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Figure 5.3. Angle between coplanarity and four spacecraft timing estimates of bow shock normals, as a function of 6gn. Coplanarity is an unreliable estimator of bow shock orientation for 6gn ^ 70° but is still only accurate to around 20° for Qsn SS

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Figure 5.3. Angle between coplanarity and four spacecraft timing estimates of bow shock normals, as a function of 6gn. Coplanarity is an unreliable estimator of bow shock orientation for 6gn ^ 70° but is still only accurate to around 20° for Qsn SS

proxy of the local plasma density. It is therefore of interest to compare four spacecraft timing estimates of the shock orientation using these different estimators. We have compared the orientations deduced from EFW (spacecraft potential) and FGM (magnetic field) timings for 26 quasi-perpendicular shocks in 2001 that could be identified cleanly in both EFW and FGM data at all four spacecraft, a subset of those used by Horbury et al. (2002). The agreement between the resulting shock normals is remarkable (Figure 5.2, right panel), with the mean angular deviation being 1.8° and the largest deviation being only 3.9°. The mean absolute difference in the deduced velocity was 2 km/s. While this comparison cannot tell us about the reliability of some of the assumptions (such as constant motion and planarity) of the timing method for quasi-perpendicular shocks, it confirms that it is not sensitive to the physical parameter used.

5.2.1.2 Large scale structure of the bow shock

As discussed above in section 5.2.1.1, Horbury et al. (2002) found close agreement between bow shock normal estimates based on four spacecraft timings and those of bow shock models such as that of Peredo et al. (1995). This implies that the bow shock stays close to the parabolic shape of the Peredo et al. model, at least under steady solar wind conditions. It also places an upper limit on the size of any large scale 'ripples' on this surface: if they were present to a significant degree, the local normals deduced from the Cluster measurements would not agree with the normal derived from the model normal.

Bow shock models have been derived in the past by estimating the parameters of a conic section from thousands of single spacecraft shock crossing locations, parameterised by upstream conditions such as ram pressure and magnetic field direction. It is therefore remarkable that these models of the large scale shape agree so closely with the local normal estimates from Cluster found by Horbury et al. (2002).

5.2.2 Large- and meso-scale shock structure 5.2.2.1 Dissipation at quasi-perpendicular shocks

Fast-mode collisionless shocks grow from magnetosonic waves when the incoming flow exceeds the fast magnetosonic speed; the wave steepens and eventually becomes a standing shock in the plasma. To stand as a steady-state shock, the plasma must dissipate small-scale structure to slow the steepening and prevent the shock from overturning; furthermore, the Rankine-Hugoniot (shock jump) relations tell us that the shock must convert incoming flow energy to electron and ion heating and magnetic field energy downstream. Early theories of energy dissipation at shocks sought a single mechanism to directly provide both small-scale dissipation and plasma heating (viz., Papadopoulos, 1985). Ion and electron heating mechanisms appear to be mostly unrelated.

At quasi-perpendicular collisionless shocks above a critical Mach number (Kennel et al., 1985), a significant fraction of incident ions are reflected within the shock transition. They gyrate in the upstream region, where the magnetic field is slightly increased to form a 'foot' before returning to the steep ramp region. Having gained energy due to the solar wind v x B electric field, they traverse the ramp and become temporarily trapped in the adjacent overshoot region (Paschmann et al., 1982; Sckopke et al., 1983; Sckopke et al., 1990). The ions ultimately convect further downstream, leading to a magnetic undershoot and series of decreasing oscillations accompanied by ion mixing and thermalisation.

The spatial scales over which the shock dissipates energy, and slows the incoming flow, are thought to be related to the nature of the dissipation mechanism itself. Hence, knowing these scales and their dependence on macroscopic plasma parameters is tantamount to knowing the dissipation physics at the shock. Quasi-perpendicular (Q^) shocks have been traditionally targeted for dissipation scale studies, including aspects such as:

1. the role/interpretation of competing dissipation mechanisms within more classical frameworks (anomalous resistivity, viscosity, Hall physics)

2. differing scales for the transition of different bulk parameters (magnetic field, density, velocity)

3. Ohm's law, including contributions from electron inertia and departures from isotropy

4. the role of stationary (DC) fields in the dissipation processes (electron kinetics, ion reflection)

5. the role of non-stationary fields in scattering and shaping the particle distributions at, and downstream of, the main shock transition.

6. the competition between dissipation and dispersion in effecting and limiting the steepening of the shock profile.

5.2.2.2 Shock ramp scales

The shock ramp is the region of steepest spatial gradients. The steepening is limited and balanced by dispersion and/or dissipation. The nature of the dissipation differs according to the strength of the shock, i.e. low or high value of the Mach number (Alfvenic MA or magnetosonic Mms). Resistive dissipation alone is enough at low Mach number, while an additional dissipation (e.g., viscosity) is required at high Mach number. Low and high MA (or Mmi) correspond to subcritical and supercritical Mach regime defined below and above a certain threshold (Tsurutani and Stone, 1985). This balance will define the width of the shock front and in particular the ramp width.

Theoretical and kinetic simulations (Leroy et al., 1982) suggested, together with previous observations, that the magnetic ramp occurs on either an ion inertial scale (c/wpi) or the gyro-radius of an ion moving at the upstream flow velocity in the downstream magnetic field. While the plasma density tends to follow the magnetic field (Scudder et al., 1986), the electric field shows fine scale features discussed in more detail in Section 5.2.3.

As a multi-spacecraft mission, Cluster was designed precisely to measure spatial scales in the magnetosphere. Typical Cluster spacecraft separations are 1001000 km which correspond to crossing times of 1-100 s for boundary (shock) speeds of 10-100 kms-1. Hence, sample speeds of 1-10 samples per second are sufficient to sample the shock transition and find a spatial transformation by the techniques discussed above and elsewhere (e.g., Paschmann and Daly (eds.), 1998).

Bale et al. (2003) used the Cluster EFW spacecraft potential as a proxy for electron density to study the ramp transition scale at approximately 100 Q± bow shock crossings. A shock speed (and normal) was found using the timing technique and then each shock profile was fitted with a hyperbolic tangent function n(x) = n0+n 1 tanh(x/X). Figure 5.4 shows an example fit at a Mms tt 3.5, @Bn ^ 81° shock.

A characteristic scale size for the shock ramp was then given to be L = n/\dn/dx\ evaluated at the middle of the ramp and this was expressed in terms of the fit coef-

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Figure 5.4. Density transition from downstream (shocked) to upstream (unshocked) states for a Mach Mms ~ 3.5, 8ßn ~ 81° shock. The green line is the hyperbolic tangent fit; red vertical lines show the density transition scale. From Bale et al. (2003).

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Figure 5.4. Density transition from downstream (shocked) to upstream (unshocked) states for a Mach Mms ~ 3.5, 8ßn ~ 81° shock. The green line is the hyperbolic tangent fit; red vertical lines show the density transition scale. From Bale et al. (2003).

ficients L = n0/n1 X. Bale et al. (2003) then showed that statistically the measured ramp scale size was proportional to vsh/Qci,2, the gyroradius of trapped ions, over a large range of Mach numbers. When compared with the ion inertial scale, L/c/mpi is seen to increase monotonically. This is the expected behaviour if the true shock ramp scale is like vsh/Qci,2, since vsh/Qci,2/c/wpi x MA. Figure 5.5 shows these trends.

Similar, supporting results (Horbury, 2004, unpublished) have been obtained using magnetic field data. Taken together, these scalings strongly suggest that the density, magnetic field, and velocity transition scales of the quasi-perpendicular shock are proportional to the gyroradius of the trapped ion population. At low Mach numbers, the two scales vsh/Qci,2 and c/rnpi are of similar magnitude and some ambiguity remains. However, the implication of this result is that dissipation at Qi shocks is related to the motion of the trapped ions. In a fluid sense, this corresponds to a viscosity term in Ohm's law associated with gradients in the ion pressure tensor as discussed above.

5.2.2.3 Overshoot/Undershoot structure

It is well known that supercritical shocks exhibit overshoot and undershoot behaviour of the magnetic field just downstream of the shock (Heppner et al., 1967; Russell and Greenstadt, 1979). Since this structure is only observed at supercriti-

Figure 5.5. Relationship between scale size and magnetosonic Mach number. L/ (vsh,n/Q.ci,2) (upper panel) is approximately constant over a large range of Mach number, while the ion inertial scaling (lower panel) increases with Mach number. From Bale et al. (2003).

cal shocks, it was suggested (Morse, 1976) and confirmed by computer simulation (Leroy et al., 1982) that the overshoot structure is associated with a reflected and heated ion population. It is also known that overshoot phenomena play a role in ion acceleration (Giacalone et al., 1991) and electron heating (Gedalin and Griv, 1999). ISEE-1 and -2 measured magnetic overshoot thicknesses using two-point timing to obtain shock speed (Livesey et al., 1982) and found that the observed thickness was ordered by the downstream ion gyroradius. However, Cluster has made the first density measurements of well-defined overshoot-undershoot structure at Q± shocks (Saxena et al., 2004). Using Cluster EFW spacecraft potential as a density proxy, 56 Q± shocks have been analysed using techniques similar to those of Bale et al. (2003).

Figure 5.6 shows typical overshoot/undershoot structure at the same Mms & 3.5, Q± shock as Figure 5.4. Subtracting the fitted hyperbolic tangent (top panel) leaves a clear 'chirp' signature associated with the overshoot/undershoot (middle panel).

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Figure 5.6. Overshoot/undershoot structure at the shock of Figure 5.4 The hyperbolic tangent trend is removed to show a 'chirp' (middle panel); red dots show the location of zero-crossings that are used to measure the overshoot wavelength. The maximum density perturbation between zero-crossings decays spatially (lower panel); blue diamonds show the maxima which are fitted to an exponential to retrieve a decay scale. From Saxena et al. (2004).

Figure 5.6. Overshoot/undershoot structure at the shock of Figure 5.4 The hyperbolic tangent trend is removed to show a 'chirp' (middle panel); red dots show the location of zero-crossings that are used to measure the overshoot wavelength. The maximum density perturbation between zero-crossings decays spatially (lower panel); blue diamonds show the maxima which are fitted to an exponential to retrieve a decay scale. From Saxena et al. (2004).

Then a zero-crossing algorithm is applied to the chirp (red dots, middle panel) to produce an estimated wavelength for the shock overshoot. Finally, the overshoot amplitude is seen to decay systematically (bottom panel). An exponential function is fitted to the maxima of 15n\ between each pair of zero-crossings (blue diamonds) to estimate a decay scale X. Both the overshoot/undershoot wavelength and decay scale are found to be organised by the gyroradius of trapped ions, vsh/Qci,2, (rather than the ion inertial length). The measured wavelength is consistent with ISEE magnetic observations, while the measurement of an overshoot exponential decay scale is a new result for Cluster.

5.2.3 Fine-scale features in the electric field

Within a collisionless shock front, energy transfer is achieved through the interaction between electric/magnetic waves and particles rather than the normal col-lisional processes that occur within common hydrodynamic shocks. The spatial scales over which these particles and fields can interact is important when trying to ascertain the energy transfer processes that may occur within the shock front. The determination of magnetic field structure and the spatial scales over which the field varies in the foot, ramp, and overshoot/undershoot regions has been intensively studied since shocks were first observed in the 1960's.

Typically, the foot width is of the order of 0.68Vsw/Q.ci where Vsw is the solar wind velocity and D.ci is the upstream ion gyrofrequency (Sckopke et al., 1983; Livesey et al., 1984). The ramp scale has been estimated to be less than an ion inertial length (e.g., see Balikhin et al., 1995, and references therein) with reports of one or two shocks whose ramp scale was of the order 0.1c/rnpi (Newbury and Russell, 1996; Walker et al., 1999). Figure 5.5 shows, however, that at larger Mach numbers the shock ramp is typically larger than an ion inertial length.

Reports of observations of the electric field, on the other hand, are very sparse. This was probably due to the lack of high quality, high time resolution measurements. Based on initial results from ISEE, Heppner et al. (1978) reported that short duration spike-like features were occasionally observed in the electric field as the satellite crossed the shock front. Further investigations of subcritical oblique shocks by Wygant et al. (1987) showed spike-like features with amplitudes up to 100 mVm-1 and a strong component along the shock normal. The observations were, however, not good enough to determine the free energy source, mode or scale size of these structures. They speculated that these waves may be either lower-hybrid or possibly Doppler shifted ion-acoustic waves. Based on spin averaged electric field measurements from ISEE, Formisano (1982) reported that the increase in the electric field observed at quasi-perpendicular shocks began just upstream of the magnetic ramp and lasted longer than the ramp crossing itself.

One key aspect to determine is the spatial scale over which changes in the electric field occur and its relation to the scale size over which changes in the magnetic field occur. Several differing points of view have been published. The first (Esele-vich et al., 1971; Balikhin et al., 1993; Formisano and Torbert, 1982; Formisano, 1982,1985; Balikhin et al., 2002; Krasnosel'skikh, 1985; Leroy et al., 1982; Liewer et al., 1991; Scholer et al., 2003) is that the spatial scales of the potential and magnetic field in the ramp region are similar whilst Scudder (1995) proposed the potential scale length is larger than that of the magnetic scale length. Others have suggested that the potential varies predominantly within iso-magnetic jumps, i.e., on a smaller scale than the magnetic field. In laboratory plasmas, such a short scale of the cross-shock electrostatic potential ('isomagnetic jump') was observed by Es-elevich (1982). This isomagnetic jump is often attributed to the ion sound subshock (see the review by Kennel et al., 1985).

Using data generated from numerical simulations, Lembege et al. (1999) analysed simultaneous measurements of the scale size of both the magnetic ramp region and the region in which the change in potential was observed. Their results showed that the scale lengths were of the same order. This view is also supported by the simulations of Scholer et al. (2003). The latter authors show that the main potential drop can occur over several ion scales in the foot region, while the steepened

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