2.2.1 Electron foreshock boundary
The electron foreshock, as sketched in Figure 2.1 occupies the region from the bow-shock to just downstream of the tangent magnetic field line. Like the ion foreshock, the electron foreshock is a very dynamic region in which bow shock reflected electrons are convected downstream toward the bow shock by the v x B electric field in the solar wind. Similar to the ions in the ion foreshock, the highest energy electrons are observed close to the foreshock boundary, while the lower energy electrons are observed further downstream due to time-of-flight effects and the solar wind electric field (Filbert and Kellogg, 1979). This results in electron beams producing a bump-on-tail distribution function (Fitzenreiter et al., 1984). The waves observed in this region, basically Langmuir waves, are typically at the plasma frequency fpe and its second harmonic, but far away from the foreshock boundary deep in the electron foreshock, these waves are often seen shifted to frequencies above and below fpe (Fuselier et al., 1985).
The Langmuir wave amplitudes in the electron foreshock were shown by Filbert and Kellogg (1979) to be largest near the tangent magnetic field lines. Etcheto and Faucheaux (1984), using data from ISEE1, reported maximum amplitudes of a few mV m^1 at the edge of the foreshock and only a few tens to a few hundreds of ^V m^1 further inside the electron foreshock. Cairns et al. (1997) also used ISEE 1 data to show that there is a slight offset of the large wave amplitude region from the boundary of the foreshock and that the largest amplitude waves are observed in a relatively narrow region with the Langmuir wave amplitudes falling off slowly at larger distances away from the foreshock.
Cluster multi-point measurements allow for the comparison of Langmuir wave amplitudes at various positions in the foreshock and near the boundary of the electron foreshock. Another important aspect of the Cluster measurements is the capture of waveforms by the WBD plasma wave receiver (Gurnett et al., 1997) in this region since only the Wind spacecraft up to now has had the advantage of looking at Langmuir waveforms in the foreshock (Bale et al., 1997). All previous studies were carried out using spectral density measurements, which underestimate the wave amplitudes due to temporal and spectral averaging (Robinson et al., 1993). Using the WBD measurements, some initial studies of the Langmuir waves observed in Earth's electron foreshock have been carried out. Sigsbee et al. (2004b) found that the characteristics of the waves were in agreement with most of the previous studies. In addition the Cluster observations were found to follow the log-normal statistics predicted by stochastic growth theory (Cairns and Robinson, 1999); however, deviations from this prediction occurred at large wave amplitudes when electric fields measured at a wide range of distances to the boundary between the electron foreshock and solar wind were included. This finding generally agrees with the results of Bale et al. (1997) and Cairns and Robinson (1997). However, Sigsbee et al. (2004b) pointed out that the slope of the power law obtained from the Cluster data was steeper.
Sigsbee et al. (2004a,b) showed that the center of the probability distributions constructed for small bins of Df, the distance from the spacecraft to the tangent field line in the x GSE direction, shifts to lower amplitudes as one goes deeper into the foreshock, as illustrated in Figure 2.2. The data plotted in this figure show that the power law tail on the distribution for all values of Df may result from the sum of the log-normal distributions at different locations, which is a new result. Stochastic growth theory may thus still be correct in explaining the observed tail distribution of amplitudes.
All of the Cluster results thus far have dealt with case studies. Future work in this area will concentrate on statistical studies of various electron foreshock crossings by Cluster under varying solar wind conditions and IMF orientations.
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