Electrostatic Electron Cyclotron Waves Generated by Low Energy Electron Beams

J. D. Menietti, O. Santolik, J. D. Scudder, J. S. Pickett, D. A. Gurnett

Abstract. We report the results of an investigation of waves observed by the Polar spacecraft at high altitudes and latitudes and at frequencies just above the cyclotron frequency. These observations are made frequently when the spacecraft is over the polar cap as well as near the dayside cusp and the nightside auroral region, and for ratios of gyrofrequency to plasma frequency, f_p/f_ce ~ 1. We investigate the role of electron beams with E 1 keV in the generation of these waves. Observed plasma parameters are used as input to a modification of the WHAMP computer code to place constraints on the free energy source and growth of these waves.

I. Introduction

Electrostatic electron cyclotron waves (EEC) have been studied for many years [cf. Mosier et al., 1973; Kurth et al., 1979a,b]. The observations have typically been obtained during crossings of the plasmasheet near the magnetic equator, where the ratio of plasma frequency, f_p, to gyrofrequency, f_g, is typically much larger than 1, and where electron pitch angles are large. Intense noise bands are frequently observed in the plasmasphere at frequencies between f_p and the upper hybrid frequency, [Mosier et al., 1973]. Original numerical studies have shown that highest growth rates occur for f_p/f_g > 1 [cf. Tataronis and Crawford, 1970]. Bernstein mode waves [Bernstein, 1958] can grow, however, for both large and small ratios f_p/f_g (cf. Krall and Trivelpiece, 1973], and the ratio f_p/f_g determines the precise location of the emission fundamental and harmonic frequencies. Young et al. [1973] have shown that a dense, warm electron distribution of electrons with df/dv > 0 can excite waves with f_g < f < 1.5 f_g and, for k 0, a temperature anisotropy T > T can excite waves between f_g and 2f_g. More recent investigations of electron cyclotron waves generated by beams and temperature anisotropies have been conducted for f > f_g [Winglee et al., 1992] and for f < f_g [Wong and Goldstein, 1994].

Gurnett et al. [1983] have reported "3/2 f_g" emission in the cusp in the DE-1 plasma wave data at a radial distance of about 3.5 R_E, at a time when f_p/f_g 1. Farrell et al. [1990] subsequently studied a number of examples of wave intensifications near the cusp for f f_g. They found that the waves most often occurred at times of large magnetic index, Kp. For specific cases, these authors found the waves to be oblique with wave normal angles varying from ~ 10 to ~ 60. The electric field intensity of these waves was typically of the order of a few to tens of µV/m, much smaller than the large amplitude (n+1/2) f_g oscillations observed by Kurth et al. [1979a,b] inside the plasmasphere.

Menietti et al. [2000] reported observations of EEC waves made by the Polar spacecraft as it traversed the northern polar magnetosphere at altitudes from approximately 7 R_E to 9 R_E. At these altitudes f_p ~ f_g and the EEC waves were observed at a relatively narrow-banded emission at frequencies just above f_g. The apogee of the Polar spacecraft over the northern polar cap during 1996 and 1997, was approximately 9 R_E, and the spacecraft often entered a region where fp fg, because the gyrofrequency falls off more rapidly with radial distance than the plasma frequency. The waves are observed near the cusp, polar cap, and both dayside and nightside auroral regions. The wave intensities were frequently on the order of 0.1 mV/m, and the measured wave normal angles varied, but could be quite large (> 60).

In this paper we investigate electrostatic electron cyclotron wave growth using a modification of the computer code, WHAMP, waves in homogeneous, anisotropic multi-component plasmas [Rönnmark, 1982]. We show that low energy electron beams often seen associated with the waves, in the presence of a cold core plasma population can provide the free energy source for wave generation.

II. Instrumentation

The Polar satellite was launched in late February 1996 into a polar orbit with apogee of about 8 R_E and a perigee of about 2.2 R_E. POLAR is the first satellite to have 3 orthogonal electric antennas (E_u, E_v, and E_z), 3 triaxial magnetic search coils, and a magnetic loop antenna, as well as an advanced plasma wave instrument [Gurnett et al., 1995]. This combination can potentially provide the polarization and direction of arrival of a signal without any prior assumptions.

The Plasma Wave Instrument on the POLAR spacecraft is designed to provide measurements of plasma waves in the Earth's polar regions over the frequency range from 0.1 Hz to 800 kHz. Five receiver systems are used to process the data: a wideband receiver, a high-frequency waveform receiver (HFWR), a low-frequency waveform receiver, two multichannel analyzers, and a pair of sweep frequency receivers (SFR). For the high frequency emissions of interest here, the SFRs and the HFWR are of special interest. The HFWR measures high resolution waveform data using three orthogonal electric field antennas and three triaxial magnetic search coils. With the 25 kHz filter the sampling rate is 71.43 kHz. The SFR has a frequency range from 24 Hz to 800 kHz in 5 frequency bands. The frequency resolution is about 3% at the higher frequencies. In the log mode a full frequency spectrum can be obtained every 33 seconds.

The Electron and Ion Hot Plasma Instrument (HYDRA) [Scudder et al., 1995] is an experimental three-dimensional hot plasma instrument for the POLAR spacecraft. It consists of a suite of particle analyzers that sample the velocity space of electron and ions between ~ 2 keV/q to 35 keV/q in three dimensions, with a routine time resolution of 0.5 seconds. The satellite has been designed specifically to study accelerated plasmas such as in the cusp and auroral regions.

III. Observations and Calculations

In Figure 1 we show a frequency-vs.-time spectrogram of the swept frequency receiver (SFR) for a 9-hour period on day 201 of 1997, when the Polar spacecraft was high over the polar cap and proceeded to the midafternoon auroral region. The frequency range shown is 200 Hz to 20 kHz. Clearly seen are whistler mode emission at frequencies f < f_g, where f_g is displayed by the white line. Also clearly seen are intense, highly electrostatic emissions in a relatively narrow band at frequencies just above f_g. Less intense, broadbanded electrostatic emission is often seen to extend well above f_g. Menietti et al. [2001] showed that these emissions are highly electrostatic and show no preferred polarization, consistent with a linear polarization measured by independent axial wave receivers on the spinning Polar spacecraft.

In Figure 2 we plot contours of the electron distribution function in velocity space for a time period when electrostatic electron cyclotron waves were observed. The distribution is collected for a two-spin period of the Polar satellite. An electron beam with E < 1 keV is seen traveling up the field line away from Earth. During this time period the electron cyclotron frequency, f_g = 2200 Hz as measured by the magnetometer on board Polar, and the plasma frequency, f_p = 2520 Hz as determined from integration of the particle data from the hot plasma instrument (HYDRA). This is a typical electron beam seen during the time of the wave observations shown in Figure 1. Electron beams with E < 1 keV and varying temperatures are observed throughout the period of Figure 1 (cf. Menietti et al., 2001, Figure 4).

In order to investigate the role of these electron beams in the generation of the EEC waves, we have used a modification of a well-tested computer code, WHAMP, Waves in Homogeneous, Anisotropic Multicomponent Plasmas [Rönnmark, 1982]. WHAMP is a computer program which solves the dispersion relation of waves in a magnetized plasma. The dielectric tensor is derived using the kinetic theory of homogeneous plasmas with Maxwellian velocity distributions. Up to six different plasma components can be included (we use up to 3 in this work), and each component is specified by its density, temperature, particle mass, anisotropy and drift velocity along the magnetic field.

In this study a modified form of the distribution function introduced by Rönnmark [1982; 1983] is used as follows:

where v_|| and v are the particle velocities parallel and perpendicular to the magnetic field, respectively; and are the parallel and perpendicular thermal velocities, respectively; v_d is the parallel drift velocity, and n_l is the density of the "l"th plasma component.

To begin our investigation we fit the distribution function of Figure 2 using a nonlinear leasts squares fitting routine. In Figure 3 we show the model fit consisting of a background and a drifting Maxwellian with fitting parameters shown in Table 1. For this distribution and all others we will present f_p = 2.520 kHz and f_g = 2.2 kHz, and the total plasma density is n_total = 7.82 x 10^-2 cm^-3. While the fit is good, the distribution of Figure 3 is not unstable to the growth of electrostatic waves.

Table 1. Fitting Parameters for Observed Electron Distribution

Background Maxwellian Drifting Maxwellian

T_|| = 85 eV T_|| = 14 eV

n_bg = 94% n_beam = 6%

T/T_|| = 0.87 T/T_|| = 0.11

E_drft = 0 E_drft = 61.5 eV

Farrell et al. [1990] have shown that low energy electron beams (T_beam/T_background ~1) with a cold background plasma are marginally unstable to EEC waves. Young et al. [1973] found that a dense warm distribution with either df/dv > 0 or a temperature anisotropy (T> T_||) can excite waves for k_|| 0 in the range f_g < f < 2.0 f_g. We confirm some of these results using a modification of the WHAMP code by introducing a distribution function with 3 Maxwellian components as described in Table 2. The energy of the cold component is well below the low energy cutoff of the HYDRA instrument. Farrell et al. [1990] note the importance of keeping the temperature of the cold component low to obtain growth of the EEC waves.

Table 2. Fitting Parameters for Electron Distribution With Large (T/T_||)_beam

Cold Maxwellian Background Maxwellian Drifting Maxwellian

T_|| = 0.03 eV T_|| = 85 eV T_|| = 26 eV

n_cold = 37% n_bg = 28% n_beam = 35%

T/T_|| = 1 T/T_|| = 1.0 T/T_|| = 3.0

E_drft = 0 E_drft = 0 E_drft = 61.5 eV

In Figure 4 we plot a contour of the distribution function for the parameters of Table 2. This distribution contains a beam, but is obviously much broader in (T/T_||)_beam than the observed distribution of Figure 2. In Figure 5 we show the results of calculations of the roots of the dispersion relation, for the parameters of Table 2, and for wave normal angle, = 50. The coordinate of the plot is wave number, k(m^-1), and the three-panels display the real frequency (bottom), cB/E (middle), and the ratio f_i/f_r of imaginary to real frequency (top). In the bottom panel we display the whistler mode which shows a magnetic component but is not a growing mode for this distribution. Also seen is an electrostatic beam mode which does have a small but measurable imaginary frequency and thus growth rate. This mode is dependent on the beam and lies between f_g and f_UH as indicated.

Table 3. Fitting Parameters for Electron Distribution With Beam

Cold Maxwellian Background Maxwellian Drifting Maxwellian

T_|| = 0.1 eV T_|| = 85 eV T_|| = 26 eV

(& 0.03 eV)

n_cold = 20% n_bg = 70% n_beam = 10%

T/T_|| = 1 T/T_|| = .93 T/T_|| = 0.11

E_drft = 0 E_drft = 0 E_drft = 61.5 eV

We have determined that distributions quite similar to those observed (Figure 2) are also unstable to the beam mode. In Table 3 we display the plasma parameters yielding the contours of the distribution function shown in Figure 6. Compared to Table 1, we see that this distribution also contains a low density beam and the temperature anisotropy of the beam distribution is the same as observed. Table 3 also indicates a cold component that is necessary for the distribution to be unstable to the beam mode. This cold component would be invisible to the plasma instrument because << E_min, where E_min is the minimum energy of HYDRA, and E_min ~ 10 eV. In Figure 7 we plot the frequency vs. wave number in the same format as Figure 5 for a wave number of = 75 and two different temperatures of the cold background, = 0.11 eV and = 0.028 eV. A beam mode that is unstable to growth is present with f_i/f_r ~ 8.5x10^-3, k = 2x10^-2 ( = 0.11 eV); and f_i/f_r ~ 2.2x10^-3, k = 2.5 x 10^-2 (= 0.028 eV). The real frequencies lie in the range f_g < f < f_UH, where they are observed. When the beam is removed, f_i becomes negative and thus damped as expected for the beam mode. We find the waves grow only for a fairly narrow range of large wave normal angles. For = 0.028 eV, this range was 50 85, and the largest growth occurred for ~ 75. Plots of f_i/f_r vs. k for the range 50   80 are shown in Figure 8.

The polarization of the emissions can be examined by plotting the time-tagged electric field components in field-aligned coordinates as was shown in Menietti et al. [2000]. However, after examining consecutive plots these authors found no consistent sense of polarization, rather there are about as many cases of apparent left hand as right hand polarization. This result is consistent with a linear polarization expected for electrostatic emission. The waves are measured independently from the three orthogonal antenna/receiver systems E_u, E_v, and E_z. A linear polarized wave will thus show a random phase difference between receivers for E_u and E_v, for instance. Thus, in transforming the signals to a field-aligned coordinate system it is expected that the polarization will show a random "circular" polarization as we observe, with no preferred handedness.

From our calculations we obtain real E_x, and both real and imaginary components of E_y and E_z, with E_z along the ambient magnetic field direction. The value of real E_y and the imaginary components of both E_y and E_z are extremely small. The ratio of E_x/E_z is consistently ~ 3.7, thus the results are consistent with nearly linearly polarized waves propagating at oblique angles with respect to the ambient magnetic field.

Table 4. Fitting Parameters for Electron Distribution W/Higher Beam Energy

Cold Maxwellian Background Maxwellian Drifting Maxwellian

T_|| = 0.03 eV T_|| = 225 eV T_|| = 266 eV

n_cold = 15% n_bg = 75% n_beam = 10%

T/T_|| = 1 T/T_|| = 1.1 T/T_|| = 0.16

E_drft = 0 E_drft = 0 E_drft = 57.5 eV

We have further investigated the role of T in the growth of the observed waves. In Figure 9 we show contours of the velocity-space electron distribution function observed by Hydra during a different time period of the same day when electrostatic electron cyclotron waves were observed near the dayside auroral region. The time period is several hours earlier than Figure 2, and the electron beam parallel temperature is about 10 times higher. In Table 4 we list the fitting parameters used to model the data of Figure 9. We again comment that the cold distribution function is essentially invisible to the Hydra instrument, being well below the low-energy limit of the instrument. However, we find that the cold component is again necessary along with the beam to obtain growth of the waves. In Figure 10 we plot the calculated real and imaginary frequencies. Of the two values of mentioned above, growth occurred only for = 0.028 eV, with a value of f_i/f_r ~ 1.3x10^-3 near k ~ 0.018, and = 75. These waves again grow in the range f_g < f < f_UH as observed. For this electron distribution, with a higher beam velocity and larger T_||, the waves were seen to grow in an even narrower range of wave normal angles, 70 85, with the largest growth near = 75.

IV. Summary and Discussion

We have presented calculations of the dispersion function for an electron distribution containing a low-energy, low-temperature beam. Our calculations are performed using a modification of the WHAMP computer code for anisotropic plasmas that can be described by summed Maxwellian distributions with drifts and temperature anisotropies [Rönnmark, 1982]. We find that using electron distributions very similar those observed near the wave observations, we obtain a moderate wave growth of waves in the frequency range, f_g < f < f_UH as observed.

The electron distributions (measured by HYDRA) are associated with electron cyclotron waves observed by PWI as intense narrowbanded emissions at frequencies near and above the local gyrofrequency. These observations have been reported in the past for DE-1 wave data by Farrell et al. [1990] and for the Polar wave data by Menietti et al. [2001]. The waves have been observed often by the Polar spacecraft plasma wave instrument near the dayside auroral region, polar cap, and nightside auroral region for northern hemisphere passes when the satellite altitude is such that f_g f_p. The electric fields are often nearly perpendicular to the ambient magnetic field. There are weak, if any, oscillations of the magnetic field present. The polarization measurements are consistent with a linear polarization.

Menietti et al. [2001] reported an association of these observations with low-energy electron beams. These results are therefore consistent with Farrell et al. [1990] who observed a correlation of similar waves seen in the cusp with beaming electrons with energies of a few hundred eV. These latter authors indicated that electron distributions containing low energy beams with T_beam/T_background ~ 1 and f_p/f_g ~ 1.28 were mildly unstable to the growth of the waves. For the latter study the beam was relatively high density with the ratio of n_beam/n_total = 0.35. Our studies indicate that the observed electron distributions, which we believe are typical of those made by PWI on northern hemisphere passes of the Polar spacecraft, contain temperature anisotropies with (T_||/T)_beam ~ 0.1. For low energy electron beams with (T/T_||)_beam ~ 0.1 in the presence of a warm background and a cold core electron distribution, the plasma is unstable to the growth of electrostatic waves in the observed frequency range (f_g < f < f_UH). For our case the beams are relatively low density, with n _beam/n_total = 0.10. We conclude that the observed low-energy electron beams are the probable free-energy source of the observed electrostatic electron cyclotron waves.

V. References

Bernstein, I. B., Waves in a plasma in a magnetic field, Phys. Rev., 109, 10-21, 1958.

Farrell, W. M., D. A. Gurnett, J. D. Menietti, H. K. Wong, C. S. Lin, and J. L. Burch, Wave intensifications near the electron cyclotron frequency within the polar cusp, J. Geophys. Res., 95, 6493, 1990.

Gurnett, D. A., S. D. Shawhan, R. R. Shaw, Auroral hiss, Z mode radiation, and auroral kilometric radiation in the polar magnetosphere: DE 1 observations, J. Geophys. Res., 88, 329, 1983.

Gurnett, D. A., et al., The Polar plasma wave instrument, Space Sci. Rev., 71, 597, 1995.

Krall, N. A., and A. W. Trivelpiece, in Principles of Plasma Physics, McGraw-Hill, Inc., New York, N. Y., pp. 410-412, 1973.

Kurth, W. S., M. Ashour-Abdalla, L. A. Frank, C. F. Kennel, D. A. Gurnett, D. D. Sentman, and B. G. Burek, A comparison of intense electrostatic waves near f_UHR with linear instability theory, Geophys. Res. Lett., 6, 487, 1979a.

Kurth, W. S., J. D. Craven, L. A. Frank, and D. A. Gurnett, Intense electrostatic waves near the upper hybrid resonance frequency, J. Geophys. Res., 84, 4145, 1979b.

Menietti, J. D., J. S. Pickett, D. A. Gurnett, and J. D. Scudder, Electrostatic electron cyclotron waves observed by the plasma wave instrument on board Polar, J. Geophys. Res., 106, 6043-6057, 2001.

Mosier, S. R., M. L. Kaiser, and L. W. Brown, Observations of noise bands associated with the upper hybrid resonance by the IMP 6 radio astronomy experiment, J. Geophys. Res., 78, 1673, 1973.

Rönnmark, K., Computation of the dielectric tensor of a Maxwellian plasma, Plasma Phys., 25, 699-701, 1983.

Rönnmark, K., Waves in homogeneous, anisotropic, multi component plasmas, KGI Report No. 179, Kiruna Geophysical Institute, Kiruna, Sweden, 1982.

Scudder, J., et al., HYDRA-A 3-dimensional electron and ion hot plasma instrument for the Polar spacecraft of the GGS Mission, Space Sci. Rev., 71, 459, 1995.

Tataronis and Crawford, Plasma Phys., 4, 231, 1970

Winglee, R. A., J. D. Menietti, and H. K. Wong, Numerical simulations of bursty radio emissions from planetary magnetospheres, J. Geophys. Res., 97, 17131, 1992.

Wong, H. K. and M. L. Goldstein, Electron cyclotron wave generation by relativistic electrons, J. Geophys. Res., 99, 235, 1994.

Young, T. S. T., J. D. Callen, and J. E. McCune, High-frequency electrostatic waves in the magnetosphere, J. Geophys. Res., 78, 1082, 1973.

Figure Captions

Figure 1. Frequency-vs.-time spectrogram showing the emissions in question as an often intense (yellow), relatively narrowband feature just above the gyrofrequency (the white line). This survey plot covers 9 hours of day 97/07/20 (day 201).

Figure 2. Contours of the velocity-space electron distribution function observed during a time period when electrostatic electron cyclotron waves were observed.

Figure 3. Contours of the velocity-space electron distribution function resulting from a model of drifting Maxwellians (equation 1). The fitting parameters are listed in Table 1.

Figure 4. Same as Figure 3 but now for the fitting parameters of Table 2, which contain a cold background and a beam with a large temperature aniostropy, (T/T_||)_beam = 3.

Figure 5. Results of the solution of the dispersion equation: The coordinate of the plot is wave number (m^-1) and the three-panels display the real frequency (bottom), cB/E (middle), and the ratio f_i/f_r of imaginary to real frequency (top). In the bottom panel we display the non-growing whistler mode and the electrostatic beam mode (f_g < f < f_UH), which does have a small but measurable imaginary frequency and thus growth rate.

Figure 6. Same as Figure 3 but now for the fitting parameters of Table 2, which are close to those observed (cf. Figure 2).

Figure 7. We plot the frequency vs. wave number in the same format as Figure 5. Note that a beam mode that is unstable to growth is present with f_i/f_r ~ 1% near k = 2x10^-2. There are two curves for the beam mode, = 0.1 eV and = 0.028 eV. The real frequencies lie in the range f_g < f < f_UH, where they are observed.

Figure 8. Plots of f_i/f_r vs. k for the range of wave normal angles 50 80 are shown for the case of = 0.028 eV and other parameters as listed in Table 3.

Figure 9. Contours of the velocity-space electron distribution function observed during a time period when electrostatic electron cyclotron waves were observed. The time period is several hours earlier than Figure 2 and the electron beam energy and parallel temperature are higher.

Figure 10. Plots of f_i/f_r in the same format as Figure 5, but now for the plasma parameters listed in Table 4 (to approximate the distribution shown in Figure 9). Note that f_i/f_r > 0 near k ~ 0.18, indicating a positive growth rate for the waves in the frequency range f_g < f < f_UH, consistent with the observations.

beam mode

whistler mode