GENERATION OF SPIKY POTENTIAL STRUCTURES ASSOCIATED WITH MULTI-HARMONIC ELECTROSTATIC ION CYCLOTRON WAVES. Su-Hyun Kim and Robert L.

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1 1 GENERATION OF SPIKY POTENTIAL STRUCTURES ASSOCIATED WITH MULTI-HARMONIC ELECTROSTATIC ION CYCLOTRON WAVES Su-Hyun Kim and Robert L. Merlino Department of Physics and Astronomy, University of Iowa, Iowa City, IA Gurudas I. Ganguli Plasma Physics Division, Naval Research Laboratory, Washington DC (Submitted to Physics of Plasmas, August 10, 2005) ABSTRACT The production of coherent, spiky electrostatic potential and electric field structures, similar to those that have been observed in the earth s auroral region, is reported. These structures are associated with coherent multi harmonic electrostatic ion cyclotron (EIC) waves in a current free plasma. A multi harmonic EIC spectrum is produced when broadband electrostatic noise, launched into the Q machine plasma from an antenna, propagates through a spatially localized region of parallel (to B) ion flow with a gradient in the direction transverse to B. The spiky potential waveforms result from a linear combination of coherent multi harmonic EIC waves, when the harmonics have comparable amplitudes and are phase locked. PACS Numbers: Mw, Ss, Tz

2 2 I. INTRODUCTION A ubiquitous feature of electric fields observed on satellites in the earth s auroral region is their spiky, repetitive nature. These spiky electric field structures appear as either unipolar or bipolar pulses in high resolution time domain waveforms of the potential difference between pairs of spheres deployed from the spacecraft. Time domain waveforms of three different hydrogen-cyclotron wave events observed with the S3-3 satellite showed examples of both narrow spectral features at a frequency just above the local hydrogencyclotron frequency (Ω + Η ) and spiky, bipolar structures with a repetition frequency just above Ω + Η. The latter were interpreted as steepened ion cyclotron waves. 1 Measurements on the Polar Satellite, which traverses the southern auroral region at altitudes of about 6000 km, showed bipolar structures in the parallel electric field in conjunction with spikes in the perpendicular electric field that occurred with an average repetition rate of 1.2 Ω + Η. 2 Data obtained from the FAST Satellite in the upward current northern auroral region showed a multi-harmonic EIC spectrum with corresponding spiky structures in both the perpendicular and parallel electric field waveforms. 3,4 An example of a spiky electric field and multi-harmonic EIC spectrum obtained by FAST is shown in Fig. 1 (a) and (b). Spiky bipolar structures in the parallel electric field signals were associated with regions of inhomogeneous intense upward ion flows with a spatial dependence consistent with a transverse shear dv di /dx 1.3 Ω + Ο, where v di is the ion flow speed along the B field, x is the coordinate transverse to B and Ω + Ο is the local oxygen gyrofrequency. 4 The wave experiment on the Swedish Viking satellite frequently detected solitary bipolar structures in the potential difference measurements on probes separated along the magnetic field. 5 It was noted that the frequently observed large amplitude EIC waves may form the seeds for

3 3 the solitary wave development. 6 Measurements using the Wideband Plasma Wave Receiver located on the four Cluster spacecraft at R E showed both bipolar and tripolar electric field structures at they crossed magnetic field lines that map into the auoral zone. 7 Gavrishchaka et al. 4 and Ganguli et al. 8 have shown theoretically that parallel ion flows with transverse shear can generate a multimode spectrum of EIC waves even in the absence of an electron drift (field aligned current). Unlike current-driven EIC waves in which the critical electron drift required to excite higher harmonics increases with harmonic number, the critical ion flow shear is approximately independent of the harmonic number. Thus a number of higher harmonics can be simultaneously excited. The plasma equilibrium considered in these studies corresponded to that encountered in the ionosphere where the entire ion population was found to be drifting along the magnetic field. Lakhina showed that a multi-harmonic ion-cyclotron instability can also be driven by velocity shear of a hot ion beam embedded in a thermal ion background. 9 The presence of a multiharmonic spectrum is a critical factor in understanding the origin of coherent electric field structures, since as Ganguli et al. 8 have argued, a linear superposition of spontaneously generated multimode EIC waves can be the seed that leads directly to the formation coherent electric field structures. If the linear combinations last long enough for the phases to get locked due to nonlinear processes, they can develop into coherent structures. The nonlinear properties of the shear-driven EIC waves were studied using a particle-in-cell code. 4 A representative time series and power spectrum showing up to five ion cyclotron harmonics is shown in Fig. 1 (e) and (f). An understanding of how these coherent electric

4 4 field structures are generated is central to the question of how electrons that produce the visible aurora get accelerated parallel to the geomagnetic field. This paper describes the results of an experiment in which coherent electrostatic potential structures associated with multi-harmonic EIC waves were produced. A multiharmonic EIC spectrum (with several harmonics all having amplitudes within 10% of the amplitude of the fundamental) was produced when a broadband white noise signal was applied to an antenna that launched electrostatic waves into a plasma containing parallel ion flow with transverse shear. An example of a multi-harmonic spectrum and corresponding spiky potential waveform is shown in Fig. 1 (c) and (d). The effects of ion flow shear on the excitation of EIC waves has been previously reported by Teodorescu et al., 10 Agrimson et al., 11,12 and Kim et al. 13 An example of a time series of an ion flow shear modified EIC wave that is less sinusoidal than an currentdriven EIC wave in a homogeneous plasma was shown by Koepke et al. 14 II. EXPERIMENTAL SETUP The experiment was performed in a double ended Q machine, 15 shown schematically in Fig. 2. A Cs + plasma was formed by contact ionization of cesium atoms on two 6 cm diameter tantalum hot plates maintained at ~ 2000 K which also emit thermionic electrons. The plasma is confined radially by a uniform magnetic field in the range of T. Typical plasma densities are ~ cm -3, with electron and ion temperatures, T e T i 0.2 ev. Both electrons and ions are magnetized and the plasma is collisionless. The hot plate sources are operated under electron rich conditions in which a potential drop of ~ 3-4 V is present in a sheath at each grounded hot plate. The ions are accelerated into the plasma

5 5 by this potential drop, acquiring a flow energy ~ 3 4 ev. A profile of the floating potential, V f, of a Langmuir probe (within a few T e of the plasma potential) which was scanned across the plasma column is shown in the lower plot of Fig. 3(a). Note that over the central portion of the plasma, where the experimental measurements were carried out, there is negligible radial (transverse to B) electric field. To produce a plasma having parallel ion flow with transverse shear, a metal ring of 8 cm outer diameter and 2.3 cm inner diameter was placed at one plasma cross section, and a metal disk of 2.2 cm diameter was placed at another cross section, as shown in Fig. 2. The ring and disk were separated axially by 88 cm. The ring and disk were both biased at ~ 4 V to collect all ions flowing to them, so that between the ring and disk, the central core contains only plasma flowing from HP2 and the outer portion only contains plasma flowing from HP1. The boundary between the inner and outer plasma is then a region of strong velocity shear. The flow profile was measured previously using a double-sided Langmuir probe. 16 When the bias on the ring and disk were raised to about 0.5 V, the ions were reflected from the ring and disk, resulting in no net flow or shear. The presence of velocity shear was also verified by observing the very low frequency (~ 1 khz) fluctuations due to the parallel velocity shear instability (also known as the D Angelo instability), discussed theoretically by D Angelo 17 and observed experimentally by D Angelo and von Goeler. 18 Fig. 3(a) shows the radial profile of ion density, n i, with the large amplitude low frequency fluctuations, V %, [ shown expanded in time in Fig. 3(b)] that f mark the locations of velocity shear. The radial extent of the low frequency oscillations provides a measure of the width of the shear region as x 7 8 mm (3-4) ρ i, where ρ i is the ion gyroradius. Assuming that the difference in flow velocity across the shear

6 6 layer corresponds to twice the ion drift acquired at the hot plate, we estimate the shear 1 parameter at B = 0.3 T as, S = ( Ω ) ( dv / dx ) ( s -1 ) -1 ( ms -1 / ci m) 2, where Ω ci is the ion gyrofrequency and v di is the ion drift speed. di III. EXPERIMENTAL RESULTS. A strip antenna, (see Fig. 2) 5 cm long and 1 cm wide, oriented with its normal perpendicular to the magnetic field was used to launch electrostatic waves in the ion cyclotron frequency range, f > nf, into the plasma. 13 The amplitude of the potential % ci fluctuations of the EIC wave was measured at several radial positions in the plasma cross section coincident with the center of the antenna. A factor of ~2 increase in the wave amplitude was observed in the regions of velocity shear as compared to the case in which no velocity shear was present. 13 This result was obtained using wave frequencies corresponding to the fundamental EIC mode and several harmonics. The increase in amplitude of the EIC waves in the region of velocity shear was interpreted in terms of the theory of Ganguli et al., 8 who showed that EIC waves can grow, even in the absence of parallel electron drift, by ion flow with transverse shear through inverse ion-cyclotron damping, as verified experimentally by Teodorescu et al. 10 and Kim et al. 13 A multi-harmonic spectrum of EIC waves was produced by applying a broadband white noise signal (random noise extending up to about 1 MHz) to the antenna. A probe was located in the region of velocity shear to record the potential fluctuations. Fig. 4 shows the power spectra of the potential fluctuations for the cases in which the transverse velocity shear was ON or OFF. When there is no shear in the plasma, a relatively flat spectrum was observed reflecting the broadband noise applied to the antenna and the background noise in

7 7 the plasma. However, when the shear was on, a multi harmonic spectrum with spectral features just above Ω ci and 7 harmonics was produced. Thus, in the presence of shear, many EIC wave modes are excited, as observed earlier, 13 and predicted by Ganguli et al. 8 Note that the amplitudes of the higher harmonic EIC waves are comparable to that of the fundamental. The time series of the potential fluctuations corresponding to the spectrum in Fig. 4 is given in Fig. 1(b). Examples of the time series of the EIC potential oscillations for three values of the magnetic field are shown in Fig. 5(a c). The time series show spiky, bipolar structures, with repetition rates just above the fundamental cyclotron frequency, Ω ci. The separation in time between the spikes was measured for many waveforms of the type shown in Fig. 5(a c) for several values of the magnetic field. The result is given in Fig. 5(d), which shows clearly that the time between the spikes is determined by the period of the fundamental EIC mode. The waveforms shown in Fig. 5(a c) are remarkably similar to electric field waveforms observed on the FAST satellite [see Fig. 1(a)]. IV. DISCUSSION As pointed out by Temerin et al., 1 harmonics can be generated in the linear Vlasov theory of EIC waves. This can occur as a result of the current driven instability of Drummond and Rosenbluth, 19 but usually this requires a very large electron drift (which are not typically observed) since the critical drift velocity increases with harmonic number. This mechanism is not operative in our experiment since there is no electron current. The ion flow gradient instability of Ganguli et al., 8 provides more easily for the generation of multi harmonic EIC waves, and the results of our experiment clearly link the observation

8 8 of multi harmonic EIC waves with ion flow shear. A multi harmonic EIC spectrum in itself however would not produce coherent spiky potential or electric field structures. As Temerin et al., 1 and Ganguli et al., 8 point out, linear combinations of the harmonics must persist long enough for the phases to get locked due to nonlinear processes and develop into spiky coherent structures. Fig. 6 provides three illustrative time series showing how the spiky waveforms can result from linear combinations of multi harmonic EIC waves. A model time series, St () = Asin( nω t+ ϕ ), where A n and ϕ n are the amplitudes and relative phases of the n n 0 n harmonics, was computed based on the spectral data of Fig. 4 with the fundamental frequency ω 0. Fig. 6(a) is the model time series using the amplitudes, A n = A exp,n, n = , where A exp,n are the actual experimentally measured amplitudes taken from the spectrum in Fig. 4, with all the phases are set equal to zero, ϕ n = 0, n = Fig. 6(b) shows two model time series provided to illustrate the effect of the amplitudes and phases of the harmonics on the structure of the time series. The grey curve uses the experimentally measured amplitude of the fundamental, A 1 = A 1,exp, with all other harmonics decreased in amplitude by an order of magnitude, A n = 0.1 A n,exp, n = , and all phases equal to zero, ϕ n = 0, n = The result is, as expected, very nearly sinusoidal since the fundamental is the dominant mode. For the black curve in Fig. 6(b), the experimental amplitudes were used, A n = A exp,n, n = , but each mode was assigned a random phase between 0 and 2π. These examples serve to illustrate that the shape of the time series depends crucially on both the relative phase and amplitude of the harmonics, although the spacing in time between the spiky features is always determined by the dominant fundamental frequency. The model time series that most closely resembles the actual time

9 9 series of Fig. 1(c) is the one shown in Fig 6(a), in which the modes are phase coherent and the harmonics are of comparable amplitude to the fundamental. The generation of spiky structures in both the electrostatic potential and electric field was also observed in the numerical particle-in-cell code of Gavrishchaka et al. 4 and Ganguli et al., 8 and arise when several coherent EIC waves simultaneously grow and saturate in amplitude. V. CONCLUSIONS We have demonstrated experimentally that coherent, spiky electric potential structures can be generated by a linear combination of a multi harmonic spectrum of electrostatic ion cyclotron waves. There have been many theoretical attempts to model these structures in terms of nonlinear waves (see, e.g., refs. 1, 20 27). These approaches generally try to describe the evolution of a single linear EIC wave as it grows nonlinearly into a finite amplitude wave. The solutions that are obtained often do resemble the observed waveforms, but this only implies that a nonlinear state is possible. This approach does not clarify the chain of physical events that leads to the formation of these nonlinear structures. In addition, they cannot explain, why in some laboratory experiments in which a multiharmonic spectrum is not observed, the fundamental cyclotron mode remains sinusoidal even at very high amplitudes (a nice example 28 is given in Fig. 10 of ref. 14). The present approach, first argued on theoretical grounds by Ganguli et al., 8 takes as its starting point the generation of a multi harmonic spectrum of EIC waves, and proceeds to show the formation of spiky potential structures self-consistently, thereby elucidating the causal relationship between the physical processes that leads to their formation. The identification of the nonlinear processes that act to ensure the necessary phase locking of the modes is

10 10 beyond the scope of this work, but is nonetheless a remaining important aspect of the problem that needs to be explored further. The present work not only serves to emphasize the key role of velocity shear in this process, but also points to the possibility that EIC waves may be generated when broadband noise produced in one plasma region (e.g. the magnetosphere), propagates into another plasma region (e.g., the ionosphere) where inhomogeneous ion flows are present. ACKNOWLEDGEMENTS This work at the University of Iowa was supported by the National Science Foundation and The U. S. Department of Energy. The work at the Naval Research Lab was supported by the Office of Naval Research. We thank M. Miller for technical support in carrying out the experiments and N. D Angelo and F. Skiff for useful discussions.

11 11 REFERENCES 1. M. Temerin, M. Woldorff, F. S. Mozer, Phys. Rev. Lett. 43, 1941 (1979). 2. F. S. Mozer, R. E. Ergun, M. Temerin, C. A. Cattell, J. Dombeck, and J. Wygant, Phys. Rev. Lett. 79, 1281 (1997). 3. R. E. Ergun, C. W. Carlson, J. P. McFadden, F. S. Mozer, G. T. Delory, W. Peria, C. C. Chaston, M. Temerin, R. Elphic, R. Strangeway, R. Pfaff, C. A. Cattell, D. Klumpar, E. Shelley, W. Peterson, E, Moebius, and L. Kistler, Geophys. Res. Lett. 25, 2025 ( 1998). 4. V. V. Gavrishchaka, G. I. Ganguli, W. A. Scales, S. P. Slinker, C. C. Chaston, J. P. McFadden, R. E. Ergun, and C. W. Carlson, Phys. Rev. Lett. 85, 4285 (2000). 5. R. Boström, G. Gustaffson, B. Holback, G. Holmgren, H. Koskinen, and P. Kintner, Phys. Rev. Lett. 61, 82 (1988). 6. H. E. J. Koskinen, P. M. Kintner, G. Holmgren, B. Holback, G. Gustafsson, M. Andre, R. Lundin, Geophys. Res. Lett. 14, 459 (1987). 7. J. S. Pickett, S. W. Kahler, L. J. Chen, R. L. Huff, O. Santolík, Y. Khotyaintsev, P. M. E. Décréau, D. Winningham, R. Frahm, M. L. Goldstein, G. S. Lakhina, B. T. Tsurutani, B. Lavraud, D. A. Gurnett, M. André, A. Fazakerley, A. Balogh, and H. Rème, Nonlin. Proc. Geophys. 11, 183 (2004). 8. G. Ganguli, S. Slinker, V. Gavrishchaka, and W. Scales, Phys. Plasmas 9, 2321 (2002). 9. G. S. Lakhina, J. Geophys. Res. 92, 12,161, (1987). 10. C. Teodorescu, E.W. Reynolds, and M. E. Koepke, Phys. Rev. Lett. 89, (2002).

12 E. P. Agrimson, N. D Angelo and R. L. Merlino, Phys. Lett. A 293, 260 (2002). 12. E. Agrimson, S.-H. Kim, N. D Angelo, and R. L. Merlino, Phys. Plasmas 10, 3850, (2003). 13. S.-H. Kim, E. Agrimson, M. J. Miller, N. D Angelo, and R. L. Merlino, Phys. Plasmas 11, 4501, (2004). 14. M. E. Koepke, C. Teodorescu, E. W. Reynolds, C. C. Chaston, C. W. Carlson, and J. P. McFadden, and R. E. Ergun, Phys. Plasmas 10, 1605 (2003). 15. R. W. Motley, Q Machines (Academic Press, New York, 1975). 16. J. Willig, R. L. Merlino, and N. D Angelo, J. Geophys. Res. 102, 27,249 (1997). 17. N. D Angelo, Phys. Fluids 8, 1748 (1965). 18. N. D Angelo, and S. von Goeler, Phys. Fluids 9, 309 (1966). 19. W. E. Drummond and M. N. Rosenbluth, Phys. Fluids 5, 1507 (1962). 20. P. K. Chaturvedi, Phys. Fluids 19, 1064 (1976). 21. P. K. Shukla and S. G. Tagare, Phys. Rev. A, 30, 2118 (1984). 22. H. L. Rowland and P. J. Palmadesso, J. Geophys. Res. 92, 299 (1987) 23. P. K. Shukla and L. Stenflo, Ann. Geophysicae 16, 889 (1998). 24. D. Jovanovic and P. K. Shukla, Phys. Rev, Lett. 84, 4373 (2000).

13 R. V. Reddy, G. S. Lakhina, S. V. Singh, and R. Bharuthram, Nonlin. Proc. Geophys. 9, 25 (2002). 26. J. F. McKenzie, J. Plasma Physics 70, 533 (2004). 27. R. V. Reddy, S. V. Singh, G. S. Lakhina, and R. Bharuthram, Proc. ISSS-7, 26 (2005). 28. The example given in Fig. 10 of ref. 14 shows a current-driven, sinusoidal EIC waveform corresponding to the case of a homogeneous plasma (no transverse parallel flow shear). The amplitude of this wave was, δn/n ~ 15%, well above the linear state. In fact, the first report of the amplitude of an EIC wave by Motley and D Angelo, Phys. Fluids 6, 296 (1965), showed a sinusoidal wave with δn/n ~ 50%.

14 14 FIGURE CAPTIONS Figure 1. Example time series and EIC wave spectra obtained by the FAST satellite, in the laboratory, and in PIC simulations. (a) Time series of the parallel electric field and (b) multi-harmonic electrostatic hydrogen cyclotron wave spectrum obtained with the FAST satellite (adapted from ref. 4). (c) Time series of the electrostatic potential and (d) multiharmonic EIC spectrum obtained in the present laboratory experiment. The vertical dashed lines in (b) correspond to the hydrogen cyclotron frequency, while those in (d) correspond to the cyclotron frequency for singly ionized cesium ions. PIC simulation code results: (e) time series of the spatial Fourier components, and (f) power spectrum for the H + cyclotron modes (adapted from ref. 4). Figure 2. Schematic diagram of the double-ended Q machine. Cesium (Cs + ) plasmas are formed on the 2 hot plates (HP1, HP2). The ring and disk (D) electrodes used to collect ions from each source and create an annular region of parallel ion flow with transverse shear. A strip antenna (A) is used to launch electrostatic waves into the plasma. Plasma parameters are monitored with a Langmuir probe (LP). Figure 3. Radial profiles of (a) ion density, n i, and (b) Langmuir probe floating potential (very close to the plasma potential), V f. The baseline for both plots is at the top of the plot. (b) Time series of the low frequency (1 khz) oscillations due to the D Angelo instability, seen on the ion density profile in the region of velocity shear.

15 15 Figure 4. Power spectra (linear plot) of the potential oscillations of a probe located in the region of velocity shear. Black plot: spectrum obtained when velocity shear was ON; Grey plot: spectra taken when the velocity shear was OFF. The vertical lines are multiples of the cyclotron frequency. Figure 5. Spiky potential waveforms observed for (a) B = 0.23 T, (b) B = 0.29 T, and (c) B = 0.34 T. (d) Measurement of the separations between the spikes for all magnetic fields investigated. Figure 6. Model time series formed by linear superposition of EIC harmonic waves, using: (a) the experimental amplitudes of Fig. 4 (A n = A n,exp, n= ) and zero relative phases for all modes (ϕ n = 0, n = ); (b) grey curve :A 1 = A 1,exp, A n = 0.1A n, exp, n= and zero relative phase, ϕ n = 0, n = ; black curve: A n = A n,exp, n = , but random phases, 0 < ϕ n < 2π, n = These model time series are to be compared with the actual time series shown in Fig. 1(c).

16 FAST Orbit 1822 (a) (b) Laboratory Data (c) (d) x x x x 10 3 time (s) Frequency (Hz) Particle-in-Cell Simulation φk (arb.) (e) P(ω) (arb.) (f) Ω i t ω/ω i Figure 1

17 B A HP1 HP2 D R LP Figure 2

18 V f n i 1 cm 1 V (a) Radial Position 0.05 V f 0 (b) Time (s) Figure 3.

19 Power (arb.) Shear ON Shear OFF x x x 10 3 Frequency (khz) Figure 4

20 Potential (V) (a) B = 0.23 T Potential (V) (b) B = 0.29 T Potential (V) Peak Separation (µs) B = 0.34 T time (ms) (c) (d) 2π/Ω ci B (T) Figure 5

21 0.02 (a) 0.01 S(t) x10-5 1x x10-4 time(s) (b) S(t) x10-5 1x x10-4 time (s) Figure 6