Phase coupling in Langmuir wave packets: Evidence of four wave interactions in solar type III radio bursts



[1] It is reported that one of the Langmuir wave packets in a type III solar radio burst is observed as a localized field structure with a short duration ∼3.2 ms and high intensity exceeding the strong turbulence thresholds. The peak intensity and short time scale well satisfy the criterion for the observed wave packet to be the collapsing Langmuir envelope soliton. The spectrum of this wave packet contains a resonant peak at the local electron plasma frequency, fpe, a Stokes peak at a frequency slightly lower than fpe, anti-Stokes peak at a frequency slightly higher thanfpe, and a low frequency enhancement below a few hundred Hz, which satisfy the frequency and wave number resonance conditions of the oscillating two stream instability (OTSI) type of four wave interaction. Here, for the first time, we apply the trispectral analysis technique, and show that the spectral components of this wave packet, namely, the beam-generated Langmuir wave, Stokes and anti-Stokes modes are coupled to each other with a high degree of phase coherency (high tricoherence). This conclusively shows that the observed characteristics of the wave packet provide evidence for the OTSI.

1. Introduction

[2] The characteristic feature of solar type III radio bursts is the fast negative frequency drift from hundreds of MHz to tens of kHz. Ginzburg and Zheleznyakov [1958] were the first to propose a two step process for the production of these bursts: (1) excitation of high levels of Langmuir waves at the local electron plasma frequency fpe = 9ne1/2by flare accelerated electron beams through bump-on-tail instability (ne is the electron density in m−3), and (2) subsequent conversion of these Langmuir waves into radio emissions at fpe and 2fpe. This hypothesis, known as the plasma hypothesis, has been confirmed by the in situ detection of electron beams [Lin et al., 1973, 1986] as well as Langmuir waves [Gurnett and Anderson, 1976] in type III burst source regions. However, the question, how does the electron beam preserve its bump-on-tail distribution over distances of 1 AU and more against quasi-linear relaxation (Sturrock's dilemma [Sturrock, 1964]), remains unanswered. Here, one should note that Sturrock did not consider the effects of re-acceleration of beam particles by Langmuir waves. For example, by treating the wave particle interaction along the beam path self-consistently, some authors [Zaitsev et al., 1972; Escande and de Genouillac, 1978] claim that the beam can survive large distances of the order of 1 AU and more.

[3] Currently, it is believed that some nonlinear process, which can pump the Langmuir waves out of resonance with the beam faster than the quasi-relaxation time is probably responsible for beam stabilization. The induced scattering off ion clouds, whose resonance version is the electrostatic decay (ESD) of initial Langmuir wave into a daughter Langmuir wave and an ion sound wave whenTe > Ti (Te and Ti are the electron and ion temperatures, respectively) is suggested as one such mechanism [Kaplan and Tsytovich, 1968]. Although some signatures of ESD are detected in type III sources [Lin et al., 1986; Gurnett et al., 1993; Hospodarsky and Gurnett, 1995; Thejappa and MacDowall, 1998; Thejappa et al., 2003; Henri et al., 2009], the time scale appears to be too long to prevent the plateau formation [Zheleznyakov and Zaitsev, 1970]. Some authors suggested that the strong turbulence processes, namely, the four-wave interactions called the OTSI [Papadopoulos et al., 1974; Smith et al., 1979; Goldstein et al., 1979], and related soliton formation and spatial collapse [Zakharov, 1972; Nicholson et al., 1978] are more effective in stabilizing the electron beam. The OTSI excites a low frequency ion sound wave, which can beat with two of the beam-excited Langmuir waves and produce down-shifted and up-shifted side bands. The spatial collapse, on the other hand, occurs due to intensification of the Langmuir wave packet in the self generated density cavity. Although some signatures of the strong turbulence processes in Jupiter's foreshock [Gurnett et al., 1981], solar wind [Kellogg et al., 1992], and source regions of type III bursts [Thejappa et al., 1993; Thejappa and MacDowall, 1998; Thejappa et al., 1999; Thejappa and MacDowall, 2004] have been reported, in these cases, high intensities which are well above strong turbulence thresholds and the sideband spectral structures were not observed simultaneously.

[4] Thejappa et al. [2012]have reported the high time resolution observations of a Langmuir wave packet associated with a type III radio burst. This wave packet is characterized by (1) side band spectral structures as well as low frequency enhancement which satisfy the frequency and wave number matching rules of OTSI, and (2) peak intensity which satisfies not only the threshold condition of OTSI, but also the criterion for the observed wave packet to be the collapsing envelope soliton. In this study, for the first time, we apply the trispectral analysis on this Langmuir wave packet, and show that its spectral components, namely, the beam-generated Langmuir wave, Stokes and anti-Stokes modes are coupled to each other with a high degree of phase coherency (high tricoherence). This conclusively shows that these observations provide evidence for the OTSI as correctly concluded byThejappa et al. [2012]. In section 2, we review the observations, and in sections 3 and 4,we present the trispectral analysis and conclusions, respectively.

2. Review of Observations

[5] Figure 1 shows the STEREO/WAVES [Bougeret et al., 2008] observations of a local type III burst, drifting fast from ∼5 MHz to the local electron plasma frequency, fpe ∼ 30 kHz, and non-drifting Langmuir wave emissions in the 27–32 kHz interval. The Time Domain Sampler (TDS) [Kellogg et al., 2009] of the SWAVES experiment, which samples the A/C electric field from 3 orthogonal antennas has resolved these Langmuir waves into intense waveforms. Each of these waveforms contains 16384 samples with an acquisition rate of 250,000 samples per second (a time step of 4 μs for a total duration of 65 ms). Figure 2 shows the most intense wave packet captured by the Ex antenna, which is characterized by the peak electric field strength EL of 56.5 mVm−1 and inline image-power durationτ of ∼3.2 ms. Since the Ey, Ez and Ex − y signals are weaker and show the same general features as the Ex signal, Thejappa et al. [2012] analyzed only the Exsignal, and justified one-dimensional treatment by assuming that these Langmuir wave fields are probably aligned along the ambient magnetic field.Figure 3 shows the spectral characteristics of this wave packet: (a) the total spectrum extending from 0 to 65 kHz, (b) the narrow spectrum, containing an intense peak (L) at fpe ∼ 30 kHz, corresponding to ne ∼ 1.1 × 107 m−3, a Stokes peak (D) at ∼29.54 kHz and an anti-Stokes peak (U) at ∼30.41 kHz, and (c) the low frequency spectrum, containing a clear ion-sound wave associated enhancement below 450 Hz. The STEREO/PLASTIC experiment [Galvin et al., 2008] has measured the solar wind speed vsw as ∼450 kms−1. The electron temperature Te during this event is assumed to be ∼105 K. Assuming that the type III electrons propagate along the Parker's spiral field lines, a frequency drift curve is fitted to the dynamic spectrum. This yielded the beam speed vb of ∼0.22c for the Radio Astronomy Explorer (RAE) density model [Fainberg and Stone, 1971] (c is the velocity of light). Since, the pitch angle scattering is known to increase the path length of electron beams by a factor of α = 1.3 to 1.7 [Alvarez et al., 1975; Lin et al., 1973], the beam speeds vb and the wave numbers of Langmuir waves kL = inline image are corrected accordingly to range from ∼0.29c to ∼0.37c, and from ∼2.2 × 10−3 m−1 to ∼1.7 × 10−3 m−1, respectively. The observed quantities yielded: (1) Debye length, λDe = 69Te1/2ne−1/2 ∼ 6.6 m, and (2) normalized peak energy density inline image = inline image ∼ 10−3, and (3) the inline image-power spatial scale of the wave packetS ∼ τvsw ∼ 219λDe.

Figure 1.

Dynamic spectrum of a local type III radio burst (fast drifting emission from ∼5 MHz down to ∼30 kHz) and associated Langmuir waves (non-drifting emissions in the frequency interval 27-32 kHz).

Figure 2.

The Langmuir wave packet observed by the Time Domain Sampler (TDS) during the type III event of Figure 1. The inline image power duration of 3.2 ms which is equivalent to the spatial scale of 219λDe is also shown.

Figure 3.

(a) The spectrum of the TDS event from 0 to 65 kHz, (b) the narrow spectrum around f ∼ fpe ∼ 30 kHz, where the L, D, and Urefer to the beam excited Langmuir wave, Stokes peak at ∼29.54 kHz, and anti-Stokes peak at ∼30.41 kHz, respectively, and (c) the low frequency spectrum which shows an enhancement below 450 Hz corresponding probably to ion-sound waves.

[6] The threshold for the OTSI [Zakharov, 1972] inline image > (kLλDe)2 is easily satisfied in this case, since inline image ∼ 10−3 and (kLλDe)2 ranges from ∼1.3 × 10−4 to ∼2.1 × 10−4. The wave packet also satisfies the criterion of a collapsing envelope soliton [Thornhill and ter Haar, 1978] inline image ≥ (ΔkλDe)2, (Δk = inline image is the wave number characteristic of the envelope), since the observed inline image ∼ 10−3 is greater than (ΔkλDe)2 ∼ 8 × 10−4 estimated for the spatial scale S of ∼ 219λDe.

[7] The observations of a strong Langmuir wave peak with upper and lower sidebands, together with low frequency enhancement were interpreted in terms of OTSI, in which, the beam driven Langmuir wave is the pump wave, and the sidebands and low frequency waves are the nonlinearly excited daughter waves. The frequency matching condition 2fL = fD + fU is easily satisfied, since the frequency shifts Δfof the down-shifted and up-shifted side bands are symmetric with respect to the Langmuir wave pump, being ∼442.5 Hz and ∼427 Hz, respectively, which are also in good agreement with the frequencies of the ion sound waves of <450 Hz. Using the expression for the Doppler's shift Δf = inline image(De)cosθ, the wave numbers De can be estimated as ∼−0.04 and ∼0.04 for Δfof ∼442.5 Hz and 427 Hz corresponding to the Stokes and anti-Stokes modes, respectively. Here,θ is the angle between inline image and vsw; θ = 0 and θ = πcorrespond to the anti-Stokes and Stokes modes propagating away from and toward the Sun, respectively. This suggests that the beam excited Langmuir waves withkLλDe ∼ 10−2 are pumped into those of forward and backward propagating daughter waves with higher wave numbers. Similarly, using the expression for the Doppler shift, q =  inline image, the wave numbers of the ion sound waves De can be estimated as ≃ 0.042 for Ω = 450 Hz and vsw = 450 kms−1, which are comparable to those of the sideband emissions. Thus, the matching condition inline image is reasonably satisfied, yielding inline image, since kL is three to four times less than q.

[8] In the present case, the inequalities inline image and inline image are easily satisfied, indicating that the wave packet is in the supersonic modulational instability regime as referred by Zakharov et al. [1985]. Here mei are the electron and ion masses, respectively, and inline image. The growth rate of the OTSI inline image ∼ ( inline image)1/2 is ∼4.3 × 10−4, whereas, the bandwidth inline image= 3(kLλDe)2 inline image is ∼3 × 10−6. Here, the spectral width inline image [Lin et al., 1986; Benz, 2002] is estimated for the velocity spread of the beam Δvb ∼ 0.1vb and the linear growth times N before the onset of OTSI of ∼9. Thus, the growth rate is much larger than the bandwidth, which indicates that the observed wave packet is monochromatic enough for excitation of OTSI.

3. Trispectral Analysis

[9] In the power spectrum estimation, the waveform is treated as a superposition of statistically uncorrelated harmonic components, and the phase relations between the spectral components are suppressed. To extract the information regarding the presence of nonlinearities, one has to use the higher order spectra, defined in terms of the higher order moments or cumulants of the data. The third-order spectrum, commonly referred to as the bispectrum decomposes the signal's third moment over frequency. Since it is related to the skewness of a signal, it is usually used to detect the asymmetric non-linearities, such as the electrostatic decay [Henri et al., 2009; Balikhin et al., 2001; Walker et al., 2003] and harmonic generation [Bale et al., 1996]. The amplitude of the bispectrum at the bifrequency (fk, fl) measures the amount of coupling between the frequencies, fk, fl and fk+l.

[10] The fourth-order spectrum is referred to as the trispectrum. It can be viewed as the decomposition of the signal's kurtosis over frequency. If the signal is unskewed and contains information about a symmetric process, such as the OTSI type of four wave interaction, it can be extracted using the trispectral analysis. The trispectral method has been developed [Kravtchenko-Berejnoi et al., 1995a] and applied to synthetic [Kravtchenko-Berejnoi et al., 1995b; Lefeuvre et al., 1995] and to simulated data [Soucek et al., 2003]. The trispectral analysis can detect the phase relationships among four Fourier components, which is the key information regarding four-wave interaction. In OTSI, the sidebands interact with the strong beam excited Langmuir waves satisfying the matching rules for wave numbers, frequencies and phases, simultaneously. For the OTSI type of four-wave interaction, the cumulant based trispectrum is given by [Kravtchenko-Berejnoi et al., 1995a]

display math

where X1, X2, X3 and X4 are the complex Fourier components of the signal corresponding to frequencies f1, f2, f3 and f4, N(1, 2, 3, 4) = E[X1X2]E[X3*X4*] + E[X1X3*]E[X2X4*] + E[X1X4*]E[X2X3*], f4 = f1 + f2 − f3, and E[] is the expectation operator. The tricoherence, which is the normalized trispectrum is more useful, because it eliminates the dependence of trispectrum on the amplitude of the signals. The expressions for cumulant based square tricoherence function can be written as [Kravtchenko-Berejnoi et al., 1995a]:

display math

A unit value for the tricoherence indicates perfect coupling, a zero value indicates no coupling, and any value between zero and one indicates partial coupling. The tricoherence quantifies the fraction of the total product of powers at the frequency quartet (f1, f2, f3, f1 + f2f3), that is owing to cubicly phase-coupled modes. The tricoherence is zero for a Gaussian process. Even for a Gaussian process, due to statistical fluctuations, the estimate of the tricoherence from a finite data record will not be zero. Therefore, the method of periodograms is used to estimate the trispectrum and tricoherence. This involves the division of the data record into M segments; an appropriate window is applied to each segment to reduce leakage; the trispectrum as well as tricoherence are computed for each segment by using the DFT; finally, the trispectrum is averaged across segments in order to reduce the variance of the estimator. As seen fromequation (1), the trispectrum estimator is symmetric with respect to permutations of its arguments f1, f2 and f3. The principal domain for the interaction of the type f1 + f2 = f3 + f4 is determined as [Kravtchenko-Berejnoi et al., 1995a] 0 ≤ f1 ≤ fN, 0 ≤ f2 ≤ f1, 0 ≤ f3 ≤ f2, and f3 ≤ f1 + f2 − f3 ≤ fN, where fN is the Nyquist frequency.

[11] In this study, we have used the segment length N = 1000 (0.004 s), number of segments M = 16 and Hamming window, and calculated the tricoherence spectrum as a function of three frequencies. Since it is difficult to visualize the results in such a 3-D space, we have fixed one of its frequencies and displayed the results in two dimensions. We have made the cross-sections of the tricoherence spectrum at frequenciesfD = 29.5 kHz, fL = 30 kHz and fU = 30.5 kHz, corresponding to the Stokes, beam-excited Langmuir, and anti-Stokes modes, respectively. InFigure 4, we present these cross sections. The cross-section of the tricoherence spectrum atfD = 29.5 kHz(top panel) shows: (1) a peak at (29.5 kHz, 29.5 kHz, 29.5 kHz); this is due to self-dependence and is of no interest, (2) a second peak at (30.5 kHz, 30 kHz, 29.5 kHz), which is due to the phase relation 2ϕL = ϕD + ϕU, where ϕL, ϕD and ϕUare the phases of the beam-excited Langmuir, Stokes and anti-Stokes modes, respectively; the maximum tricoherencet2 ∼ 0.53 in this case, and (3) a third peak at (30.5 kHz, 30.5 kHz, 29.5 kHz); this appears to be due to the interaction of the sidebands themselves, and it is of no interest in the present case. The cross-section of the tricoherence spectrum atfL = 30 kHz (second panel) shows (1) a peak at (30 kHz, 30 kHz, 30 kHz), which is due to self-dependence, and (2) a second peak at (30 kHz, 30.5 kHz, 30 kHz), which is due to the phase relation 2ϕL = ϕD + ϕU; in this case, the maximum t2 ∼ 0.5. The tricoherence cross-section atfU = 30.5 kHz (bottom panel) does not show any peaks. Thus, the significant tricoherences corresponding to 2ϕL = ϕD + ϕUin the cross-sections of the tricoherence spectrum at Stokes and beam-excited Langmuir wave frequencies provide evidence for the OTSI type of four-wave interaction.

Figure 4.

The cross-sections of the tricoherence spectrum of the TDS event ofFigure 2at Stokes, beam-excited Langmuir and anti-Stokes frequencies (top to bottom), respectively.

4. Conclusions

[12] Thejappa et al. [2012] had reported the high time resolution observations of a very coherent and intense Langmuir wave packet in the source region of a local type III radio burst. This wave packet is characterized by (1) a spectrum containing (a) a resonant peak at the local electron plasma frequency, fpe, (b) Stokes peak at a frequency slightly lower than fpe, (c) anti-Stokes peak at a frequency slightly higher thanfpe, and (d) low frequency enhancement corresponding to ion sound fluctuations; the frequencies and wave numbers of these spectral components satisfy the resonance conditions of oscillating two stream instability (OTSI), (2) peak intensity, which is well above the threshold for strong turbulence processes, and (3) short time scale of a few ms. These observations were interpreted in terms of OTSI. In this study, for the first time, applying the trispectral analysis on this Langmuir wave packet, it is shown that the spectral components of the wave packet are coupled to each other with a high degree of phase coherency (high tricoherence), which provides crucial support for the interpretation of these observations in terms of OTSI.


[13] The research of T. G. is supported by the NASA grants NNX08AO02G and NNX09AB19G. The SWAVES instruments include contributions from the Observatoire of Paris, University of Minnesota, University of California, Berkeley, and NASA/GSFC. We thank M. J. Reiner for calculating the electron beam speed. We also thank the referees for very constructive comments and helpful suggestions.

[14] The Editor thanks Donald Gurnett and an anonymous reviewer for their assistance in evaluating this paper.