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-1500 1000 ■■ SOO 0 500 1000 1500 V Para (km/sec)

Figure 2.18. Ion distribution from CIS/CODIF on Cluster 1 on February 3, 2002, averaged over the interval 04:03 - 04:10UT, indicating the existence of a hot, diffuse, ion population. The backstreaming ions fill the plot; the red region to the right corresponds to the unresolved solar wind. The existence of a hot ion distribution is necessary to produce the observed waves. (Figure provided by M. Scholer).

ities, which would have the same polarisation in the spacecraft time series. The waves were not specifically identified before the launch of Cluster at least not for their remarkably clear polarisation.

2.3.1.3 Oblique ULF waves

The left handed (in the spacecraft frame) quasi-monochromatic ULF foreshock waves with 30 s period are excited by back streaming ions through the right-hand resonant ion-ion beam instability. The linear growth rate of this instability is maximum for parallel propagation. This is in contrast to experimental observations, which have consistently shown that such waves propagate obliquely to the field. Eastwood et al. (2004) have recently shown that for large IMF cone angles, although the 30 s waves are observed to propagate obliquely, their propagation direction is confined to the plane defined by the x-GSE direction and the magnetic

This has been interpreted as a non-local effect. When the IMF cone angle is large, the waves are convected across energetic particle gradients by the solar wind. The upstream waves are excited by the reflected ions in the quasi-perpendicular region and then refracted as they enter the quasi-parallel region (Hada et al., 1987). As shown in 2-D simulations of a quasi-parallel shock by Krauss-Varban and field.

Omidi (1993) and Scholer et al. (1993), the waves excited far upstream by the diffuse ions are also refracted in the region of large diffuse ion gradients closer to the shock.

However, the term refraction should be used with care: firstly, the wavelength is comparable to the scale over which the superthermal particle distributions change, and secondly, the propagation of the waves through the inhomogeneous medium is altered by the convective plasma motion.

2.3.1.4 Dispersion relation of low frequency waves

One way to identify wave modes is by examining the dispersion relations directly. However, in the case of the foreshock, a correction must be made for the Doppler shift. The rest frame frequency is given by

where mrest, a>sc, k and Vsw are the angular frequency in the plasma and spacecraft frames, wave vector, and solar wind velocity, respectively. The spacecraft velocity being of the order of a few km s-1 is neglected with respect to the solar wind speed.

Hoppe et al. (1981) and Hoppe and Russell (1983), based on ISEE1 and ISEE2 magnetic field data, determined plasma frame frequencies and wave numbers of magnetohydrodynamic waves in the foreshock. Dudok de Wit et al. (1995), Ba-likhin et al. (1997a,b) analysed AMPTE-UKS and AMPTE-IRM magnetic field data. Using MVA and projected wave numbers they derived approximate dispersion relations in the plasma frame assuming that the quasi-monochromatic wave assumption holds for the propagation direction.

Narita et al. (2003) used magnetic field and CIS/HIA plasma data from Cluster 3 for the time interval 1700-1730 UT on February 20, 2002 to study the dispersion relation of low frequency waves in the foreshock. They applied the wave telescope/k-filtering technique (Pincon and Lefeuvre, 1991; Motschmann et al., 1996; Glassmeier et al., 2001) to determine the wave-vectors k. After determination of the wave-vector corresponding to the largest wave power, the Doppler shift was calculated with the help of Eq. (2.1) using the fluid velocity moment. Applying principles of the optics of quasi-monochromatic waves (Fowler et al., 1967; Arthur et al., 1976; Born and Wolf, 1980), the ellipticity of polarisation in the frequency domain was found, being -1 for left-handed, 0 for linear, and 1 for right-handed waves.

Figure 2.19(a) displays the rest frame frequencies and their associated wave numbers with the sign of ellipticity overplotted, where wave number k is projected to the mean propagation direction dkB = 24° in the plasma rest frame. Right- and left-handed polarisations are represented by diamonds and plus signs, respectively. The frequency is normalised to the proton cyclotron frequency Qcp = 1.1 Hz, and the wave number is normalised to (Va/^cp)-1 = 10.3 km-1. The propagation direction is nearly aligned to the magnetic field direction at various frequencies.

The insert zooms into the dispersion relation at —0.2 < m/ilcp < 0.4 for and 0.0 < kVA/Ocp < 0.3.

Two different branches of the dispersion relation show up from this analysis, one being slightly curved and connected to (m, k) = (0,0), the other being a straight line connected to (m, k) = (—Qcp, 0) and intersecting the former branch. Both right and left handed polarisations are identified on both branches. The dispersion of these waves is consistent with the ion-ion interaction picture between the bow-shock reflected ion beam and the solar wind ion flow.

Figure 2.19(b) shows the theoretical cold plasma dispersion branches for a cold beam to plasma density ratio nb/n = 0.001 (i.e., 0.1 %), MA = 5.6 for the Alfven Mach number of the beam, and quasi-parallel propagation (dkB = 24°). The wave propagation angle dkB has been chosen to be the same as the mean propagation angle in the observation. Due to the anomalous Doppler effect, a right-handed wave exists at resonant frequency m = kVb — Qp propagating along the ion beam. Four other wave modes turn up, all of them result from the single ion, cold plasma system: R+, L+, L—, R—, where '+' stands for a forward propagation with respect to the magnetic field and '-' for a backward propagation, and 'R' for a right-hand polarisation and 'L' for a left-hand polarisation. At and around a crossover frequency between 'R+' and 'Res.', stationary nonlinear structures in abeam plasma system are found (Sauer et al., 2001; Sauer and Dubinin, 2003). They behave like solitons and so-called oscillitons.

Application to the results shown in Figure 2.19(a) yields that the major curved branch corresponds to the R+ branch in Figure 2.19(b). The straight branch also corresponds to the resonant branch. However, not only right-handed but also left-handed polarisations are found in the observations, and some waves even deviate from the major branch around and above the proton cyclotron frequency. The curvature of the major branch is also slightly different from the R+ branch. These disagreements must be resolved in the future by, for example, comparing different methods of polarisation analysis, testing various models of dispersion relation, including hot beams in a ft ~ 1, and investigating more events.

2.3.1.5 Statistics of low frequency waves

The Cluster spacecraft provide the unique opportunity to undertake a statistical study of LF wave structure and propagation in the foreshock based on FGM magnetic field and CIS/HIA plasma data.

Various distributions of the foreshock wave properties like frequencies, wave numbers, phase velocities, propagation directions, and polarisation have been investigated by Narita et al. (2004) who for the statistical study selected intervals of wavelengths more than ~ 200 km from the small spacecraft separation phase (February 3 - June 17, 2002) when the separation distance was ~ 100 km.

The wave telescope/k filtering technique was applied to 36 wave events selected in order to find the wave vectors k for wave frequencies < 0.5 Hz in the spacecraft

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