List of Figures
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.
1.2 The magnetic field profile from a supercritical quasi-perpendicular collisionless shock simulation, showing the important regions in the shock transition.
1.3 A map of Bx taken from a simulation with qBn=88° and V in » 5.66 vA. The maximum value of Bx is approximately 2 B0 and the profile of |B| is superimposed.
2.1 The model flare morphology of Shibata [1995] and Yokoyama & Shibata [1996]
2.2 The dominant plasma wave modes propagating parallel to the magnetic field in a cold hydrogen plasma. The w axis is labelled with the ion cyclotron (wci), electron cyclotron (wce) and electron plasma (wpe) frequencies.
2.3 Velocity diagram in (v||, v^) showing the velocities acquired by the transfer to the de Hoffman-Teller frame from the initial plasma frame (IPF) and normal incidence frame (NIF). The dashed line indicates the set of shock frames and all velocities are assumed to be in the coplanarity plane.
2.4 Velocity space diagram showing velocities relevant to diffusive acceleration in a parallel shock. The upstream initial plasma frame (IPF), downstream plasma frame (DPF) and normal incidence/de Hoffman-Teller frame (NIF/HTF) are shown in grey. Particle distributions at the average energy are shown in black, labelled by the number of passes through the shock, N. The solid lines represent the part of the distribution that will travel back through the shock, whilst the dotted lines represent the particles that will escape from the shock.
2.5 The magnetic topology of the solar wind's interaction with the Earth's bow shock (BS) and magnetopause (MP). The electron foreshock is shaded grey.
3.1 A diagram of the simulation box and initial conditions that we use in order to generate a shock wave.
3.2 Graphs illustrating the conservation properties of electron trajectory integrators. The intensity of the boxes represents particle number density on a logarithmic scale. The fourth order Runge-Kutta scheme is shown with Dt = 0.1 We-1 (red) and Dt = 0.02 We-1 (green). The implicit leapfrog scheme is shown with Dt = 0.1 We-1 (blue).
3.3 Labelling of quantities within a grid cell.
3.4 Graph comparing the Bx field component interpolated using different interpolation schemes. The interpolation methods used are bilinear (black), bicubic (blue) and splines (red).
4.1 The geometry of a collisionless shock, as used in our simulations and discussion of electron acceleration mechanisms. The magnetic field B, inflow velocity V0 and shock normal [^(n)] are co-planar. In this geometry, the reference frame is such that V0 and [^(n)] are anti-parallel.
4.2 Diagram of all E and B components for q Bn=88° , V in=4vA for a simulation where the upstream magnetic field lies in the plane of the simulation.
4.3 Diagram of Alfvén speed, density and the magnitudes of E and B components for q Bn=88° , V in=4vA for a simulation where the upstream magnetic field lies in the plane of the simulation.
4.4 Diagram of all E and B components for q Bn=88° , V in=4vA for a simulation where the upstream magnetic field points out of the plane of the simulation.
4.5 Diagram of Alfvén speed, density and the magnitudes of E and B for q Bn=88° , V in=4vA for a simulation where the upstream magnetic field points out of the plane of the simulation.
4.6 This hodogram illustrates the lack of correlation between the Bx and Bz field components at the overshoot.
4.7 Graph showing how the power in the shock ripples increases with the size of the overshoot. The points from simulations with rippling are labelled with their value of q Bn.
4.8 Graph showing how the power in the ripples increases with q Bn for an inflow speed of 4vA.
4.9 These figures show the how the square of Bx, integrated over the y direction, varies with x for a variety of different upstream parameters. The value of the total magnetic field integrated over y is shown for comparison. The < Bx2 > quantity is an indicator of the power in the ripples at a given x. The location of a kink in the graph is indicated by a dashed line.
4.10 Graphs of the local flow speeds in units of the local fast mode and Alfvén speeds variety of shock transitions. The location of the kink in the ripple amplitude shown in Figure 4.9 is indicated by a dashed line.
4.11 These slices of the Bx field (top) were taken in the shock frame at the positions indicated (bottom). The colour scale is the same as that used for the Bx component in Figure 4.2.
4.12 The Fourier plots (left) show the w-k distribution of power for the Bx slices (Figure 4.11) using a logarithmic scaling. The w-power plots (right) show slices through the w-k diagrams at k=1 (8th Fourier mode). Figure 4.13 shows equivalent diagrams for regions inside the shock.
4.13 The Fourier plots (left) show the w-k distribution of power for the Bx slices (Figure 4.11) using a logarithmic scaling. The w-power plots (right) show slices through the w-k diagrams at k=1 (8th Fourier mode). Figure 4.12 shows equivalent diagrams for regions outside the shock.
4.14 Chart correlating the speed of ripples to other significant wave speeds at q Bn=85° . The crosses correspond to the measured best c2 fit to the ripple speed, the open circles to the overshoot Alfvén speed and the solid triangles to the downstream Alfvén speed.
4.15 Fourier transforms of the three magnetic field components of a field slice running along the shock overshoot of a q Bn=88° , V in=4vA simulation.
4.16 Fourier transforms of the Bx field component for simulations with different numerical parameters. These graphs do not show the whole range of the Fourier transform, as only the region with |k| < 2 was used to fit a dispersion relation. Outside this range, any signal becomes lost in noise.
4.17 A sketch of our model for a shock surface wave. The surface mode consists of a pair of exponential solutions which meet at the top of the overshoot. At a point upstream of the overshoot, the local flow velocity exceeds the local fast mode wave speed and the surface mode's exponential solution (dotted line) decays into an evanescent wave.
5.1 Velocity space diagram showing adiabatic reflection for a shock with q Bn=88° , V0=4vA. The reflected population is shown in red and the population that transmits is in blue.
5.2 Velocity space diagram showing adiabatic reflection for a shock with q Bn=85° , V0=4vA. The reflected population is shown in red, the population that transmits is in blue and the population that never interacts with the shock is black.
5.3 Velocity space diagram showing adiabatic transmission for a shock with q Bn=88° , V0=4vA. The reflected population is shown in red and the population that transmits is in blue.
5.4 The variation of |B| along a field line running along the shock front.
5.5 The distribution of initial pitch angle, a, for electrons that end the simulation upstream (dashed) and downstream (solid), for B0 in the simulation plane (top) and out of the simulation plane (bottom).
5.6 Electron differential energy spectra in the upstream plasma frame (solid lines downstream, dashed lines upstream) with B0 out of the simulation plane (bottom) and B0 in the simulation plane (top).
5.7 Graphs showing the degree to which magnetic moment is conserved, plotting the logarithm of the ratio of final to initial magnetic moment, log([(m final)/(m final)]), against initial pitch angle, a init, for all electrons in the simulation.
5.8 Graphs comparing the parallel and perpendicular components of the velocity at the end of the simulation for electrons that escape upstream (top) and downstream (bottom), for a variety of simulation parameters.
5.9 Energy, magnetic moment m, pitch angle a, total magnetic field |B| and spatial trajectories for two electrons whose acceleration is consistent with adiabatic theory.
5.10 Energy, magnetic moment m, pitch angle a, total magnetic field |B| and spatial trajectories for two electrons that become trapped in the shock structure.
5.11 Energy, magnetic moment m, pitch angle a, total magnetic field |B| and spatial trajectories for two electrons that show unusual behaviour. We believe that Figure (a) is subject to numerical inaccuracies for t >~2 Wi-1.
5.12 Graph showing how the electron in Figure 5.11(a) is trapped in the shock transition twice.
6.1 A diagram showing the features of the downstream (solid line) and upstream (dashed line) accelerated electron energy spectrum. The shaded region represents the portion of the upstream spectrum which may be approximated by a power law. The high energy shoulder to the upstream spectrum is probably a numerical artefact.
6.2 Graphs showing the upstream (dashed) and downstream (solid) final electron energy distribution in the qBn=88° , V in=4vA, Einit=100 eV shock. The time step in the electron test particle code Dte, was varied to show the numerical dependence.
6.3 Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions for qBn=88° and a range of inflow velocities V in.
6.4 Graphs showing (a) the percentage of electrons reflected upstream and (b) the power law exponent of the electron energy spectrum. Both quantities are plotted against the inflow velocity of plasma into the simulation box, V in, for shocks with qBn=88° .
6.5 Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions at a fixed inflow speed of 4vA and a range of angles q Bn.
6.6 Graphs showing (a) the percentage of electrons reflected upstream and (b) the power law exponent of the electron energy spectrum. Both quantities are plotted against the value of the angle q Bn, for shocks with V in=4vA.
6.7 Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions for a variety of initial energies, Einit, in a shock with V in = 4vA and q Bn = 85° . The high energy "shoulder" to the distributions appears to be a numerical artefact.
6.8 Graphs showing (a) the percentage of electrons reflected upstream and (b) the power law exponent of the electron energy spectrum. Both quantities are plotted against the logarithm of the initial electron energy, \qopname \relax olog(E init), in a shock with V in = 4vA and q Bn = 85° .
6.9 The distribution of initial pitch angle, a, for electrons that end the simulation upstream (dashed) and downstream (solid), for three values of Einit, in a shock with V in = 4vA and q Bn = 85° .
6.10 Graphs showing the upstream (dashed) and downstream (solid) final electron energy distributions for a variety of conditions typical of the solar corona.
6.11 Time sequence of the electron energy spectrum for all particles in the solar coronal shock simulation with qBn=88° , V in=4vA, Einit=1 keV.
Page Maintained By :
Rob Lowe
Last Revision : 1st March 2003