Evidence for electrostatic decay in the solar wind at 5.2 AU



[1] The Unified Radio and Plasma wave Experiment (URAP) on the Ulysses spacecraft provides in situ observations of Langmuir waves and ion-acoustic waves in the solar wind. The observations presented in this paper were obtained at 5.2 AU from the Sun. Low-frequency (20–200 Hz in the spacecraft frame) electric field signals are observed coincident in time with the most intense Langmuir waves. The low-frequency wave signals are identified as long-wavelength ion-acoustic waves. These observations provide evidence for the decay of Langmuir waves into daughter Langmuir and ion-acoustic waves (the electrostatic decay process) in the solar wind.

1. Introduction

[2] This paper presents evidence for one of the most important non-linear wave-wave interactions, the electrostatic decay instability, observed in the solar wind at a distance of 5.2 AU from the sun by the Ulysses Unified Radio and Plasma Wave Experiment (URAP). This non-linear wave-wave process involves the decay of Langmuir waves into daughter Langmuir and ion-acoustic waves. The main consequences of this process include the cascade of the Langmuir waves towards lower wave numbers (higher wavelengths). In the case of repeated cascades, this process can lead to the formation of the Langmuir condensate in the spectral region of negligible Landau damping, i.e., very small wave numbers kL, where the weak turbulence approximation breaks down. For eventual damping of the waves piled up in the Langmuir condensate, strong turbulence processes such as modulational instability [Vedenov and Rudakov, 1965] and Langmuir collapse [Zakharov, 1972] have to be invoked.

[3] When electron temperature Te is much higher than the ion temperature Ti, the electrostatic decay process usually precedes the Langmuir soliton formation and Langmuir collapse processes. In the case of type III radio bursts, the electrostatic decay process is believed to be responsible for moving the beam-excited Langmuir waves from the spectral regions of resonance with the beam toward lower wave numbers and subsequent stabilization of the electron beam so that it can travel very large distances (>1 AU). Since the secondary Langmuir wave spectrum generated during this process is oppositely directed with respect to the beam excited Langmuir waves, they can coalesce with each other by producing electromagnetic emission at 2fpe, where fpe is the electron plasma frequency. Other models, which may provide alternative ways to solve the problem of beam stabilization are: (1) inhomogeneous quasilinear relaxation models, which invoke that the Langmuir waves excited at or near the front of the beam are later reabsorbed by the slower electrons ensuring the beam propagation over large distances and retension of the form of the distinct beam [see, e.g., Magelssen and Smith, 1977; Grognard, 1982; Kontar, 2001], (2) scattering of Langmuir waves out of resonance with the beam by density fluctuations either through anomalous diffusion when kskL [Goldman and DuBois, 1982] or through angular diffusion when kskL [Muschietti et al., 1985] (2π/ks is the scale length of the density fluctuations and 2π/kL is the wavelength of the Langmuir wave), and (3) stochastic growth theories [Robinson, 1992].

[4] The temporal association of ion-acoustic waves with Langmuir waves is one of the observable signatures of the electrostatic decay. Therefore, several type III burst associated in situ wave data sets obtained by ISEE 3 [Lin et al., 1986], Galileo [Gurnett et al., 1993] and Ulysses [Kellogg et al., 1992; Thejappa et al., 1993, 1995, 1996, 1999; Thejappa and MacDowall, 1998] were examined for such temporal associations. For example, Lin et al. [1986] have reported the observations of 30–300 Hz long wavelength ion-acoustic wave electric fields in association with Langmuir waves at 1 A.U., which were interpreted in terms of the decay of Langmuir waves into daughter electromagnetic waves at fpe and ion-acoustic waves (electromagnetic decay instability). Subsequently, the same observations were interpreted in terms of electrostatic decay instability [Robinson et al., 1993]. Gurnett et al. [1993] reported the observations of highly modulated Langmuir waves during a type III radio burst, which were interpreted in terms of beating between the beam excited Langmuir waves and the secondary Langmuir waves generated during the electrostatic decay, even though ion-acoustic waves were not observed during this event.

[5] The occurrences of low-frequency electromagnetic waves, corresponding either to whistlers or to electromagnetic lower-hybrid waves, in association with type III related Langmuir waves have been reported in Ulysses data [Kellogg et al., 1992; Thejappa et al., 1995]. These observations were interpreted in terms of either wave-wave interactions [Kellogg et al., 1992], or simultaneous excitation of the low-frequency electromagnetic waves and Langmuir waves by the electron beams [Thejappa et al., 1995]. The observations of ion-acoustic waves as well as whistlers in close association with a local type III burst associated Langmuir waves have been reported by Thejappa and MacDowall [1998]. These authors argue that whistlers and Langmuir waves are probably excited directly by the type III electron beam, whereas the ion-acoustic and Langmuir waves are coupled through the electrostatic decay process. Since Langmuir waves were also observed to be strongly turbulent, those observations were interpreted in terms of coexistence of weak and strong turbulence processes in the source regions of type III radio bursts. Thejappa et al. [1999] have reanalyzed the in situ wave observations reported by Thejappa et al. [1993] and found that Langmuir envelope solitons sometimes accompany ion-acoustic wave signals, confirming their earlier findings of coexistence of weak and strong turbulence processes in type III burst source regions [Thejappa and MacDowall, 1998]. Here we note that the conclusion about the coexistence of weak and strong turbulence processes was based on (1) the observed inverse correlation between the width of the wave packet and peak intensity (a signature of modulational instability) and (2) the observed temporal association of the Langmuir waves and ion-acoustic waves (a signature of the electrostatic decay instability).

[6] In this paper, we present new observations where ion-acoustic waves are detected in association with Langmuir waves, whose intensity levels are well below strong turbulence thresholds. We show that the intensity levels of the Langmuir waves and the observed frequencies are consistent with the electrostatic decay instability. This is the first time where clear evidence for the electrostatic decay process has been observed as far away as 5.2 AU from the Sun.

2. Observations

[7] This paper is based on the in situ wave data of January 10, 1999, obtained in the solar wind by the Unified Radio and Plasma Wave Experiment (URAP) on Ulysses spacecraft. In this study, we use the data from (1) the Radio Astronomy Receiver (RAR), which measures the wave electric fields from 1 to 940 kHz with ∼128 s time resolution, (2) the Plasma Frequency Receiver (PFR), which measures the wave electric fields from 0.5 to 35 kHz with a 16 s time resolution, and (3) the Wave Form Analyzer (WFA), which measures the wave electric and magnetic fields from 0.08 to 448 Hz with a time resolution of 64 s. The details of URAP instrumentation can be found in Stone et al. [1992]. We also use the electron temperature (Te), ion temperature (Ti), electron density (ne), solar wind speed (vsw) and electron distribution data from the Ulysses SWOOPS experiment [Bame et al., 1992]. We give the relevant parameters in Table 1.

Table 1. Plasma and Wave Parameters
Solar wind density, ne4.5 × 104 m−3
Solar wind velocity, Vsw415 km s−1
Electron temperature, Te8.8 × 104 K
Ion temperature, Ti2.5 × 103 K
Debye length, λD97 m
Electron plasma frequency, fpe1.9 kHz
Ion plasma frequency, fpi44.5 Hz
Electron thermal speed, VTe1.2 × 106 ms−1
Ion sound speed, cs2.8 × 104 ms−1
Langmuir wave peak amplitude, EL3.5 × 10−5 Vm−1
Normalized peak energy density, equation image2 × 10−7
Ion-acoustic wave peak amplitude, Es7.6 × 10−7 Vm−1
Normalized peak energy density, equation image5 × 10−9
Beam velocity, vb (estimated)1.5 × 107 ms−1
Beam width, Δvb/vb (assumed)0.3

[8] Figure 1 shows the spectral display (dynamic spectrum) of the wave data, where the degree of darkness is proportional to the logarithm of the intensity. The top two panels show the RAR Hi-band (52–940 kHz) and the Lo-band (1.25–48.5 kHz) electric field data. The fast drifting emission feature from ∼940 kHz down to ∼20 kHz is a type III radio burst. The PFR electric field measurements are shown in the third panel. The field enhancements seen from ∼06:45 to ∼8:30 UT both by the PFR (third panel) and RAR (second panel) in a narrow band around fpe ∼ 2 kHz correspond to Langmuir waves. In the fourth through seventh panels, WFA data are presented. During the Langmuir wave activity, the fourth panel shows the presence of the electric field enhancements in the 20–200 Hz range, whereas the sixth panel does not show any of such wave magnetic field enhancements in the same frequency range. This implies that the observed low-frequency electric field signals correspond to electrostatic wave modes. The ion-acoustic waves are the only appropriate electrostatic waves at these low frequencies.

Figure 1.

Dynamic spectrum of remote radio and in situ wave activity in the interval 4:00–16:00 UT on January 10, 1999. The data presented in the top two panels are from the Radio Astronomy Receiver (RAR), the third panel is from the Plasma Frequency Receiver (PFR) and the bottom three panels are from the Wave Form Analyzer (WFA). The type III burst and correlated Langmuir and long wavelength ion-acoustic waves are marked in the figure. The dashed line shows the leading edge of the type III burst (see text).

[9] The background levels across the observing range of the URAP instruments vary by orders of magnitude. For this reason, the data are displayed in units of dB normalized to the background at the frequency of the data, permitting the detection of much weaker events than would be possible if an absolute calibration were used. The peak electric field values of the wave activity detected by the PFR and the WFA (electric field) can be seen in Figure 3, along with the background values. The peak electric field value for the type III burst is 2 × 10−9 V m−1 Hz−1/2 detected at 45 kHz. The background level for the WFA (magnetic field) at 28 Hz is 1.8 × 10−13 Tesla Hz−1/2; no event is detected.

[10] Figure 2 shows the time profiles of (1) type III radio burst at 44.8 kHz (top panel), (2) Langmuir waves ∼2.2 kHz (second panel), and (3) ion-acoustic waves at 28 Hz (third panel) and (4) the lack of any magnetic field enhancements at 28 Hz (fourth panel). It is clear from these time profiles that the type III radio burst is characterized by a rapid rise followed by a smooth and slow decay, whereas Langmuir waves (see, panel 2 from ∼0700–0800 UT) are very irregular and impulsive, i.e., the difference between the peak and the average electric field intensities is very large. Here we note that the frequency of ∼2.2 kHz corresponding to the Langmuir wave signals of panel two is slightly higher than the local electron plasma frequency ∼1.9 kHz corresponding to the solar wind electron density (ne) of ∼4.5 × 104 m−3 (Table 1), measured by the Solar Wind Plasma Experiment (SWOOPS) [Bame et al., 1992]. The normalized peak energy density of these Langmuir waves is, WL/(nekBTe) = ϵ0EL2/(nekBTe) ≃ 2 × 10−7 for the observed peak amplitude of the electric field signals EL ∼ 3.5 × 10−5 V m−1, ne = 4.5 × 104 m−3, Te = 8.8 × 104 K and Boltzmann Constant of kB = 1.3807 × 10−23 J K−1. As seen from the third panel, the impulsive low-frequency electric field bursts (∼28 Hz) are almost coincident in time with intense Langmuir wave bursts (panel 2). The enhanced noise from ∼09:30 to ∼12:00 UT is due to a known source of interference. The lack of any magnetic field enhancements (see fourth panel) during the electric field enhancements at 28 Hz, confirms the electrostatic nature of the observed low-frequency signals. The temporal coincidence of low-frequency waves and intense Langmuir wave bursts indicates that they are probably coupled with each other through some non-linear process.

Figure 2.

Time profiles of the radio and in situ plasma wave phenomena observed on January 10, 1999: (a) type III radio emission at 44.8 kHz, (b) Langmuir wave electric fields at 2.2 kHz, (c) ion-acoustic waves at 28 Hz and (d) lack of any wave magnetic field enhancements at 28 Hz. The enhanced background from ∼09:30–∼12:00 UT in the third panel is due to interference.

[11] As far as the instrumental constraints are concerned, the following is worth mentioning. During this Langmuir wave event, the Ulysses data rate is 512 bps, which has some consequences for the time resolution of the URAP instruments. The PFR is in fast scan mode, which means that all 32 frequencies are sampled every 0.5 seconds during a 16 second window to provide the peak value at each frequency. The precise time(s) of the peak values are not determined in fast scan mode. Due to telemetry restrictions, PFR data are only telemetered for alternate 16 second windows. Meanwhile, the WFA data are obtained using discrete samples over 4 successive cycles of a wave at the desired frequency. This restriction was required due to the limited data processing capability available at the time. Consequently, the “dead time” for a given WFA frequency is considerable. For example, at 28 Hz “dead time” is 83%. The combination of these effects is that, only approximate one-to-one time correspondence can be made between the PFR and WFA data sets.

[12] Figure 3 presents the electric field spectra obtained by both the WFA and PFR during 07:13:20–07:14:20 UT. The observed spectral peaks during this time interval at ∼2.2 kHz and ∼100 Hz clearly demonstrate the simultaneous occurrence of Langmuir and ion-acoustic waves. The dotted line in the figure is the instrumental background.

Figure 3.

Spectral plots of electric fields observed by the WFA and PFR during this correlation. The dotted line corresponds to instrumental threshold. The spectral peak in these figures at ∼2.2 × 103 Hz corresponds to Langmuir waves, whereas the spectral peak at ∼100 Hz corresponds to long wavelength ion-acoustic waves. The spectral peak at ∼1 kHz is due to interference, which can be also seen in the dynamic spectrum presented in Figure 1.

[13] The observed frequency, fs corresponding to the peak enhancement of the ion-acoustic waves can be used to estimate their wave numbers ks using the relation

equation image

where vsw is the solar wind speed and equation image is the ion sound speed. Since vswcs, the measured frequency is strongly influenced by the doppler shift. For the observed values of vsw = 415 km s−1, and cs = 28 km s−1 (calculated for the observed values of Te = 8.8 × 104 K and Ti = 2.5 × 103 K (see, Table 1)), we obtain ks ∼ 1.4 × 10−3 m−1. The normalized energy density of these waves is Ws/(nekBTe) ∼ ϵ0Es2/(nekBTe)(1/(ksλD))2 ∼ 5 × 10−9, for the observed peak intensity Es of ∼7.6 × 10−7 Vm−1 and the Debye length λD ∼ 96 m.

3. Source of Langmuir Waves

[14] Electron beams are the only known sources of Langmuir waves in the solar wind. Three kinds of events are believed to produce such beams: suprathermal electrons escaping from solar flare sites, electron beams accelerated by interplanetary (IP) shocks, and electrons in magnetic holes. The solar radio emissions associated with the first two sources are known as type III and type II radio bursts, respectively. They are emitted at the fundamental and second harmonic of the electron plasma frequency. Langmuir waves in magnetic holes, first detected by the Ulysses spacecraft [Lin et al., 1995], do not produce any detectable radio emissions. This is probably because the volume of a “hole” is small, typically passing the spacecraft in less than a minute [Winterhalter et al., 1994].

[15] There are no magnetic holes at the time of the Langmuir waves being studied (Figure 4). Examination of high time resolution (1 s) magnetic field data confirms that no short duration magnetic holes are present (R. Forsyth, private communication, 2000). There are no shocks observed by Ulysses for 8 days before and 2.5 days after this event, which would seem to rule out IP shocks as a likely source for the electrons. The intense type III burst seen in Figures 1 and 2 is the result of a flare; a GOES C 1.2 x-ray flare was reported starting at 5:40 UT. Thus, energetic electrons escape from the flare site approximately 1.5 hrs before the Langmuir wave event onset; however, there are major problems with associating this flare/type III burst and the Langmuir waves. The length of an Archimedean spiral for vsw = 415 km s−1 to a distance of 5.19 AU is 14.6 AU. For electrons to travel along this spiral and arrive from the Sun in 1.5 hours, they would have to have an average velocity of 4.1 × 105 km s−1; such a superluminal velocity is physically impossible. Even if we propose a field line from the source to Ulysses that is somehow substantially shorter than the expected Archimedean spiral, the type III burst does not comply with expectations. It may be seen from Figure 1 that any extrapolation of the leading edge of the type III burst reaches the local plasma frequency, i.e., the vicinity of the spacecraft, much later than the occurrence of Langmuir waves. Bursts with constant exciter speed usually have their leading edges tracing straight lines on plots scaled as time vs. (1/frequency). The dashed line shown in Figure 1 is derived from a straight line approximation to the leading edge of the burst at frequencies below 50 kHz when the burst is plotted in 1/frequency space (not shown). Other frequency ranges yield different extrapolations, but all cause the drift of the leading edge of the burst to reach the plasma frequency hours after the Langmuir bursts occur. Furthermore, there is no evidence of “local” type III emission at frequencies near fpe; the type III radiation has a lower limit of 35 kHz at the time of the Langmuir waves.

Figure 4.

Ulysses solar wind, wave, and magnetic field data for 19990110, 5:00–11:00 UT. The top 2 panels show solar wind density and velocity. The third panel shows the URAP PFR data. Note the occurrence of a short duration burst of Langmuir waves at 10:30. The bottom 3 panels show magnetic field magnitude and direction; the angles theta and phi correspond to an RTN coordinate system. It may be seen in the field amplitude that a magnetic hole occurs at 10:30, corresponding to the time of a burst of Langmuir waves.

[16] If we examine the electron measurements from Ulysses, the story remains uncertain. At energies above 40 keV, HI-SCALE measurements do not show evidence of an electron enhancement (not shown). At energies below 1 keV, SWOOPS data are ambiguous. There is some suggestion of a beam at 7:06 UT (Figure 5) in a single distribution function; however, the beam is very dense and cold - very atypical of electron beams that have been observed in the solar wind. It is possible that the detailed characteristics of this beam are distorted by the small count statistics due to the low solar wind density. In any case, the relative density of the beam in this observed distribution function is such that a beam mode interaction with broad-band wave activity would be expected [Onsager and Holzworth, 1990]. Typical type III electron beams are observed in the range of 1 keV to 40 keV [see, e.g., Lin et al., 1981, 1986]; unfortunately, no instrument on Ulysses makes electron measurements in this energy range.

Figure 5.

Representations of the electron distribution function at 7:06 UT from the Ulysses SWOOPS instrument. (a) Natural log of the electron flux shown using a color scale plotted versus velocities v and v (in km s−1). The data have been smoothed to provide resolution that is comparable to the instrument angular resolution. (b) Three-dimensional view of the same distribution function with the noise floor set at ln(flux) = −67.0.

[17] We might consider that these waves are not Langmuir waves, but this is the only mode that produces narrow band emission close to the electron plasma frequency. In fast solar wind (vsw > 600 km s−1), ion-acoustic waves can be Doppler shifted to or above the plasma frequency [MacDowall and Kellogg, 2001]; however, vsw is too low here for this to occur. Furthermore, the burst of Langmuir waves that does occur in a magnetic hole later in the day at 10:30 UT, shows clearly that the waves occurring from 6:50 to 7:40 UT have the characteristic bandwidths of Langmuir waves (see Figure 4). We are forced to conclude that we have detected Langmuir waves, but that we cannot identify the event that excited them.

4. Discussion

[18] The dispersion relations of Langmuir (L) and ion-acoustic (S) waves are

equation image
equation image

where kL and ks are the respective wave numbers and vTe is the electron thermal speed. Wave-wave interactions relevant in the present context are: (1) electrostatic decay of a pump Langmuir wave (L) into another Langmuir wave (L′) and an ion-acoustic wave (S), (2) oscillating two stream instability or four-wave instability, in which two pump Langmuir waves L1 and L2 decay into two daughter Langmuir waves L3 and L4 via the exchange of an ion-sound wave, and (3) electromagnetic decay instability, where the pump Langmuir wave (L) decays into a daughter electromagnetic wave (T) and an ion-acoustic wave (S). These processes are characterized by the existence of a threshold amplitude which must be exceeded by the pump wave in order that the instability may occur. In the following, we will analyze each of these processes in the context of present observations.

4.1. Electrostatic Decay Instability

[19] The resonance conditions for electrostatic decay are:

equation image
equation image

Substituting the respective dispersion relations (2) and (3) in equation (5), one obtains

equation image

where k0 = 2 ωpecs/3vTe2. Note that k0 depends solely on the parameters of the ambient plasma. Thus, for fpe ∼ 1.9 kHz, vTe ∼ 1.2 × 106 m s−1, and cs ∼ 2.8 × 104 m s−1, we obtain: k0 = (4π/3)fpecs/vTe2 ∼ 1.7 × 10−4 m−1. From equation (4), one can write

equation image

where θ is the angle between equation image and equation image. From equations (6) and (7), we obtain

equation image

The maximum possible value of the daughter ion-acoustic wave number is [Melrose, 1982; Cairns and Melrose, 1985]

equation image

and the daughter Langmuir wave is directed almost opposite to the pump wave. Using equation (9) we can estimate the wave number of the Langmuir waves as kL ∼7.8 × 10−4 m−1 for the measured values of ks ∼ 1.4 × 10−3 m−1 and k0 ∼ 1.7 × 10−4 m−1. Thus the probable speed of the electron beam responsible for the observed Langmuir waves is ωpe/kL ∼ 1.5 × 107 m s−1.

[20] If the pump Langmuir waves are assumed to be coherent, the threshold condition for electrostatic decay is:

equation image

where ΓL and Γs are the damping rates of Langmuir and ion-acoustic waves, respectively. Landau damping by the negative slope of the electron beam distribution (∂fb/∂v) is the main damping mechanism for the Langmuir waves. Since equation image is not available in this case, ΓLpe can be estimated by using the observed peak energy density of Langmuir waves, WL = ϵ0EL2 as [Melrose et al., 1986]

equation image

where Δvb is the velocity dispersion in the beam. For the observed values of WL ≃ 1.1 × 10−20 J, ne = 4.5 × 104 m−3, and assumed values of Δvb ≃ 0.3vb = 4.6 × 106 m s−1, we obtain ΓLpe ≃ 7.9×10−8.

[21] The relative damping of ion-acoustic waves is [see Robinson, 1997]:

equation image

where me and mi are the electron and ion masses, respectively. For Te = 8.8 × 104 K and Ti = 2.5 × 103 K, we obtain Γss ≃ 2.9 × 10−2.

[22] Thus the threshold 4(ΓLΓs/(ωpeωs)) ≃ 9.2 × 10−9, estimated using these observed and assumed values is well below the observed normalized energy density of Langmuir waves WL/(kBneTe) ≃ 2 × 10−7. This indicates that the threshold condition is easily satisfied for the electrostatic decay instability, even if we assume that the observed Langmuir waves are coherent in nature.

[23] The threshold for electrostatic decay derived using the wave kinetic equations by assuming that the Langmuir waves are not coherent is [Cairns et al., 1998]:

equation image

This expression differs by a factor of 2 from that of Cairns et al., because we use γL as the amplitude growth rate (for example, growth rate of the bump-on-tail instability), instead of energy growth rate as used in their original derivation [see, Robinson et al., 1993]. Here we note that this expression and that of Cairns et al. [1998] are equivalent, since the energy growth rate is twice that of the amplitude growth rate. For the observed parameters, the threshold amplitude of the Langmuir wave electric field is estimated as ∼2 × 10−5 V m−1, which is well below the observed peak values of 3.5 × 10−5 V m−1. This again indicates that the electrostatic decay is a viable process even under random phase approximation.

[24] If the electrostatic decay has taken place, the initial wave action of the pump Langmuir waves should be at least equal to that of the daughter ion-acoustic waves, which can be expressed as NsNL. Here these occupation numbers are defined as NL = (2π)3WL/(ωLkL2ΔkLΔΩL), and Ns = (2π)3Ws/(ωsks2ΔksΔΩs), where ΔkL and Δks are the wave number spread and ΔΩL and ΔΩs are the solid angles of the corresponding waves. This inequality can be written as Es2 ≤ (ωsL)(ks/kL)2kskL) (ΔΩs/ΔΩL)(ksλD)2EL2. For ωs = kscs, cs2 = (Te + 3Ti)/mi, λD = VTepe, ωL = ωpe, ΔΩs ∼ 4ΔΩL, ks ∼ 2kL, and Δks ∼ 2ΔkL, this inequality can be simplified as:

equation image

For this particular case of Te = 8.8 × 104 K, Ti = 2.5 × 103 K, ks = 1.4 × 10−3 m−1, and λD = 96 m, we get a very simple expression: EsEL/10. This inequality is easily satisfied for the observed peak values of Es ∼ 7.6 × 10−7 V m−1 and EL ∼ 3.5 × 10−5 V m−1. This implies that the observed ion-acoustic waves could be the daughter products of the electrostatic decay instability. Thus, the energy constraints as well as the threshold conditions indicate that the observed Langmuir and ion-acoustic waves are probably coupled through the electrostatic decay process.

4.2. Oscillating-Two-Stream (OTSI) or Four-Wave Instability

[25] The conditions for the occurrence of the oscillating-two-stream (OTSI) or four-wave instability are (1) the frequency of the pump Langmuir waves (ωL) should be less than or equal to the local electron plasma frequency (ωpe), and (2) the energy density of the pump Langmuir waves should exceed a certain threshold [Sagdeev, 1979]:

equation image

where the average spread in wave numbers of the Langmuir wave packet ΔkL can be estimated as [cf. Lin et al., 1986]:

equation image

Here the number of linear growth times N of Langmuir waves before onset of the instability can be estimated from Nlnp ∼ 6.2, where p ∼ 5 × 102 is the ratio of the peak electric field amplitude (Figure 2) to the thermal background which is given by the RAR background levels. Thus for Δvb/vb ≃ 0.3, and N ∼ 6.2, we obtain ΔkL/kL ≃ 1.7 × 10−2 and ΔkL ≃ 1.3 × 10−5 m−1 for kL ∼ ωpe/vb ∼ 7.8 × 10−6 m−1, and subsequently, for λD ∼ 96 m and we obtain 3(ΔkLλD)2 ∼ 5 × 10−6. The observed normalized peak intensity of Langmuir waves, WL/(kBneTe) ∼ 2 × 10−7 is well below this threshold value, indicating that the oscillating two stream instability is not relevant in the present context. Here we note that we have not included the contribution of quasi-linear relaxation to the spectral width of Langmuir waves. The inclusion of such a contribution (which is an unknown quantity) will rise the threshold for the strong turbulence processes. This will still support the conclusion that the strong turbulence processes may not occur in the present circumstances.

[26] The characteristic wave number of the low-frequency fluctuations associated with OTSI can be estimated as: ks02= WL/(12λD2) and the wave number corresponding to 0.1-level is k0.1 = 4.5(2ks0/π) [Lin et al., 1986]. Therefore, the Doppler-shifted frequency corresponding to this value is equation image ∼ 0.6 Hz. No electric field enhancements are observed at this frequency. It is also possible that the WFA did not capture these weak waves at these very low frequencies. Moreover, during this event, the Fast Envelope Sampler (FES) of the URAP experiment did not detect any intense field structures of scale lengths of ∼100λD, which are the main signatures of the OTSI (four-wave instability) [Thejappa et al., 1993, 1999; Thejappa and MacDowall, 1998]. This indicates that the observed low-frequency fluctuations can not be accounted for by the modulational instability processes.

4.3. Electromagnetic Decay Instability

[27] The resonance conditions for this instability are

equation image
equation image

where ωT = ωpe + kT2c2/2ωpe2 is the dispersion relation and kT is the wave number of the electromagnetic wave and c is the speed of light. In this process, kTkL and according to the resonance conditions, we obtain kLks ∼ 1.4 × 10−3 m−1. This value can be used to estimate the speed of the electron beam vb = ωpe/kL ∼ 1.5 × 107 m s−1. In the case of coherent wave assumption, the threshold for this instability is

equation image

Here ΓTpe is the relative damping of the electromagnetic waves, which is mainly due to collisional damping, νepeequation image, where ln Λ ≃ 21.6 − 0.5 ln ne + ln Te [Melrose, 1986]. For the values presented in Table 1, we can estimate νepe ∼ 5.4 × 10−11. Using equation (12), we estimate Γss ∼ 2.9 × 10−2 for the observed Te and Ti values. These damping rates yield the threshold equation image ∼ 6.2 × 10−12, which is well below the observed energy densities. However, the electromagnetic decay is probably a convective instability. In such a case, because of the high group speed of the electromagnetic waves, they easily escape the region of instability, which inhibits the growth of the ion-acoustic waves. This makes it difficult to account for the observed correlated waves in terms of electromagnetic decay. Even if the electromagnetic decay instability is operative in this case, it should yield the fundamental radio emission at the local electron plasma frequency, fpe. However, the expected local electromagnetic emission at fpe is not observed in this case.

[28] In the case of random phase approximation, the threshold for the electromagnetic decay is [Robinson et al., 1994]

equation image

This expression also differs by a factor of 2 from that of Robinson et al. [1994], because here we use γL as the amplitude growth rate instead of as energy growth rate. For the observed values, this threshold can be estimated as 2.3 V m−1, which is well above the observed values. This indicates that the electromagnetic decay instability does not occur in the present case.

5. Conclusions

[29] The observations of electric field signals (20–100 Hz) in close association with Langmuir waves provide good evidence for the occurrence of non-linear wave-wave processes at distances as far away as 5.2 A.U from the Sun. The electrostatic decay process, i.e., the decay of the pump Langmuir wave into daughter Langmuir and ion-acoustic waves is most consistent with the observations. The threshold of this instability is well below the observed Langmuir wave intensities for both coherent and random phase approximations. The daughter ion-acoustic waves would produce Doppler-shifted frequencies in the observed range.

[30] For the electromagnetic decay instability which is characterized by the decay of the pump Langmuir wave into a daughter electromagnetic wave at the fundamental of the electron plasma frequency, fpe and an ion-acoustic wave [Tsytovich, 1970; Melrose, 1980; Lin et al., 1986], the conservation of momentum requires that the wave number of the ion-acoustic wave ks should be approximately equal to that of the Langmuir wave (kL), since the wave number of the electromagnetic wave will be very small. In such a situation, the requirement that the number density of the daughter ion-acoustic waves should be less than that of the pump Langmuir waves puts a constraint on their respective electric field values as Es2 ≤ 2.1 × 10−6EL2. This inequality is not satisfied in the present context. As pointed out by Lin et al. [1986], the convective effects may severely limit the generation of transverse waves by this process. Moreover, the threshold for the electromagnetic decay instability derived using the random phase approximation is well above the observed intensities. Therefore, the simultaneous occurrence of Langmuir and ion-acoustic waves is probably not due to electromagnetic decay instability.

[31] The threshold of the oscillating two stream (OTSI) or four-wave instability is well above the observed energy densities, indicating that strong turbulence processes are probably not present during this event, even though such processes are often observed to occur in the type III source regions near 1 AU [Thejappa et al., 1993, 1995, 1996, 1999; Thejappa and MacDowall, 1998].

[32] Electrostatic decay was invoked to explain beam stabilization as well as second harmonic radio emission by several authors [see Goldman, 1984]. The present observations indicate that this process appears to be important for certain events. Moreover, the non-linear processes appear to be different for different events which depend on physical conditions of the ambient plasma.


[33] The URAP investigation is a collaboration of NASA Goddard Space Flight Center, the Observatoire de Paris-Meudon, the University of Minnesota, and the Centre d'etudes des Environnements Terrestre et Planetaires (CETP). We acknowledge A. Balogh and D. McComas for making other Ulysses data sets available through the NSSDC. The research of GT is supported by the NASA grants NAG57145, NCC-255 and NAG56059. E. S. and J. L. acknowledge support from the NASA Heliospheric Guest Investigator Program.

[34] Shadia Rifai Habbal thanks Paul J. Kellogg and two other referees for their assistance in evaluating this paper.