Shear Alfvén wave radiation from a source with small transverse scale length

Transcription

1 PHYSICS OF PLASMAS VOLUME 7, NUMBER 10 OCTOBER 2000 Shear Alfvén wave radiation from a source with small transverse scale length D. Leneman, W. Gekelman, and J. Maggs University of California at Los Angeles, Room 15-70, 1000 Veteran Avenue, Los Angeles, California Received 14 March 2000; accepted 13 July 2000 Shear Alfvén waves are studied in the kinetic and inertial regimes. The waves are launched from an antenna which is on the order of the electron collisionless skin-depth, c/ pe, in size. The experiment is performed in the LArge Plasma Device, LAPD W. Gekelman et al., Rev. Sci. Instrum. 62, at the University of California, Los Angeles, using a new antenna design that modulates parallel plasma electron current. The plasma is 100 skin-depths in diameter and 3.6 Alfvén wavelengths long. The results include the calculation of the wave currents based on measurements of the wave magnetic field. Differences in the perpendicular phase velocity of the wave in the kinetic regime as compared to the inertial regime are also reported. Results generally compare favorably with predictions of a theory which includes collisional damping and kinetic electron dynamics with fluid ions American Institute of Physics. S X I. INTRODUCTION Alfvén waves are known to be very important in the dynamics of magnetized plasmas in space 1 4 and in laboratories on earth. 5,6 They transport electromagnetic energy, may accelerate plasma particles, and are produced in many magnetized plasma environments involving changes in plasma currents or magnetic field configuration. There are two basic Alfvén waves in the magnetohydrodynamic MHD limit. One is compressional, where plasma density oscillates in phase with magnetic field strength and, for a plane wave, the wave number vector, k, wave magnetic field, B, and the background magnetic field, B 0 are coplanar. The other is the shear mode which has oscillations in B that are perpendicular to B 0. Shear Alfvén waves propagate at frequencies below the ion cyclotron frequency ( ci ), while the compressional wave can propagate at frequencies above and below this value. The currents associated with the shear wave are carried by both electrons and ions, but these two species perform different roles. The electrons carry the wave currents which are parallel to B 0 and the ions carry the perpendicular currents though the polarization drift. The experiments described in this article are concerned with shear Alfvén wave radiation from sources with transverse scale comparable to the electron collisionless skindepth,, which is the speed of light divided by the electron plasma frequency, c/ pe. When the source is of this scale, there are important differences in the properties of the shear wave depending on the ratio of the Alfvén wave phase speed, v A, to the electron thermal speed, v e, in the plasma. Here v e (2kT e /m e ) 1/2, where T e and m e are the electron temperature and mass. The Alfvén wave phase speed is defined as v A v A0 1 / ci 2 1/2. 1 Here is the wave frequency, v A0 B 0 /(4 n i m i ) 1/2, n i is the ion density, m i is the ion mass, and B 0 B 0.Ifv A /v e is much smaller than 1, the wave is termed kinetic, 7,8 i.e., the plasma electrons respond as a fluid to the wave parallel electric field. If v A /v e is much larger than 1, the wave is termed inertial, 9,10 i.e., the electron inertia determines their response to the wave fields. In a quasi-neutral plasma, m i /m e 8 n e kt e m i /B 0 2 m e v e /v A0 2, where is the electron beta. Equation 2 indicates m i /m e is a convenient parameter to use for determining how kinetic or inertial the shear Alfvén wave is. One motivation behind this research effort is that shear waves launched with small sources have an electric field parallel to B 0, E. This field can, in turn, interact with particles in the plasma. Another motivation is to study shear wave radiation as it may spontaneously occur in a plasma. In this experiment the radiation is unaffected by the machine boundaries so the findings described here might be helpful in the interpretation of some low-frequency ( ci ) space plasma phenomena. In space plasmas there are field-aligned current filaments with transverse size as small as the electron collisionless skin-depth Alfvén wave sources of this size are predicted and observed to have a profound impact on the radiation observed. For example, the wave energy will spread radially away from the flux tube of the source as it propagates axially. There are at least two possible scenarios for the generation of localized currents in space. A fluctuating electric field, aligned with the background magnetic field, and localized on the skin-depth scale, would generate such a structure. This electric fieldcouldbecausedbyanother disturbance. Another possibility is the transient, localized electric field associated with the formation or decay of double layers thought to exist in the auroral ionosphere. 21 Such a transient structure would have a broad frequency spectrum. Each frequency component less than ci would radiate its own shear Alfvén wave structure with its own X/2000/7(10)/3934/13/$ American Institute of Physics

2 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with characteristic radial spreading. If the source current circulates and closes back on itself as in the model described in this article the magnetic field associated with the wave structure will be radially confined. If the current closes at a very large radius, the wave structure will be more extensive, exhibiting a 1/r radial decay in amplitude at large r values. Such structures may also naturally occur at the edge of laboratory plasmas. 6 Magnetic field measurements on TO- KAMAK edge plasmas show 5 waves with high correlation lengths along the magnetic field and short correlation lengths across it. These are most likely shear waves and may play a significant role in edge plasma physics. This article is organized as follows: first, an overview of the theory is presented. Next the experimental arrangement including a brief discussion of the plasma device is described. The data are then compared with the theoretical predictions. Comparisons are made for the dispersion relation, the shape of radial profiles of the wave magnetic field magnitude, the radial effect of damping, radial wave propagation, and the wave current structure. We then present a summary of the results. II. THEORY AND EXPECTED WAVE BEHAVIOR Before describing the experiment, a brief description of the relevant theory 16,17 is in order. The theory describes shear Alfvén wave radiation from one disk-shaped source with radius a. The disk is oriented so that the normal of its surface is aligned with the background magnetic field. The field-aligned current density at the source axial position z 0) is modeled as constant, for radial distances, r, less than a; and 0, for r a. The amplitude of the ac flowingtothe disk is I 0. The azimuthal symmetry of the model implies that the wave magnetic field, B, will be azimuthally symmetric and in the azimuthal direction. Radial and axial components of B are not expected unless ci. The spatial dependence of the azimuthal component of B, B,isgivenby 16 B r,z 2I 0 sin k a J ca 0 k 1 k r exp ik k z dk. 3 Here J 1 is the first-order Bessel function and k and k are the parallel and perpendicular components of the wave number vector. The dispersion relation, 17 which includes damping, is 1 i v A /v e 2 2 Z k 2. 4 Here Z ( ) is the derivative of the plasma dispersion function 22 with respect to its argument, (1 i ), /(k v e ), and c /, where c is the dominant collision rate. The function k (k )ineq. 3 can be found from Eq. 4. The inclusion of the plasma dispersion function properly describes the electron motion along the magnetic field for all values of the wave parallel phase velocity. The dispersion relation is simplified in the inertial or kinetic limit when is set to 0. In the inertial limit is large, Z ( ) 1/ 2, and the dispersion relation becomes /k v A 1 (k ) 2 1/2. In the kinetic limit is small and Z ( ) FIG. 1. Theoretically predicted shear Alfvén wave magnetic field magnitude is shown normalized to its source s ac amplitude, and as a function of radial distance from the field line threading the azimuthally symmetric source. The radial wave structure varies with plasma beta. 2. Therefore, /k v A0 1 ( / ci ) 2 (k s ) 2 1/2 where s is (T e /m i ) 1/2 / ci and is the relevant spatial scale for plasmas with m i /m e 1. In the experiments reported here, an attempt is made to cover a significant portion of, space and evaluate the affects of these parameters on the wave. Both inertial and kinetic waves radiated from single-disk antennas exhibit a 1/r radial dependence of B at large r. The radius at which this dependence begins is called the cone radius r c. The radial dependence of B for r r c can be quite complex, as in the theoretically predicted curves for m i /m e 100 and 0.01 in Fig. 1, where r c 10 cm and z equals 3 parallel wavelengths ( ). However, shear waves for all values of exhibit a single peak in B (r) when z is sufficiently small or when m i /m e 1. There is one exception: B (r) in inertial waves, with sufficiently small damping, exhibits two sharp features associated with the source edge. These source edge features become less prominent as the axial distance from the source increases and are not visible in Fig. 1. Secondary peaks, which are not source edge features, appear at different radial positions depending on FIG. 2. Source current density as a function of r when disk current reaches maximum amplitude. The source current is axial, positive for r a d, negative for a d r a p, and zero for r a p.

3 3936 Phys. Plasmas, Vol. 7, No. 10, October 2000 Leneman, Gekelman, and Maggs FIG. 3. Side view of the LArge Plasma Device LAPD at UCLA. whether the shear wave is in the inertial or kinetic regime Fig. 1 ; as the axial distance from the source increases, so does the number of peaks assuming very small damping. In the inertial regime the secondary peaks appear at larger radii than that of the main lobe, but in the kinetic regime, the secondary peaks appear at radii smaller than that of the main lobe. In these experiments, waves are radiated using a dualdisk antenna described later rather than a single-disk antenna. The dual-disk antenna does not create a wave radial profile with a 1/r fall off at large r because of the antenna design. The theory for a single-disk source must be modified to model the double-disk source. In a modified theory used here, the disks broadcast out of phase and all the ac flowing out of one disk returns through the plasma to the other through an annular area coplanar with the disks. The inner radius of the annulus is denoted by a d the actual radius of the disks and the outer radius is denoted by a p. Figure 2 shows the model current profile of the source. The wave field is calculated by summing two integrals of the form given by Eq. 3, where in one integral a a d and in the other a a p. The two integrals are subtracted since one disk is radiating 180 out of phase with respect to the other. The relation of I 0 in Eq. 3 to the amplitude of the ac flowing to the antenna disks, I ant,is I 0 I ant / 1 a d /a p 2. 5 Results of this model will be compared with data later. The effect of collisional damping on the structure of the shear wave is incorporated into the theory where it is parametrized by. Two common types of collisional damping in laboratory devices are electron ion Coulomb collisions and electron collisions with neutral gas He, for example. In this experiment the Coulomb collision frequency, ei, is dominant since it is at least ten times the electron-neutral collision frequency. The published theory does not include the velocity dependence of Coulomb collisions but, nevertheless, predicts that greater damping smoothes out sharp features, broadens peaks in B (r), and moves them out to larger radii. If is large enough, a peak in B (r) can appear outside the predicted cone radius. The ratio of the perpendicular group velocity to the parallel group velocity, v g /v g, for the inertial and kinetic limits is given in Eqs. 6 and 7, respectively, v g v g k 2 /v A 1 k 2 1/2, v g v g k s 2 /v A0 1 / ci 2 k s 2 3/2. The minus sign appearing in Eq. 6 indicates that the inertial wave is a backward wave in the perpendicular direction. Experimentally, the wave energy is observed to spread radially outward, i.e., in the positive r direction, as it must. Therefore, the radial phase velocity for the backwards inertial wave is expected to be inwards in the negative r direction. FIG. 4. Typical plasma density profile measured with a Langmuir probe from averages over 20 plasma pulses. The error bars represent the standard deviation of the average. 6 7

4 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with mode. Operating this way considerably increases the lifetime of the cathode coating since data cannot be acquired continuously, in any event, due to probe positioning delays. The plasma in the 9.3 m long section beyond the anode is current-free and quiescent ( n/n 2% over the time of a typical experiment. Therefore the antenna placement and the measurement locations are all in this region. The cathode-emitted beam electrons which make it past the anode are fully pitch angle scattered and, in the present experiment, their mean-free path for inelastic collisions ranges between 5 and 15 m. Langmuir probes show no signs of a bump-on-tail electron distribution in the non-currentcarrying region. The background magnetic field, B 0, is produced by electromagnets surrounding the 10 m cylindrical section of the machine. The field is solenoidal and aligned along the axis of the vacuum vessel. The magnets are sufficiently separated to allow easy access to the ports and to accommodate a wide range of angular manipulation of the probe apparatus, while at the same time maintaining a nearly uniform field 2% at the edge of the plasma for high-quality experiments. FIG. 5. Antenna detail. The disks are closely spaced but are electrically isolated from each other. The mesh holes are aligned to achieve maximum transparency to the plasma. III. EXPERIMENTAL SETUP A. The LAPD The plasma device used for these experiments is called the LArge Plasma Device LAPD 23 andisdepictedinfig. 3. The LAPD vacuum vessel has an overall length of 11 m. The experimental region is a 1 m diameter, 10 m long cylinder. A 1 m long end section supports the cathode assembly of an electron beam source and has large pump-out ports for the high vacuum system. The vacuum system consists of a pair of turbomolecular pumps backed by rotary vane pumps, the chamber is back filled with He. The energy source for the ionization required to produce the plasma consists of an electron beam accelerated by a potential of up to 55 V. Electrons are drawn out of a heated sheet of BaO-coated nickel and accelerated by the electric potential applied between the Ni sheet the cathode and a wire grid the anode placed 0.7 m downstream. Since the emissivity of the coating decreases on the millisecond time scale, 24 plasma is most efficiently produced in the pulsed B. Plasma characteristics The LAPD is capable of producing plasmas with electron density, n e cm 3 and T e up to 15 ev. In the experiments described, the parameter regime was restricted to n e cm 3 and T e 12 ev. The cathode coating technique used in these experiments produces plasmas with spatial variations see Fig. 4. Typical transverse spatial variations in the electron density over the middle 20 cm of the plasma are 7% and 10% in the electron temperature. The highest transverse spatial variation for n e is 10% and for T e it is 15%. Axial variations are smaller: the radially averaged n e typically varies less than 3% over 7 m. The radially averaged T e monotonically decreases with distance from the electron beam source, typically dropping by 14% over 7 m. After the voltage on the cathode is turned off, this gradient in T e decreases considerably. These data are derived from Langmuir curves averaged over an ensemble of 20 samples because the plasma varies slightly from pulse to pulse. In 20 separate plasma pulses the Langmuir characteristic was recorded at the same time after the start of each pulse and at the same position in the chamber. The error bars shown in Fig. 4 represent the standard deviation of the ensemble. The wave field data to be presented also consist of averages over 20 separate pulses of FIG. 6. During the experiment the termination plate is at the plasma floating potential. When the transistor switch is closed, the dc bias holds the antenna disks up near the plasma potential. The inductor suppresses any net ac which might flow to the antenna; and the center tapped transformer adds the ac signal to the dc bias. The RF current is fed 180 out of phase to the two disks via a coaxial cable.

5 3938 Phys. Plasmas, Vol. 7, No. 10, October 2000 Leneman, Gekelman, and Maggs TABLE I. and values for each experimental case studied. The third column is included because v A differs from v A0 due to finite wave frequency,. The last column is the average slope of (r). For a given, as m i /m e decreases, so does / r. In case 5 the average slope is negative. Case no. m i /m e v A /v e / r ( /cm) % % % % % % % % % % 3 4 near the antenna is not much larger than the random spatial variability shown in Fig. 4, and decreases by a factor of 2 for every axial meter away from the wave source. Nevertheless, it is located on the flux tube defined by the source and broader in the transverse direction than the random spatial variations. FIG. 7. db/dt probe. The probe has a small profile in the plasma and measures all three components of db/dt along one background magnetic field line. plasma. It is experimentally determined that this average is sufficient to characterize the wave properties in spite of the pulse to pulse variability of the plasma. In practice there is also a systematic perturbation of the plasma by the wave launching antenna. It produces a radial depression in plasma density and electron temperature which extends along its flux tube. The contrast of the depression C. Wave source The shear Alfvén wave antenna is constructed from two identical Cu wire mesh disks see Fig. 5. Their radius, a d,is 0.50 cm while the electron skin-depth in the experiments ranges from 0.34 to 0.45 cm. The disks are glued together with high-temperature, vacuum compatible epoxy such that the meshes line up for maximum transparency. The layer of epoxy between them assures electrical isolation. Current is fed to the disks via a coaxial cable where each conductor services one disk. The combination is supported by a small metal shaft which is 0.3 cm in diameter. The shaft is insulated from the antenna disks and the chamber wall. For reasons presented later, a dc bias voltage is placed in series between the copper plate that terminates the plasma and the two disks, as in the circuit schematized in Fig. 6. FIG. 8. Measurement geometry. The layout of the r, measurement locations is shown in part a. Part b is a perspective drawing depicting the perpendicular measurement planes along the z axis and their relation to the antenna position. For each set of plasma conditions a minimum of four planes of data is acquired. FIG. 9. Measured versus calculated wave phase for each case and z position. Here c z/v A, where v A is definedineq. 1. The slope of the line fit is 0.97.

6 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with FIG. 10. Experimental B magnitude radial profiles along all spoke angles for case 2 at z 283 cm (1.1 ). The antenna dc bias allows the production of large sinusoidal current fluctuations. If a small, sinusoidally timevarying voltage is applied to a disk whose dc voltage is held at the plasma floating potential, it will collect an excess of electrons on positive swings and an excess of ions on negative swings. If the amplitude of the ac voltage is increased, the current collected in the negative swings saturates because FIG. 12. Calculated B and experimental B radial profiles at several z positions for m i /m e 1.0 and 1.0 case 2. FIG. 11. Calculated B and experimental B radial profiles at several z positions for m i /m e 6.8 and 1.0 case 1. the ion current is limited by the ion sound speed. The resulting ac resembles the rectified current of a diode. Applying a positive dc bias to the disk allows the use of larger ac voltages without deformation in the sinusoidal current because the oscillating voltage will now modulate primarily electron current. This technique makes it possible to produce oscillating current with peak-to-peak amplitude near the electron saturation current. In the dual-disk design, the dc voltage is applied to both disks and ac voltage is applied so that the electron current flowing to each disk is modulated 180 out of phase. For a given dc bias, the ac amplitude is limited to a value which ensures that the negative swing stays above the plasma floating potential. In practice the dc bias is set in order to reasonably minimize the effects of drawing current to the antenna. Since drawing a current produces a density and electron temperature depression in the plasma, the bias is set so the product n e T e in the depression is not less than 0.6 of n e T e in the surrounding plasma. In most cases the bias is about 2.5T e V where T e is in ev. The ac voltage amplitude is chosen so that the higher harmonics in the received signals are below 10% of the fundamental. The combined requirements of ac

7 3940 Phys. Plasmas, Vol. 7, No. 10, October 2000 Leneman, Gekelman, and Maggs FIG. 14. Calculated B and experimental B radial profiles at several z positions for m i /m e 1.0 and 8.2 case 4. FIG. 13. Calculated B and experimental B radial profiles at several z positions for m i /m e 0.5 and 1.0 case 3. and dc voltages necessitate the use of a center tapped transformer see Fig. 6. The dual construction of the antenna is necessary for a near-zero net ac flowing in the feed. In previous theoretical and experimental studies 16,17,20,25 a single-disk antenna was modeled and used, and a net ac flowed in a conductor in the support shaft of the antenna. The radial current element also radiated a wave which was superimposed on the radiation from the disk. Since this current is not modeled in the theory, it is difficult to make conclusions concerning the radial effects of damping, radial propagation, and the detailed structure of the wave. Although there is no net ac flowing to the two-disk source, an Alfvén wave is launched because the disks still modulate dc plasma currents: as one disk draws more electrons, the other draws correspondingly fewer. To balance the antenna current system a choke is used see Fig. 6 to reduce the net ac in the coaxial feed to the acceptable level of 3% to 4% of the ac to either disk. The current flowing to the antenna is digitized during the experiment in order to monitor the constancy of the source over 24 h data runs. The antenna is located at the radial center of the plasma, 0.3 m from the anode, in the experimental region of the LAPD. Its center defines the position of the magnetic field aligned aligned z axis (r 0), as well as the position of the plane where z 0. D. Alfvén wave detector In the present study only the Alfvén wave s magnetic field, B, is measured. Three small wire coils are used to measure db/dt as a function of time t and position in cylindrical coordinates, r, and z. The coils are mutually orthogonal see Fig. 7 in order to measure the three components of db/dt which lie along their axes. Since the launched and received signal is sinusoidal, it is immaterial that db/dt is measured instead of B. The voltage measured between the coil leads can be contaminated by the influence of the wave electric field. To reduce this effect, each coil consists of two oppositely wound multi-turn loops and the respective signals are subtracted to reject the common mode. 26 The coils and their coaxial cables are housed in a metal cover sufficiently thick to shield out the wave electric field. A slit is cut in each coil cover in order to prevent the formation of eddy currents. Figure 7 shows how the three sets of paired loops are arranged, supported, and fed with coaxial cables. Each loop pair has 116 windings, an inner diameter of 0.4 cm, an outer diameter of 0.6 cm, and is 0.3 cm tall. The probe can resolve spatial changes of 0.5 cm scale. The three coils are positioned along and centered on a common B 0 field line. This arrangement is preferable because the wave radiation is expected to have a 1 cm scale in the perpendicular direction and a 1 m scale in the parallel direction. The coils and their covers are not sup-

8 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with FIG. 15. Calculated B and experimental B radial profiles at several z positions for m i /m e 0.23 and 7.5 case 5. FIG. 16. Experimental and theoretical wave phase as a function of r for several z positions and for m i /m e 6.8 and 1.0 case 1. The positive slope indicates radially outward propagation. ported by their leads nor are the leads exposed to the plasma as Fig. 7 suggests. A vacuum compatible epoxy is used to insulate the leads as well as to attach the coil covers to the support structure and hold their mutual orthogonality to within 0.01 rad. The fully assembled db/dt probe was calibrated using a long thin wire with a known ac flowing in it. The probe was positioned over a range of distances from 0.5 to 200 cm and tested over a range of frequencies from 100 khz to 1 MHz. TABLE II. Values of plasma current source outer diameter, a p, and range of values of the damping parameter used in the theoretical fits to the data in each case. The required damping parameter,, atz 283 cm for cases 1 and 2 is 0.2. Case no. a p (cm) Axial range of E. Measurement technique and data acquisition The db/dt of the launched Alfvén wave is measured at 235 positions in several planes perpendicular to B 0 in the plasma volume. At each position the experiment is repeated 20 times, and the received db/dt signal is averaged and stored on disk for later analysis. The pattern of measurement locations is the same in each plane. It consists of a fivespoked acquisition grid depicted in Fig. 8 a. The density of measurement points is higher near the center in order to better sample the fine structure of the radiation pattern. The r, arrangement of the measurement locations affords efficient monitoring of symmetry over. The probe location is set by a computer-controlled drive mechanism, which manipulates the support shaft of the probe from outside the vacuum vessel. Where the current density, j (j 1/ 0 ÃB), is to be calculated, nine data planes spaced as closely as /4 are acquired. The accuracy of alignment of the probe coil axes with respect to B 0 is 0.02 rad. Therefore measurements of the axial component of B have a maximum inaccuracy of 2% of the magnitude of the perpendicular component of B. The accuracy in positioning the probe is 0.02 cm in the horizontal direction of Fig. 8 a. When a continuous series of

9 3942 Phys. Plasmas, Vol. 7, No. 10, October 2000 Leneman, Gekelman, and Maggs FIG. 17. Experimental and theoretical wave phase as a function of r for several z positions and for m i /m e 1.0 and 1.0 case 2. Inthiscase the direction of radial propagation depends on r. points is sampled, as in the series of points in a spoke, the precision in the vertical direction of Fig. 8 a is 0.04 cm. Given the wave structures encountered, the positioning inaccuracy has a noticeable effect only where r is less than about 0.5 cm. There are other possible sources of inaccuracy: pulse-topulse differences in the plasma, Alfvénic noise at the frequency of the launched wave, reflections at machine boundaries, and perturbations due to the probe itself. The effect of plasma variations and background Alfvén waves is reduced by averaging over a 20 pulse ensemble. In this experiment there were no detectable reflected waves from either the end plate, the anode or the cathode. The db/dt probe is large enough that it could impede wave currents; this was tested by detecting the wave with a second probe. It was found that the presence of the first probe did not significantly change the signals received at the second probe. The plasma conditions are varied in order to observe the wave in two different collisionality regimes different values of. By varying the plasma, the Alfvén wave is observed in the kinetic, inertial, and intermediate regimes. Five different plasma conditions, or cases, are studied. Table I shows the parameters and assigns numbers for each case. Since is comparable to ci in this experiment, the wave s parallel phase speed, v A, is not equal to v A0 see Eq. 1, therefore FIG. 18. Experimental and theoretical wave phase as a function of r for several z positions and for m i /m e 0.5 and 1.0 case 3. The positive slope indicates radially outward propagation. There is a clear trend towards inward propagation as m i /m e decreases from 6.8 Fig. 16 through 1.0 Fig. 17 to 0.5 this figure. v A /v e is physically more meaningful than m i /m e (v e /v A0 ) 2 and is included in the table. The errors listed in the table are all due to the axial gradient of T e. Three values of m i /m e are studied where is approximately 1. Additionally, two cases with a high value of are examined. Since data from different cases must be compared at the same scaled distance (z/ ) from the source, and since axial probe access is limited to regularly spaced ports, it is desirable that is the same value in each case, 2.50 m. This value is chosen because it affords an attainable and interesting range in, space. Since ei depends on n e and T e, 27 moving in, space involves one or more of the experimental parameters n e, T e, B 0,and. In principle, since the LAPD s operation imposes a constraint on n e and T e,and since,, and are fixed for each case, there are four constraints on n e, T e, B 0, and. So, they are uniquely determined and all must be changed to obtain a desired or. IV. OBSERVATIONS AND COMPARISON WITH THEORY Figure 9 is a scatter plot of the measured wave phase, m, versus a calculated phase, c, for each case and z po-

10 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with FIG. 19. Experimental and theoretical wave phase as a function of r for several z positions and for m i /m e 1.0 and 8.2 case 4. A positive slope indicates radially outward propagation. FIG. 20. Experimental and theoretical wave phase as a function of r for several z positions and for m i /m e 0.23 and 7.5 case 5. While not all curves have a negative slope, there is a clear trend towards inward propagation as m i /m e decreases from 1.0 Fig. 19 to 0.23 this figure. sition sampled. The wave phase was measured by fitting a sine curve with phase as a free parameter to the wave magnetic field data. This phase is then averaged over the five spoke angles and over about a 3 cm radial interval centered at about r 7 cmtogive m for each z position. The quantity c is calculated according to c z/ v A0 (1 ( / ci ) 2 ) 1/2, where a plane wave is assumed (k 0). The best fit to the data is a straight line with a slope of 0.97, where a slope of 1 would indicate exact agreement of measurement and calculation. This plot indicates that the observed disturbance axially propagates with the phase of a shear Alfvén wave over a wide range of densities and background magnetic fields. To compare the observations with theory we consider the radial dependence of the amplitude of the wave magnetic field. Given the symmetry of the source, and from theoretical considerations, the wave radiation pattern is expected to be azimuthally symmetric. Figure 10 shows the magnitude of the azimuthal component of the wave magnetic field, B,as a function of r along each of the spoke angles, at a distance of 1.1 from the antenna. A similar degree of azimuthal symmetry is observed in all the cases studied. Accordingly the data are reduced by averaging over the five spoke angles. The observed departure from symmetry is likely due to asymmetries in the source current density which probably arise from spatial variations in plasma density observed near the disks of the antenna. Figures show azimuthally averaged B (r), B (r) solid line, and the best theoretical fit to these data dashed line at various axial positions for the five cases studied. Notice that the vertical scale changes as z varies. In the fits, a p and are free parameters. Here is treated as a free parameter because the Krook collision operator does not include any velocity dependence and, as such, is only a crude model for Coulomb collisions. The radius a p is free because mechanical constraints made the measurement of the magnetic field near the antenna impractial. We constrain a p to have one value in any particular case, whereas is allowed to change axially within a case. We find that for z,the theoretical profile is much more dependent on a p than. At farther positions the reverse is true. Without exception, this two-disk model produces radial profiles which drop to zero faster than 1/r, and the radius at which they peak depends on a p and. Thisradiusiscomparable to those observed in the experiments when a p is between2and3cmand is comparable to the values in

11 3944 Phys. Plasmas, Vol. 7, No. 10, October 2000 Leneman, Gekelman, and Maggs FIG. 21. Wave current density as a function of r and z calculated from the curl of measured B at an arbitrary time and averaged over for case 1 ( m i /m e 6.8). a The magnitude of the vectors is rescaled to take out the axial and radial decrease in amplitude. b The magnitude is rescaled for the axial decrease in amplitude only so the structure where r 1 cm and z 400 cm is visible. c Current density calculated from theory using the same parameters that produce the fits for case 1 ( m i /m e 6.8). This figure can be compared to part a since the vectors have been rescaled axially and radially in the same way. Table I. This is an indication that the model accurately describes the source. Table II shows the parameters required to make these fits. The required values for used in the fits correspond reasonably well to the experimental values see Table I. Onlyincases1and2wherez 283 cm do the required values differ by more than a factor of 2 from the experimental value. Overall, the fits are reasonable for cases 2 5. In case 1, the most kinetic, no values of a p and gave satisfactory fits for axial positions beyond z 283 cm. The cause of this discrepancy may be the radial depression in T e causedbythe antenna, which could have a pronounced effect in the kinetic case. Including the axial and radial variation in n e and T e may improve the theoretical model. The shapes of the profiles also suggest that the putative source return current should taper off to zero as opposed to being sharply cut off at a p. The axial decay of the wave agrees well with the predictions of the model used to describe the two-disk antenna. In fact, the axial decay is not very sensitive to the values of calculated on the basis of measured experimental parameters. This insensitivity arises from the antenna design. As mentioned earlier, the wave structure is dependent on the model parameter a p which defines the extent of the return currents in the vicinity of the two-disk antenna. This parameter plays a key role in determining the rate of axial decay in the first few wavelengths and results in a rate of decay much larger

12 Phys. Plasmas, Vol. 7, No. 10, October 2000 Shear Alfvén wave radiation from a source with than that caused by either collisional or collisionless damping. By comparing the measured radial profiles in Figs. 12 and 14, the effect of different damping parameters 1.0 and 8.2 can be seen. As expected, at each axial location, the radial profile peaks at a larger r and the profile is broader for the higher case Fig. 14. The rate of spreading depends on / ci ; but in both these cases, the ratio is the same, Theory predicts that the phase of the kinetic wave exhibits outward radial propagation, while the inertial wave exhibits inward radial propagation. 16,17 confirmation of this prediction is illustrated in Figs , which show the theoretical dashed line and azimuthally averaged experimental solid line radial dependence of the wave phase, (r). The theoretical curves are generated using the same parameters used in the fits to the profiles discussed in Sec. III B. Figure 16 shows very good agreement between theory and experiment and the measured propagation is clearly outward in the positive r direction. In Figs. 17 and 18 the agreement is poor, but the transition away from the outward propagation of Fig. 16 where m i /m e is highest is evident. Figure 19 shows good agreement between theory and experiment, but in Fig. 20 there is only agreement for 0.25 cm r 2.5 cm and z equals 94 and 188 cm. Nevertheless, there is a clear trend towards inward in the negative r direction radial propagation as m i /m e decreases. The last column of Table I shows the average slope of (r), / r, for all cases. The slope / r is taken from r 0.5 cm to the point where the amplitude profile drops to twice the background noise level. Among cases with the same value of, the decrease in / r, and the trend towards negative values, as decreases is evident. The wave current density is calculated from the curl of the experimentally measured wave magnetic field. Figure 21 a shows r j which is rj(r,,z) averaged over at an arbitrary time for case 1. There are clear vortex patterns which give a coherent picture of how the currents close. The j field is rescaled in z to take out the axial decay of the wave amplitude and multiplied by r to take out the radial dilution inherent to the cylindrical geometry. The rescaling in z is accomplished by normalizing j to its maximum amplitude at each axial location. This technique is employed to make it easier to see the circulation patterns. Figure 21 b displays j rescaled in z only in order to show the parallel current for r 2 cm. The enhanced parallel current in the region where r is less than about 1 cm and z 400 cm corresponds to the extra peak in B (r) appearinginfig.11. Figure 21 a can be compared with Figure 21 c which shows the current density calculated from theory using the same parameters that produce the fits in Fig. 11. The vectors have been rescaled in the same way as those in Fig. 21 a. There is surprising agreement in light of the poor fits to the radial profiles in Fig. 11. V. CONCLUSION From the data and calculations presented several conclusions can be made. 1 Measurements of wave phase as a function of axial position confirm that the disturbances observed in the present experiments are shear Alfvén waves, and that their axial phase velocity satisfies the finite frequency correction given by v A v A0 1 / ci 2 1/2. 2 The azimuthal component of the wave field is symmetric within experimental limitations. The most likely experimental reason for the asymmetries observed is variations in plasma conditions local to the wave source. In addition, the wave magnetic field is predominantly in the azimuthal direction. 3 A secondary peak at smaller radius characteristic of the kinetic shear Alfvén wave has been observed. In the most kinetic case studied case 1 where m i /m e 6.8) the observed wave exhibits a peak structure in the radial profile of B at small r, which theory predicts to occur. The subsidiary peaks for kinetic shear waves are predicted to only appear at lower r values than the main peak while secondary peaks for inertial waves should appear only at r values larger than that of the main peak. Therefore the extra peak appearing in case 1 inside the main peak is a distinctive mark of the kinetic wave. 4 The wave structures spread radially with increasing axial distance from the source. For four of these five cases the wave structure spreads at the rate predicted by theory. For case 1 the most kinetic it spreads faster than predicted. 5 When is increased, the peaks of the amplitude profiles move radially outward and broaden. 6 The radial propagation of the shear Alfvén wave has been observed. The radial phase velocity decreases in the transition from the kinetic to the inertial regimes where the inertial wave is a radially backward wave. All of the above statements are consistent with the single-disk theory by Morales et al. 16,17 The following statements are consistent with the double-disk superpositon model. 7 All observed radial profiles drop off faster than 1/r. The rapid radial decay indicates the reversal of axial currents as the radius increases. 8 The wave amplitude undergoes an axial decrease more rapid than the decay expected from damping. 9 The Alfvén wave current density calculated from the observed wave magnetic field explicitly displays oppositely flowing axial currents. The current density field alsoshows that the exterior axial currents return to the core axial currents via successive radial structures to complete closed current paths which lie within half a parallel wavelength. ACKNOWLEDGMENTS This work was supported by the Office of Naval Research and the National Science Foundation ATM. The authors gratefully acknowledge valuable discussions with Dr. George Morales and Dr. Steven Vincena. 1 J. Piddington, Sol. Phys. 38, J. Hollweg, M. Bird, H. Volland, P. Edenhofer, C. Stelzried, and B. Seidel, J. Geophys. Res. 87,

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