Langmuir “snakes” and electrostatic decay in the solar wind



[1] When Langmuir waves are driven by an electron beam to large amplitudes, they can undergo electrostatic (ES) decay to smaller wave numbers via a series of backscatters. Truncated ES decay, where the number of backscatters is reduced due to damping, is modeled here using the three-dimensional ES Zakharov equations. Langmuir beats develop in “snake”-like structures parallel to the electron beam direction and are most evident when decay is truncated to a single backscatter. From these results, an analytic form is derived and shown to be consistent with some of the waveforms and spectra observed by STEREO in the source regions of type III solar radio bursts. The agreement between the model and observations provides strong evidence for ES decay and Langmuir “snakes” parallel to the electron beam and so the ambient magnetic field.

1 Introduction

[2] In the solar wind, Langmuir waves associated with type III solar radio bursts are generated by electron beams produced during solar flares. Electron beams generate Langmuir waves via the bump-on-tail instability [Scarf et al., 1971; Gurnett and Anderson, 1976], which are subsequently converted to fundamental and harmonic radio emission [Ginzburg and Zhelezniakov, 1958]. Proposed mechanisms for the generation of radio emission involve electrostatic (ES) and electromagnetic (EM) decay [Melrose, 1980; Lin et al., 1986; Cairns and Robinson, 1995], linear mode conversion [Field, 1956], emission from Langmuir eigenmodes [Ergun et al., 2008; Malaspina et al., 2010], modulational instabilities [Papadopoulos et al., 1974], and wave packet collapse [Thejappa et al., 2012].

[3] Previous analytic and observational work has shown that the conditions required for modulational instabilities are rarely satisfied [Cairns and Robinson, 1998; Cairns et al., 1998]. Similarly, there is a lack of observational evidence for wave packet collapse [Nulsen et al., 2007; Graham et al., 2012a, 2012b]. Langmuir waveforms consistent with eigenmodes of density wells are commonly observed [Ergun et al., 2008] and may account for some of the radio emission [Malaspina et al., 2010], but many of the waveforms and spectra observed are inconsistent with localized eigenmodes. Langmuir beating and split spectral peaks are often observed near the local plasma frequency, suggestive of ES decay occurring [Cairns and Robinson, 1992; Hospodarsky and Gurnett, 1995; Henri et al., 2009]. A similar process to ES decay is scattering off thermal ions (STI). Although STI and decay produce similar product Langmuir waves, analytic work [Cairns, 2000; Mitchell et al., 2003] shows that decay should be favored over STI for typical solar wind conditions. Non-thermal ion-acoustic waves consistent with decay are often observed [Lin et al., 1986; Robinson et al., 1993; Cairns and Robinson, 1995].

[4] In this letter, we model ES decay using the Zakharov equations [Zakharov, 1972; Robinson, 1997] and show that intense beating of driven and backscattered Langmuir waves develops in “snakes” parallel to the mean wave vector of driven Langmuir waves and thus parallel to the electron beam. An analytic expression is derived for the electric field observed during a transit of one of these snakes. We show that specific Langmuir waveforms observed by Solar Terrestrial Relations Observatory (STEREO) [Bougeret et al., 2008] during type III radio bursts, which have two well-defined spectral peaks and appear to be localized, are consistent with these snakes of Langmuir beats. Moreover, the Doppler-shifted frequency differences observed between the spectral peaks near the plasma frequency in these events are consistent with ES decay, providing evidence for ES decay and Langmuir snakes in the solar wind during type III bursts.

[5] The outline of the paper is as follows. In section 2, we model ES decay by solving the Zakharov equations. In section 3, we derive an analytic expression for the observed waveform of a transit through the snake. In section 4, we apply this expression to Langmuir waveforms observed in type III bursts. Section 5 is the summary.

2 Simulations of Truncated Backscatter

[6] Truncated backscatter of Langmuir waves is modeled here using the ES Zakharov equations. In dimensionless form, these equations are [Zakharov, 1972; Robinson, 1997]

display math(1)
display math(2)

where E is the electric field envelope, δn is the density perturbation from the mean, cs=(1+3Ti/Te)1/2 is the ion-acoustic speed normalized to the cold-ion limit, and inline image and inline image are the Langmuir and ion-acoustic damping terms, respectively. In these equations time, length, δn, and E are expressed in units of inline image, (9mi/4me)1/2λD, (4me/3mi)n, and (16nkBTeme/3miε0)1/2, respectively. Here me and mi are the electron and ion masses, ωp is the angular electron plasma frequency, n is the unperturbed electron number density, λD is the Debye length, Te is the electron temperature, kB is Boltzmann's constant, and ε0 is the permittivity of free space. We use the temperature ratio Ti/Te=0.2, which gives inline image, where α=0.6 and k is the wave number, corresponding to strong damping of ion-acoustic waves [Robinson and Newman, 1989].

[7] We solve the Zakharov equations in three dimensions on a 1283 grid with dimensions of (1200λD)3, using the code in Graham et al. [2011]. The numerical integration is optimized using the phase-removal scheme of Skjaeraasen et al. [2011]. In our simulations, damping is applied to Langmuir waves at all k at a rate γ0=−6×10−4ωp so that ES decay is truncated to a single backscatter. This damping rate is consistent with the damping rates of backscattered Langmuir waves at 1 AU [Robinson and Cairns, 1995]. However, similar Langmuir beats develop for a range of γ0 [Robinson et al., 1992], and for γ0=0 [Robinson and Newman, 1989; Graham et al., 2012c]. Langmuir waves are driven in k-space at a rate given by

display math(3)

where Γb is the maximum driving strength, kB is the driving wave number in the kx direction, and w0 is the driver width perpendicular to the electron beam. In type III bursts, the range of beam-driven wave numbers is greater in the perpendicular direction than the parallel direction [Robinson et al., 1993]. In solar wind and foreshock plasmas, the electron beam is parallel to the magnetic field B, so beam-driven Langmuir waves will have fields and wave vectors roughly parallel to B. We use the driving parameters Γb=3×10−4ωp, kb=0.105kD, and w0=0.025kD, where kD is the Debye wave number, corresponding to a narrow electron beam (Δvb/vb≈0.1) with speed vb≈10ve [Robinson et al., 1993]. These parameters are reasonable: a type III beam with relative beam density of 10−6 and a beam speed equal to 20ve, where ve is the electron thermal speed, has kb=0.05kD and Γb≈4×10−4ωp for a Maxwellian plasma and beam with equal electron temperatures.

[8] Figure 1 shows energy density isosurfaces of the Langmuir “snakes” that develop. The Langmuir snakes are clearly defined, having much larger energy density than the system-wide mean, and remain close to parallel to the driving wave vector kb over large length scales. Some snakes extend the entire length (1200λD) of the simulation in the x direction. The periodic beating within each snake is due to the driven Langmuir waves and backwards propagating backscattered Langmuir waves beating together. Since background damping is strong, which reduces spectral broadening, and kbw0, Ey and Ez have much weaker field strengths and have much lower wave numbers than Ex, so E remains approximately parallel to the electron beam velocity vb. The snake widths are ∼100λD, as seen in Figure 2a. When the driver has a finite width ww0 parallel to vb, the Langmuir “snakes” still extend distances much greater than the perpendicular length scale.

Figure 1.

Isosurfaces of constant energy density. Axes are labeled in simulation grid points.

Figure 2.

Energy density contours and density perturbation contours of truncated backscatter. (a) Contours of W. (b) Density perturbation contours (δn<0 dark, δn>0 light). (c) Electric field in the x direction (black) and density perturbations δn/n (red) observed along the red line in Figures 2a and 2b.

[9] For w0=0.025kD, the global periodic behavior observed for narrow w0 [Robinson and Newman, 1989; Robinson et al., 1992] is lost due to the decrease in the coherence length of the driver. In this case, the variations in time of the mean energy density and the energy of the driven Langmuir waves are much less extreme than in the w0=0 limit. At all times, there are multiple Langmuir snakes in the simulation volume for these parameters. Langmuir “snakes” are also observed in two- and three-dimensional simulations for a range of Γb when γ0=0, and a narrow driver is used [Robinson and Newman, 1989; Robinson et al., 1992; Graham et al., 2012c].

[10] Figures 2a and 2b show contour plots of the Langmuir energy density and density perturbations, respectively. The density perturbations produced by ES decay occur in the snakes where Langmuir beating is most intense, as seen by comparing Figures 2a and 2b. The Langmuir energy density fluctuates periodically due to Langmuir beating, and the density perturbations vary approximately sinusoidally in the x direction.

[11] In the solar wind at 1 AU and Earth's foreshock, the angle θ between B and the solar wind flow vsw is typically about 45°. Therefore, if the snakes shown in Figure 2 occur, we expect a satellite to pass through them at an angle ∼45° if its speed is much smaller than the solar wind speed. Figure 2c shows the observed electric field parallel to B and the density perturbations observed along the trajectory denoted by the red line in Figures 2a and 2b. When passing through a Langmuir snake, the field and density perturbations undergo intense fluctuations; outside the snake, the fields and density perturbations are small.

3 Analytic Form

[12] In this section, we develop a simple analytic form for the electric field associated with Langmuir snakes. We assume the Langmuir waves are driven in three dimensions by a finite width driver given by ((3)) with γ0=0. In the steady-state limit, when the linear growth rate of the beam driven waves is matched by the rate of ES decay, Wb∝Γ(k) [Robinson, 1997], where Wb is the energy density of the beam-driven Langmuir waves. Assuming the steady-state limit, the electric field of driven Langmuir waves in k-space is approximately in the kx direction, with

display math(4)

where AL is the amplitude of the electric field in k-space. We then inverse Fourier transform ((4)) to obtain

display math(5)

where A0 is the amplitude of the driven Langmuir waves in real space.

[13] We make the following assumptions: (1) The backscatter spectrum is given by an analogous equation to ((4)), except the waves are at kx=−kL+k0, where inline image, inline image is the ion-acoustic speed, and inline image. (2) The satellite is assumed to transit along a straight line, as in Figure 2, so we assume the satellite remains in the z=0 plane throughout transit. (3) We assume that the magnitude and k-space width of the backscatter spectrum can differ from those of the driven Langmuir waves. (4) Second and higher order backscatters are not included. (5) High frequency oscillations, given by the linear dispersion relation inline image, near ωp are included.

[14] With the above assumptions, the solution becomes

display math(6)

where subscripts 0 refer to the beam-driven Langmuir waves and subscripts 1 to the first backscatter. The amplitudes A0,1 depend on the distance in the z-direction of the center of the snake from the z=0 plane and the relative field strengths of the driven and backscattered Langmuir waves.

[15] We now reformulate ((6)) in terms of Doppler-shifted Langmuir wave frequencies for comparison with solar wind observations. In the solar wind, we assume STEREO is stationary, with the solar wind and Langmuir event convecting past at the solar wind speed vsw (STEREO's orbital speed is ∼30 km s−1vsw∼400 km s−1). The relative position of the event to the satellite is x=−vswt| cosθ| and y=vswt| sinθ|−y0, where θ is the angle between vsw and B at the time of the event, and y0 is the position of the center of the snake in the direction perpendicular to B.

[16] In the satellite frame, Doppler-shifted frequencies are observed. So ((6)) becomes

display math(7)

where the Doppler-shifted frequencies of the beam-driven Langmuir waves (inline image) and the first backscatter (inline image) are

display math(8)
display math(9)

The Doppler-shifted frequency difference inline image between the driven Langmuir waves and the first backscatter, assuming they propagate in opposite directions along ±B, is [Cairns and Robinson, 1992; Henri et al., 2009]

display math(10)

Here kb=2πfp/vb and vb is the electron beam speed. The Doppler-shifted frequency of the ion-acoustic waves produced during ES decay is inline image [Cairns and Robinson, 1992; Henri et al., 2009].

[17] The radial function in ((7)) needs not be a Gaussian, as demonstrated in Figure 2c where more complicated forms can be seen. However, it is shown below that a Gaussian is often a good approximation. Therefore, ((7)) is generalized to

display math(11)

where g0,1 are arbitrary radial functions perpendicular to B, and φ is the phase difference between the driven and backscattered waves. Equations ((8))–((11)) describe the waveform expected when passing through a Langmuir snake. Since the radial function in ((11)) is arbitrary, the driver does not have to be a Gaussian for the model to apply.

4 Application to Type III Bursts

[18] We now show that ((8))–((11)) reproduce many of the observed waveforms, providing evidence for ES decay and Langmuir snakes in type III source regions. We use electric field waveforms observed by the Time Domain Sampler (TDS) receiver of the SWAVES instrument onboard STEREO [Bougeret et al., 2008]. SWAVES consists of three 6 m orthogonal monopole antennas. The TDS data used in this letter are obtained over a time interval of 65 ms at a cadence of 4 μs, corresponding to 16,384 samples per event for each component of E. Therefore, oscillations at the local fp∼20 kHz are easily resolved.

[19] The voltages from each antenna are converted to electric fields, and the coordinate system is rotated to find electric field components E and E parallel and perpendicular, respectively, to the local B. For the events considered below E is approximately parallel to B over the recorded time interval. STEREO records the local vsw and B so vsw and cosθ are known from direct measurements. However, ve is not directly measured by STEREO, so we use the nominal values inline image and inline image, corresponding to Te≈1.5×105 K and Te/Ti≈3. The electron beam speed is estimated by vb=D/(tt0), where D is the distance to STEREO from the Sun along a Parker spiral given by equation ((2)) of Robinson and Cairns [1994], and tt0 is the time between the start of the type III burst radio signature and the observed Langmuir event. Since the electron beam can decelerate and the beam can travel distances different from the Parker spiral length, the uncertainty in vb is large (∼30%). Based on the predictions in sections 2 and 3, Langmuir “snake” events should exhibit distinct spectral peaks near fp, and a waveform that both exhibits periodic beating and also drops to the background level in the time interval of the TDS data.

[20] The first event we consider was observed on 19 December 2011 at 13:50:55.160 UT by STEREO A. The waveform and spectrum of E are shown in Figures 3a and 3c, respectively. Figures 3b and 3c show the fit of ((8))–((11)) to the observed waveform. We find very good agreement between the fitted and observed waveforms and spectra. The fitted parameters are inline image and inline image (when fitting we treat inline image, or equivalently inline image, as a free parameter). By assuming the electron beam propagates along a Parker spiral at a constant speed, we obtain vb/c=0.093 and estimate inline image for nominal ve and vs based on ((10)). Since inline image is primarily determined by vb, the uncertainties in the estimated inline image are ∼30%. Thus, we find excellent agreement between the observed and estimated inline image, well within the 30% uncertainty estimate.

Figure 3.

Electric field, spectrum, and density perturbations of the Langmuir event observed on 19 December 2011 at 13:50:55.160 UT by STEREO A. (a) Observed and (b) fitted waveforms E. (c) Observed spectrum (black) and fitted spectrum (red). (d) Inferred δn/n. The fit parameters are: inline image, inline image, inline image, inline image, φ=π, and y0=6380 m.

[21] Figure 3d shows the inferred density perturbations, calculated from the low frequency electric field using the method in Henri et al. [2011]. The density perturbations are localized to where the field is most intense, consistent with simulations, and have the same features as Figure 2b. The density perturbations have frequency inline image, so inline image, as expected for ES decay.

[22] Figure 4 shows another example of a waveform consistent with the above theory, observed by STEREO B on 22 January 2011 at 10:27:12.789 UT. The fitted waveform agrees very well with the observed waveform. Similarly, the observed and fitted spectra agree well, as seen in Figure 4c. The electric field is weaker than in Figure 3, so the inferred density perturbations are much smaller, as shown in Figure 3d.

Figure 4.

Electric field, spectrum, and density perturbations of the Langmuir event observed on 22 January 2011 at 10:27:12.789 UT by STEREO B. (a) Observed and (b) fitted waveforms E. (c) Observed spectrum (black) and fitted spectrum (red). (d) Inferred δn/n. The fit parameters are inline image, inline image, inline image, inline image, inline image, φ=π, and y0=8590 m.

[23] For this second event, inline image based on fitting of ((8))–((11)) to the observed waveform and spectrum, while we independently estimate vb/c=0.082 and an expected inline image. The difference between the observed and estimated inline image may be due to vb being overestimated or ve or vs differing from the nominal values, so this example is not inconsistent with ES decay. The density perturbations between 20 ms and 40 ms have frequency inline image, consistent with ES decay.

[24] In both examples, best fits of g0 and g1 to the observed waveforms and spectra are Gaussians. The events in Figures 3 and 4 have characteristic lengths inline image perpendicular to B of approximately 80λD and 130λD, respectively.

[25] The radial functions, g0 and g1 in ((11)), do not necessarily have simple Gaussian functions; in principle, more complicated waveforms and radial functions can occur due to differences in electron beams, spatially dependent driving, or density changes. Figure 5 shows two examples that are consistent with ES decay but do not have such simple g0,1. The event in Figures 5a and 5b has electric field decreasing to the noise level away from the center similar to the behavior in Figures 3 and 4. The spectrum has two peaks near fp with observed inline image. The expected inline image, for | cosθ|=0.65 and an estimated vb/c=0.087. So the results are consistent with ES decay.

Figure 5.

Waveforms and spectra of two split spectral peaks events observed by STEREO A. (a and b) 21 March 2011 at 04:08:36.297 UT. (c and d) 04 October 2010 at 17:08:06.293 UT. Waveforms of E in Figures 5a and 5c. Power spectra of E near fp in Figures 5b and 5d.

[26] In Figures 5c and 5d, the intensity of Langmuir beating does not appear to drop to the noise level at the edges of the TDS interval. For this event | cosθ|=0.99, meaning that the solar wind passes the satellite very nearly along the magnetic field line. So this event may correspond to the satellite remaining with a Langmuir snake a much longer time. The observed inline image, in excellent agreement with the expected inline image, corresponding to an estimated vb/c=0.089.

[27] The agreement between the observed and fitted waveforms and spectra in Figures 3 and 4 provides strong evidence for both ES decay and well defined snakes of Langmuir beats occurring during type III bursts. The agreement between the observed and estimated inline image provides further evidence of ES decay. Furthermore, in each of the examples, the low frequency electric fields have frequencies inline image, providing strong independent evidence of ion-acoustic waves produced via ES decay. Combining the nominal Te=1.5×105 K with STEREO observations of Ti, we estimate Ti/Te=0.2, 0.1, 0.2, and 0.5 for the events in Figures 3, 4, 5a–5d, respectively. For these conditions, ES decay is expected to dominate STI [Cairns, 2000; Mitchell et al., 2003].

[28] More generally, for a sample of 596 STEREO/TDS events observed during type III bursts, ∼40% have two or more distinct spectral peaks near fp, suggestive of decay. Approximately, 12% of these events (or ∼5% of the total sample) are consistent with Langmuir “snakes”. Similar waveforms were observed in Ergun et al. [2008] and Henri et al. [2009].

[29] The range of angles θ between vsw and B for events with features consistent with Langmuir “snakes” is 16°≤θ≤76°; however, the mean and standard deviation are 45° and 14°, respectively. According to our model, Langmuir “snakes” are unlikely to be identified for small θ because the satellite travels nearly parallel to the “snake”, so the radial function cannot be clearly observed. For θ close to 90°, inline image is significantly reduced, meaning that the spectral peaks of L and L often cannot be distinguished.

5 Summary

[30] In this letter, we have presented large-scale simulations of three-dimensional truncated backscatter and compared the results with waveforms observed by STEREO. The key results are

  1. [31] Langmuir beating often occurs along snake-like structures parallel to B, based on simulations of the three-dimensional Zakharov equations.

  2. [32] An expression for the electric field observed during the transit of a snake under conditions for truncated backscatter was derived and compared with observations from STEREO. Snake-like structures are able to account for some of the Langmuir waveforms observed during type III bursts.

  3. [33] Good fits of the theoretical expression to the data are found. Each of the events fitted well has a relatively localized waveform with beats and two peaks in its power spectrum, consistent with the Doppler-shifted frequency differences expected for ES decay.

  4. [34] In these events, density perturbations associated with ES decay are observed when the fields are maximal, consistent with numerical simulations and theory, and have inline image.

[35] These results provide evidence for ES decay and the formation of Langmuir snakes parallel to B being two important nonlinear processes occurring in type III source regions. These Langmuir snakes may be a source of radio emission either by electromagnetic decay or coalescence of the driven Langmuir waves with the first backscatter.


[36] The authors are grateful to O. Skjaeraasen, the primary developer of the code used in section 2. The authors are grateful to the SWAVES team for data access and support. This work was supported by the Australian Research Council and an Australian Postgraduate Award.

[37] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.