Earth's foreshock is the region upstream from the Earth's bow shock that is magnetically
connected to the bow shock and contains both solar wind plasma and also charged particles
coming from the bow shock. As described in the last section, protons and other solar wind
ions specularly reflected from the bow shock have gyrocenter velocities directed into the
upstream plasma for 
degrees. These are not the only particles that may stream
back into the solar wind from the bow shock. Instead, the following classes of plasma particles
can also move into the solar wind from the bow shock:
All charged particles moving in the foreshock, no matter whether they are from the undisturbed
solar wind or the bow shock, feel the solar wind's convection electric field and must move with
the 
drift velocity, as well as their parallel velocity and their gyrovelocity.
Accordingly, all electrons and ions leaving the bow shock are constrained to lie downstream from the
magnetic field line tangent to the bow shock (Figure 13.13). Generalizing Figure 13.13 to consider
particle motions in other planes parallel to the plane containing 
and 
,
the upstream boundary to the foreshock is the locus of field lines tangent to the bow shock in these
planes, while the bow shock comprises the downstream boundary (Figure 13.14).
Figure 13.13: The location of the foreshock relative to the bow shock, together with
definitions for important coordinates and velocities [Cairns and Robinson, 1999].
Figure 13.14: Foreshock structure in 3-D [Lacombe et al., 1988].
The drift also causes the development of beam features in particle
distributions in the foreshock. This can be seen in Figure 13.13: since the gyrocenters of
all particles move in straight lines with 
, it is
clear that particles with larger 
move more nearly parallel to and are
found upstream from particles leaving the same point with smaller . Put another way,
differences in lead to dispersion in position. Consider next the parallel speeds of
particles reaching a position 
in the foreshock, where R and 
are defined in
Figure 13.13.
Simple geometry immediately shows that the minimum speed of a particle reaching
that point from the bow shock is given by [Filbert and Kellogg, 1979; Cairns, 1987]
with a maximum speed of c. This minimum speed, the ``cutoff speed'' corresponds to particles
reaching from
the tangent point itself. (In more detail, the minimum speed corresponds to particles leaving the
shock along a line tangent to the bow shock and passing through (R,x) [Cairns, 1987], but
equation (13.16) is a good approximation under most circumstances.)
Ignoring self-generated
wave fields, the particle distribution can be constructed using the Vlasov equation and shown to
have a sharp cutoff at 
(Figure 13.17 below).
Qualitatively, this ``cutoff'' distribution
looks like a bump-on-tail distribution and can be expected to drive wave growth via a bump-on-tail
instability. According to Eq. (13.16) and simple geometry, 
is a strong function of position
in the foreshock, so that the beam speed must vary substantially with position (Figure 13.15).
Figure 13.15: Lines of constant in the foreshock [Cairns, 1987], showing that and so the
speed of beams varies substantially with position in the foreshock, and that the only regions
with large and fast beams are very close to the foreshock boundary.
Before proceeding to describe the plasma waves driven by cutoff distributions in the foreshock, we
remark that unstable particle distributions are often produced in space plasmas by such ``time-of-flight''
effects in which the combination of a localized source of particles and a convection
electric field & associated drift (or other plasma drift) leads
to constraints on the parallel speed of particles able to
reach specified locations. Examples include the magnetosheath, cusp, and plasma sheet in Earth's
magnetosphere, as well as interplanetary travelling shocks and the lunar foreshock.
Figure 13.16 [Fitzenreiter et al., 1990] shows the electron distributions observed as well as those predicted (Figure 13.17) assuming mirror reflection at the bow shock and the above cutoff effects.
Figure 13.16: ISEE-1 observations near the leading edge of the
foreshock (left), deeper in the foreshock (centre) and deep in
the foreshock (right). The top panels show the 2-D electron distributions,
the middle panels the differences 
,
and the bottom panels the reduced distributions 
. The vertical lines show the predicted value of for the
observation location. Beams and loss-cone features are visible, as is very good agreement
between and the observed beam speeds.
Figure 13.17: Predicted electron distribution functions corresponding to the observations in
Figure 13.16 [Fitzenreiter et al., 1990]. These are constructed by following particle paths
using the Boltzmann equation with no source/loss terms and including mirror reflection
and magnetospheric leakage at the shock. Note the cutoff distribution at and
loss cone features. The bottom panels show the effects of limited instrumental
resolution.
Note that the vertical lines at predicted by Eq. (13.16) show a
clear separation between undisturbed solar wind electrons at lower parallel speeds and
particles streaming away from the bow shock at higher parallel speeds. There is also
clear evidence for plateaued bump-on-tail distributions (cf. Figure 10.4). The symmetric
``horns'' in the distribution at significant 
and partial hole in the
distribution near 
above are consistent with the formation of
a loss cone due to magnetic mirror reflection at the bow shock (cf. Lecture 2).
Figure 13.18 [Cairns et al., 1997] illustrates the Langmuir waves excited near 
by cutoff distributions of electrons. In addition, ion beams are also
produced by ion reflection at the bow shock. These ion beams drive high levels of ion
acoustic waves which are also observed in Figure 13.18. It can be questioned how the ``ion
acoustic'' waves in Figure 13.18 are produced at frequencies of 
kHz which
are much larger than the ion plasma frequency ( 
kHz here). The answer is
``Doppler shift''. The observed wave frequency 
where 
is the relative velocity between the observer and the wave.
In this situation 
and 
, so that
the observed wave frequency is almost entirely Doppler shift. Figure 13.18 also
shows radiation generated near 
, presumably by the same processes that
produce similar radio emissions in type III solar radio bursts.
Figure 13.18: ISEE-1 grayscale dynamic spectrogram showing Langmuir waves in the foreshock and
solar wind, ion acoustic waves in the foreshock, radiation generated in the
foreshock, type III solar radio bursts, and Auroral Kilometric Radiation (AKR).
This interval is discussed more by Cairns et al. [1997].