LANGMUIR WAVE AMPLITUDES AND THE ELECTRON DISTRIBUTION FUNCTION NEAR THE SOLAR WIND-FORESHOCK BOUNDARY
S. D. Bale, G. Chisham, D. Burgess and S. J. Schwartz
Astronomy Unit, Queen Mary & Westfield College, London E1 4NS, U.K., E-mail: email@example.com
Recent theoretical work on foreshock Langmuir wave generation has emphasized the importance of proximity to the solar wind/electron foreshock boundary. Waves generated near the boundary decouple quickly from the tenuous beam and evolve by wave-wave interactions; deeper in the foreshock, the effect of damping and beam bunching should be more pronounced. We compare observed electron distributions and wave intensities to study the effect of various cutoff distributions. The distribution of wave amplitudes is consistent with stochastic growth theory which implies that the distribution function should only be observed as unstable very infrequently.
The existence of large amplitude Langmuir waves in the solar wind, upstream of the bow shock, is usually explained by the existence of a time-of-flight electron distribution. The flux of electrons streaming away from the shock is carried anti-sunward by the solar wind; only electrons above a certain `cutoff' velocity are capable of reaching a given point (Filbert and Kellogg, 1979; Cairns, 1987). The saturation mechanism of these Langmuir waves has been the subject of much study and conjecture. The waves may decay nonlinearly, for example by reactive parametric or modulational instability (which may terminate in Langmuir wave collapse), or they may damp kinetically on the thermal electron distribution or by trapping beam electrons (Muschietti et al., 1996).
We have recently shown (Bale et al., 1996) that the peak-to-RMS ratio for these intense waves is relatively small. Hence, the largest amplitude waves tend to be very nearly monochromatic; at smaller amplitudes, the waveforms break up and form choppier wavepackets. This result disagrees with the analysis of Robinson and Newman (1991) who suggested that intense Langmuir waves were in a state of collapse and that large, spikey electric fields were being averaged away systematically by instrumental effects. A more likely decay scenario involves the reactive parametric decay process LL' + S, where the initial Langmuir wave decays to a backward propagating Langmuir wave and a ion acoustic wave.
Recent Vlasov simulations of Langmuir wave generation and decay (Goldman et al., 1996) have suggested that the distinction between `warm' and `cold' beams is quite important to resulting wave saturation mechanism. Warm beams tend to generate waves that damp on the thermal electron distribution while cold beams generate waves that may `detach' from the particle distributions and decay nonlinearly.
The probability distribution of wave amplitudes has recently been calculated for upstream Langmuir waves (Bale et al., 1996; Cairns and Robinson, 1996) and found to have a form consistent with the stochastic growth theory (SGT) of solar type-III radio bursts (Robinson, 1992). In this model, an electron beam associated with a solar active region propagates through the solar wind at marginal stability. The electrons encounter small patches wherein the distribution becomes unstable and Langmuir waves grow; the beam then reforms by advection. Quantitatively, wave growth is considered to be a random variable; that is where G is the wave growth or number of e-foldings. Since G~ln E, the resulting distribution goes as P(E)~1/E. This model has been developed for, and successfully applied to, the problem of the persistence of solar type-III bursts to distance of 1 AU.
Langmuir waves are observed copiously by the WAVES experiment on WIND. In particular, we use the Time Domain Sampler (TDS) instrument; the TDS samples electric field waveforms at rates up to 120,000 sample/s. It has 90 dB of dynamic range and returns 2048 data points in each event with the peak centered in the interval. The instrument is described in Bougeret et al. (l995). The data shown below come from early December, 1994 when WIND made many passes across the solar wind-foreshock boundary.
Figure 1 shows the maximum amplitude of Langmuir waves, as measured by the TDS instrument, as a function of distance (Diff) from the solar wind-electron foreshock boundary. The spacecraft position in a foreshock coordinate system is estimated as in Filbert and Kellogg (1979); that is, a line along the magnetic field direction is extended from the spacecraft position toward a model paraboloid bow shock. If this line intersects the model shock, it is translated in the Xgse direction until it is just tangent to the shock. Diff is then the distance of this translation and Dist is the distance from the tangent point to the point directly upstream of the spacecraft, along this new, translated line. We use a paraboloid shock
spacecraft locations within the electron foreshock, where the flux of electrons from the bow shock is observed. This distribution of amplitudes is well known (viz. Filbert and Kellogg, 1979; Etcheto and Faucheux, 1984) and is thought to be evidence of the time-of-flight beam distribution.
Figure 1: Langmuir wave amplitude as a function of distance from the solar wind-foreshock boundary, from WIND on 1994 1202 and 1994 1203.
As mentioned above, the TDS was sampling at a very high rate 120,000 s/s) and fully resolves the plasma frequency for these events. In Bale et al. (l996), we showed that the largest amplitude waveforms were very monochromatic and lack structure on small scales. This seems to preclude the importance of Langmuir wave collapse as a saturation mechanism (Robinson and Newman, 1991).
Figure 2 shows the estimated cutoff velocity as a function of maximum wave amplitude. In the time-of-flight model, the cutoff velocity is given by vc = (Dist/Diff) vsw cos; in Figure 2, we have used vsw = 350 km/sec. Electrons with speeds below vc cannot kinematically reach the spacecraft and this gives rise to a positive slope in the distribution function. It can be seen that the largest amplitude events occur with cutoff velocities on the order of 1000-10,000 km/sec; this is consistent with the peaks and shoulders observed in reduced electron distribution functions near the boundary (next section and, for example, Fitzenreiter et al. (l984)).
Figure 2: Langmuir wave amplitude as a function of cutoff velocity. The cutoff velocity is calculated as vc = (DIST/DIFF) vsw cos using a solar wind speed of vsw = 350 km/sec
In Figure 3, we show 2 electron distributions and reduced distributions sampled near the solar wind-foreshock boundary by the Ampte UKS satellite on 29 August, 1984. The distributions are measured during a 5 second integration and are sequential. The distribution in the first panel shows connection to the bow shock, evidenced by the enhancement in the -v|| direction. In the reduced distribution, this appears as a slight bump at v = 5.103km/s; this is a common observation near the foreshock-solar wind boundary (e.g. Fitzenreiter et al., 1984). In the next panel, the enhancement at -v|| is somewhat less prominent and there is a slight enhancement near +v|| = 4.103km/s. These distributions are concomitant with intense Langmuir waves observed by the plasma waves instrument on Ampte-UKS. The free-energy features in electron distributions are only present for, at most, 2 consecutive 5 second distributions. The short-lived nature of the unstable features in the reduced distributions has been noted before (Fitzenreiter et al., 1984), as has the existence of a backward enhancement, toward the bow shock. In the simulations of Goldman et al. (l996), a similar feature evolves due to the damping of the daughter wave in the parametric decay process L L'+S. Although upstream electron distributions show marginally unstable features for short intervals, they do not typically show evidence of very high velocity, 'cold' beams.
Fig. 3. Caption included with image
Taken together, the 1/E Langmuir wave amplitude distribution (Bale et al., 1996) and the nonobservation of persistent, unstable features in the electron distribution function indicate that the stochastic growth model of wave generation is plausible near the solar wind- foreshock boundary.
Furthermore, the fact that the largest amplitude waves are very monochromatic, implies a cold beam source for the linear instability. Very narrow, high velocity beams are not generally observed near the foreshock edge, the peaks in the reduced distribution function generally are fairly broad (v/v ~ 0.25). More detailed study of the probability distribution and cutoff velocity dependence should help elucidate the wave generation processes.
We acknowledge useful discussions with M. V. Goldman. We thank the WIND/WAVES team (P.I.: M. L. Kaiser, NASA/GSFC) for access to TDS data and the MFI data processing team at the Laboratory for Extraterrestrial Physics, NASA/GSFC for use of the key parameter magnetic field data. This work is supported in part by PPARC (UK) grant GR/J88388.
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