Observations show that the shock structure varies primarily with the
fast mode Mach number 
and with the angle
between the upstream field 
and the local shock
normal. In particular, for quasi-perpendicular regions of the shock
( 
degrees) there is a transition between ``laminar''
(smooth) and ``turbulent'' magnetic profiles in the domain 
,
with laminar profiles at low 
(Figure 13.7).
Figure 13.7: The top panel shows a laminar bow shock profile while the
bottom panel shows a turbulent shock profile [Livesey et al., 1984;
Gosling and Robson, 1985].
Laminar profiles are by definition smooth, except for upstream trains of standing whistler
waves (which are related to dissipation processes at the shock). Turbulent
shocks have a well-defined ``shock foot'', where the magnetic field smoothly
increases over a scale length 
, and an ``overshoot''
behind the ramp where the field increases above the Rankine-Hugoniot
prediction on a scale (sometimes with additional
ripples) before finally reaching the average field level predicted by the
Rankine-Hugoniot conditions. Particle detectors show that the foot and
overshoot regions are associated with gyrating beams of
solar wind protons reflected at the shock (Figure 13.8).
Figure 13.8: Ion velocity distributions observed downstream from a
marginally supercritical shock with 
degrees
and almost perpendicular to the instrument's measurement plane
[Schopke et al., 1983]. The dashed circle in the second frame shows
the predicted locus for specularly reflected ions.
Laminar shock transitions sometimes have an upstream wave train but do not have significant levels of gyrating ions. The production of these gyrating ions is intrinsic to the both the structure and dissipation processes active at supercritical shocks, as explained more below. The definition of a ``supercritical'' shock is that gyrating ions represent the primary dissipation process active at the shock.
Quasi-parallel shocks are very turbulent with fluctuations 
and an extensive foreshock region, sometimes to such a degree that it
is difficult to identify where the shock transition occurs (Figure 13.9).
Figure 13.9: Schematic illustration of the quasi-parallel and quasi-perpendicular
regions of Earth's bow shock and the foreshock region upstream of the shock
[Greenstadt and Fredericks, 1979]. Note
that the foreshock region upstream of the quasi-parallell shock is extremely
turbulent, often making it difficult to identify the shock transition itself. .
Large fluxes of gyrating ions are often observed throughout a huge volume near a quasi-parallel shock transition.
The development of reflected, gyrating beams of solar wind ions, with associated creation of foot and overshoot regions and thermalization of the downstream particles is also associated with the self-consistent development of an electrostatic potential across the shock (the ``cross-shock potential''). These phenomena can be explained as follows [Gosling et al., 1982; Schwartz et al., 1983; Goodrich, 1985; Scudder et al., 1987].
(1) During one gyroperiod solar wind protons travel about 5000 km (at 
), while
the electrons travel 
km, and the shock ramp is 
km thick. The ions thus
see the ramp as an abrupt discontinuity during their gyromotion while the
electrons complete numerous gyroperiods as they cross the ramp.
(2) A self-consistent electrostatic potential is encountered at the shock ramp, which slows and resists the ion flow (but accelerates the electron flow).
(3) A
fraction ( 
) of the solar wind protons have insufficient normal
velocity to overcome the potential barrier and are specularly reflected
at the magnetic ramp; that is, their perpendicular velocities are reversed as
shown in Figure 13.10.
Figure 13.10: Schematic from Gosling and Robson [1985]. (a) Sketch of the trajectory of
an ion specularly reflecting off a shock with 
degrees. (b) Idealized
2-D ion velocity distributions at several distances from the shock ramp. Specularly
reflected particles move along a circle of radius 
centred
at the bulk flow velocity. The dashed lines are aligned parallel to the shock
surface.
(4) The specularly reflected protons move so as to
gain energy from the convection electric field and develop a large gyrospeed 
given by
as well as a different gyrocenter velocity. Here 
is the initial
speed in the local shock frame.
(5) The gyrocenter velocity is directed downstream for quasi-perpendicular shocks
( 
degrees) but upstream for quasi-parallel shocks [Gosling et al., 1982].
(6) For quasi-perpendicular shocks, then, the reflected protons gyrate upstream for a partial gyro-orbit before encountering the shock again, now with sufficient normal speed to overcome the potential barrier and pass downstream as a component with large apparent thermal energy. The current associated with this motion causes the magnetic field to increase in the downstream region. The spatial extent of the foot and overshoot are then of order the gyroradius of the specularly reflected ions.
(7) The potential develops as a result
of the different motions of electrons and ions. Starting from the electron momentum
equation, for a perpendicular shock with the normal in the 
direction and the
magnetic field in the 
direction, the momentum equation for the electron fluid
is
whence after neglecting the electron inertia term one may finally write
The cross-shock potential 
thus results from the gradients in pressure and magnetic
field energy across the shock (an ambipolar potential) as well as the effects of the net
ion drifts associated with the gyrating ions (second term). The cross-shock
potential is primarily oriented in the shock's normal direction.
Order of magnitude estimates are that
These estimates explain the characteristic size of the cross-shock potential and the amount of ion heating downstream from the shock. Moreover, the above balance between the upstream flow energy and the downstream temperature enhancement also explains qualitatively how the overall Rankine-Hugoniot conditions for the shock transition can be obeyed on the large scale, as they must be despite very different microphysics.
Figure 13.11 shows the development of gyrating ion beams in hybrid simulations (fluid electrons and particle ions), while Figure 13.12 shows the corresponding spatial profiles of the magnetic field strength and the cross-shock potential [Leroy et al., 1983].
Figure 13.11: Ion phase space plots 
and 
from hybrid shock simulations
[Leroy et al., 1982; Goodrich, 1985]. Here x is the coordinate along the shock normal
and the magnetic field is in the z direction, 
, and
. The times for each panel are (a) t = 0, (b) 
,
(c) 
, (d) 
, and (e) 
.
Note the development of the gyrating ion beams.
Figure 13.12: Plots of the magnetic field and cross-shock potentials as a function of x and
time for the simulations in Figure 13.11 [Leroy et al., 1982; Goodrich, 1985]. Note
the development of the foot and overshoot in the potential and magnetic field.
Qualitatively the gyrating ions comprise a second ion population moving with significant relative speed to the core solar wind ions, especially perpendicular to the magnetic field. The associated gradients in the particle distribution function imply the possibility of instabilities redistributing this free energy and ending up with a single, broadened and thermalized ion distribution in the magnetosheath. Figures 13.4 and 13.8 show this process observationally [Schopke et al., 1983].
A qualitatively important point is that the cross-shock potential and formation of gyrating ion beams involve reversible physics in the sense that no dissipation, energy loss, or entropy change is involved. How can this lead to an irreversible shock transition with a change in entropy between the upstream and downstream states? The answer is that wave-particle interactions associated with unstable particle distributions lead to dissipation and an increase in entropy. Nevertheless, much of the apparent ``heating'' of the electrons and ions across the shock can be understood in terms of ``reversible'' physics and the operation of the cross-shock potential.
A final remark is that for quasi-parallel shocks the gyrating ions have gyrocenters directed upstream from the shock, leading to an extended upstream region filled with gyrating ions and the large-amplitude MHD waves they drive. These upstream ``foreshock'' regions are then very turbulent with much evidence of Fermi acceleration and wave-particle interactions.