Many plasmas may be considered to be collisionless in the sense that Coulomb interactions are so rare that it is the global fields in the plasma that govern the dynamics. The mean free path in the solar wind near the Earth, for example, is of the order of the Earth-Sun distance. Shocks in collisionless plasmas are observed to be a source of highly energetic electrons. These electrons are seen in locations as diverse as the Earth's bow shock and supernova shocks. The exact electron acceleration mechanism, however, remains poorly understood.

We investigate structure within quasi-perpendicular collisionless
shocks. This means that the angle between the upstream magnetic field and the
normal to the shock plane, q_{Bn}, is greater than 45 degrees. Such
shocks show ripples in the density and magnetic field moving along the shock
ramp. We use a two-dimensional hybrid code, which models the plasma as ion
macroparticles and an inertialess electron fluid, to calculate the fields. We
then study electron acceleration by integrating electron trajectories within
these fields.

The theory of adiabatic reflection assumes that
electrons move in a magnetic field that is stationary and one dimensional. The
gyroradius of the electrons must also be much smaller than the shock thickness.
In this case, the perpendicular magnetic moment of an electron, (mu), is conserved:

where *m* is the electron's mass, *v*(perp) is the component of its
velocity perpendicular to the magnetic field and *B* is the magnetic field
strength.

We work in the de Hoffmann-Teller frame, in which
the motional electric field (-**V**×**B**) is zero. This means that
we can also consider an electron to conserve its energy. The velocity required
for this transformation from the shock frame is:

where **n** is the normal to the shock plane and
a subscript 0 denotes upstream quantities, so that **V**o is the
upstream flow velocity. I shall denote quantities in the de Hoffmann-Teller
frame with a tilde and unit vectors with a hat. The components parallel and
perpendicular to **B**o of **v**, an electron's velocity in
the upstream plasma frame, are:

Both perpendicular magnetic moment and energy are conserved,
but there will be some electrons that cannot cross the shock whilst conserving
both. This leads to a reflected fraction. Since electrons are constrained to
move along field lines, the reflected electrons have the parallel component of
their velocity reversed in the de Hoffmann-Teller frame. It is important to
note that is negative, so
transforming back to the upstream plasma frame results in an accelerated
reflected population:

Our electron simulations start with an initial test particle
distribution that is a uniformly covered spherical shell in velocity space. The
figure below shows the possible destinations for an upstream electron.

This theory allows us to calculate the energy and
fraction of electrons reflected upstream and compare them with our simulation
results. Adiabatic reflection has been found to be unable to accelerate
electrons to the highest observed energies unless q_{Bn}
is very close to 90 degrees.

We simulate a shock by reflecting homogeneous
plasma, moving at constant velocity, off a stationary perfectly conducting
barrier. This is a standard method for launching a shock, which is physically
simple, but provides a clean shock once the shock front is clear of the
barrier.

The diagram above shows two dimensional structure at
the shock front in the form of ripples. These results were obtained for
conditions similar to those in the Earth's bow shock and at q_{Bn} of 85 degrees.
We find that the strength of the ripples increases with both q_{Bn}
and the plasma inflow speed. The *y*-*t* slices demonstrate that these ripples are moving
in both directions along the shock front. By acting as moving magnetic mirrors,
these ripples may accelerate electrons more than adiabatic reflection alone.

We investigate electron behaviour using two orientations
of the upstream magnetic field, **B**o. With **B**o
lying in the plane of the simulation, electrons move along a field
line and feel the full 2-D field structure. We also attempt to mimic a 1-D
simulation by directing **B**o out of the simulation plane, so that the electrons feel little variation along
a field line.

This figure shows logarithmic spectra of electron
energies in the upstream plasma frame at q_{Bn} = 85 degrees. The downstream
population is represented by solid lines and the upstream population by dashed lines.

Initial electron energy: 0.1 keV

Upstream energy range predicted by theory: 0.15 - 0.39 keV

Bo direction |
Fraction | |

upstream | downstream | |

theory | 87% | 13% |

out | 74% | 23% |

in | 47% | 52% |

The above figure shows electron distributions that are
broadly consistent with adiabatic theory when **B**o is directed out of the simulation
plane. The distributions when **B**o is in the simulation plane,
however, differ. Observations downstream of the Earth's bow shock reveal a
population of electrons with a power law distribution of velocities at high
energies. Our simulations produce spectra that are consistent with a power law
when **B**o is directed in the plane of the simulation. This indicates that shock surface
ripples are important in understanding the electron acceleration process.

The upstream and downstream distributions are similar when **B**o is in the simulation plane and
electrons are allowed to feel the spatial variations in the magnetic field.
This suggests that reflections occur within the shock to such an extent that
the two distributions become mixed and therefore alike. This additional
scattering can be attributed to the 2-D structure.

In future, we hope to improve the electron test particle code by
the addition of cubic spline interpolation for the fields. We also propose to
study the effect of downstream waves on the formation of shock front ripples by
artificially damping downstream wave activity. An obvious criticism of this
kind of study is the tight restriction placed on q_{Bn} required to produce appreciable
rippling. We propose to consider the effects of inhomogeneities in the upstream
field. Larger scale inhomogeneities will be examined by adding a bend to the
upstream field. Smaller scale effects can be studied by introducing upstream
waves. This will more accurately resemble the conditions seen at actual shocks
and may increase the range of q_{Bn} over which ripples are significant.

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Last Revision : 3rd September 1999