Chapter 6
Parametric Survey of Shock Accelerated Electrons

This chapter brings together the study of shock ripple properties in Chapter 4 with our demonstration in Chapter 5 that such shock structure can affect electron acceleration. We look at a variety of different shocks and discuss how the properties of the shock structure affect electron acceleration. We also investigate how the acceleration process is affected by the initial energy of the electron distribution and by the inclusion of relativistic effects and discuss how our results relate to observations of accelerated electrons at the Earth's bow shock. We also investigate how electrons might be accelerated by shocks in solar flares and compare our results with the observations discussed in Chapter 2. We find that there is significant scope for further work on these subjects.

6.1 Spectral Properties

Figure 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.

We showed in Chapter 5 that the conventional view of adiabatic acceleration of electrons at collisionless shocks may, when the shock is supercritical, be supplemented by an additional population of trapped electrons. In some cases the trapped population may dominate the accelerated electron energy spectrum and Figure 6.1 shows, in schematic form, a spectrum from acceleration by shock ripples. This diagram is based on the energy spectra shown in Figure 5.6 for simulations with B₀ in the simulation plane. The downstream spectrum is relatively simple, with a peak near the injection energy and an extended power law tail. This tail results from the trapping of electrons which according to adiabatic theory are expected to reflect.

The upstream spectrum is more complicated. In order for an electron with pitch angle a to escape upstream its kinetic energy must exceed

E_esc(a) =

é
ë

æ
è

1 -

V₀² sin⁴q_{B_n}

c² cos²q_{B_n}cos²a

ö
ø

-1/2

- 1

ù
û

m_e c²

(6.1)

where m_e c² is the rest mass energy of the electron. The upstream spectrum has a low energy cut-off at E_esc(a = 180^°), the minimum energy required to escape upstream. Above this energy, the upstream flux rises rapidly as electrons with a wider range of pitch angles meet the escape energy requirement. The upstream flux reaches a maximum once it reaches a similar value to the downstream flux. Above this energy, in the shaded region of the spectrum in Figure 6.1, the power law decline of electrons at a given energy dominates over the increasing fraction that are able to escape upstream. The upstream and downstream fluxes in this region of the spectrum are broadly similar and may both be characterised by a power law. At the highest energies the downstream flux continues to follow a power law. The upstream spectrum, however, exhibits a "shoulder" which corresponds to the electrons that pass through the shock transition twice and undergo trapping in both passes, as seen in Figures 5.11(a) and 5.12. In Section 6.1.1 we show that this shoulder is probably a numerical artefact.

Although Figure 6.1 shows the general form of the energy spectrum for trapped electrons, conditions may not always favour trapping and we may also see evidence of adiabatic electron behaviour. In Chapter 5 we saw that adiabatic electrons have a significantly different spectrum to that for trapped electrons. The adiabatic downstream electrons with high m are preferentially accelerated and the maximum acceleration possible, given ideal conditions, is

E_final

E_init

= 1 + 2

V_HT

(6.2)

The highest energy gain that can be achieved by the shocks studied in this chapter would accelerate an electron from E_init = 100 eV to E_final » 590 keV. Although a given shock will not necessarily be able to accelerate transmitted electrons to this energy, any electrons that exceed it must result from a process that is not adiabatic.

The adiabatic upstream population has a flat spectrum between two energies:

f(E) =

E_max-E_min

\thinspace

F_r

F_tot

(6.3)

for E_min £ E £ E_max and f(E) = 0 otherwise. Comparison with Figure 6.1 shows that the trapped and adiabatic spectra are therefore quite different and it should be possible to distinguish between the two.

This thesis has so far examined how our simulations allow us to study the relaxation of two constraints required by adiabatic theory: the conservation of m and the stationary shock front. A further assumption of adiabatic theory breaks down when the electron gyroradius becomes comparable with the ion inertial length (r_g » v_A W_i^-1) since the gyroradius is no longer negligible compared to the width of the shock. The relationship between an electron's gyroradius, r_g, and its perpendicular velocity, v_^ is

r_g =

gm v_^

B e

(6.4)

where g is the electron's Lorentz factor. In the upstream regions of Earth's bow shock, the gyroradius becomes comparable with the ion inertial length at an energy of around 30 keV, which is not relativistic. At the overshoot, however, a magnetic field strength of B=4B₀ would require an energy of 400 keV for the gyroradius to be comparable with the ion inertial length. At this energy g = 1.8, so we are in the relativistic regime. The scale length that is important for electron trapping at ripples will be different, however, since the distance between reflections is smaller than the shock thickness and hence less than an ion inertial length. Figure 5.12 shows reflections on scales as low as 0.1 v_A W_i^-1. At the overshoot r_g » 0.1 v_A W_i^-1 at an electron energy of around 5.5 keV, so any features in the electron energy spectra, if they exist, would be seen above this energy.

These characteristic spectra and energies illustrate the properties that we should look for in our simulation results. We should be able to characterise a spectrum as being a result of either adiabatic reflection or trapping. We also know at what energies we might find features that are the result of a relativistic transition or the electron gyroradius becoming comparable with ion scales. We therefore have a basis for comparing the results from simulations with different initial conditions, which allows us to conduct a parameter survey.

6.1.1 Numerical dependence

Figure 6.2: Graphs showing the upstream (dashed) and downstream (solid) final electron energy distribution in the q_Bn=88^°, V_in=4v_A, E_init=100 eV shock. The time step in the electron test particle code Dt_e, was varied to show the numerical dependence.

In Section 4.5.1 we showed that the properties of the shock fields produced in our simulations were not significantly affected by the numerical resolution of the simulation grid. Chapter 3 showed that our electron test particle code is able to integrate electron trajectories well in a magnetic bottle configuration. In shock fields, however, we do not expect electrons to conserve their magnetic moment. When particles are confined in a magnetic trap, the reflection time depends on the parallel velocity of the particles and the size of the trap. We are interested in the properties of the tail of the electron distribution, which are more likely to have a reflection time that is comparable with the time step in the electron code, Dt_e. It is therefore important to understand how changing Dt_e can affect electron trajectories in shock fields.

Dt_e	power law exponent
	upstream	downstream
0.1 W_e^-1	-1.73	-1.93
0.05 W_e^-1	-2.07	-2.16
0.025 W_e^-1	-2.06	-2.18
0.0125 W_e^-1	-1.96	-2.02

Table 6.1: The power law exponent of the upstream and downstream distributions for a variety of simulations with q_Bn=88^°, V_in=4v_A, E_init=100 eV and varying time steps in the electron test particle code, Dt_e.

Figure 6.2 shows the final electron energy distributions for simulations run with different time steps, Dt_e, in the electron test particle code. The position and height of the upstream shoulder is very sensitive to Dt_e. For the simulations with Dt_e=0.1 W_e^-1 the shoulder is so dominant that it becomes difficult to determine a power law exponent for the upstream distribution. As Dt_e becomes smaller, the shoulder starts to appear at progressively higher energies and also becomes smaller in size. This dependence of the shoulder properties on a numerical parameter, and the fact that the shoulder reduces in size as the numerical accuracy increases, shows that the upstream shoulder is largely, if not completely, a numerical artefact.

In the case of the Dt_e=0.05 W_e^-1 simulation, the upstream shoulder starts to appear at an energy of around 4 keV. At this energy, the distance travelled by an electron in a time step Dt_e is approximately 0.017 v_A W_i^-1. This is an order of magnitude smaller than the grid spacing, Dx = 0.2 v_A W_i^-1, so the numerical inaccuracies are not related to the resolution of the numerical grid.

The shoulder is probably caused by the fact that the most energetic electrons reflect very rapidly and have short reflection lengths (Figure 5.12). For these rapidly reflecting electrons, it seems that the reflection time becomes comparable with Dt_e. The electron can then artificially escape from the magnetic trap between two ripples, although their energy and magnetic moment appear to be conserved. In the case of Figure 5.12 the electron can be seen to escape into another, adjacent, magnetic trap at around t = 2.0 W_i^-1. The downstream distribution has no such shoulder and does not appear to be dependent on Dt_e.

Although the Dt_e=0.0125 W_e^-1 simulation shows considerably less evidence of the upstream shoulder, it is prohibitively expensive to use in a parameter survey because of the computational time required. In theory, an adaptive time step could be used to reduce the total computational time, since each electron spends a relatively small period of time experiencing the high field gradients that can lead to numerical inaccuracies. Our implementation of the bicubic field integrator, however, requires a significant amount of computational time in each electron time step to calculate field derivatives at each grid point. This means that just one electron interacting with the shock would slow down the simulation significantly. Since at least one electron is interacting with the shock throughout the majority of the simulation, this means that we would need to redesign our field interpolation algorithm in order for an adaptive time step to be practical. The inclusion of an adaptive time step is therefore an avenue for future research.

Table 6.1 shows how the power law exponent of the upstream and downstream distributions determined by a c² fit depends on Dt_e. The interval used for the fit corresponds to the shaded region in Figure 6.1. In both the upstream and downstream the power law exponent of the electron distribution does not have any significant dependence on Dt_e. We therefore assume that, although the upstream distribution function is unreliable above the energy at which the shoulder appears, all other parts of the energy distribution are sufficiently free of numerical artifacts. For the remainder of our simulations we have used an electron time step of Dt_e=0.05 W_e^-1, which has a much lower computational cost than the Dt_e=0.0125 W_e^-1 simulation. This time step allows us to simulate enough of the distribution to be able to calculate the power law exponent and restricts the upstream shoulder to a relatively small fraction of the total simulated particles. Importantly, the power law exponents shown in Table 6.1 do not show a significant difference between Dt_e=0.05 W_e^-1 and the smaller time steps.

6.2 Changing Initial Conditions

In Chapter 5 we showed how the process of adiabatic reflection depends strongly on the angle q_{B_n}. We also saw in Chapter 4 that this angle, along with the plasma inflow speed, V₀, has a significant effect on the amplitude of rippling in the shock structure. We will therefore investigate how changing these parameters can have an impact on the shocked electron spectrum.

V_in	q_{B_n}	M_A	áB₁ ñ/B₀	áB₂ ñ_max/B₀	Var(B_x)/B₀	E_esc(a = 180^°)
1.4 v_A	88^°	2.68	2.22	2.25	0.00238	60 eV
2 v_A	80^°	3.30	2.79	2.70	0.0469	3.5 eV
2 v_A	85^°	3.34	2.80	2.66	0.0198	15 eV
2 v_A	88^°	3.35	2.76	2.63	0.0290	94 eV
2 v_A	89^°	3.35	2.79	2.67	0.0163	380 eV
4 v_A	80^°	5.65	4.55	3.87	0.689	11 eV
4 v_A	85^°	5.67	4.54	3.93	1.01	43 eV
4 v_A	88^°	5.67	4.67	3.86	1.14	270 eV
4 v_A	89^°	5.68	4.49	3.71	1.30	1.1 keV
6 v_A	88^°	8.21	6.12	3.95	3.35	570 eV

Table 6.2: The properties of a sample of ten shocks.

The properties of the sample of ten shocks used to look at this variation of ripple properties, all of which have b = 0.5 and B₀ directed in the simulation plane, are shown in Table 6.2. The quantity áB₁ ñ is the value of the magnetic field at the top of the overshoot, averaged over the y direction (i.e. averaged over in the shock plane). Var(B_x) is the variance of the B_x field component, also calculated over y direction, and is a good measure of the power in the shock ripples. áB₂ ñ_max is the maximum value of the averaged downstream magnetic field and, when subtracted from áB₁ ñ, provides a measure of the size of the overshoot. The parallel escape energy, E_esc(a = 180^°), represents the energy required in a direction parallel to the magnetic field in order for an electron to escape upstream.

V_in	t_init	t_final	upstream boundary
1.4v_A	15 W_i^-1	37.5 W_i^-1	x £ -25 v_A W_i^-1
2v_A	15 W_i^-1	37.5 W_i^-1	x £ -25 v_A W_i^-1
4v_A	10 W_i^-1	37.5 W_i^-1	x £ -25 v_A W_i^-1
6v_A	10 W_i^-1	27.0 W_i^-1	x £ -10 v_A W_i^-1

Table 6.3: Time intervals over which the electron simulations are run.

The electron test particle simulation conditions are identical to those used in Chapter 5, except that the start and end times of the electron simulation will change to reflect the differing shock speeds. As V_in varies, so does the time interval over which it is appropriate to run the electron simulations. The starting time, t_init, was determined by the time it takes the shock to clear the influence of the reflecting barrier. The finishing time, t_final, was restricted by the time at which electrons at the upstream escape boundary would be outside the simulation box. In addition, for V_in=6v_A, the upstream escape boundary was moved. These times and boundary conditions are shown in Table 6.3.

In common with the simulations in Chapter 5 the number of particles simulated is not constant between simulations, again due to computational resource constraints. Also, in those cases where increased resolution appeared to be of interest, additional simulations were run with different random seeds in order to increase the particle count.

We present the results of our simulations as logarithmic electron energy spectra at t_final. We have attempted to measure the power law exponent of the energy spectra in the region of the spectrum corresponding to the shaded area in Figure 6.1. This was done by using a c² fit over the region of the logarithmic spectrum that was approximately linear. We also calculated the fraction of electrons that had reflected and passed upstream by t_final. Any electrons remaining in the vicinity of the shock were assumed not to have reflected and, for the shocks with inflow velocities of V_in ³ 4v_A they amount to, at most, 2% of the total number of particles simulated. For the lower Mach number shocks, however, around 10% of simulated particles had yet to escape from the shock by t_final. These electrons do not tend to be energetically significant, however, because they are not usually still interacting with the shock. Such electrons are more likely to simply have velocities which cause them to keep up with the movement of the shock in the x direction.

6.2.1 Inflow speed

Chapter 4 shows that ripple amplitude is particularly sensitive to the size of the overshoot, which is in turn very sensitive to the shock Alfvén Mach number, M_A. This obviously makes the speed of plasma inflow into the simulation box, V_in, a good candidate for our parametric study. We may see an additional dependence because the time taken for a field line to convect through the shock falls with increasing M_A. The duration of the interaction of electrons with the shock transition will be correspondingly smaller, which could result in reduced electron acceleration.

Figure 6.3: Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions for q_Bn=88^° and a range of inflow velocities V_in.

$ref_frac_vs_v0$

Figure 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 q_Bn=88^°.

The energy spectra for a sample of shocks with varying V_in are shown in Figure 6.3. All shocks had q_Bn=88^° and the simulations were run with an initial electron energy E_init=100 eV. Table 6.2 shows that the shocks with inflow velocities of 1.4v_A and 2v_A have little, if any, overshoot and have a correspondingly small amount of rippling. We would therefore expect them to produce predominantly adiabatic electron acceleration. Figure 6.4 shows that the reflected fraction for these simulations are consistent with the adiabatic predictions. The upstream power law exponents are very steep, which is also consistent. According to adiabatic theory, the downstream electrons will be accelerated by drift along the motional electric field to, at most, 600 eV. This is consistent with the simulations, although the downstream spectrum may also be broadened by a low level of leakage from upstream, in addition to variations in the cross-shock potential. The V_in=2v_A shock produces a very low flux of high energy electrons. This could be the result of weak rippling, consistent with an adiabatic energy spectrum dominating at lower energies. At the higher energies we do not expect to see adiabatic electrons, so the component of the energy spectrum produced by trapped electrons (shown in Figure 6.1) becomes visible.

The shock with an inflow velocity of 4v_A has appreciable rippling and produces an energy spectrum consistent with the schematic of Figure 6.1. The power law exponents of -2.1 in the upstream and -2.2 in the downstream are comparable, with a shoulder appearing in the upstream spectrum above around 5 keV. We believe that this shoulder is a numerical artefact, but we note that its appearance corresponds to the energy at which r_g » 0.1 v_A W_i^-1 and the electron gyroradius becomes comparable with reflection scales. In common with the V_in=1.4v_A and V_in=2v_A shocks, the low energy upstream cut-off is consistent with that predicted in Table 6.2.

The V_in=6v_A shock has particularly strong rippling. This inflow velocity is too high for electrons to reflect adiabatically, although there is a small population of upstream electrons with energies above around 10 keV. This energy is substantially higher than the cut-off energy of 570 eV predicted in Table 6.2. Since no electrons are expected to reflect, this could indicate that the population has been trapped and brought downstream, being accelerated to above 570 eV in the process. The electrons could then have reflected and undergone trapping and acceleration for a second time on their second transition through the shock. This energisation process is identical to that proposed for electrons in the upstream shoulder to the distributions seen in other simulations and the existence of this upstream population could therefore be a numerical artefact. Additionally, Figure 6.4(b) shows that the downstream spectrum is significantly steeper than that at V_in=4v_A, despite the stronger rippling. This suggests that the field line convection time is significant in determining the slope of the accelerated spectrum, although the ripple amplitude is likely to dominate in determining the proportion of electrons that experience trapping.

It seems that the trapping process is strongest in the transition between low inflow speeds, where there is very little rippling and adiabatic reflection dominates, and high inflow speeds, where electrons are unable to gain sufficient energy to reflect back upstream. Gosling et al. [1989] state that interplanetary shocks lack the high energy tail to the electron distribution that is seen at the bow shock, or they at least have a substantially reduced flux. Such shocks also have a lower Mach number than the bow shock. Our results provide an explanation for such behaviour, since the production of the high energy tail is very strongly dependent on Mach number and, in the case of sub-critical shocks, no energetic electrons are produced at all. The Mach number dependence of trapping might also explain the intermittent nature of Type II radio emission from interplanetary shocks, which is due to the interaction of reflected electrons with the foreshock. The Mach number of interplanetary shocks varies due to the inhomogeneity of the solar wind, so trapping is a reflection process that could have the necessary intermittent nature.

The V_in=2v_A shock illustrates a difficulty with identifying specific features of the trapped spectrum in regions where adiabatic acceleration is dominant. These measures would be interesting since the behaviour of electrons in shocks with varying Mach numbers is likely to be dependent on both the ripple amplitude and the field line convection time. Further work might consist of generating a range of shock simulations with inflow speeds between 2v_A and 6v_A and running electron simulations with a larger number of particles so that the power law tail extends to energies at which adiabatic electrons are no longer present.

6.2.2 Shock geometry

In adiabatic theory the reflected fraction and final energy of the electron distribution are highly sensitive to q_{B_n}. We have seen that the properties of ripples are, in fact, only very weakly dependent on q_{B_n}, so a study of the effect of q_{B_n} on the electron population is interesting.

Figure 6.5: Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions at a fixed inflow speed of 4v_A and a range of angles q_{B_n}.

$ref_frac_vs_tbn$

Figure 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_{B_n}, for shocks with V_in=4v_A.

The energy spectra for a sample of shocks with varying q_{B_n} are shown in Figure 6.5. All shocks had V_in=4v_A and the simulations were run with an initial electron energy E_init=100 eV. Table 6.2 shows that the shocks all have appreciable but varying amounts of rippling, in addition to very similar values of M_A and overshoot size. The spectra exhibit an upstream shoulder which we discuss in Section 6.1.1 and believe to be a numerical artefact. The energy at which the shoulder starts appears to increase with q_{B_n}. Apart from the q_{B_n} =85^° shock, these energies are somewhat below the energy of 5.5 keV at which we suggested that the electron gyroradius might become significant.

All three of the shocks with q_{B_n} between 80^° and 88^° exhibit a low energy cut-off to the upstream distribution that is consistent with the predictions made in Table 6.2. The q_{B_n} =89^° shock is not expected to produce upstream electrons, yet has a small upstream population. The upstream spectrum has an anomalously high upstream cutoff, in common with the V_in=6v_A shock described in the previous section, and is therefore likely to be a numerical artefact. The downstream spectrum is significantly steeper than expected, but appears to become shallower at higher energies. This would be consistent with the domination of adiabatic electrons at low energies.

Figure 6.6(b) shows that the upstream and downstream spectra have comparable power law exponents that are consistent with the range -2 to -4 observed by Gosling et al. [1989] and Anderson [1981]. The energy spectra become shallower as q_{B_n} increases. This could be due to the fact that the strength of the ripples increases with q_{B_n}, resulting in stronger trapping. It could also be due to the fact that, as q_{B_n} approaches 90^°, the field lines become nearly parallel to the plane containing the ripples, so electrons will find it more difficult to escape from the shock front and the time spent by an electron being accelerated at the shock front will show a corresponding increase. It is likely that both of these factors are involved, and additional simulations would be required to determine the exact nature of the relationship. At q_{B_n} = 89^°, however, the reflected fraction becomes so small that there is no source for trapped electrons. The downstream power law energy spectrum is much steeper than at q_{B_n} = 88^°, which suggests that the acceleration is dominated by adiabatic behaviour.

Figure 6.6(a) shows how the fraction of electrons that are reflected compares with the adiabatic prediction. As q_{B_n} rises, both the measured and predicted reflected fractions fall, which suggests that the size of the trapped population depends on the adiabatic reflected fraction. When q_{B_n} =80^° and 85^° the actual reflected fraction is a significant proportion of the expected adiabatic reflected fraction. At q_{B_n} =88^°, however, the higher upstream cut-off energy means that a larger proportion of trapped electrons are convected downstream. As the low energy cut-off becomes less significant, we expect about half of the trapped population to escape upstream. When q_{B_n} = 89^° no electrons are expected to reflect and the number of trapped electrons is correspondingly small.

A key result of this study is that, in contrast to adiabatic theory, the properties of trapped electrons are more sensitive to the shock's Alfvén Mach number, M_A, than the angle q_{B_n}. Although the high energy power law spectrum does get shallower as q_{B_n} increases, it is still possible to accelerate electrons by a factor of at least 100 with q_{B_n} =80^°. In contrast, the most that the adiabatic theory described in Section 5.1.1 can achieve is acceleration by a factor of 2.3.

6.2.3 Electron energy

In order for the proposed trapping mechanism to produce an energy spectrum consistent with bow shock observations, the shocked energy spectrum at high energies must be consistent with an unbroken power law. This means that, for all values of the initial electron energy, the power law exponent of the high energy tail must be a constant. If the exponent were to vary, a broken power law would result. It is therefore necessary to examine how the shocked electron spectrum changes with initial energy.

Figure 6.7: Graphs showing the upstream (dashed) and downstream (solid) final logarithmic electron energy distributions for a variety of initial energies, E_init, in a shock with V_in = 4v_A and q_{B_n} = 85^°. The high energy "shoulder" to the distributions appears to be a numerical artefact.

$ref_frac_vs_energy$

Figure 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, log(E_init), in a shock with V_in = 4v_A and q_{B_n} = 85^°.

In Section 6.1.1 we suggested that electrons with very short reflection times could cause a numerical artefact in the high energy section of the upstream electron energy spectrum. This is of particular concern in this section, where increased initial electron energies, E_init, will lead to shorter reflection times. The results in this section should therefore be regarded as preliminary, pending further work to characterise any possible numerical dependencies.

The energy spectra for a sample of simulations with varying E_init are shown in Figure 6.7. In this section, we will look at acceleration in a single set of shock fields, with V_in = 4v_A, q_{B_n} = 85^° and B₀ directed in the simulation plane. At E_init ³ 10 keV and E_init = 10 eV, the highest and lowest initial energies, the shocked energy spectra are dominated by electrons behaving adiabatically.

Figure 6.8(a) shows the measured and predicted values of the reflected fraction plotted against E_init. The adiabatic behaviour at E_init = 10 eV is caused by a lack of a reflected population, which means that there is no source of trapped electrons. At initial energies of E_init ³ 10 keV, however, the adiabatic reflected fraction is around 80% and we should therefore expect to see a large trapped population. As E_init increases, a larger proportion of the electrons are reflected without being trapped and, by E_init = 1 keV, the reflected fraction is in line with adiabatic expectations. In the simulations with E_init ³ 10 keV, therefore, we do not see a significant trapped population because most reflected electrons move along field lines faster than the shock structure, so they feel a quasi-static field. The spectra are consequently dominated by the adiabatic component. For this shock the minimum energy required for an electron to escape upstream is E_esc(a = 180^°) = 43 eV. Further work is required to determine the fraction of electrons that escape upstream, although it is likely to depend on both E_init and E_esc(a = 180^°), as well as the wavelength and velocity of the ripples. At E_init > 1 keV the reflected fraction is actually greater than adiabatic expectations. Some of this increased fraction may be explained by the fact that the presence of ripples will cause the maximum value of the magnetic field that is felt by electrons to be higher than the average value used in our calculations. Also, as discussed in Chapter 5, numerical scattering can cause artificial perpendicular heating which will lead to an increase in the reflected fraction.

Figure 6.9: The distribution of initial pitch angle, a, for electrons that end the simulation upstream (dashed) and downstream (solid), for three values of E_init, in a shock with V_in = 4v_A and q_{B_n} = 85^°.

Figure 6.9 shows the distribution of initial pitch angle, a_init, for the upstream and downstream electron populations in simulations with three different values of the initial energy E_init. Electrons with initial pitch angles that should cause them to reflect are transmitted downstream only if they have undergone trapping. As initial energy increases, the trapped population is increasingly restricted to electrons with initial pitch angles close to 90^°. This suggests that electrons with higher parallel velocities have a higher chance of being able to escape upstream fast enough to avoid feeling the magnetic structure in the shock ramp.

The power law exponents of the upstream and downstream distributions are shown against E_init in Figure 6.8(b). The E_init = 100 eV spectrum shows a case where the trapped population dominates and can be approximated by a single power law with comparable exponents in the upstream and downstream cases. The 300 eV and 1 keV energy spectra have broken power laws in both the upstream and downstream. It is possible that this is the result of an adiabatic population dominating at low energy and a trapped population dominating at high energy. Alternatively, it is possible that the high energy shoulders are related to the upstream shoulders seen in previous simulations and are therefore a numerical artefact. Although it is difficult to measure the power law exponents for these spectra, we have attempted to calculate the exponent of the low energy power law in the upstream and both power law exponents in the downstream. Figure 6.8(b) shows power law spectra that steepen significantly at E_init ³ 10 keV and E_init = 10 eV. These simulations are dominated by adiabatic electrons and are not therefore representative of an extended power law tail caused by trapping.

The spectral indices between 100 eV and 1 keV could be consistent with an energy independent exponent for the trapped population, but the results are inconclusive. The dominance of adiabatic electrons at other energies means that the overall spectral profile is not constant with E_init. In order to duplicate the observed spectra at the bow shock, therefore, we would need to conduct simulations with a continuous spectrum of initial energies, E_init, in order to simulate the unshocked energy spectrum of the solar wind. If, as seems possible, the power law exponents for the trapped population are independent of E_init, adiabatic electrons would dominate only at low energies and we would still expect to see a high energy tail to the shocked electron distribution which could be characterised by a single unbroken power law.

6.3 Solar Flare Shocks

We saw in Chapter 2 that electrons in solar flares are accelerated to energies as high as 100 keV on time scales of less than 10-20 ms [Aschwanden et al., 1995]. The electron energies can extend as high as tens of MeV with power law exponents as shallow as 3. This is consistent with the spectral parameters derived from X-ray data, described in Chapter 2, which have power law exponents in the range of 3 to 10. In order to study electron acceleration in a solar flare shock, the Alfvén speed and magnetic field strength need to be changed significantly. Neither of these quantities have an impact on the shock structure as they are both normalised quantities in the hybrid code. The initial electron energy must also change. This is a parameter of the electron test particle code and, by applying an appropriate normalisation, flare shock simulations are equivalent to bow shock simulations. The exception to this comes at relativistic energies when the higher Alfvén speed in solar flares may cause relativistic effects to become significant where they would not in the bow shock. We will also assume an inflow velocity of V_in=4v_A in order to ensure the presence of rippling.

Figure 6.10: Graphs showing the upstream (dashed) and downstream (solid) final electron energy distributions for a variety of conditions typical of the solar corona.

Figure 6.11: Time sequence of the electron energy spectrum for all particles in the solar coronal shock simulation with q_Bn=88^°, V_in=4v_A, E_init=1 keV.

As discussed in Chapter 2, the plasma in the region of a flare shock is likely to have an Alfvén speed v_A » 160 km s^-1 and a magnetic field strength of around 100 mT. If we assume a coronal temperature of 2 ×10⁶ K, this gives a thermal electron energy of around 300 eV. This is equivalent to an electron energy in bow shock conditions of around 40 eV. At the overshoot the electron gyroradius, r_g, reaches 0.1 v_A W_i^-1 at an electron energy of around 38 keV and 1.0 v_A W_i^-1 at an energy of around 1.5 MeV. In contrast to the bow shock simulations, we can therefore expect to be able to investigate effects due to gyroradius and relativistic effects.

Graphs of the electron energy spectra for simulations run with these solar coronal conditions are shown in Figure 6.10. Figure 6.10(a) represents a simulation in which E_init=300 eV, which is typical of a thermal electron. With the modified Alfvén speed, this shock with q_{B_n} = 85^° and V_in=4v_A now has an upstream escape speed E_esc(a = 180^°) = 310 eV. This simulation produces a spectrum with a broken power law in both the upstream and downstream. This is the result of adiabatic behaviour dominating at low energies and the production of high energy electrons by trapping and, as such, produces a spectrum similar to Figure 6.3(b).

In Figure 6.10(b), we have used an initial energy of E_init=1 keV. This is in the thermal tail of the electron energy distribution and corresponds to an energy of 140 eV in the bow shock. We use the shock simulation with q_{B_n} = 88^° and V_in=4v_A as, in our bow shock studies, this was most successful at producing an accelerated population that was dominated by trapping. In solar coronal conditions this shock has an upstream escape speed of E_esc(a = 180^°) = 1.9 keV, which is consistent with our results. The downstream spectrum has a power law exponent of -2.1, whilst the upstream spectrum below the shoulder has an exponent of -2.2.

Figure 6.10(c) shows results from a simulation with the same parameters as Figure 6.10(b), but with relativistic effects removed from the electron code. This was achieved by setting the Lorentz factor g = 1 in the electron equation of motion. In order to produce electrons an electron spectrum with a tail running into the relativistic regime, both of these simulations were run with a total of 360,000 particles, which is more than any of the other simulations in this thesis.

Although there are no features in the downstream spectrum, the upstream shoulder appears at around 30 keV, which is consistent with the energy of 38 keV at which we predicted that the electron gyroradius may become significant. It remains unclear whether the upstream shoulder is entirely a numerical artefact, or whether it has a physical analogue that is related to the size of the electron gyroradius. Other than the upstream shoulder, the power law spectra in Figures 6.10(b) and 6.10(c) continue to have a constant exponent down to the numerical resolution of the simulation. We have therefore been unable to find any impact of the relativistic transition, at around 500 keV, on the electron spectrum.

Figure 6.11 shows a time sequence of the electron energy spectrum for all particles in the relativistic simulation with E_init=1 keV. We can see that electrons with energies of 100 keV start to appear after t=2 W_i^-1 and energies of 1 MeV appear at around t=10 W_i^-1. Using the quantities described in Chapter 2 we find an ion cyclotron time W_i^-1 » 0.1 ms for solar coronal conditions. The acceleration time for 100 keV electrons is therefore around 0.2 ms and for 1 MeV electrons is around 1 ms.

We have seen that electron trapping can be effective in producing a highly energetic power law tail in the shocked electron energy spectrum when starting from thermal electron energies and using a shock that is 5^° from perpendicular. If we increase the value of q_{B_n} to 88^° and look at higher initial electron energies we are able to produce a power law tail with an exponent of around -2 extending up to relativistic energies. The time scale of this acceleration is around 1 ms and is consistent with observational constraints. In order to ensure the presence of ripples, however, we have assumed that the inflow velocity can be 4v_A, giving a shock with an Alfvén Mach number of M_A = 5.67. This was necessary in order to make the shock supercritical and therefore stimulate the production of ripples. The simulations of Yokoyama & Shibata [1996], however, suggest that a Mach number of M_A = 1.6 is more likely. This means that an essential factor in determining the importance of trapping in solar flare shocks is whether they are capable of reaching supercritical Mach numbers and therefore producing a significant overshoot to permit the formation of ripples.

6.4 Summary

Adiabatically reflected electrons appear to act as the seed population for trapping. Those shocks which are not expected to produce any adiabatic reflection are relatively poor at producing a trapped population. This means that shocks with a high inflow speed that result in a very high upstream escape velocity are unable to keep electrons in the shock structure for long enough to accelerate them. Shocks with inflow speeds that are so low as to make them either subcritical or slightly supercritical, conversely, either do not generate ripples or produce rippling that is too weak to allow significant trapping. It therefore seems that the trapping process is strongest in the transition between subcritical shocks with low inflow speeds and shocks with inflow speeds that are so high that electrons are unable to reflect.

We found that the dependence of the accelerated electron spectrum on q_{B_n} was surprisingly weak. This contrasts strongly with adiabatic theory, where significant electron acceleration can only take place when the shock is within a few degrees of perpendicular. We found that at very large values of q_{B_n} the upstream escape velocity became too large for any electrons to reflect. Below this cut-off, as q_{B_n} decreased, the fraction of electrons that were reflected increased. The power law exponents of both the reflected and transmitted distributions, however, became progressively steeper as q_{B_n} was reduced. Although the high energy power law spectrum does get shallower as q_{B_n} increases, it is still possible to accelerate electrons by a factor of at least 100 with q_{B_n} =80^°, compared with the factor of 2.3 that adiabatic theory predicts.

It does not seem that either relativistic effects or electron gyroradius changes are significant. This could be due to the fact that these transitions only come into play at very high energies compared to the initial injection energy. The only feature that appears in the energy spectrum at these energies is the "shoulder" to the upstream distribution. This appears, however, to be a numerical artefact rather than having a physical basis.

The acceleration time for electrons in our simulations is very short. Significant acceleration is produced within 2 W_i^-1 and the bulk of electrons have escaped the shock after 10 W_i^-1. This corresponds to acceleration times of 0.2 to 1 ms in solar flares and 2 to 10 s at the Earth's bow shock. This rapid acceleration is consistent with observations.

Although we have shown that trapping can produce a power law energy spectrum, our results do not allow a conclusive comparison with the observed spectra. The problem is that if two power law distributions are combined, the result is a "broken" power law distribution with two different values of the power law exponent on each side of the "break". We have shown that the power law exponent can depend on both M_A and q_{B_n}. The dependence on of the exponent on M_A should not be of concern, since the flow speed will be reasonably constant along the shock. The value of q_{B_n} at the bow shock, however, varies considerably. We mentioned in Section 5.1 that theories of acceleration at curved shocks show that electrons travel considerable distances along the shock, so electron distributions at a given point will have experienced a large range of q_{B_n}. We showed in Section 5.4.1, however, that trapped electrons move considerably less along the shock as a result of their trapping, so this q_{B_n} variation is unlikely to be an issue. We have studied acceleration with a single initial energy, E_init, whilst the actual unshocked plasma will obviously have a spectrum of energies. It remains unclear whether the power law exponent is dependent on E_init. There is some evidence that the trapped population could produce a power law exponent that is independent of initial energy, although this would require further work to be convincing.

The Earth's bow shock is supercritical, so we would expect ripples to be generated there. There are also observations of reflected electrons, so the trapping process will have the required seed population. The trapping process is therefore likely to be significant at Earth's bow shock, particularly as the weak dependence on q_{B_n} means that trapping can take place over a large part of the quasi-perpendicular shock surface. The applicability of trapping to solar flares, however, will depend on whether supercritical shocks can be generated.

Page Maintained By : Rob Lowe
Last Revision : 1st March 2003

Chapter 6 Parametric Survey of Shock Accelerated Electrons