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 52242

Gurudas I. Ganguli

Plasma Physics Division, Naval Research Laboratory, Washington DC 20375

(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: 52.35 Mw, 94.20 Ss, 94.30 Tz

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 hydrogen-

cyclotron frequency (ΩΗ+) and spiky, bipolar structures with a repetition frequency just

above ΩΗ+. 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 ΩΗ+.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 dvdi/dx⊥ ≈ 1.3 ΩΟ+, where vdi is the ion flow speed along the B field, x⊥ is

the coordinate transverse to B and ΩΟ+ 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

the solitary wave development.6 Measurements using the Wideband Plasma Wave

Receiver located on the four Cluster spacecraft at 4.5−6.5 RE 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 multi-

harmonic 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

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 multi-

harmonic 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 current-

driven 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 0.2 – 0.4 T.

Typical plasma densities are ~ 1010 cm-3, with electron and ion temperatures, Te ≈ Ti ≈ 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

by this potential drop, acquiring a flow energy ~ 3 – 4 eV. A profile of the floating

potential, Vf, of a Langmuir probe (within a few Te 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’Angelo17 and observed experimentally by

D’Angelo and von Goeler.18 Fig. 3(a) shows the radial profile of ion density, ni, with the

large amplitude low frequency fluctuations, fV% , [ shown expanded in time in Fig. 3(b)] that

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) ρi, where

ρi is the ion gyroradius. Assuming that the difference in flow velocity across the shear

6

layer corresponds to twice the ion drift acquired at the hot plate, we estimate the shear

parameter at B = 0.3 T as, 1( ) ( / )ci diS dv dx−⊥= Ω ≈ (2.2×105 s-1)-1×(2 × 2000 ms-1/3×2×10-3

m) ≈ 2, where Ωci is the ion gyrofrequency and vdi is the ion drift speed.

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, cif nf>%

, into the plasma.13 The amplitude of the potential

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

the plasma. However, when the shear was on, a multi−harmonic spectrum with spectral

features just above Ωci 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, Ωci. 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

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,

0( ) sin( )n nnS t A n tω ϕ= +∑ , where An and ϕn are the amplitudes and relative phases of the

harmonics, was computed based on the spectral data of Fig. 4 with the fundamental

frequency ω0. Fig. 6(a) is the model time series using the amplitudes, An = Aexp,n, n = 1. . .

8, where Aexp,n are the actual experimentally measured amplitudes taken from the spectrum

in Fig. 4, with all the phases are set equal to zero, ϕn = 0, n = 1. . .8. 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, A1 = A1,exp, with all other harmonics decreased in

amplitude by an order of magnitude, An = 0.1 An,exp, n = 2 . . . 8, and all phases equal to

zero, ϕn = 0, n = 1. . .8. 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, An = Aexp,n, n = 1. . . 8, 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

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 multi-

harmonic spectrum is not observed, the fundamental cyclotron mode remains sinusoidal

even at very high amplitudes (a nice example28 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

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

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, 105001

(2002).

12

11. 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

25. 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

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) multi-

harmonic 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, ni, and (b) Langmuir probe floating potential

(very close to the plasma potential), Vf. 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

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 (An = An,exp, n= 1. . . 8) and zero relative

phases for all modes (ϕn = 0, n = 1 . . . 8); (b) grey curve :A1 = A1,exp, An = 0.1An, exp, n=

2. . . 8 and zero relative phase, ϕn = 0, n = 1 . . . 8; black curve: An = An,exp, n = 1. . . 8, but

random phases, 0 < ϕn < 2π, n = 1. . . 8. These model time series are to be compared with

the actual time series shown in Fig. 1(c).

FAST Orbit 1822

Laboratory Data

(c)

-0.4

-0.2

0

0.2

0.4

0 100x10-6 200x10-6

time (s)

10-4

10-3

0 125 x 103 250 x 103

Frequency (Hz)

(d)

Particle-in-Cell Simulation

Figure 1

(a) (b)

ω/Ωi

(f)

P(ω

) (ar

b.)

Ωit

(e)

φ k (a

rb.)

HP1

HP2

LP

A

D R

Figure 2

B

-0.05

0

0.05

0 0.002 0.004 0.006Time (s)

Radial Position

1 V

ni

Vf

Vf

(a)

(b)

Figure 3.

1 cm

0 100 x 103 200 x 103 300 x 1030

0.0005

0.001

Pow

er (a

rb.)

Frequency (kHz)

0.5

1.0

0

Shear ONShear OFF

0 100 200 300

Figure 4

-0.5

0

0.5

Pot

entia

l (V

)B = 0.23 T

(a)

-0.2

0

0.2

Pot

entia

l (V

)

B = 0.29 T

(b)

-0.5

0

0.5

0 1.0 2.0

Pote

ntia

l (V)

time (ms)

B = 0.34 T

(c)

20

30

40

0.2 0.25 0.3 0.35

Pea

k S

epar

atio

n (µ

s)

B (T)

(d)

2π/Ωci

Figure 5

-0.01

0

0.01

0 5x10-5 1x10-4 1.5x10-4

S(t)

time (s)

(b)

-0.02

-0.01

0

0.01

0.02

0 5x10-5 1x10-4 1.5x10-4

S(t)

time(s)

(a)

Figure 6