Experimental Measurements of the Propagation of Large Amplitude Shear Alfv n Waves

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1 Experimental Measurements of the Propagation of Large Amplitude Shear Alfv n Waves Walter Gekelman, S. Vincena, N. Palmer, P. Pribyl, D. Leneman, C. Mitchell, J. Maggs, all at the Department of Physics and Astronomy, University of California, Los Angeles Published in Plasma Physics and Controlled Fusion, 4, ppb15-b6 (000) Experiments on the edge plasma of Tokamaks have discovered magnetic fluctuations which are highly correlated along the magnetic field, and are correlated with scale size of the electron inertial length (δ = c/ω pe ) across the field. They are, in all probability, shear Alfv n waves. The FREJA, FAST, and Interball satellites have frequently encountered density striations in the auroral ionosphere. These can be narrow, also on the order of δ. Intense wave activity has been measured within these structures and tentatively identified as shear inertial (V A > V the ) Alfvén waves. These waves have been studied in great detail in the Large Plasma Device (LAPD) at UCLA. The plasma, which is 10m in length and 500 ion Larmor radii in diameter (He (λ m), Ar λ 10m), 1.5 kg, 40 cm plasma diameter, n=1-4.0x10 1 cm -3, fully ionized) supports, Alfvén waves. Our initial investigations, which will be briefly reviewed, involved low amplitude (δb wave /B ) shear waves launched by modulating a skin depth size current channel and have examined the wave characteristics in the kinetic (V A < V the ) and inertial regimes, and in magnetic field gradients. Launching higher power waves (δb wave /B ) waves with the use of a helical antenna has extended these studies. Both shear Alfvén waves ( ω < ω ci ) and compressional Alfvén waves have been investigated. Below f ci the wave fields slowly spread across the background magnetic field and the current associated with it forms a rotating spiral. The higher power wave causes a localized density perturbation when δb wave /B 0 exceeds 10-3 even when the wave propagates below the cyclotron frequency. The perturbation is measured using Langmuir probes as well as laser induced fluorescence (LIF) signal from Ar II ions. We present data of the wave propagation in which the temporal history of the vector magnetic field was acquired at 0,000 spatial locations. The data is used to calculate 3D wave currents, wave phase fronts and energy propagation. In helium the wave pattern is more complex than in argon. We also present the space and time evolution of the density perturbations associated with the wavein an Ar plasma. LIF data was used to directly measure the ion motion in the Work supported by the U.S Department of Energy, the Office of Naval Research and the National Science Foundation. 1

2 electric field of the wave, ion polarization currents and the motion of the ions as they form the density non-uniformities. Introduction Alfvén waves are the fundamental means by which information about local changes in plasma current and magnetic fields are communicated within a magnetoplasma. There are two very different modes of electromagnetic propagation at low frequencies, a compressional wave in which density and field strength vary and a shear wave in which only the direction of the magnetic field changes. The compressional wave has been used in heating schemes for thermonuclear plasmas. These waves may also play a significant role in edge plasma turbulence in fusion devices. Correlation measurements 1 of magnetic field fluctuations in Tokamaks have revealed structures with transverse dimension of order of the electron skin depth across the background magnetic field and much longer correlation lengths along it. In all likelihood these are shear waves. They are a source of concern in fusion plasmas in which energetic alpha particles may occur. These particles can destabilize toroidal Alfvén modes, which, in turn, can ill affect the particle confinement. Alfvén were observed early on in the solar wind, Coleman saw propagating MHD waves in the solar wind satellite data and interpreted them as Alfvénic turbulence, and have been studied extensively from then on 3. They are involved in the heating and energzation of solar coronal loops 4, and in the physics of extragalactic jets 5. There has been a number of interesting observations concerning shear Alfvén waves in the Earth s auroal ionosphere by the Freja and FAST satellites, both which have high temporal resolution, which means they probe small spatial scale lengths. Louran et al. 6 using Freja data for an auroral crossing region observe an Alfvén wave (with phase velocity derived from a measurement of E/B) of scale size of m or a distance of order of the electron inertial length. A deep ( δn n 070. ) density depression is simultaneously observed. The authors conjecture is that a highly nonlinear Alfvén wave is responsible for the density depletion. Electron skin depth sized inertial shear waves were observed by the FAST satellite 7. These were also observed in conjunction with deep density perturbations and regions of electron current. The observed density perturbations were much larger than theories 17,18 they used could predict. These observations lead to many intriguing questions. Is a significant component of the parallel electric field observed in the auroral ionosphere due to Alfvén waves? Do the waves cause the field aligned density

3 perturbations, or are they refracted into and interact with pre-exisitng cavities? We report a series of laboratory experiments on these issues. Shear and Compressional Alfvén waves In the MHD limit, for infinite plasma the Alfvén wave has two distinct branches, which we will call the shear mode and fast mode. The shear mode only exists below the ion cyclotron frequency and the fast mode both above and below it. The fast mode, which is sometimes called the compressional wave, since it is accompanied by a density perturbation has the same phase velocity across and along B, and in the MHD limit its dispersion relation is ω B = VA =, where ρ m is the mass k 4 πρm density. The shear wave on the other hand propagates only along the magnetic field and its dispersion is given by ω = V A. The experiments discussed involve the shear k wave for the most part. It has been long recognized that the shear wave can develop a transverse wavenumber and, therefore, a parallel electric field. If the parallel phase velocity of the wave is within the electron distribution function Landau damping is important and these waves are best treated using a kinetic formalism. The dispersion ω () 1 ε ( k ) ε = k ε c ω ( 1 I0( )) : () ε ( ) ; ( 3) ε ω ζ ζ ω λ λ = = + pe p+ e + with Z ω Ω λ relation is 8 : Here ω p+ and Ω + are the ion plasma frequency and gyrofrequency, and where ζ = ω/k a and λ + = (k a + /Ω + ). The average electron thermal speed is denoted by a = (T e /m e ) 1/ with T e the electron temperature, measured in ergs, and m e the electron mass. The ion thermal velocity a + is defined similarly with T + and M + denoting the ion temperature and mass. Z'( ζ ) is the derivative of the plasma dispersion function with respect to ζ, and I 0 is the modified Bessel function of order zero. In a plasma in which Te > Ti, in the regime where v A < a, the electrons respond adiabatically to the wave, and the dispersion may be written as + + 3

4 ω ω ( 4) v ( 1 k ),, = c /, c = (T / M ) = A ϖ + ρs ϖ = ρ s s Ω+ s e + k Here ρ s is the ion sound gyroadius, and c s the ion sound, or ion acoustic wave speed. The correct terminology for this branch of the wave is the kinetic shear wave. This wave has a parallel electric field given by kk kak (5) E = ρs ρs E = E 1 1 ϖ ( 1 ϖ )( 1 ϖ + k ρ ) / When the wave phase velocity is much large than the electron thermal velocity, the electron plasma beta, β e = ( 8πnkT B ), is much smaller than the mass ratio (the reverse is true in the kinetic case) and the dispersion relation becomes: ( 6) ω k v ( 1 ), = c A ϖ δ ( 1 k δ ) ω = + pe This wave is properly termed the inertial shear Alfvén wave and the electron inertial length, δ, appears in the dispersion relationship, rather than ρ s. In the limit ω << ω, k ci = 0, these both reduce to the MHD limit. In some MHD formulations of these waves, both δ and ρ s appear in the same dispersion relation. This is incorrect because they represent opposite limits, and the fact that distribution functions are not part of MHD. The dispersion relation alone is not enough to determine the three dimensional structure of these waves. This is governed by the radiation properties of the antenna, or disturbance that produced these waves. We have studied the propagation of shear Alfvén waves in both the kinetic and inertial regimes when the waves were launched by fluctuating current filaments with transverse dimension of order δ 9. The antenna was a small circular Cu mesh aligned with its normal along B 0, the background magnetic field, and of transverse dimension δ. The experimental setup for both these waves, and the larger amplitude ones to be discussed later is shown in figure1. The waves launched using the disk exciter, in both the kinetic and inertial case had λ >> λ and moved across the background magnetic field at a small angle 1 0. The wave magnetic field is zero on axis, rises rapidly as a function of radial distance from the exciter and then decays. In the inertial case there is a radial location beyond which all the wave current is zero and the wave magnetic field drops as 1/r. In the kinetic regime the radial dependence may include secondary maxima 10, and a propagation cone is not as clearly defined, but the magnetic field drops off outside the region of main current flow. Figure a is a schematic of the disk exciter and a conical region in which most of the parallel current is confined. A sketch of the radial dependence of the magnetic field Ω + s 1/ 4

5 averaged over several wave periods is shown on a plane at an axial distance from the exciter. Figure b shows an experimental measurement of the wave field in plane with one axis parallel to the background magnetic field. The wave is linear with B wave-max = 1 mg, B wave max 6 = Here the plasma was Helium and λ = 7m.. B 0 Figure 1. Experimental setup. A timing circuit locked to a primary clock, which controls the discharge, triggers and arbitrary waveform generator. This outputs a programmed waveform, which is amplified, goes through a transformer and then fed to a disk exciter or helical antenna. Magnetic fields are measured with a three-axis probe. The signal is amplified and sent to a bank of digitizers (up to 5 GHz /channel). Temporal records for each position are written to a computer disk. 5

6 Figure a) A schematic of the cone-like propagation of shear Alfvén waves. In the inertial case the parallel wave currents are confined within an expanding cone with axis aligned along the background magnetic field. On any transverse plane the magnetic field has a null point on axis. r b) Data of the wave magnetic field in a plane containing the background magnetic field B= B 0 k ˆ. The wave is in the inertial regime, and the disk exciter is located 94 cm, on axis, to the rear of plane. The black stripe is a D version of the B(r) line in a. The pattern is symmetric about r=0 6

7 Helical Antenna: The largest peak-to-peak current that can be drawn to a disk exciter, in theory, is the electron saturation current. When this is exceeded probe rectification sets in and the current waveform is no longer sinusoidal. To produce a large amplitude Alfvén wave with a disk exciter the following method was used. A positive pulse is applied to the exciter and electron current is drawn. The rf wavepacket used to launch the shear wave is used to modulate the D.C. current. For the disk exciters the radius was of order of half c the collisionless skin depth (δ = 5. mm ), and the current drawn ( Ip p = je sat A) ω pe places an upper bound on the wave magnetic field. What is more troublesome is that a channel of depleted density always forms whenever a large DC current is drawn to an electrode in a magnetoplasma. This occurs because electron heating and depletion of electrons within the channel causes a change in the plasma potential, which acts to drive ions out. The density drops, and because the system has inductance, the electron drift velocity goes up to maintain the current. This results in further heating, which reinforces the process until a steady state is reached 11. To avoid these difficulties, an antenna with a large (400 A pp ) circulating r.f. current was employed. The antenna was shielded from the plasma and fed by a floating transformer as shown in figure 1. The system used the antenna inductance as part of a resonant circuit, which had to be modified for use at different frequencies. The antenna consists of two helical Cu wires one half of a wavelength (45 cm) long for a shear f Alfvén wave in helium at = 076., B 0 = 1.5kG. The ends were joined with a thin f ci ring of Cu, which was 10 cm in diameter. In Argon, the ion cyclotron frequency is 1/10 that of He, the parallel wavelength is 10m, and the antenna is, for all practical purposes, straight. Since the dimensions of this exciter were much large than δ, we did not expect to see "Alfvén cone" like radiation patterns. A sinusoidal rf burst lasting 40 µsec with antenna current or 600 A p-p was applied. Data of the vector wave magnetic field was acquired on 15 planes with their normal parallel to the background magnetic field. Data at 048 time steps ( δt =400 ns ) and 105 locations, spaced 1 cm apart on a rectangular grid, was acquired on each plane. An example is shown in figure 3. Unlike the patterns from a disk exciter, there are clearly two current channels roughly 4 cm in diameter and 1.6 cm apart. The wave fills almost the entire plasma column in a plane orthogonal to B 0. The largest wave magnetic in the volume is was 1.6 Gauss, which corresponds to δb wave B0 3 10, where 7

8 me = The pattern is intriguing because, when viewed in time, it appears MAr to be a rotating spiral. The wave vectors in the center are due to the constructive interference of the fields from the two current channels. This reminiscent of interference patterns previously observed in a different tsituation 1 which involved the intereference of two Alfvén waves. Fig 3. Data on a plane.64 meters from the end of the antenna. The magnetic vector field is color coded in accord with the field strength. Data was acquired at each location an arrow is drawn. The background magnetic field is directed into the page. The enhanced magnetic field between the current channels does not appreciably change orientation as the wave goes by, but at every half wavelength, when the field become small, the currents rotate in the electron diamagnetic direction (clockwise in this view). The spatial morphology of the wave is shown in the next figure 4. The largest field, in red, occurs at the wave maxima. The spiral pattern is clearly visible. The wave currents flow where the field is small. This is clearly seen at dz=3.96. Here the currents flow in the two elliptical blue patches next to the field maxima at approximately :00 and 7:00. The pattern in Helium is much more complex and consists of a higher m mode 8

9 involving up to five current channels. These patterns will be discussed in far more detail in a future publication. r Fig 4. The transverse scalar magnetic field B = Bx + By of a shear Alfvén wave. Six of the 15 planes on which data was acquired are shown here. The spatial extent (dx=41cm, dz=5 cm) is the same as that in the previous figure. The data is acquired at t=173 µsec (8.5 wave periods) after the turn on of the rf. In Ar the currents associated with this wave are not very complicated. They are shown in the next figure 5. The wave current is calculated from the volume vector data set using r 1 r j= B. At the low frequency of this wave the displacement current is negligible. 4π Current flowing towards the observer is shown in red, that flowing away in blue. The field-aligned current is carried by the electrons and cross-field current by the ions via the polarization drift. The long stretches of field aligned current reflect the10-m parallel v wavelength of this wave. This wave is in the kinetic regime with the. Since the VA cyclotron frequency in Ar is it is low ( fci = 57 khz, the damping is relatively large, ν γ = collision 0, which is reflected in observed wave amplitudes which drop by a factor ν wave of two, 6.3 meters from the antenna. Nevertheless the antenna produces what is considered a large amplitude shear wave. For example, the wave magnetic field in these 9

10 experiments is in 1- orders of magnitude larger than the linear waves studied in the disk exciter experiments. 10

11 Fig 5. Currents of a shear Alfvén wave launched by a helical antenna. The wave magnetic field on a plane orthogonal to the background magnetic field is shown to guide the eye. These waves are associated with spatially localized density perturbations. The process, which leads to the perturbation, is not yet fully understood; the data is presented and then is discussed in a subsequent section. Figure 6 shows the density perturbation δn, on three n planes as well as the vector magnetic field and its magnitude. The magnetic field was measured on a 41cm x 5 cm grid (105 locations). At each position, each component of the vector field was averaged over 10 shots. The magnetic field amplitude is largest in the center and the regions where the wave current channels are clearly visible. The density perturbation, which is localized to the central region, was measured on a 5 X 41 position grid with.5 mm spacing. The transparent plane is a surface of background density, when the wave is not present. The density displayed here was derived from the ion saturation current drawn to a 1mm ( V bias = -66 V, kt e = 6.5 e.v.) Langmuir probe. In a separate experiment the Langmuir current voltage curve was rapidly (10 µsec), swept at several times and it was verified that there was no significant change in the electron temperature in the region and that the ion saturation current was a measure of the density. In this picture the ion data was smoothed over a half wave period to eliminate high frequency noise. In a temporal sequence of this data one sees the density on the closest plane (dz = 0.66m) is affected first, and then see disturbance propagate along the device. One can also see the density on the closest plane oscillate at the wave frequency with an amplitude of order 5-10% as well. As the sinusoidal oscillations progress the DC density perturbation grows. 11

12 Figure 6. Plasma density perturbation in three planes at a time t = 4 µsec after the antenna is turned on. The largest density perturbations are 10%. The vector magnetic field, along with its magnitude displayed on the bottom are shown for dz=3.3 meter. The grid spacing in the x and y directions is 1 cm. The axial propagation of the density perturbation can be estimated by measuring the time difference between the onset of the density perturbation on the first plane at dz=0.66m and the furthest, dz=3.3m. This time difference is 70 µsec and the propagation speed of order 104 cm/sec, or 5% of the ion acoustic speed. As the density perturbation is spatially non-uniform and is not invariant along z this is speed is a rough estimate. Examination of the temporal waveforms at different axial and radial locations show different parts of the density perturbation seem to arrive at different speeds, and in some cases these may exceed the ion sound speed. After the tone burst is over the density perturbation gradually goes away and peaks and valleys of the perturbation wander around the transverse planes ( at v < cs ) as they decay. The ion motion has been directly measured using Laser Induced Fluorescence (LIF) with a tunable dye laser using the ArII spectral line at nm13. The laser light was introduced to the plasma with an optical fiber, which illuminated a volume of 1 mm3. The observed signal, from the decay of a metastable state (3dG9/), is observed at nm by a second fiber capped with a grin lens and positioned at aright angle to the first. If one assumes that the metastable density is directly proportional to the ion density this measurement yields the ion 1

13 distribution function. We record the LIF signal as a function of time (t = 819 µs, dτ = 1.6 µs) at 0 wavenumbers spanning the ion line. 13

14 Figure 7 Laser Induced Fluorescence measurements in the center of the density perturbation and 5 cm to the side. Shown is the ion distribution function as well as the frequency of oscillation of f i (v) The light was originally digitized at 00 ns/sample and then smoothed with a running average over 8 samples. The signal amplifier, however has a 100 khz bandwidth which gives an effective time resolution of 10 µs. The data contains information on the ion temperature as well motion of the ion distribution function in velocity space, and along the probe laser beam in real space. Figure 7a shows the density perturbation on a 4cm X 4cm plane 3.3 m from the exciter (this was a different data run from the one shown in figure 6). The magnetic field pattern on this data run was measured as well, and found to be the same as that in figure 6. The laser intersected the plane at (x,y)=(0,0), the center of the plane and at (x,y)=(5,0), a location 1 cm to the right of the edge. Figure 7b shows the ion distribution function at an instant of time after the perturbation has developed. While the wave is present the ion distribution function is observed to rock back and forth in velocity space about v=0, and the amplitude of these oscillations is used to determine the AC drift velocity; in this case vdrift = 018., v thi is the ion thermal speed for 1.14 ev. ar.gon. A time series of the motion of a point vthi near the peak is Fourier analyzed and the frequency spectrum is displayed in fig. 7b. The frequency spectrum has a sharp peak at 49 khz that of the Alfvén wave. Further away from the center of the wave pattern, fig 7b, we observe slighter hotter ions with a smaller AC drift velocity. Also present at x=5 cm is a DC component of the ion drift. This corresponds to motion of ions involved in the formation of the density perturbation moving at vion = 01. vthi = 10 4 cm/s =.05 c s. The LIF data may be used estimate the perpendicular electric field of the wave. There are two components of the E ion drift the EXB drift, v = de EXB and the ion polarization drift, vp = 1. The latter is B0 ωcb dt responsible for the closure of the Alfvén wave currents across the magnetic field as shown in figure 5. The two are out of phase and if magnetic field measurements are made at the point where LIF data is taken they can be distinguished from one another. This will be done in future experiments in which we will do LIF in an entire plane. Since f=0.86f ci in this experiment these drift velocities will be comparable in magnitude. For the drift velocity in 7b, E = 045. V/cm. One may use equation 5 to predict the parallel electric field of these waves, E = 11 mv/cm. Finally a double-sided Langmuir probe with two faces, each 1 mm, was placed in the wave current with the normal to its face along the machine axis and parallel/antiparallel to B 0. If the probes are biased in the electron saturation regime the difference in current to them is the net field aligned current. The current and voltage to this probe is shown in figure 8 as it is swept from ion saturation to electron saturation current over five cycles of the wave. The parallel current measured at this location was j = 0. 5 A/cm p-p. The current measured with this method is spatially non-uniform; it is close to zero in the central region between the two current channels (figure 5) and as large as 0.7 A/cm, which corresponds to A-C density perturbations of order 5%. 14

15 Figure 8. I-V characteristic curve of the difference current to a small double-sided probe place in the Alfvén wave current. Th voltage to each face is swept from -45V to +80 V over a 00 µs interval. The wave period is 0 µsec. Discussion of results Shear Alfvén waves come in different flavors depending upon the method of excitation. Low amplitude shear waves excited by electron skin depth scale oscillating currents form cone-like structures, which slowly spread across the magnetic field. They produce no significant density perturbation and the wave magnetic field is transverse unless f f 14 ci, The magnetic field of these waves is localized to the center of the plasma column. The wave produced by the helical antenna fills the entire plasma column in the transverse direction and although it obeys the same dispersion relation its spatial pattern and associated current system are very different. The wave magnetic field was 15

16 measured at both high ( b wave B ) and low b B wave amplitudes and the wave field looks the same (figure 3,4) in both cases. At the higher amplitude a density perturbation as high as δn n 15% developed in the region of the largest wave magnetic field. There is no observable DC current or electron heating associated with this wave. Because of the importance of Alfvén waves in the auroral ionosphere and their observation in, presumably, field aligned density cavities, several calculations of ponderomotive force effects have been recently published 15,16,17,18,19. These calculations, most of which are for the inertial shear wave, predict incredibly small perturbations ( δn n 10 6 ) for our plasma parameters. Shukla and Stenflo 0 have calculated the ponderomotive force of a kinetic shear Alfvén wave and found that the parallel electron ponderomotive force drives density depressions and the parallel ion ponderomotive force produces density enhancements, both of which are observed in this experiment (figure 6). This theory, which neglects the compressional branch of the wave, predicts a one- percent perturbation for our experimental parameters. We have observed with probes and LIF, that close (dz < 3.3m < λ) to the exciter, a significant AC density oscillation exists while the wave is present. At this frequency the compressional wave is cut off since the column diameter is much less than the wavelength. This does not mean that the compressional wave does not exist, it is evanescent. The imaginary part of the wavenumber should scale as the reciprocal of the plasma column diameter (40 cm) and is easily observable at 66 cm from the antenna, the lowest plane in figure 6. To test for effects arising from the compressional wave we have extended these measurements to above the ion cyclotron frequency and observed density perturbations, which seem to get smaller at increasing frequency. The wave magnetic field pattern is dramatically different above the ion cyclotron frequency. We have also applied a current pulse to the helical antenna and observed density perturbations. A pulse contains a spectrum of frequencies and can launch field line resonances 1 as well as evanescent compressional waves. These measurements will be reported in a forthcoming publication, but we take them as evidence that the compressional wave plays a role in the production of the density perturbation. Finally Hollweg has done a fluid calculation for the AC density perturbations due to the kinetic Alfvén wave using a dispersion relation which mixes the electron inertial length and the ion sound gyroradius, but considers compressional effects. It predicts an AC perturbation of order 14%, which is in accord with our observation. The mechanism for formation of a DC density perturbation, and any feedback mechanism between the large AC density fluctuations and the ponderomotive force, must be determined with future experiments and theory. 16

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