processes and also represents cold electrons which stream into and through the end cell from other regions of the tandem mirror. The second component is a mirror confined energetic population which represents the ECRH generated hot electrons. The distribution function, f H , of these electrons is taken to be a relativistic Maxwellian in energy of temperature T H with a step function distribution in pitch angle, 8, with a loss cone angle 8,:

494 C. J. LASNIER er al.

where E = ymoc2, g = moc2/k,TH, and K , is the modified Bessel function of the second kind. Additionally, 8 is the pitch angle between v and B, x = cos 6, xl = cos e,, and H is the Heaviside step function. Note that this distribution function is normalized in momentum space; nevertheless, for mirror applications it is more convenient to express the distribution function in energy and pitch angle variables (ELLIS et al., 1982; LASNIER and ELLIS, 1986).

2.2.3. Axial profiles. The axial profile of the magnetic field B(i ) is taken to be that of the actual TMX-U configuration (SIMONEN, 1981). Output of the EFFI code for TMX-U is used in our radiation transport code.

The axial profile of the hot electron density is modeled as a Gaussian,

where io is the axial position of the plug midplane and TH and 8, are assigned a single value, independent of i.

For the cold electrons two cases are employed. The first case assumes an axially uniform n,, and this is used to model the plasma during the active pulse time, i.e. the time when ECRH, ICRH, neutral beams, centre cell, etc., are all activated. During this time a variety of cold plasma sources lead to a more or less uniform distribution of cold electrons. The second case assumes the cold electrons have the same profile shape as the hot electrons [equation (3)]. This is used to model the ‘afterglow’ phase of the discharge, which follows termination of the active phase. In the afterglow phase the highly energetic hot electrons (T , 2 100 keV) are mirror trapped and are lost only by infrequent collisions. The cold electrons in this phase are generated by impact ionization of residual gas and it is reasonable to assume that the axial profile of the cold electrons is similar to the hot.

2.3. Description of the model

2.3.1. Radiation transfer equation. The radiation intensity seen by a detector is determined by the equation of radiation transfer (BEKEFI, 1966b). This equation has been discussed in some detail in TSAKIRIS and ELLIS (1983) and here we only repeat the definitions for completeness. The intensity is given by

where S , is the source function defined by

andj , is the emission coefficient, rT is the absorption coefficient, and nT is the ray refractive index. The subscript T stands for the total of all components. The optical

depth z is given by z = r,dz where the integral is from the detector at z = 0 to a


point, z , in the plasma. Finally, the radiation temperature T,(w) is defined as

2.3.2. Emission and absorption coefficients. Equation (4) requires knowledge of the emission and absorption coefficients as functions of z which are assumed to be functions of the local plasma parameters and are calculated as indicated below.

(a) Hot electrons.-The emission and absorption coefficients for the hot electrons [j,(w), a,(w)] have been given in LASNIER and ELLIS (1986), equations (6) , and will not be repeated here. However, two points are important for understanding the experimental results:

(1) Both j , and GI, are proportional to the hot electron density n, so that denser plasmas both emit and absorb more, which also implies that the optical depth T increases with n,.

(2) The frequency widths of U, and j , are determined primarily by Doppler broadening, with A o l w - (PI,) where (PI,)' - 2kBTl,,",c2 and TI is the parallel temperature. Thus, for relativistic temperatures a, and j , are very broad. In particular, there is substantial emission and absorption far from o /w , = 1. This is very different from the cold electron case.

We note that these forms of x, andj, are valid in the tenuous plasma limit, which is met for the densities and temperatures of the hot electrons considered here. Were the hot electron density higher (> lo'* ~ m - ~ ) one would have to calculate x, from a relativistic dispersion relation and j , from relativistic kinetic theory.

(b) Cold electrons.-The absorption coefficient for cold electrons GI,(O) is calculated from the dispersion relation for whistler mode propagation (ELLIS et al., 1982; ELLIS et al., 1983), and is valid at arbitrary densities. This is necessary because the cold electrons often do not satisfy the tenuous plasma condition. As with the hot electrons we note two points:

(1) The absorption coefficient and optical depth increase with cold electron density. (2) The line profile of x,(o) is again determined by Doppler broadening, but since

the temperature is so low the line width is quite small. This has the effect of causing cold plasma absorption to be localized very close to the region where w/w, = 1.

Concerning the emission coefficient, the cold plasma emission is calculated using Kirchoff s law (TSAKIRIS and ELLIS, 1983). However, this contribution is negligible.

(c) Total coeficients.-The total emission coefficient is simply

and this is jH(o) to a very good approximation. Similarly we use

which is a good approximation as long as the hot electrons remain in the tenuous plasma limit.

2.3.3. Index of refraction. For the hot electrons we have used n = 1, consistent

496 C. J. LASNIER et al.

with the tenuous plasma assumption. As mentioned in LASNIER and ELLIS (1986), we have solved the relativistic whistler mode dispersion relation for some test cases and confirmed that n = 1 is a good approximation for our parameters. For the cold electrons the index of refraction is calculated from the nonrelativistic whistler mode dispersion relation, which is valid at arbitrary density. Finally, nT is simply equated to the real part of the cold electron index of refraction (or 1 if no cold plasma is present).

2.3.4. Scale lengths. The validity of the radiation transfer equation requires that the quantities change over distances much greater than a wavelength. For TMX-U the magnetic field scale length is greater than, or of the order of, 1 m which clearly satisfies the condition. Likewise the hot electron density scale length aH always exceeds the wavelength by at least one order of magnitude. In addition, strict application of the equation requires that the absorption be ‘weak’ in the sense that k , < k,, where k,(k,) is the imaginary (real) part of the wave vector. All of the data presented in this paper, with one exception, satisfies this condition, largely because we are operating in the low density regime for the energetic electrons or far from resonance for the cold electrons. The one exception is the 35 GHz cold plasma absorption data (Fig. 8) for which k , - k,. However, for such cold plasma absorption we have shown previously (ELLIS et al., 1983) that the radiation transport equation gives results in good agreement with experiment and in good agreement with the predictions of Budden tunneling, which does not require k , < k,.

2.3.5. Computer code. These features have all been incorporated into a large computer code (WECE) which solves the radiation transfer equation numerically, producing I T ( o ) , for the TMX-U magnetic field and the specified hot and cold electron parameters. The code can be run in ‘shot simulation mode’ in which the experimentally measured density versus time behaviour and other assumed inputs are used to drive the WECE code and produce a simulation T R ( t ) which can be compared to the actual experiment. This has proven to be extremely valuable and is used extensively in this paper.

2.4. Some important modeling predictions

2.4.1. Attaining black body emission. We first consider the conditions necessary to attain z > 1 in the TMX-U plasma, as this will determine the utility of WECE as a temperature diagnostic. We consider a case in which no cold electrons are present and we calculate the radiation temperature T R versus the hot electron density nH for a variety of TH values; the results for a frequency of 12.5 GHz are shown in Fig. 1. The curves show quite clearly saturation of the radiation temperature, demonstrating that the emission has become optically thick. In addition, the saturated radiation temperature is very close to TH, indicating that emission levels very close to black body have been attained. The small deviation from black body levels is due to the fact that the distribution is not a Maxwellian, rather has a nonthermal loss cone feature. Moreover, Fig. 1 shows that densities of a few times 10” ~ m - ~ are sufficient to attain black body conditions for T H - 100 keV. Larger values of uH would have lowered the required density and vice versa. We note that the minimum electron


200

- >, Y Y

I-= 100

0

f = 12.5 G H z TH (keV )

I I I I I 2 4 6 8 IO

nH (lo'Ocm-3)

FIG. 1.-WECE radiation temperature versus hot electron density for various hot electron temperatures. Saturation of T, at a value T, 1 T, indicates black body levels attained.

OL = 45", oH = 30 cm: n, = 0.

cyclotron frequency in the TMX-U end cell, coco, is 271 x 14.0 GHz so that Fig. 1 represents a case for which w/wc0 z 0.9. This condition (o /wc0 < 1) is referred to as 'operating a a frequency below the bottom of the magnetic well', and it can have important influence on reducing the effects of cold electrons, as is discussed below.

Although Fig. 1 represents a particular (though reasonable) choice of parameters, we have done extensive modeling for other choices, with the same basic conclusion: rather modest hot electron densities (nH 5 5 x lo1' ~ m - ~ ) are sufficient to attain optical thickness and the radiation temperature is quite close to black body levels.

2.4.2. T h e effects of cold electrons. When cold electrons are considered, the problem becomes somewhat more complicated. This occurs primarily because the cold electrons can absorb the hot electron whistler emission. Now, the absorption line width for cold electrons is determined by Doppler broadening, with

* - (3)' = 1.4 x 10-3J'T, (eV), WC

(7)

so that for T, d 100 eV the line width is very small and cold electron absorption occurs in the immediate vicinity of the w/w, = 1 surface. The location of this absorption surface with respect to the hot electrons has a primary influence in determining the effect of cold plasma absorption, and in a fixed magnetic geometry the location of w = U, depends only on frequency. The second important deter- minant of cold plasma absorption is, of course, the strength of the absorption, which depends on the density and temperature of the cold plasma and the magnetic field gradient at the w = w, surface [see ELLIS et al. (1983) for further discussions]. These two effects make the cold plasma absorption problem strongly frequency and profile dependent, but two generally important effects are clear:


100

20

0' 1

IO8 lo9 do IO' ' FIG. 2.--WECE radiation temperature versus cold electron density for fixed hot electron parameters at three frequencies: 12.5 GHz (solid line). 15 GHz (dashed line), 18 GHz (dotted

line). Or. = 45", uH = 30 cm, T, = 100 eV.

(a) If one operates with w < w,,, then no cold plasma absorption exists inside the mirror cell.

(b) If one operates with w > w,,, then a cold plasma absorption layer is between the antenna and some portion of the hot electrons. This will reduce the detected intensity, unless the hot electrons have become optically thick closer to the antenna than the cold plasma resonance.

These points are illustrated in Fig. 2 which displays radiation temperature TR versus cold electron density n, for fixed hot electron parameters and three frequencies. The hot electron parameters were chosen to be optically thick and black body when no cold electrons were present. Note firstly that the f = 12.5 GHz curve (solid line) is horizontal, indicating virtually no cold plasma absorption, as discussed in (a) above. The other two frequencies, which correspond to w > coco, show very substantial reductions in T, for quite modest cold electron densities, as discussed in (b). The fact that these two curves cross at some density is a result of the details of the geometry and the absorption. In addition, the fact that TR is not exactly the same at all three frequencies for very low n, is due to the fact that the emission is not exactly black body because of the loss cone. Figure 2 illustrates the importance of operating with w < U,,, in the presence of cold plasma. Later in this paper data will be presented which demonstrate the effects of cold plasma absorption very clearly.

In addition to absorption, cold electrons of high density can lead to another effect which we refer to as 'cold plasma suppression' of the emission from the energetic electrons. Consider a case where the emission and absorption coefficients are determined by the energetic electrons, but the index of refraction nT can be strongly influenced by high density cold plasma. In equation (5) we see that the index of refraction appears as n;' and for the whistler mode nT > 1 and increases with density. Thus, if a substantial density of cold plasma is superimposed on a radiating hot component, and the hot component parameters, and thus j , and SI,, remain un- changed, then the source function is reduced, with a corresponding reduction in the intensity. This effect is not strongly frequency dependent and thus operating at


o < coco does not alleviate it. Experimental evidence demonstrating this effect will be presented later in the paper.

This somewhat unusual behaviour is due to the fact that we are operating in a distinctly non-thermal environment where nT can be specified independently of j , and r y . If we were considering a medium in thermal equilibrium at temperature To thenj,, UT, and n~ would be constrained by Kirchoff’s law so that the source function would equal the blackbody intensity &(U, TO) where

and the intensity inside the medium would equal &Bo(co, TO) (BEKEFI, 1966h). The radiation transfer equation then guarantees that the intensity outside the medium (where n~ = 1) is simply &(U, TO), independent of n,, as it must be.

3. D E S C R I P T I O N O F A P P A R A T U S In this section we describe the experimental configuration and the key components

of the emission detection system. In addition we very briefly list the other TMX-U diagnostics which were incorporated in our analysis.

3.1. TMX-U TMX-U is a large tandem mirror with a solenoidal central cell bounded by two

MHD stable non-axisymmetric end cells; the machine has been described in detail elsewhere (SIMONEN, 1981). One of the major features of TMX-U is the creation of a thermal barrier and a potential barrier in the end cells to reduce axial ion loss. To produce the thermal barrier configuration it is necessary to generate an energetic electron population in the end cell, which is done using ECRH at 28 GHz. The ECRH is applied at two locations, one near the end cell midplane at 2nd harmonic resonance and one outboard from that (further from the centre cell) at fundamental cyclotron resonance. Figure 3 shows the magnetic field on axis and the corresponding cyclotron frequency profile of the East end cell of TMX-U. Also indicated are the locations of the ECRH sources and the additional diagnostics of interest to this work. The horizontal scale, z , denotes the distance from the centre of the centre cell (the axial centre of the entire machine) and the East end cell is the region from z = 400 to 700 cm. A somewhat larger scale view, including the end cell magnet set, is shown in Fig. 4; the horizontal coordinate z is the same as in Fig. 3. Note here that the end cell is defined as the region between the two high field C-coils; the end fan is the region from the outer C-coil to the machine end wall.

The measurements in this paper are concerned with axially propagating whistler ECE which is detected outside the end cell in the end fan region. Figure 4 shows the location of the microwave antenna used for our measurements in relation to the East end cell.

3.2. Microwme apparatus

3.2.1. Dish antenna and waveguide. The microwave emission was detected by a large (60 cm diameter) spherical dish antenna which is described in considerable


In ter ferometer ECRH (Fundamental) Diamagnetic Loop

I 25 F- Center

In ter ferometer ECRH (2nd Harmonic) X-ray Spectrometer

- N I (3 U

0) 0 U-

Diamagnetic Loop I I I I I , I I 0 0

350 400 450 500 550 600 650 700 750 z (cm)

FIG. 3.-Axial profile of magnetic field and electron cyclotron frequency in the East end cell of TMX-U. Horizontal coordinate z is with respect to the centre of the centre cell. Locations

of diagnostics and ECRH also indicated.

PARALLEL ECE ANTENNA (WECE) >

EAST END

WALL

OUTER C-COIL V

CENTRAL CELL,-+---- EAST END CELL- EAST END FAN -4 400 700 1075

z (cm) FIG. 4.--Schematic of TMX-U East end cell magnet configuration and WECE antenna

location. Axial coordinate z the same as Fig. 3.


detail elsewhere (LASNIER, 1985). Briefly, the antenna was designed using Gaussian beam optics (GOLDSMITH, 1982), so that it produced an initially converging beam with a Gaussian beam waist at the outer C-coil. From that point the beam diverged quite weakly (divergence half angle of power of 1.3') producing a reasonably narrow beam pattern at the location of the hot electrons in the adjacent end cell. Such a focusing, near field, system is far better than a conventional high gain horn. The antenna pattern was carefully measured in the laboratory and was crudely checked after installation in TMX-U. The narrow beam pattern was very important in that it limited our detection to axially propagating emission and dramatically reduced any problems with stray radiation getting into our system. In addition, the end fan region was so isolated from the end cell (the C-coil hole being the only major connection) that stray radiation would have difficulty entering the end fan. In fact, we did not detect any substantial emission which could not be adequately explained as axially propagating.

The microwave signals were carried to the diagnostic racks via circular waveguide of 2.2 cm diameter, employing mitre bends throughout. Mitre bends were employed because the 3 0 4 0 GHz emission was overmoded in this size waveguide and mitre bends provided much better transmission and polarization preservation at these frequencies. The mitre bends also performed quite well at the lower frequencies (12- 18 GHz) and were chosen as a compromise to accomodate two frequency bands in one compact size waveguide.

At the diagnostic racks the signals were guided to commercial rectangular waveguide using standard transitions in the conventional manner. All waveguide components were calibrated in situ using commercial noise tube sources. Because the antenna was not polarized, and because of polarization scrambling in the long waveguide run, we calibrated for a variety of input polarizations and power averaged the results. The net result of the calibration uncertainties could be as large as k 2 dB.

3.2.2. Microwaue receiuers. The microwave signals were detected by state of the art high sensitivity superheterodyne receivers in two frequency bands. The first receiver operated at 12.4-18 GHz (KU Band) with an I F of 60 MHz, IF bandwidth 27 MHz. This receiver could be swept in frequency in a minimum time of 10 ms and had a minimum detectable signal of less than -90 dBM. The second receiver operated from 29 to 40 GHz (KA-band) with an IF of 300-600 MHz and a similar sweep speed and minimum detectable signal. Both receivers could be operated in swept frequency or fixed frequency mode, with the latter being used most frequently. The receivers were calibrated with in-band calibrated noise tubes and calibrated attenuators.

3.2.3. Radiation temperature. The measured intensity is presented in terms of radiation temperature [equation (12)], which is found from the detected voltage of the receiver, V,, by means of the relation

where Af = receiver bandwidth, g, = receiver sensitivity, g, = waveguide attenu-


ation, and g, = antenna filling factor. We have discussed our measurements of g, and g,. A calculation of g, is more difficult because one needs to accurately know the radial distribution of the hot electrons and the antenna pattern as installed on the machine. A crude measure of the hot electron radius was obtained using a vari- able radius limiter with the conclusion that for some conditions the profile was approximately parabolic with a radius of 15 cm. As for the antenna pattern, we have the measurements made in the laboratory. The final result of this is that we estimate that g, is approximately 0.5 in the 12.5-18 GHz range with an estimated accuracy of *33% and approximately 1.0 in the 3 M O GHz range with an estimated accuracy of +ox, -33%.

We note the fact that g, < 1 does allow the possibility of stray radiation entering our system. However, two comments are worth making concerning this point. Firstly, it was required that our antenna be located against the end fan wall, with a maximum diameter of -60 cm (these requirements were imposed by the TMX-U geometry and machine operation considerations). Given these constraints our spherical dish antenna produced very nearly the smallest possible beam pattern in the end cell because of its design as a focusing gaussian beam device. Thus g, was maximized for the TMX-U case. In addition, the C-coil which separated the end cell from the antenna (see Fig. 4) also acted to reduce stray radiation. This point was discussed in more detail in Section 3.2.1.

The total uncertainty in the measured radiation temperature is a combination of the uncertainties in calibrating the microwave system and predicting the value of g,. Combining our two stated accuracies assuming uncorrelated errors yields the result that T, is accurate within a factor of approximately 1.7, which may also be stated as -t 2.3 dB. In percentage terms the accuracy is thus + 70%, - 41%. Ideally, one would like to calibrate the system using a large black body, but this was not possible and moreover would not eliminate the uncertainty in g,.

3.3. 0 ther diagnostics The other diagnostics of interest are shown on Fig. 3. We list briefly their

characteristics. 1. Microwave interferometer-140 GHz, measures the line integral of the electron

density at the end cell midplane. 2. Diamagnetic loops-measure the plasma diamagnetism (and thus radially

averaged energy density) at the axial locations. These can also provide an estimate of the axial extent of the hot electrons.

3. X-ray spectrometer-used to detect bremsstrahlung X-rays with a sensitivity in the range 50-700 keV and a time resolution of approximately 40 ms or greater, depending on parameters. This instrument was working sporadically (and as later discovered somewhat improperly) during the course of the WECE measurements and could only provide indications of the nature of the energetic electrons.

4. E X P E R I M E N T A L R E S U L T S A N D C O M P A R I S O N W I T H T H E O R Y The experimental results are presented in this section, along with the comparisons

with our theoretical model. The results are divided into two cases: (a) those conditions for which the detected intensity is dominated by hot electrons alone; (b) conditions for which the cold electron component makes an important contribution.


We will treat case (a) first and then move on to the more complicated, but also more interesting, case (b).

In applying the WECE model to the experiment it is often necessary to supply, as inputs, parameters which we have not measured. In such cases we have made reasonable choices, which are consistent with the pertinent physics and consistent with any diagnostic information available. We have also adopted the policy of adjusting an unmeasured parameter to produce best agreement between the measured and predicted intensity.

It is also worth noting here that each TMX-U discharge has two phases: an active phase and an afterglow phase, discussed briefly in Section 2.2. The active phase is the time during which ECRH, neutral beams, gas puffing, etc., is activated. It is the high density phase, and is the phase of interest for tandem mirror physics. Following the termination of inputs to the plasma (which usually occurs at a time -60 ms in the discharge), most of the cold plasma decays away, but the highly energetic mirror trapped electrons in the end cells remain for more than half a second, until the magnetic fields are terminated. We refer to this latter time as the afterglow phase and it is characterized by having each end cell as a more or less ‘stand alone’ hot electron mirror cell. The electrons during the afterglow phase are predominantly energetic electrons with a low density of cold electrons generated by hot electron ionization of neutral gas. We present WECE measurements for both the active and afterglow phases. The afterglow phase is particularly nice for WECE measurements because the effects of cold plasma are minimized.

4.1. Emission dominated by hot electrons There are many discharges in TMX-U in which a very substantial number of

energetic electrons is created in the end cells. If the WECE detection frequency is set at 12.5 GHz (w/oc0 = 0.89) for these cases then the detected intensity is little affected by cold plasma, because no cold plasma resonance exists inside the end cell (see discussion in Section 2.4). Conditions of this type allow us to study the nearly pure hot electron whistler emission.

We begin by presenting evidence that the WECE emission becomes optically thick and approximately black body for a high enough hot electron density. In Fig. 5 we show a typical strong hot electron discharge, where Fig. 5(a) shows the ECRH heating pulse, Fig. 5(b) the electron line density, Fig. 5(c) the plasma perpendicular stored energy W,, and Fig. 5(d) the WECE signal at 12.5 GHz. For this shot W- has almost no contribution from energetic ions generated by neutral beams. How- ever, the line density has a substantial contribution from cold (or warm) electrons. In Fig. 5(b) a significant portion of the increase from 32 to 45 ms is due to the generation of cold electrons; this is confirmed by the lack of a break in W, at 32 ms and the rapid decay of an equivalent portion of the signal following ECRH termination, with virtually no change in W,. If one crudely accounts for this cold electron contribution, the time history of the hot electron density from approximately 17 to 45 ms is that of a steady increase. Not also that W- shows a similar increase during this time period, though quantitative comparisons are difficult due to the uncertainty in nH. Turning to the WECE signal, we see a steady rise up to approximately 30 ms, followed by saturation. This saturation in the presence of a rising W_ and rising nH is explained by the emission becoming optically thick at the time of saturation. In fact,

504 C. J. LASNIEK et al.

- 0

1.0 ys

o_ 2 0.5

< O

1 c

6 4 N

-

I

f 12.5 GHz (d)

Time (ms)

FIG. 5.-Typical TMX-U discharge in which a large fraction of hot electrons is generated and the WECE emission becomes black body. (a) ECRH heating pulse at the end cell midplane. (b) the midplane line density, (c) the near midplane perpendicular stored energy. (d) the detected WECE radiation temperature. The sharp spikes on the WECE signal are

instability generated noise, discussed in the text.

no other viable picture can explain the three traces. This conclusion is supported by two other observations: (a) the electron density at 30 ms is a few times 10" cmW3 and a substantial fraction of this is hot plasma; such a hot density should become optically thick according to our modeling (see Fig. 1); (b) the average hot electron energy obtained from traces 5(b) and 5(c) is in reasonable agreement with the WECE radiation temperature. Similar observations for many other shots also support this conclusion. As discussed earlier, measurements of the bremsstrahlung X-ray spectra during the period of WECE measurements were plagued with difficulties. What limited data exists is of an uncertain quality, but it clearly shows mean X-ray energies in the range of the radiation temperatures we observe and for some subset of the data a rough proportionality between mean X-ray energy and radiation temperature. We note that Fig. 5 shows sharp spikes on the WECE signal. These spikes are bursts of radiation generated by a microinstability of the ECRH heated hot electrons and detected by our antenna; they are of the same frequency as the detected WECE emission. Such bursts appear only for strong hot electron discharges and only during the ECRH pulse. For many cases with instability bursts a quiescent or 'thermal' pulse shape can be easily distinguished.

In Fig. 6 we demonstrate that discharges where WECE is dominated by energetic electrons can be well modeled by the WECE code. In Fig. 6(a) we show the WECE signal and in Fig. 6(b) the corresponding line density for such a discharge. For this case the cold electron density was the dominant fraction (as determined by the


diamagnetic loop), but the hot electron density was sufficient to determine the emission. We used the WECE code to model the discharge of Fig. 6 by specifying a cold plasma density which models that of Fig. 6(b) and a hot plasma density which rises linearly in time during the ECRH pulse (from t 10 to 60 ms); these are shown in Figs 6(d) and (e). We assumed a constant TH given by the measured T R at saturation and the absolute magnitude of nH was adjusted for best fit. The predicted radiation temperature [Fig. 6(c)] versus time is in very good agreemenrt with the experimental trace, Fig. 6(a). The rise in T R at 60 ms is a cold plasma effect and is very well reproduced; this is discussed in more detail below. Numerous simulations of this type have been performed and the results can be summarized by saying that if we assume T H = TR (saturated) and use hot and cold electron densities which are consistent with the other diagnostics then the modeled WECE intensity (both the magnitude and time history) is in very good agreement with the experimentally measured intensity.

4.2. Emission f o r gases with substantial cold plasma effects In this subsection we present experimental results and modeling for cases in which

cold plasma has a large effect on whistler emission from the hot electrons. These results are important for two reasons. Firstly, they offer a critical test of the WECE model with respect to the observed emission. In fact, the cold plasma results provide the strongest evidence that the WECE model is the proper one. Secondly, cold plasma effects limit the diagnostic capabilities of WECE emission and a detailed predictive capability in this area is of great importance.

As discussed in Sec. ILD, cold electrons can affect the hot electron whistler emission through cyclotron absorption and cold plasma suppression (increase of nT) . Cold plasma absorption can be eliminated by operating at o < coco, but this is not true for suppression. This effect is demonstrated in Fig. 7, for which the detected frequency was 12.5 GHz (o/oc0 = 0.89). Figures 7(a) and (b) show the experimental radiation temperature and line density for a ‘weak’ ECRH shot in which only a very low density of hot electrons was generated in a higher density cold plasma. Note that the maximum radiation temperature is only - 2 keV, indicating the low hot electron density (optically thin emission) and also note that the line density is due almost entirely to cold electrons. The most prominent feature of the TR trace is a substantial dip which is coincident with the large burst of cold plasma shown in trace 7(b). This dip in T R is caused by the cold plasma and can be well modeled by the WECE code. This is demonstrated in Figs 7(c)-(e) which show the inputs (nH, n,) and the output (TR) for the code. The n, trace was taken from the experimentally measured density, TH was fixed at 4 keV, and nH was adjusted to produce a best fit. The rapid rise of nH to a low value is typical of such discharges in which hot electron production ceases when any substantial cold electron density is generated. The important point to note is that the pronounced dip is well produced and the rise at 70 ms is also shown in both the data and simulation. The dip in TR is not due to a decrease of uH or TH caused by cold plasma because the T, signal recovers most of its value wken the density decreases (22-30 ms) and also the time scales of the decrease in line density and increase in TR are the same and can be adequately modeled. It is possible that the cold plasma has caused an ejection (or cooling) of hot electrons and a decrease of ECRH efficiency, which is then reversed when the


'0 20 40 60 80

Time (ms)

FIG. 6.-Measurements and corresponding modeling for a TMX-U discharge in which cold electron effects on emissions are small. (a) Measured WECE radiation temperature; (b) corresponding measured line density, predominantly cold electrons; (c) the predicted radiation temperature; (d) the input cold electron density, from (b); (e) the input hot electron density with magnitude adjusted to produce best agreement in predicting TR. TH measured

from (a), O L = 45", aH = 30 cm, T, = 100 eV.

cold plasma decreases. However, it is highly unlikely that the levels and time scales would correspond so well. Unfortunately neither the diamagnetic loop nor the X- ray system had the sensitivity to detect these levels of hot electrons. Moreover, there are more intense hot electron discharges which show cold plasma suppression (and absorption) and which show no effect on the diamagnetic loop. However, the cold plasma effects are not as clear because the hot and cold density cannot be as clearly separated; Fig. 7 is a particularly graphic example. Finally, later experiments also employed a higher frequency perpendicularly viewing ECE system in conjunction with a WECE system. Again, no changes were observed on the perpendicular signal when cold plasma effects were seen on the WECE signal. These data are not used because the large focusing dish antenna was not part of the WECE system. We do note that cold plasma can have a real effect on the hot electrons; the slow decrease of T, seen in Fig. 7(a) is due to such a loss of hot electrons and has not been included in the model. Such effects are limited to relatively cool hot electron temperatures and are somewhat rare. Well defined dips or 'wells' in TR can always be explained by cold plasma effects on the WECE radiation rather than modifications of the hot electrons themselves. We note that a similar but less dramatic example of cold plasma suppression was shown in Fig. 6 and good agreement between the experiment and the simulation was obtained in that case also.


2 - 5,

P I 1

0 N 2 E nu

3 <

I

2 1

0 3

2 - z F 1

0 08

E 0.6

e 0 04

1 e

- m

." U

= 02

0

- I C

m ?$

OE

0

c

20 40 60 80

Time (ms)

FIG. 7.-Measurements and corresponding modeling for a TMX-U discharge in which only a low density of energetic electrons is generated and cold plama suppression is evidenced. (a) Measured WECE radiation temperature; (b) corresponding measured line density, predominantly cold electrons; (c) the predicted radiation temperature; (d) the input cold electron density, from Fig. 7(b); (e) the input hot electron density with magnitude adjusted to produce best agreement in predicting TR. TH = 4 keV, O L = 45', uH = 30 cm, T, = 100 eV.

We now turn to a case for which cold plasma absorption was most important, which is shown in Fig. 8. For this case the detected frequency was 35 GHz, locating the cold plasma resonance quite far from the midplane. This shot was characterized by an initial burst of cold plasma (8-20 ms) and a second smaller burst of cold plasma (45-70 ms). Coincident with the latter burst is a dip in TR, which is caused by cold plasma absorption and is the object of interest. The WECE model inputs and output are shown in Figs 8(c)-(f), where tzC is taken from experiment, and TH from emission at 12.5 GHz. The hot electron density in Fig. 8 is a low density uniform component assumed to exist out beyond the 35 GHz resonance. This density has been adjusted for best fit, but the nature of the absorption dip does not depend critically on the hot electron parameters. We see very good agreement between the measured dip i.1 TR and the modeled dip.

A very graphic demonstration of the cold plasma absorption effect was accomp- lished by conducting an experiment in which gas was puffed into the end cell during the afterglow phase of the discharge. During this phase the plasma consists mostly

508 C. J. LASKIER et al.

Time (ms)

FIG. 8.-Measurements and corresponding modeling for a TMX-U discharge in which emission at 35 GHz was strongly affected by cold plasma absorption. (a) Measured WECE radiation temperature; (b) corresponding measured line density, predominantly cold electrons; (e ) the predicted radiation temperature; (d) the input cold electron density, from Fig. 8(b); (e) the input hot electron density at the 35 GHz resonance, adjusted to produce best agreement in predicting TR: (0 input hot electron temperature. QL = 4 5 O , T, = 100 eV.


of mirror trapped hot electrons combined with the untrapped cold electrons generated by hot electron impact ionization of residual gas. If in addition, gas is puffed into the end cell during this time it will generate a burst of cold plasma which will decay away with a streaming loss time. One expects this cold plasma to lead to absorption of the hot electron WECE emission for o > coco. This is shown very clearly in Fig. 9 where the experimental [9(a)] and simulation [9(b)] radiation temperatures and cold density [9(c)] at a receiver frequency of 35 GHz are displayed. The simulation hot electron parameters were chosen as for Fig. 8 and the cold plasma pulse was chosen for best fit. We see that the measured dip is well produced in the simulation. Moreover, the fact that the experimental radiation temperature recovers its level following the gas burst shows that the dip is not due to a loss of hot electrons, but rather an absorption of the whistler emission.

The most conclusive data demonstrating the effects of cold plasma absorption were obtained by measuring the frequency spectrum of the WECE emission at various times during the afterglow phase. As discussed above, during this phase the electron component is dominated by energetic electrons with a cold component being generated by ionization of the residual gas. As time proceeds in the afterglow the hot electron density remains more or less fixed, but the residual gas is pumped away and correspondingly the cold plasma component decreases. Thus we expect cold plasma effects to be manifested at all times, but with a decreasing influence as time proceeds. In Fig. 10 we show measured WECE emission (data points) in the frequency range 12.5-18 GHz taken at three times during the afterglow using the receiver in swept frequency mode. The emission is close to black body at the lowest frequency but then falls strongly as cold plasma absorption comes into play, reaching a minimum for o somewhat bigger than oca. From this point TR increases because the cold plasma absorption moves to a region where the magnetic field scale length is decreasing. As time progresses the cold plasma effects decrease. The predictions of the WECE code are shown as lines on Fig. 10. For the simulations the hot electron values were chosen to produce the experimentally measured conditions at 12.5 GHz and the cold plasma parameters and uH were adjusted for best fit. In order to fit the data, we used a cold electron component which was Gaussian in shape, with the same profile as the hot electrons, and an additional cold, lower density, streaming, axially uniform component to represent the cold electron losses. The values used are shown on the figure caption, where n,, is the central cold density and n,, the streaming cold density. We see that the agreement between the experimental spectra and the model is very good in almost every way. There can be no question but that the important elements of the spectrum are well predicted by the WECE code.

The measurements and modeling presented in this subsection on cold plasma effects rely on the assumption that the cold plasma does not substantially alter the hot electron parameters. This point has been discussed in some detail earlier with respect to Fig. 7 and many of those arguments apply to Figs 8, 9, and 10 as well. In addition we note a number of other points.

(a) For the data of these figures an estimate of the Coulomb scattering time for the energetic electrons is far too long for cold plasma to cause any changes in nH and TH, at least on the time scale of the changes in TR.

(b) For the data of these figures the ECRH was off when the radiation temperature

510

120

C. J. LASNIER et al,

I I I I

- -

m

9

- 0.5 'f U l.oI '120 160 200 240 280 320

Time (ms)

FIG. 9.-Measurement and simulation of afterglow phase WECE emission for a case where gas was puffed into the end cell midplane. (a) Measured WECE radiation temperature; (b) predicted radiation temperature from the WECE code; (c) cold electron density input to the WECE code, adjusted to produce best agreement in predicting TR. TH = 30 keV, nH =

4.2 x lo8 ~ m - ~ at the 35 GHz resonance, OL = 45", T, = 100 eV.

Whistler mode electron cyclotron emission from energetic electrons in TMX-Upgrade 51 1

increased following a decrease in n, (see also Fig. 6). With no power input this cannot be caused by heating or generation of new hot electrons.

(c) The spectra shown in Fig. 10 can in no way be explained by cold plasma cooling or expulsion of hot electrons, especially given the low densities involved.

(d) Changes in the radial profile of the hot electrons, caused by injection of cold plasma, would probably not produce the effects observed. The antenna pattern is larger than the hot electron cloud and these cases are all optically thin so that T, is proportional to an area integral of nH. A changing radius without loss of particles would not change TR, at least to first order. Moreover, point (a) above applies also to changes in the radial profile. Nevertheless, no conclusive data exists to absolutely rule out changes in the radial profile, though it is extremely unlikely they could explain the observed results.

4.3. Diagnostic implications Based upon the data presented here and the predictions of a model which has

been well confirmed it is now possible to evaluate the diagnostic capabilities of whistler emission for hot electrons in mirror end cells. Our conclusions in this area are the following:

(a) An effective hot electron temperature can be measured if the cold electron density is not too high and if the receiver frequency satisfies w < coco; even relatively small cold plasma densities cause substantial absorption for co > coco.

(b) The hot electron density necessary to attain black body levels is geometry dependent, but in general is quite modest for most mirror devices.

(c) Large cold plasma densities, typical of many discharges, reduce the diagnostic capabilities of the whistler view.

(d) The afterflow phase of hot electron mirrors will usually be characterized by low cold electron densities and WECE is very useful during this time. If the afterglow hot electrons are typical of the active phase hot electrons then WECE can be a useful diagnostic independent of the cold plasma effects during the active phase.

5. D I S C U S S I O N A N D C O N C L U S I O N S

5.1. Some observations A number of points are worthy of further discussion at this time. Firstly, in

applying our simulation model to the experiment it was found that a number of important inputs to the code could not be measured in the present TMX-U device. We adopted the view that we would make reasonable assumptions about the missing quantities and in many cases adjust them so as to obtain a best fit prediction for the radiation temperature. In all cases shown here the conjectured input data were consistent with reasonable physical expectations for mirror devices and consistent with any diagnostic information available. The important point is that virtually all our WECE data could be explained by the theoretical model with some inputs taken from experimental data and sensible assumptions about unmeasured quantities.

The second point concerns the fact that the WECE model assumes a hot electron density distribution which is Maxwellian. While no definite X-ray measurements exist for this data, later X-ray measurements show that the electrons with energies greater than -30 keV have an essentially Maxwellian energy distribution with a


single temperature. Similar results are reported for other ECRH heated mirrors (SMATLAK, 1984). Moreover, theoretical calculations of WECE from distributions which contain a 'hot tail' show conclusively that for TMX-U-like parameters, hot tail densities would have to exceed 20% of the bulk hot density to have a substantial effect on the measured radiation temperature. (These results will be published sepa- rately). There is no evidence in the X-ray data, or any other diagnostic, to lead us to believe that such a hot tail exists.

The third point concerns the possible effects of refraction in our data, which has been ignored in the modeling. The first thing to note is that almost all of our data is presented for low densities o2 9 oi,(nT - 1) for which refraction should not be an issue. Moreover, the whistler wave propagates along B and thus along the axial density gradient, so that one does not expect substantial refraction to occur. Finally, in our much more controlled experiments on a small device (ELLIS et al., 1982; ELLIS et al., 1983), no refractive effects were observed for densities exceeding these we are dealing with here, up to values of o2 ~ k : aie.

5.2. Conclusions Two important general conclusions have been reached as a result of this study: (a) The axially propagating emission in TMX-U is well explained on the basis of

whistler mode electron cyclotron emission, including the effects of hot and cold electrons and the complexities of radiation transport in situations where the electron density and magnetic field profiles (and thus the emission and absorption profiles) were quite inhomogeneous.

(b) Whistler mode electron cyclotron emission can be used to measure an effective hot electron temperature in mirror devices for conditions in which the cold electron density is not too high.

Acknor~'ledgements-The authors are grateful to C. CELATA and T. LEID for helpful discussions, D. MARTIN for technical support. and the entire technical staff of TMX-Upgrade for their able and helpful assistance. This work was supported by the U.S. Department of Energy and the National Science Foundation. The computer calculations were supported in part by the Computer Science Center of the University of Maryland.

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BOOSKE J. H., GETTY W. D., GILGENBACH R. M. and JONG R. A. (1985) Phj'sics Fluids 28, 3116. BOYD D. A., STAUFFER F. J. and TRIVELPIECE A. W. (1976) Phys. Rev. Lett. 37, 98. COSTLEY A. E., HASTIE R. J., PAUL J. and CHAMBERLAIN J. (1974) Phys. Rev. Lett. 33, 758. DANDL R. A. et al. (1964) Nucl. Fusion 4, 344. EFTHIMION P. C., ARUNASALEM V. and HOSEA J. C. (1980) Phys. Reo. Lett. 44, 396. ELLIS R. F., TSAKIRIS G. D. and BOYD D. A. (1982) Phys. Rec. Lett. 48, 93. ELLIS R. F., TSAKIRIS G. D. and BOYD D. A. (1983) Phys. Fluids 26, 1528. FUJIWARA M. et al. (1985) Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th Int.

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Whistler mode electron cyclotron emission from energetic electrons in TMX-Upgrade 5 13

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Q u o s B. H. et al. (1985) Physics Fluids 28, 1503. SIMONEN T. C. (1981) Proc. IEEE, 69(8), 935. Sihioxm T. C. et al. (1985) Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th Int.

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