Figure 5.12. The relationship between the amplitude AE of the electric field spikes and Shocks closer to 90° show a higher range of amplitudes. From Walker et al. (2004).
in the case of the Earth's bow shock, to relative motion of the bow shock with respect to the Earth in response to changes in solar wind conditions.
In the early 1980s, in the response both to new observations of the Earth's bow shock and computational capabilities, collisionless shock physics matured rapidly (e.g., Kennel et al., 1985, and accompanying papers). Indications of non-stationarity were found in low frequency oscillations of the ion flux at the bow shock (Vaisberg et al., 1984, 1986a,b) and at the bow shock of Uranus (Bagenal et al., 1987). Kinetic hybrid simulations (Leroy et al., 1981, 1982) for parameters typical at the Earth's bow shock (MA = 8 and = & = 0.6, where MA is the Alfven Mach number, fiej is the ratio of the thermal and magnetic pressures, and 'e' and 'i' refer to electrons and ions respectively) showed that the shock structure varies with time. For example, the maximum value of the magnetic field exhibits temporal variations with a characteristic time of the order of the ion gyroperiod, the magnitude of these variations being about 20%.
Quest (1985) modelled high Mach number perpendicular shocks (MA = 22, & = 0.1). In the absence of electron resistivity the ion reflection process is periodic, alternating between periods of 100% ion reflection and 100% ion transmission. As a result, a periodic shock front reformation was observed rather than a stationary structure. Quest (1986) extended these preliminary simulations to perform a systematic study of high Mach number perpendicular shocks. For & = 0.1 he revealed that the previously found (Leroy et al., 1982) tendency of a shock to become increasingly time-dependent as MA increases was also observed for MA > 10 and resulted in cyclical wave breaking for MA > 20. In addition, for & = 1 and MA > 10 a non-trivial dependence of the shock front structure on the resistivity was found.
Krasnosel'skikh (1985) and Galeev et al. (1988c) proposed models that attribute the shock front instability to the domination of nonlinear effects over dispersion and dissipation. Non-stationary whistler wave trains, which had been previously suggested (Galeev et al., 1988c,b,a), were reported in observations of the Earth bow shock onboard Intershock-Prognoz-10 and AMPTE UK spacecraft (Krasnosel'skikh et al., 1991). Recently, theoretical work and 1D full particle simulations have been used to analyse this mechanism in detail. Its application to obliquely propagating shocks has revealed a critical Mach number above which these non-stationarity processes operate (Krasnosselskikh et al., 2002).
Lembege and Dawson (1987b) have shown that the non-stationarity of the shock front can be due to the cyclic self-reformation of the shock front, and have recovered fluctuation levels of 20% in the magnetic field at the overshoot amplitude and in the density of reflected ions. They analysed this self-reformation in detail for an exactly perpendicular low-beta non-resistive shock in 1D full particle simulations and showed that this reformation persists even for a moderate (still supercritical) Mach number (MA = 2 — 4). This non-stationary process persists over an angular range below dBn = 90° as long as the density of reflected ions is high enough to feed the reformation (Lembege and Dawson, 1987a). Lembege and Savoini (1992) confirmed the previous results with the help of 2D full particle simulations and showed that reformation continues to occur even when finite resistivity effects due to cross field current instabilities are included self-consistently. In addition, the shock front appears to be rippled rather than uniform for both perpendicular and oblique planar shocks. Moreover, the reformation is expected for relatively low ion pi (i.e. relatively cold upstream plasma) and/or high Mach number shocks, but disappears as Pi reaches relatively high values as shown by both 1D hybrid (Hellinger et al., 2002) and 1D full particle simulations (Scholer et al., 2003; Hada et al., 2003).
The problem of shock front stationarity described above gives several indications about possible manifestations of these effects in observations. Most of the results indicate that the characteristic timescale of the shock front variations is of the order of one ion gyroperiod or less, related to either the physics of the whistler mode expected to dominate the overall transition and/or the overturning due to non-steady ion reflection. This time period is comparable to and often shorter than that required to obtain full ion and electron distributions by Cluster. As noted above, shock motion can complicate this matter. Indeed, most of what we know about the position and shape of the bow shock is based on statistical studies together with modelling (e.g., Peredo et al., 1995, and references therein); the detailed response of the bow shock position to fluctuations in the upstream solar wind conditions has not been practical prior to the multi-spacecraft approach of Cluster.
In the following sections we provide an overview of the key Cluster results in this area. Evidence for intrinsic non-stationarity comes by studying variations of the shock profile, and by inferences on the variability of the ion reflection pro-cess(es). Cluster has also addressed directly the motion of the bow shock.
The near-simultaneous measurement of the shock profile by four spacecraft allows us to study spatial and temporal variability in ways that have not previously been possible. Different physical parameters such as the density, electric field and magnetic field would be expected to vary in different ways. The variability of one of these, the magnetic field, through the quasi-perpendicular shock was considered briefly by Horbury et al. (2001).
By considering the magnetic field profile through a nearly perpendicular supercritical shock (QBn w 86°, plasma P w 0.1, Alfven Mach number MA w 4.8), Horbury et al. (2001) could identify structures which were stationary (i.e., phase standing) relative to the main shock ramp, and others that were not. The shock is shown in the top panel of Figure 5.13: the four profiles look superficially very similar. When the profiles are synchronised at the time of the crossing (Figure 5.13, bottom panel), some other features become visible. In particular, the shock overshoot, undershoot and subsequent oscillations in the magnetic field magnitude are fairly well synchronised between the four spacecraft, implying that this field magnitude structure does not vary significantly over the spacecraft separation (around 600 km) or the time differences between the shock passages of the different spacecraft (up to 30s).
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