Observational Evidence for Langmuir Wave Collapse in the Source Region of a Solar Type III Radio Burst

High-time-resolution in situ wave observations show that Langmuir waves associated with solar type III radio bursts often occur as coherent localized one-dimensional magnetic-field-aligned wave packets with short durations of a few milliseconds and peak intensities well above the strong turbulence thresholds. In this paper, we report observations of a wave packet obtained by the time domain sampler of the STEREO WAVES experiment, which is unique in the sense that it is the most intense wave packet ever detected in association with a solar type III radio burst, with a peak intensity Et ∼ 107 mVm−1. We show that this wave packet provides evidence for (1) oscillating two-stream instability (OTSI), (2) a collapsing soliton formed as a result of OTSI, (3) the formation of a soliton–caviton pair, and (4) excitation of second and third harmonic electromagnetic waves. We also show that the peak intensity and spatial width satisfy the threshold condition for this wave packet to be the collapsing Langmuir wave packet formed as a result of nucleation processes even when δnb > δnp, where δnb and δnp are the levels of background and ponderomotive-force-induced density fluctuations, respectively. Thus, these observations provide unambiguous evidence for the spatial collapse of Langmuir waves in the source region of a type III radio burst, and the observed spectral evidence for OTSI and the ponderomotive-force-induced density cavity strongly suggest that OTSI is mostly likely responsible for the collapse of the observed wave packet.


Introduction
The purpose of this paper is to report evidence for spatial collapse of Langmuir waves observed by the time domain sampler (TDS) of the STEREO WAVES experiment (Bougeret et al. 2008) in the source region of a solar type III radio burst. The observations consist of the most intense localized magnetic-field-aligned Langmuir wave packet ever detected in association with a solar type III radio burst. We show that although the peak intensity and width of this wave packet satisfy the threshold conditions for nucleation processes, as well as the four-wave interaction called the oscillating twostream instability (OTSI), the spectral evidence for OTSI, the ponderomotive-force-induced density cavity, and the second and third harmonic spectral peaks likely corresponding to the second and third harmonic electromagnetic waves excited as a result of wave-wave interactions, strongly suggest that the OTSI is most likely responsible for the spatial collapse of this Langmuir wave packet. The solar flare accelerated electron beam, while propagating radially outward in the solar atmosphere excites electron plasma waves at the local electron plasma frequency, = [ ] f n Hz 9 pe e 1 2 , by the beam-plasma instability (n e is the electron density in m −3 ), and these waves are converted into the fundamental and harmonic radio emissions of the type III burst by some wave-particle and wave-wave interactions (Ginzburg & Zheleznyakov 1958). The nonlinear plasma processes responsible for saturation of these high-intensity Langmuir waves are of great interest for understanding not only the solar type III radio emissions, but also the beam-plasma interactions encountered in many laboratory and space plasma environments (Benz 2002;Gurnett & Bhattacharjee 2017).
Although in situ observations of the distributions of energetic electrons (Lin et al. 1981(Lin et al. , 1986Ergun et al. 1998) and associated Langmuir waves (Gurnett & Anderson 1977;Lin et al. 1986;Thejappa et al. 1993) in the source regions of solar type III radio bursts have improved our understanding of the complex physics behind this fascinating radio phenomenon, there remain several unresolved issues, such as: (1) the survival of the electron beam over large distances against the quasilinear relaxation (Sturrock's dilemma (Sturrock 1964)), and (2) conversion of Langmuir waves into type III radio emissions at the fundamental and higher harmonics of the electron plasma frequency, f pe (Thejappa et al. 1993). The induced scattering off background ions, which acts as the electrostatic decay (ESD) of beam-excited Langmuir waves into daughter Langmuir and ion sound waves in non-isothermal plasmas, has been suggested to resolve some of these issues (Tsytovich & Shapiro 1965;Kaplan & Tsytovich 1968). Signatures of ESD in type III bursts have also been reported (Lin et al. 1986;Gurnett et al. 1993;Hospodarsky & Gurnett 1995;Thejappa et al. 2003;Henri et al. 2009). However, theory predicts that when the intensities of Langmuir waves exceed the strong turbulence thresholds, the most relevant processes are the OTSI (Papadopoulos et al. 1974;Goldstein et al. 1979;Smith et al. 1979), formation of Langmuir solitons (Nicholson et al. 1978), and Langmuir collapse (Zakharov 1972;Bardwell & Goldman 1976;Goldman 1984); these processes can stabilize the beam by scattering the Langmuir waves toward higher wavenumbers with a faster rate. These strong turbulence processes are also shown to be very efficient in conversion of Langmuir waves into electromagnetic waves at the fundamental and higher harmonics of the electron plasma frequency, f pe (Papadopoulos et al. 1974;Papadopoulos & Freund 1978;Goldman et al. 1980;Akimoto et al. 1988). In spite of many years of in situ high-time-resolution measurements of the wave and beam parameters (Lin et al. 1986;Gurnett et al. 1993;Thejappa et al. 1993Thejappa et al. , 1999Thejappa & MacDowall 1998;Henri et al. 2009), evidence for spatial collapse of Langmuir waves has been proven to be extremely elusive.
The OTSI is a parametric four-wave interaction, 1 , in which two Langmuir waves L 1 and L 2 couple with an ion sound perturbation (S) and produce two oppositely propagating anti-Stokes U 1 and Stokes D 1 sidebands. The OTSI is initiated by the density cavity created in the ambient plasma by the ponderomotive force of the spatially non-uniform wave packet. In this case, the wave packet gets trapped inside the self-generated density cavity, intensifies and collapses to a state of stable Langmuir soliton in 1D. Numerical simulations (Doolen et al. 1985) have shown that a process called the nucleation instability can arise due to accumulation of background Langmuir wave energy in the pre-existing density cavities, which can lead to the faster attainment of the threshold for wave collapse, i.e., Langmuir collapse in inhomogeneous plasmas may take the route of nucleation instability.
The in situ wave data obtained in the source regions of type III bursts by the ULYSSES spacecraft have provided evidence for some strong Langmuir turbulence processes, such as the envelope solitons (Thejappa et al. 1993(Thejappa et al. , 1999, coexistence of weak and strong Langmuir turbulence processes (Thejappa & MacDowall 1998), and ion sound waves radiated as a consequence of burntout cavitons left behind by the collapse events (Thejappa & MacDowall 2004). Superior high-time-resolution in situ wave data from the STEREO WAVE experiments (Bougeret et al. 2008) and the higher-order spectral techniques (Thejappa et al. 2012a) have enabled us to successfully identify the signatures of OTSI (Thejappa et al. 2012a(Thejappa et al. , 2012b(Thejappa et al. , 2012c(Thejappa et al. , 2013a, and three-and four-wave interactions (Thejappa et al. 2012b), which have been confirmed by numerical simulations (Sauer & Sydora 2016). However, the wave packets undergoing the actual collapse have not been identified yet. In one of our earlier studies (Thejappa et al. 2013b), the observations of the collapsing wave packet and its associated density cavity were reported. The wave packet presented in this study differs from that of our previous study in three different ways: (1) it is much more intense, with E t ∼107.4 mVm −1 in comparison with 76 mVm −1 of the previous one, (2) its spectrum contains the signatures of OTSI, as well as harmonics unlike the previous event, and (3) it satisfies the condition for being the collapsing wave packet even when δn b >δn p , where δn b and δn P correspond to the background and ponderomotive-force-induced density fluctuations, respectively.
In this study, we describe some new observations. These consist of a wave packet, which is unique in the sense that for the first time it provides unambiguous evidence for the spatial collapse of Langmuir waves, where the path to collapse could be via either OTSI or nucleation instability. In Section 2, we describe the observations and in Section 3, we present the analysis of these observations, and in Section 4, we present the discussion and conclusions.

Observations
The observations are from the TDS of the WAVES experiment (Bougeret et al. 2008) of the STEREO B spacecraft. This experiment uses three mutually orthogonal monopole antennas with an effective length of ∼1 m. The high-and lowfrequency receivers of this experiment can detect radio emissions over a wide frequency range from 16 MHz to 2.5 kHz. In Figure 1, we present the dynamic spectrum, where the fast-drifting emission from 5 MHz down to 20 kHz is the type III burst, and the non-drifting emissions in the frequency interval 14-16 kHz are those of Langmuir waves. For the electron density of the solar wind n e (m −3 ), if we assume the Radio Astronomy Explorer density model (Fainberg & Stone 1971): where n 0 =5.52×10 13 , a=2.63, and r is the solar altitude in units of R e , we can convert frequency drifts of type III bursts df dt into velocities of the corresponding electron beams β (units of velocity of light c) traveling along the spiral magnetic field lines using the formula (Papagiannis 1970) Here, f is the angle of exciter direction to the Sun-spacecraft line, and f is the midpoint of frequency interval df, and the mode of emission is assumed to be the second harmonic. For the current event, we estimate v b as ∼0.3c by assuming that the pitch angle scattering increases the path length traveled by energetic electrons by a factor of α=1.7 (Lin et al. 1973;Alvarez et al. 1975;Fokker 1984).
TDS of the WAVES experiment provides the high-timeresolution in situ measurements of all three-wave electric field components. It has resolved the Langmuir waves seen in Figure 1 into several waveforms, one of which happens to be the most intense wave packet ever detected in a type III radio burst so far. This wave packet contains 16,384 samples acquired at the rate of 125,000 samples per second, i.e., its total duration is ∼131 ms. We have converted the electric potentials of this wave packet measured by the X-, Y-, and Z-antennas into electric field components in the spacecraft frame using the transfer matrix given by Bale et al. (2008). In Figure 2, we present the time profiles of these E X , E Y , and E Z components, with peak amplitudes of 57.8 mVm −1 , 55.7 mVm −1 , and 77.1 mVm −1 , respectively.
We have transformed these E X , E Y , and E Z field components from the spacecraft frame into a more useful magnetic-field-aligned coordinate system with the X-, Y-, In Figure 3, we present these E P ,Ê 1 andÊ 2 components aligned respectively along b,b v, and´b v b, with peak amplitudes of ∼105.9 mVm −1 , ∼12.5 mVm −1 , and ∼28 mVm −1 , respectively. The inequalities  E E 1 and  E E 2 imply that the observed wave packet is mostly a one-dimensional magnetic-field-aligned wave packet. Therefore, we mainly focus on the analysis of the E P component. The peak amplitude of the wave packet is The STEREO/PLASTIC (Galvin et al. 2008) and IMPACT magnetic field (Acuna et al. 2008) experiments measured the solar wind speed as v sw ∼590 km s −1 and the ambient magnetic field as B∼6.84 nT. As far as the electron and ion temperatures T e and T i are concerned, we have assigned a typical value of 10 5 K for T e and assumed that T e /T i ∼3; the measurements of T e are not available in this case. For electron density n e , we use a value of ∼3.1×10 6 m −3 , estimated from the measured f pe ∼15.8 kHz (see the FFT spectrum of the wave packet in the top panel of Figure Table 1.

Analysis
In the context of the present observations, we examine the following nonlinear processes: (1) OTSI, (2) a collapsing Langmuir soliton formed as a result of OTSI, (3) a density cavity created by the wave packet, (4) a collapsing Langmuir wave packet formed as a result of nucleation instability, and (5) excitation of harmonics.

Oscillating Two-stream Instability
The dispersion relation that describes all the relevant instabilities is (Zakharov et al. 1985) w q w w q w w where q  are the angles between k L and  k q L , W ( ) G q , is the Green function, and Ω and q are the frequency and wavenumber of the ion sound wave. Zakharov et al. (1985) have shown that the solution of this equation depends on two and k L , i.e., if we know these two parameters, we can pinpoint the mechanism responsible for saturation of the given wave packet. As seen in Figure 4, the saturation mechanisms are divided into five regimes: (I) electrostatic decay instability, (II) modulational instability, (III) subsonic modulational instability, (IV) supersonic modulational or OTSI, and (V) modified decay instability. A detailed description of these instabilities can be found in (Zakharov et al. 1985;Robinson 1997). Since the observed values~´-6 10  Figure 4), the relevant saturation mechanism is the supersonic modulational instability, which is also known as the OTSI. The threshold condition for OTSI and soliton formation Here, we note that this placement in region IV is specific to this event. Although the assignment of 10 5 K for T e due to a lack of measurements, and the estimation of the beam speed v b from the frequency drift of the type III burst, could have introduced some uncertainty in this placement, the observed spectral signatures of OTSI (discussed below) strongly support this placement. We also note that the wave packets observed in association with type III bursts mostly lie in region I (Lin et al. 1986;Gurnett et al. 1993;Hospodarsky & Gurnett 1995;Thejappa et al. 2003;Henri et al. 2009), less often in regions II and III (Thejappa et al. 1999), and very rarely in region IV (Thejappa et al. 2012c(Thejappa et al. , 2012a(Thejappa et al. , 2012b(Thejappa et al. , 2013a. As far as we know, there are no reports of wave packets lying in region V. Most likely, this is because for the wave packets to lie in region V, they should have large values of W n T L e e , as well as k L , i.e., slower electron beams, which is less likely for type III bursts.
The OTSI is a parametric four-wave interaction, where two beam-excited Langmuir waves, L 1 and L 2 , with frequency and wavenumber ( f k , pe L ), couple with an ion sound perturbation (S) with frequency and wavenumber (Ω, q) and produce two counter-streaming anti-Stokes and Stokes sidebands, U 1 and D 1 , with frequencies and wavenumbers , respectively. If this process is responsible for saturation of the observed wave packet, its FFT spectrum should contain peaks corresponding to beam-excited Langmuir waves (L), two sidebands U 1 and D 1 , and ion sound waves (S), with frequencies and wavenumbers of the corresponding waves satisfying the following resonance conditions: To verify these conditions, we have computed the FFT spectrum of the E P component. The logarithmic spectrum in a narrow frequency range from 14 to 17 kHz (top panel of Figure 5) clearly shows an intense peak (L) at~f f pe 15.82 kHz, likely corresponding to beam-excited Langmuir waves and two secondary peaks, likely corresponding to sidebands U 1 and D 1 atf 15.94 kHz U1 andf D1 15.74 kHz, respectively. The linear spectrum in the frequency interval from 0 to 500 Hz (bottom panel of Figure 5) also shows a general enhancement below 200 Hz with a peak at ∼100 Hz, likely corresponding to ion sound waves. Thus, the differences in the frequencies and -= f f 120 Hz U p e 1 agree reasonably well with the frequencies of the low-frequency enhancement with the peak at ∼100 Hz, i.e., the observed modes easily satisfy the frequency requirement.
We can verify the wavenumber resonance condition by estimating the combined frequency shift Δf suffered by the sidebands as a result of dispersion, as well as solar-windinduced Doppler shifts. From the modified dispersion relation of Langmuir waves, we can write the expression for the dispersion-related frequency shift Δf 1 of the sidebands with respect to f pe as De 2 We can also write the expression for the frequency shift suffered by these waves due to solar wind induced Doppler shift as De where θ is the angle between the wavevector k L of the wave packet and the solar wind v sw , which, in the present case is the angle between the magnetic field and the solar wind since  k b L . Therefore, for the sideband emissions propagating parallel and anti-parallel to k L , the angles between their wave vectors and the solar wind are 68 • and 112 • , respectively (the observed angle between the magnetic field and the solar wind is ∼68°), i.e., these two angles serve as the lower and upper bounds for Δf 2 . Thus, the expression for the total frequency shift is In Figure 6, we plot the frequency shift Δf versus kλ De for the observed parameters of v sw ∼590 km s −1 , f pe ∼15.82 kHz and λ De ∼12.4 m. As seen from this figure, the observed frequency differences -f f 80 Hz

pe D1
and -f f U p e 1 120 Hz of the sidebands propagating parallel and anti-parallel to the ambient magnetic field correspond to kλ De of ∼0.118 and ∼0.051, respectively. The fact that these values are much higher than the l~´k 1.37 10 L De 2 corresponding to the beam-excited Langmuir waves implies that they are most likely pumped to higher wavenumbers by OTSI. We can estimate the upper limit of the wavenumbers of ion sound waves as For v sw ∼590 km s −1 , θ∼68°, Ω=100 Hz, and λ De ∼ 12.4 m, we obtain q∼2.8×10 −3 m −1 and qλ De ∼3.5× 10 −2 . The value q∼2.8×10 −3 m −1 is also greater than k L ∼1.1×10 −3 . Thus, the frequencies and wavenumbers of the observed sidebands and low-frequency enhancements satisfy the resonance conditions (7) and (8). This implies that the observed sidebands and low-frequency enhancement probably correspond to the daughter waves excited by the OTSI, whereas the pump wave L corresponds to the beamexcited Langmuir wave. As far as the spectral peaks D 2 and U 2 are concerned, they likely correspond to the secondary Stokes and anti-Stokes modes, respectively, excited as a result of coupling of ion sound waves with wave vectors of » q 2 with beam-excited Langmuir waves. The ion sound waves with » q 2 probably are due to the coupling of primary sidebands and the beam-excited Langmuir waves. Similar observations of the wave packets with secondary sidebands have been previously reported (Thejappa et al. 2012b).

Collapsing Langmuir Soliton Formed as a Result of OTSI
The time profile of the total electric field, 2 , of the wave packet (Figure 7) clearly shows that the observed waveform is a localized intense wave packet with a e 1 -power duration of ∼11.6 ms. By assuming that the wave packet is stationary and the duration is mainly due to its Doppler shift in the solar wind, we can convert the duration τ e into spatial scale Δ e using the following relationship: Here, θ is the angle between the electric field and the solar wind, which in the present case is the angle between the magnetic field and solar wind since  E B. Thus, we obtain l D~206 e De for the observed τ e ∼11.6 ms and v sw ∼ 590 km s −1 , θ=68°, and λ De =12.4 m.
The wave packet can be identified as the Langmuir soliton formed as a result of OTSI if it satisfies the following criterion (Antipov et al. 1978;Nezlin 1993): in the dispersion relation (9) causes the wave packet to spread. The collapse and spreading of the wave packet balance each other when the width of the wave packet . Different regimes of instabilities of nonlinear Langmuir waves, defined as follows: regime I: electrostatic decay instability; regime II: modulational instability; regime III: subsonic modulational instability; regime IV: supersonic modulational instability, which is also known as the oscillating two-stream instability (OTSI); and regime V: modified decay instability. The present event, shown as the filled circle, lies in regime IV. reaches a critical value, and the corresponding field structure is usually referred to as the Langmuir soliton. Thus, the observed wave packet probably is still collapsing toward such a stable Langmuir soliton.

Density Cavity Created by the Wave Packet
The ponderomotive force of an intense Langmuir wave packet is known to create a density cavity of depth We can verify this using the measured dn n e e during this wave packet. For this purpose, we can use the following relationship (Kellogg et al. 2009;Henri et al. 2011): where T ph =3 eV is the photoelectron temperature, and the change in the spacecraft potential δΦ sc is the voltage measured by the X-, Y-, and Z-antennas in the frequency band from 100 to 2000 Hz. In Figure 8 Morales & Lee (1978) (Figure 4 in their paper). According to these authors, soliton-like structures radiate the Langmuir as well as ion sound waves, i.e., the observed density fluctuations can be interpreted as the ion sound waves radiated by the Langmuir soliton.

Collapsing Wave Packet Formed as a Result of Nucleation Instability
The nucleation instability arises as a result of accumulation of background Langmuir waves in the pre-existing density cavity. Because of such an accumulation of background Langmuir waves, the threshold for collapse may be attained at a faster rate through this route in comparison with that of OTSI. One can examine whether the observed wave packet satisfies the condition for it to be a collapsing wave packet formed as a result of nucleation instability. When the level of background density fluctuations δn b is much less than that of ponderomotive-force-produced density fluctuations δn p , i.e., δn b <δn p , the threshold condition for the wave packet to be a collapsing wave packet is (Robinson 1997): De where k B is the Boltzmann constant and Δ is the e 1 -level spatial width of the wave packet. In the opposite limit, i.e., when δn b >δn p , this threshold condition is modified as (Robinson 1997) Figure 7, the timescale τ e =11.6 ms is equivalent to the spatial scale Δ e ∼206λ De . For the observed parameters of n e ∼3.1×10 6 m −3 , T e =10 5 K and T e /T i =3, we obtain ion sound speed V s =4.1×10 4 ms −1 and electron thermal speed V Te ∼1.23×10 6 ms −1 . By substituting these values on the right sides of inequalities (18) and (19), we obtain the threshold values as ∼33.7 mVm −1 and ∼88 mVm −1 , when δn b <δn p and δn b >δn p , respectively. This implies that the observed wave packet with a peak intensity E t ∼107.4 mVm −1 easily satisfies the threshold condition for it to be the collapsing wave packet, even when the level of background density fluctuations (δn b ) is higher than that of the self-compression-produced density fluctuations (δn p ).
Here, we note that condition (18) is derived by assuming that the self-focusing and ponderomotive-force-induced compression of the wave packet exceed the dispersion-related broadening. This condition is similar to the condition for the formation of envelope solitons, namely, . In this case the background fluctuations do not have any influence on the collapse. Furthermore, the spectrum of the wave packet probably contains the sidebands as well as enhancement related to ion sound waves. This implies that when n b <n p , the nucleation instability is equivalent to the modulational instability of regime II of Figure 4. On the other hand, condition (19) is derived by assuming that the level of background density fluctuations is higher than the ponderomotive-force-induced density fluctuations. In this case, the background Langmuir waves get captured by these fluctuations, which eventually may lead to collapse, therefore the spectrum is not expected to contain any signatures of sidebands and ion sound waves. Thus, the wave packet formed as a result of nucleation instability when n b >n p cannot be placed in any region in Figure 4.

Harmonics
The logarithm spectrum of the E P component in the frequency interval from 12 to 60 kHz ( Figure 9) shows harmonic spectral peaks at f pe , 2f pe , and 3f pe , where f pe =15.8 kHz is the electron plasma frequency. The FFT spectrogram of the wave packet ( Figure 10) shows that the enhancements in the fundamental, second, and third harmonic emissions occur simultaneously, which suggests that these emissions are probably related to each other. As we discussed earlier, the spectral peak at ∼f pe corresponds to the beam-excited Langmuir waves. On the other hand, we can identify the 2f pe spectral peak with the second harmonic electromagnetic wave T f 2 pe , excited as a result of the three-wave interaction, since these modes satisfy the resonance conditions De pé -7.1 10 3 . As far as the spectral peak at 3f pe is concerned, it could be identified as the third harmonic electromagnetic wave generated as a result of the wave-wave interaction (Zheleznyakov & Zlotnik 1974;Zlotnik 1978) This identification requires the verification of phase coherence between these modes, which can be carried out using the bispectral analysis techniques (Thejappa et al. 2013a(Thejappa et al. , 2013b. We have computed the bicoherence spectrum of the E P component of the wave packet and found it to contain intense peaks at (15.8, 15.8) kHz and at (31.6, 15.8) kHz (Figure 11), with normalized peak bicoherences of ∼0.95 and ∼0.9, respectively. These bicoherences are indicative of threewave interactions of ( + 1 2 pe , since a second harmonic Langmuir wave cannot be excited by such a wave-wave interaction. Therefore, the peak at 2f pe in the FFT spectrum probably corresponds to the electromagnetic wave generated as a result of the merging of two oppositely propagating Langmuir waves, i.e., the spectral peak at 2f pe probably corresponds to T f 2 pe excited as a result of three-wave interaction, 2 pe , as suggested by Papadopoulos et al. (1974). Similarly, the merging of a Langmuir wave with a second harmonic electromagnetic wave, i.e., +  L T T f f 2 3 pe pe , probably is responsible for the spectral peak at 3f pe , i.e., the spectral peak at 3f pe corresponds to an electromagnetic wave, as suggested by Zheleznyakov & Zlotnik (1974).
The mean volume emissivity of a source located at a distance R in a mode I can be defined as (Thejappa et al. 2012a): where E I is the peak amplitude of the electromagnetic wave of the mode I, ΔΩ is the solid angle, and D = For E t =107.4 mVm −1 , the spatial scale L N =10 8 m, and n e =3.1×10 6 m −3 , we obtain Q∼2.8×10 −2 , whereas the observed ratio is 5.2×10 −2 .

Discussion and Conclusions
In this study, we have presented observations of Langmuir waves obtained by the STEREO WAVES experiment in the source region of a solar type III radio burst. We have focused mainly on one of the waveforms because of its uniqueness in the sense that it is the most intense one-dimensional magneticfield-aligned Langmuir wave packet ever detected in association with a solar type III radio burst. The peak intensity E t ∼107.4 mVm −1 of this wave packet is much higher than the present record of 76 mVm −1 (Thejappa et al. 2013b). We have also shown that although the measured wavenumber k L of this wave packet is greater than l ( ) place this wave packet in the supersonic modulational instability regime in the dispersion diagram (Zakharov et al. 1985).
Although the high-resolution wave data were searched for signatures of spatial collapse of Langmuir waves for a long time, to our knowledge this is the first time that evidence for such a phenomenon has been observed. The spatial collapse in the present case should be understood as a collapse toward a stable structure called the Langmuir soliton, formed as a result of balance between the nonlinearity-induced self-compression and the dispersion-induced spreading of the wave packet. In this way, it differs from the spatial collapse of a threedimensional wave packet.
There are two routes for collapse: (1) the OTSI, which is responsible for the sidebands and ion sound waves, as well as  . Logarithmic spectrum of the E P component of the wave packet from 12 to 60 kHz. The frequencies corresponding to spectral peaks are f pe , 2f pe , and 3f pe , where f pe is the electron plasma frequency. soliton-caviton pair, and (2) the nucleation instability, which is related to the trapping of background Langmuir waves in the density cavities left behind by previous collapses or ambient density fluctuations. We have shown that (1) the FFT spectrum of the wave packet contains the signatures of OTSI, namely: (a) an intense peak L at the local electron plasma frequency, f pe , likely corresponding to the beam-excited Langmuir waves, (b) two relatively weaker peaks at f pe ±Ω, likely corresponding to the up-and down-shifted sidebands (anti-Stokes (U 1 ) and Stokes (D 1 ) modes), respectively, and (c) a low-frequency peak at Ω likely corresponding to ion sound waves; the frequencies and wavenumbers of these waves satisfy the resonance conditions of OTSI, (2) a ponderomotive-force-induced density cavity is observed in association with this wave packet, as expected of OTSI; the OTSI is expected to lead to the formation of the Langmuir soliton and an associated density cavity; and (3) the peak intensity and width of the wave packet are consistent with this wave packet collapsing toward a stable soliton, since the self-compression due to nonlinearity dominates over the spreading due to dispersion. We have also  shown that the observed wave packet also satisfies the threshold condition for it to be a collapsing wave packet formed as a result of nucleation instability, even when the level of density fluctuations δn p associated with the ponderomotive force of the wave packet is less than that of the background fluctuations δn b . As far as we know, this is the first time that evidence for a collapsing wave packet formed as a result of nucleation instability has ever been reported in the source region of a solar type III burst. However, the signatures of OTSI in the FFT spectrum of the wave packet, and the occurrence of a ponderomotive-force-induced density cavity in association with the wave packet, strongly favor the OTSI as the route for Langmuir collapse.
We have also shown that (1) the FFT spectrum of this wave packet contains (a) a resonant peak at the local electron plasma frequency, f pe , (b) a slightly weaker peak at 2f pe and (c) a very weak peak at 3f pe ; (2) the time-frequency spectrogram indicates that these harmonic components are coincident in time; (3) the bicoherence spectrum of this 1D wave packet contains the characteristic signatures of three-wave interactions, namely (a) a peak at ( f pe , f pe ), which is indicative of a perfect phasecoupling between two oppositely propagating Langmuir waves and a second harmonic electromagnetic wave, and (b) a peak at (2f pe , f pe ), which is indicative of phase-coupling between the Langmuir, the second harmonic electromagnetic and the third harmonic electromagnetic waves; and (4) the Langmuir waves involved in the excitation of the second and third harmonic electromagnetic waves can be either the sideband emissions excited by the OTSI or the trapped Langmuir waves in the density cavity created by the ponderomotive force.
Thus, we conclude that (1) the wave packet presented in this study provides the first observational evidence for spatial collapse of Langmuir waves, where the route for collapse is probably via OTSI, and (2) the strong turbulence processes, namely OTSI, formation of Langmuir solitons, and Langmuir collapse, play critical roles in beam stabilization as well as the excitation of electromagnetic emissions of solar type III radio bursts, when the intensities of Langmuir waves are high.