Collisionless shocks are observed throughout the Universe in situations as diverse as solar flares, the Earth's bow shock and supernova shock fronts. Although these phenomena have been well explored, shock structure as a result of kinetic effects is less well studied. This thesis represents a study of the magnetic field structure that arises in collisionless shocks as a result of kinetic effects and the effect that this structure can have on electron acceleration at the shock front.
Plasma is defined as being an ionised gas in which electromagnetic forces play an important role. In the limiting case where Coulomb interactions may be entirely neglected, the plasma is said to be collisionless and particle distributions do not become equilibrated through particle-particle interactions. Highly non-Maxwellian distribution functions may therefore exist for extended periods and be significant in the dynamics of the system. For example, the solar wind near the Earth has a mean free path of around 1 AU for thermal electrons and 10 AU for thermal protons. This means that particle distributions created near the Sun may still be observed by Earth orbiting spacecraft. In a collisionless plasma, it is the interaction with fields in the plasma that dominates particle behaviour.
Any theoretical treatment of plasmas must therefore use Maxwell's equations to describe the evolution of the magnetic and electric fields, B and E. Plasma models differ in the way that they couple these equations describing fields to the particles in the plasma. A magnetohydrodynamic (MHD) treatment, for example, consists of a set of conservation equations for continuity, momentum and energy, in conjunction with an induction equation to model electrical conductivity. These equations describe the plasma in terms of its currents, density, pressure and flow velocity, which in turn couple with Maxwell's equations.
A fluid treatment of plasmas, however, is not well suited to treating environments where the shape of the particle distribution function has a significant impact on the dynamics of the system. This can be overcome by using a kinetic description, where it is the distribution of particles in phase space that is modelled. This can be accomplished by treating the plasma as a compressionless "fluid" in phase space. Alternatively, the evolution of the distribution can be studied by looking at the behaviour of individual particles. The introduction of this additional complexity, however, can make kinetic descriptions significantly more complex than fluid models.
Super-thermal electrons are common throughout the Universe. Blandford [1994] reviews the diversity of regions in which populations of high energy electrons are observed. Many of these populations are associated with collisionless shocks. There is, for example, direct evidence of electrons with energies of hundreds of keV at the Earth's bow shock and tens of GeV at the bounding shock waves of supernova remnants. Other electron populations are clearly not a result of shock acceleration. For example, the keV electrons in the terrestrial aurorae are accelerated by an electric field component that is parallel to the magnetic field.
A more complicated picture arises in solar flares, which accelerate significant numbers of ions and electrons to super-thermal energies. The collisionless nature of the plasma in the solar wind means that energetic particles can escape and be detected directly by satellites near the Earth. The particles can also be detected indirectly from the radio waves, X-rays and g-rays that they emit. The most energetic of these electromagnetic diagnostics suggest that small quantities of ultra-relativistic particles are produced. It is not yet clear, however, which acceleration mechanism is responsible for this particle acceleration. Indeed, it is possible that a number of different acceleration processes are significant in solar flares.
Cosmic rays incident upon the Earth's atmosphere provide evidence for even more dramatic acceleration. The origin of the most energetic cosmic rays, with energies around 1020eV, is not clear. There are suggestions, however, that the shock waves at supernova remnants might be responsible for cosmic rays with energy <~1015eV. It is important, therefore, to better understand how collisionless shock waves might be able to transfer such a large amount of energy to a small population of particles.
In the absence of an electric field, the Lorentz force on a charged particle as a result of the magnetic field is always perpendicular to the direction of travel. For this reason, the magnetic field acts as a centripetal force and does no work. This is important because, in the absence of an electric field, we can consider a charged particle to conserve its energy. We can consider this energy to be divided into the component E|| that represents the velocity of the particle along a field line and the component E^ that represents the circular motion about the field line and a result of the magnetic centripetal force.
As the magnetic field felt by a particle changes, so does the centripetal force and hence the associated E^ energy component. If this magnetic field varies slowly over a gyroperiod then
m =
E^
B
(1.1)
is a conserved quantity. This may apply even when there is an electric field, provided that it also changes slowly. Magnetic moment may therefore be conserved even when particle energy is not. A particle that conserves m in this way is said to be "adiabatic".
When the magnetic field varies, the concepts of energy and magnetic moment conservation can be combined to produce magnetic mirroring. The effect of adiabatic behaviour on particle trajectories can be demonstrated by differentiating m with respect to distance along the field line, x. If energy is also conserved, then
m
¶v||
¶t
= - m
¶B
¶x
(1.2)
A change in B along the electron's trajectory therefore acts like a force on the its guiding centre. If both the magnetic moment and the change in B are sufficiently large, the force can reverse the particle's direction of travel and the field acts like a magnetic mirror. A particle will mirror if it has insufficient energy to conserve its magnetic moment in the maximum magnetic field, Bmax, so that the condition for mirroring is
m ³
E||+E^
Bmax
(1.3)
Since mirroring is effectively a collision between the particle and magnetic field, it can result in a transfer of energy between the two. For a frame in which the magnetic mirror is moving, this results in an energy change that forms the basis for Fermi acceleration.
The idea of accelerating charged particles by reflecting them off magnetic field structures was first proposed by Fermi [1949]. Fermi's paper examined the reflection of charged protons from interstellar magnetic clouds, in an attempt to produce cosmic rays. The mechanism has, however, since been extended to cover the reflection of charged particles from any type of magnetic scattering centre. The exact nature of the acceleration depends on the properties of the magnetic field and the particle that it scatters, so various types of Fermi acceleration have been suggested.
In a fast Fermi process, particles receive a single energy increase from one magnetic reflection. This might correspond, for example, to a single reflection at a shock front. Such acceleration mechanisms are very rapid, since the interaction can occur over a very short period of time. The amount of energy that can be gained by a fast Fermi process is, however, limited by the approach speed between the accelerated particle and the scattering centre.
In a first order Fermi process, a charged particle is accelerated by repeatedly passing through a cycle in which it always gains energy. This can happen if the particle is reflected between approaching scattering centres, or if the particle is repeatedly reflected on either side of a collisionless shock. The process is first order because the energy gain depends in a linear manner on the time between reflections. First order Fermi processes can therefore produce unlimited energy gain if they are allowed to continue indefinitely. The fact that the energy gain is first order also means that such mechanisms produce energy gain rapidly.
Second order Fermi acceleration occurs when charged particles are accelerated by scattering off randomly moving magnetic mirrors. There is no first order energy gain, because particles can lose energy by their interactions, as well as gaining it. To second order, however, there is an energy gain because head-on accelerating collisions are more likely than decelerating ones due to the higher relative velocity. Such acceleration is stochastic in nature and corresponds, for example, to the acceleration that occurs in a randomly moving wave field. In common with first order processes, second order Fermi acceleration can produce unlimited energy gain, given sufficient time. The fact that the process is second order, however, means that the energy gain is significantly slower than an equivalent first order process.
A shock wave will arise when any flow that is travelling faster than the local information speed slows to below that speed. In hydrodynamics, the speed at which information travels is the sound speed, so a shock wave, which compresses the fluid and heats it, will form when a supersonic flow becomes subsonic. The free energy from the change in the flow speed is dissipated in heating the fluid, so the process is irreversible. The shock represents a boundary for downstream sound waves and, in some descriptions, the shock transition is represented by a discontinuity. In practice, friction and viscosity mean that the shock has a finite width of the order of the collisional mean free path.
Shocks in plasmas are complicated by the fact that the information speed is determined by the dominant wave mode. In MHD, there are three fundamental wave modes and correspondingly three different types of shock. Fast and slow shocks correspond to the fast and slow magnetosonic wave speeds respectively. In fast shocks, the magnetic field increases when moving downstream across the shock transition, whereas in slow shocks the magnetic field decreases. Intermediate shocks relate to the Alfvén wave speed and cause a 180° change in the direction of the magnetic field. In an isotropic plasma, however, they do not cause any density change and do not form a shock. Fast shocks are common, for example planetary bow shocks are fast shocks. The magnetic field increase at the shock front also makes them the most suitable for particle acceleration, so fast shocks are frequently described in theories of particle acceleration. Slow shocks are less commonly observed, but are important in many magnetic reconnection theories, for example the model proposed by Petschek [1964].
In a kinetic collisionless plasma, it is possible for energetic beams of particles to be generated at the shock front and travel upstream against the flow. Collisionless shocks are therefore unique in that particles can propagate faster than the dominant information speed and generate structures, or "foreshocks", before the shock transition. In addition, the shock structure can no longer be mediated by the collisional mean free path. In this section, we will look at the processes that determine the magnetic structure of a collisionless shock transition. We will also describe the reference frames that are frequently used when studying shock physics. First, we discuss how an MHD approximation of the plasma can be used to describe a shock wave.
MHD can describe how the field and plasma parameters on either side of the shock are related in terms of a unique transformation from upstream (region 0) to downstream (region 1). These equations, the Rankine-Hugoniot relations, describe the transition of the magnetic field B, flow velocity U, density r and pressure p.
The relations are obtained by writing the equations of MHD in conservative form, assuming that the shock is planar, steady and at rest. We use co-ordinates in which the shock normal lies parallel to the x-axis. The first three equations, which relate to the fields, are obtained from Maxwell's equations and Ohm's law:
[Bx]10 = 0
(1.4)
[Ux Bz - Uz Bx]10 = 0
(1.5)
[Ux By - Uy Bx]10 = 0
(1.6)
The remaining relations describe the plasma. The continuity equation provides a single relation, whilst the three components of the momentum equation provide a further three:
[rUx]10 = 0
(1.7)
é ë
rUx2 + p +
By2
2m0
+
Bz2
2m0
ù û
1
0
= 0
(1.8)
é ë
rUx Uy +
Bx By
m0
ù û
1
0
= 0
(1.9)
é ë
rUy Uz +
By Bz
m0
ù û
1
0
= 0
(1.10)
The energy equation, assuming a polytropic equation of state with exponent g, gives the final Rankine-Hugoniot condition:
é ë
1/2 rUx U2 +
g
g-1
p Ux + Ux
By2 + Bz2
m0
-
Bx
m0
(By Uy + Bz Uz)
ù û
1
0
= 0
(1.11)
These equations are useful in predicting the equilibrium state of the plasma downstream of the shock in terms of the upstream state. They define a plane that contains the shock normal and the upstream and downstream magnetic field and flow vectors. This is known as the coplanarity theorem and can dramatically simplify the analysis of shocks. By taking the high Mach number limit, the Rankine-Hugoniot relations also set an upper limit to both the compression ratio and magnetic field jump at the shock.
Although the Rankine-Hugoniot transformation is unique in the sense that the equations have a single solution, the equations do not completely describe the changes made to the plasma. There is no information, for example, on how each of the particle species in a plasma are affected, or which wave modes may be excited. Such details require a kinetic description of the shock transition.
Figure 1.1: Velocity diagram in (v||, v^) showing the initial plasma frame (IPF), normal incidence frame (NIF) and de Hoffman-Teller frame (HTF). The dashed line indicates the set of shock frames and all velocities are assumed to be in the coplanarity plane.
There are a variety of reference frames that are relevant to shock physics. The frames that we will look at all lie in the coplanarity plane. Figure 1.1 shows how these frames relate to each other in (v||, v^) velocity space.
Shock frames are defined by the fact that the plane of the shock is at rest. The origins of all shock frames therefore form an infinite plane in velocity space, such that the velocity between the origins of two shock frames is always perpendicular to the shock normal. We define qBn to be the angle between the shock normal and the upstream magnetic field, so this plane of shock frames intersects the shock coplanarity plane along a line that is at an angle qBn to the v|| axis.
The initial plasma frame (IPF) is the frame in which the upstream plasma is at rest. The upstream flow velocity vector between the IPF and any shock frame is therefore the inflow velocity in that shock frame. The normal incidence frame (NIF) is the shock frame in which this velocity vector is parallel to the shock normal. The inflow velocity in the NIF, V0, defines the Mach number of the shock.
In ideal MHD, where the plasma is considered to be perfectly conducting, the electric field is equal to U ×B. Thus a frame transformation can create an electric field, known as the motional electric field, that is perpendicular to the magnetic field. This will result in particle drift and individual particles cannot be considered to conserve their energy. The de Hoffman-Teller frame (HTF) is the shock frame in which the plasma velocity vector is parallel to the magnetic field. In this frame the motional electric field is zero and particle energy is therefore conserved. The HTF is therefore uniquely defined as the shock frame with v^ = 0, as shown in Figure 1.1.
In order to examine the structure upstream of a collisionless shock, we need to be able to determine whether a given particle will escape upstream. Energetic particles are tied to their magnetic field line, so escape upstream depends on whether the particle has sufficient velocity along the field line to overcome the inflow into the shock. For shocks with qBn » 90°, the necessary speed can be very large. In most reference frames, this calculation is complicated by the effects of E×B particle drift.
The de Hoffman-Teller frame can be used to determine easily whether a given particle will escape upstream. There is no E×B drift in this frame, and the fact that it is a shock frame means that the inflow velocity does not need to be known. If an particle has a positive parallel velocity component in the HTF, it will be carried downstream by the flow. If, however, v|| is negative, the particle will be travelling along the field line away from the shock with sufficient speed that it can escape upstream.
In contrast to the description of a shock given by the Rankine-Hugoniot relations, which only relate downstream and upstream quantities, observations of collisionless shocks in space suggest that there is a significant amount of structure in the shock transition. The ions in the plasma carry the majority of the energy and momentum that needs to be dissipated by the shock, so shock structure is governed almost exclusively by ion physics. The electrons in the plasma have less inertia and are therefore more mobile, so they produce charge neutrality. The physics of the shock transition is reviewed by Burgess [1995], Baumjohann & Treumann [1996] and Treumann & Baumjohann [1997].
An important factor in the nature of shock structure is the ease with which downstream ions can be transmitted back upstream through the shock. This is a function of the angle qBn between the shock normal and the magnetic field. In the extreme case of a parallel shock, with qBn
= 0°, particle movement through the shock is easiest since the particle's parallel velocity need only exceed the plasma inflow speed. In a perpendicular shock, with qBn
= 90°, the magnetic field has no component parallel to the shock normal, so a downstream particle that is tied to a field line can never escape back upstream. For intermediate values of qBn, the shock is said to be oblique and it becomes useful to create a distinction between shocks with qBn < 45°, which are called quasi-parallel, and shocks with qBn > 45°, which are called quasi-perpendicular.
In a quasi-parallel shock, significant numbers of ions can be reflected at the shock front and escape upstream to form an ion beam. This beam is unstable and interacts with the inflowing plasma to produce waves in the foreshock region. Simulations of quasi-parallel shocks show that the structure at the shock front is complex and mediated by time dependent turbulent structures. This means that there is no single steady shock surface and the foreshock contains short large amplitude magnetic structures (SLAMS) that propagate towards the main shock surface.
The nature of the quasi-perpendicular shock transition is strongly dependent on the inflow Mach number. For low Mach number shocks, the necessary dissipation can occur through the interaction of ions with turbulence at the shock front. In this case, the dissipation is confined to a narrow ramp of small, but finite, width that is related to an ion gyroradius. The amount of dissipation that can be produced by this mechanism is, however, limited by the narrow width of the shock ramp and the fact that the Rankine-Hugoniot relations set an upper limit the magnetic field jump at the shock. Above a critical inflow speed, therefore, the Rankine-Hugoniot relations predict that adiabatic ion heating would produce supersonic downstream flow. Shocks with inflow speeds above this critical Mach number are said to be supercritical and require additional dissipation, which is provided by ion reflection.
Figure 1.2: The magnetic field profile from a supercritical quasi-perpendicular collisionless shock simulation, showing the important regions in the shock transition.
The magnetic field profile of a supercritical quasi-perpendicular collisionless shock, as produced by a kinetic plasma simulation, is shown in Figure 1.2. In supercritical shocks, the reflection of ions at the shock front is responsible for the dissipation of energy. The energy gained by a magnetic reflection depends on the approach speed of the shock and the ion, so this mechanism is able to transfer more energy as the inflow speed increases. As these reflected ions gyrate back into the shock ramp, they extend the lower end of the shock, forming a shock "foot" whose width depends on their reflected gyroradius. The energy is finally dissipated in an irreversible manner when the ions interact with wave turbulence downstream of the shock. We saw in Section 1.3 that the condition for magnetic reflection depends on the maximum value of the magnetic field. In order to reflect enough ions to provide the necessary dissipation, the shock also forms a magnetic overshoot, so the maximum magnetic field in the shock transition is actually higher than that suggested by the Rankine-Hugoniot relations. The bimodal nature of the overshoot in Figure 1.2 is typical of such simulations and is likely to be related to the details of the reflected ion trajectories.
Rankine-Hugoniot Equations 1.5 and 1.6 require the electric field components perpendicular to the shock normal to be equal on either side of the shock. The finite width of the shock transition, however, means that quasi-perpendicular shocks are able to support an electric field that is parallel to the shock normal. The electric field arises in order to balance the force on electrons from the pressure gradient across the shock and thus maintain quasi-neutrality.
Chapter 2 reviews the present state of knowledge in the area of electron acceleration in solar flares. We discuss the evidence supporting reconnection in flares and the fragmented nature of the energy release process. We review the observations of highly energetic electrons in flares and describe the dominant theories for electron acceleration. We then discuss how shocks in flares may have similarities to the Earth's bow shock and argue that understanding observations of energetic bow shock electrons may provide us with a better understanding of the acceleration process for electrons in solar flares.
Chapter 3 discusses the various methods available to us to simulate plasmas numerically and focus in detail on the hybrid scheme for plasma modelling. In particular, we look at the CAM-CL algorithm, a 2-D version of which we use for all shock simulations. We discuss test particle codes and present our results comparing various schemes for integrating trajectories and interpolating fields.
Chapter 4 presents our study of ion-scale ripples. Simulations of quasi-perpendicular collisionless shocks show ripples in the density and magnetic field moving along the shock ramp. Using our 2-D hybrid code, we simulate these ripples (Figure 1.3) and determine how their properties are affected by shock parameters. We have investigated the movement of ripples along the shock overshoot using Fourier techniques and believe that the rippling is consistent with a surface mode.
Figure 1.3: A map of Bx taken from a simulation with qBn
=88° and Vin » 5.66 vA. The maximum value of Bx is approximately 2 B0 and the profile of |B| is superimposed.
Chapter 5 introduces our work on electron populations at a quasi-perpendicular collisionless shock. We observe magnetic structures that are time dependent and cannot be approximated as one-dimensional. Using our hybrid simulation and a relativistic test particle code, we examine the effect of ripples on electron acceleration. We propose a new mechanism for electron acceleration based on Fermi acceleration by field aligned structure within the shock transition. We show how trapping by this two dimensional structure can cause electrons to be convected downstream with the magnetic field, despite having magnetic moments which suggest that they should be reflected upstream. These electrons undergo considerable Fermi acceleration during the shock transition, which may explain observations of an energetic population of electrons downstream of Earth's bow shock.
Chapter 6 investigates the resulting electron energy spectra and compares the simulated power law indices with those derived from in situ measurement downstream of Earth's bow shock. Our simulations produce differential energy spectra with a power law tail whose slope is consistent with the observed power law index at the Earth's bow shock.
Chapter 7 summarises our results and conclusions in addition to making suggestions for further work that might be motivated by this thesis.
In the context of work-in-progress, Appendix A discusses the challenge of writing parallel simulation codes. We discuss how we are applying the concepts of large Beowulf clusters and parallel programming to the implementation of a parallel 3-D hybrid plasma code. Such a code would allow us to model ripple generation in more detail and, using the Fourier techniques applied to the 2-D case, determine all components of the ripples' wave velocity.
Page Maintained By :
Rob Lowe
Last Revision : 1st March 2003
