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Fundamentals of Non-relativistic Collisionless Shock Physics IV. Quasi-Parallel Supercritic


Fundamentals of Non-relativistic Collisionless Shock Physics: IV. Quasi-Parallel Supercritical Shocks
R. A. Treumann? and C. H. Jaroschek??
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Department of Geophysics and Environmental Sciences, Munich University, D-80333 Munich, Germany Department of Physics and Astronomy, Dartmouth College, Hanover, 03755 NH, USA ? Department Earth & Planetary Science, University of Tokyo, Tokyo, Japan 1. Introduction, 2. The (quasi-parallel) foreshock; Ion foreshock, Ion foreshock boundary region; Di?use ions;Low-frequency upstream waves; Ion beam waves; The expected wave modes; Observations; Di?use ion waves; Electron foreshock; Electron beams; Langmuir waves; stability of the electron beam; Electron foreshock boundary waves; Nature of electron foreshock waves; Radiation; Observations; Interpretation; 3. Quasi-parallel shock reformation; Low-Mach number quasiparallel shocks; Turbulent reformation; Observations; Simulations of quasi-parallel shock reformation; Hybrid simulations in 1D; Hybrid simulations in 2D; Full particle PIC simulations; Conclusions; 4. Hot ?ow anomalies; Observations; Models and simulations; Solitary shock; 5. Downstream region; 6. Summary and conclusions.
PACS numbers: Keywords:

arXiv:0805.2579v1 [astro-ph] 16 May 2008

I.

INTRODUCTION

At a ?rst glance it is surprising that a change in the shock-normal angle ΘBn by just a few degrees from, say, ΘBn = 50? to ΘBn = 40? should completely change the character of the supercritical shock. We have seen in the last chapter, when having discussed the conditions for re?ection of ions from the shock ramp that such a change in the shock properties is theoretically predicted. The critical shock-normal angle ΘBn = 45? does indeed separate two completely di?erent phases of a supercritical shock. At this angle the shock experiences a ‘phase transition’ from the quasi-perpendicular to the quasi-parallel shock state, with the shock-normal angle ΘBn having the property of a ‘critical control parameter’. We have already learned that the reason for the di?erent behaviour of the two shock phases is that in a quasiperpendicular shock all re?ected particles from the foot region of the shock return to the shock after not more than a few gyrations when they have picked up su?cient energy in the upstream convection electric ?eld E1 = ?V1 × B1 to ultimately overcome the shock ramp potential, pass the shock ramp and to merge into the downstream ?ow. We have not discussed what happens to the accelerated ions in the downstream region as this is not of primary importance in the shock formation mechanism which to good approximation depends only on the upstream conditions. This question will be treated in a separate section on shock-particle acceleration. In contrast, in a quasi-parallel shock the combined geometries of the upstream magnetic ?eld and generally curved shock surface prevent the shock-re?ected particles from immediate return to the shock. The reason is that their gyro-orbits, after having su?ered re?ection from the shock ramp, lie completely upstream, outside the shock ramp, such that they do not touch the shock ramp again after re?ection. Since, in addition, their upstream velocities have a large component parallel to the upstream magnetic ?eld, which increases the more the shock-normal turns parallel to the upstream magnetic ?eld, the re?ected particles are enabled to escape upstream from the shock along the magnetic ?eld thereby forming fast upstream particle beams. A quasi-parallel supercritical collisionless shock thus populates the upstream space with a re?ected particle component. This population moves a long distance away from the shock along the magnetic ?eld. Was the upstream ?ow, in the case of the quasi-perpendicular shock, completely uninformed about the presence of the shock up to a distance of the mere width of the shock foot so, at the quasi-parallel shock, it receives a ?rst signal of the presence of a shock already at quite a large upstream distance when the ?rst and fastest re?ected particles arrive on the magnetic ?eld lines that connect the ?ow to the shock. It is, in fact, only these particles that can inform the ?ow about the presence of a supercritical shock, because any low-frequency plasma wave cannot propagate far upstream for supercritical Mach numbers M > Mc , while any electromagnetic radiation that is generated at the shock has frequency ωrad > ωpe . For it the plasma ?ow presents a vacuum. The upstream ?ow recognises the re?ected particles in its own frame of reference

? Electronic

address: treumann@issibern.ch

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FIG. 1: Schematic one-dimensional pro?le taken along the nominal instantaneous shock normal of a supercritical quasi-parallel shock as seen in the magnetic ?eld component Bz . This is the analogue to the quasi-perpendicular shock pro?le. It shows the main features in the vicinity of the quasi-parallel shock transition: the large amplitude upstream waves with the turbulent ?uctuations on top of the waves, the formation of shocklets, i.e. steep ?ank formation on the waves exhibiting small-scale ?uctuations on top of the wave, which act already like small shocks, very-large amplitude pulsations (magnetic pulsations or SLAMS) which turn out to be the building blocks of the shock, multiple shock-ramps at the leading edges of the pulsations belonging to diverse ramp-like steep transitions from upstream to downstream lacking a clear localisation of the shock transition (Note that the entire ?gure is, in fact, the shock transition, as on this scale no clear decision can be made where the shock ramp is located.), and their attached phase-locked whistlers. Not shown here are the out of plane oscillations of the magnetic ?eld that accompany the waves. Also not shown is the particle phase space.

as a high-speed magnetic-?eld aligned beam. Thereby a beam-beam con?guration is created which leads to a number of beam-driven instabilities. These excite various plasma waves that ?ll the space in front of the shock and modify its properties. Figure 1 shows a sketch of the magnetic pro?le of a supercritical quasi-parallel shock which contrasts the pro?le of a quasi-perpendicular shock that has been given in the previous chapter. The quasi-parallel magnetic shock pro?le is much stronger distorted than that of a quasi-perpendicular shock, such that it becomes di?cult to identify the location of the genuine shock ramp on the pro?le. The main di?erence between quasi-perpendicular and quasi-parallel shocks is that quasi-perpendicular shocks possess a narrow ? 1rci wide foot region that is tangential to the shock surface, while quasi-parallel shocks possess an extended foreshock region. Interestingly, in curved shocks which arise, for instance, in front of spatially con?ned obstacles, both phases of a supercritical shock can co-exists at the same time, being spatially adjacent to each other. An example is shown in Figure ?? in the sketch of the curved Earth’s bow shock. Dealing with quasi-parallel shocks means to a large extent dealing with the processes that are going on in the foreshock. It will thus be quite natural to start with a discussion of the properties of the foreshock. This discussion will occupy a substantial part of this chapter. However, before continuing we point out that in spite of the strict distinction between the quasi-perpendicular and quasi-parallel shocks there is also a close relation between the two. Both, being supercritical, can exist only because they re?ect ions; and both possess an upstream region in front of the shock transition that is populated by the re?ected ions. That this region is narrow in the case of quasi-perpendicular shocks is a question of the ions being tied to the magnetic ?eld, which also holds in the case of the quasi-parallel shock. At quasi-perpendicular shocks the re?ected ions do readily return to the shock. At the quasi-parallel shock they ultimately do also return to the shock, but only after having been processed far away from the shock in the foreshock, having coupled to the ?ow, having passed several stages in this processing, and having become the energetic component of the main ?ow. As such they ?nally arrive at the shock together with the stream. In between their main duty was to dissipate the excess energy, which they possessed when arriving for the ?rst time at the shock and which could not be dissipated in the narrow

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FIG. 2: Comparison of average bulk plasma parameters in quasi-perpendicular and quasi-parallel shocks [after 16]. The ?gure shows (on the left) the mean magnetic ?eld B, bulk ?ow velocity V , average magnetic ?uctuation amplitude at ultra-low frequencies, and (on the right) plasma density N , density of high energy ions Nih , and electron temperature Te . The shaded regions are the downstream parts of the AMPTE IRM crossings of the bow shock. The data have been obtained by normalising the time with respect to crossing the nominal shock ramp by using the measured normal component (not shown) of the bulk ?ow velocity. This for the many observations included in this ?gure implies a stretching (or squeezing) of each individual shock crossing, causing some uncertainty, in particular for quasi-parallel shock crossings as there the shock ramp is not well de?ned. However, this ?gure serves for an immediate overview of the di?erences in both types of shocks.

shock-ramp transition region. This could been achieved only in the broad extended foreshock which, seen from this point of view, is already the shock. It belongs inextricably to the quasi-parallel shock transition. Here, a substantial fraction of the energy of the incident ?ow is dissipated in a way which is completely di?erent from the ?ow being shocked. It is these dissipation processes that cause the main di?erence between the quasi-perpendicular and quasi-parallel states of a collisionless supercritical shock. To stress the analogy with phase transitions a little further, we may say that quasi-perpendicular shocks are in the solid – or ordered – shock state, while quasi-parallel shocks are in the ?uid – or partially disordered – shock state.

II.

THE (QUASI-PARALLEL SHOCK) FORESHOCK

Quasi-parallel shocks are abundant in space because in most cases when shocks develop the ?ow direction is independent of the direction of the magnetic ?eld. Moreover, as pointed out earlier, when a bow shock forms around an obstacle (planet, magnetosphere, moon ...) this bow shock is curved around the obstacle, and the shock is quasiperpendicular only in a certain region on the shock surface that is centred around the point where the upstream magnetic ?eld touches the shock tangentially. Farther away the shock turns gradually to become quasi-parallel. On the other hand, when a shock survives over a long distance in space as for instance in supernova remnants then it sweeps the upstream magnetic ?eld and pushes it to become more tangential to the shock surface. In this case the shock is about quasi-perpendicular. We will later provide arguments that any quasi-parallel supercritical shock on the small scale close to the shock surface, i.e. on the electron scale, behaves quasi-perpendicularly while on the larger ion scale it remains to be quasi-parallel. This has consequences for the di?erences in the dynamics of electrons and ions during their interaction with quasi-parallel shocks. Figure 2 shows at one glance the main di?erences in the (average) bulk plasma parameters between quasiperpendicular and quasi-parallel shocks as measured with the AMPTE IRM spacecraft at many crossings of the bow

4 shock. The data used in this ?gure have been stapled, averaged and plotted with respect to the time normalised to the shock ramp crossing. For such a normalisation one uses the shock-normal upstream velocity to recalculate the time. This procedure is not very certain for quasi-parallel shocks since – as we will see later – the shock ramp is ill de?ned in a quasi-parallel shock. However, for a simple comparison of the main di?erences this uncertainty is less severe. The shaded area in the ?gure corresponds to the downstream region. Shown are – in pairs of quasiperpendicular/quasi-parallel values – the magnetic ?eld B, bulk velocity V , average ?uctuation amplitude in the ultra-low frequency waves |b|, plasma density N , high-energy ion density Nih of energy > 15 keV, and electron temperature Te . The general conclusion from this ?gure is that all quantities in the quasi-perpendicular case exhibit a much sharper transition than in the quasi-parallel case. Moreover, the quasi-perpendicular averages are quieter than those of the quasi-parallel case. Also, in general, the quasi-parallel levels are higher than the quasi-perpendicular. In almost all cases the pre-shock levels are enhanced in the quasi-parallel shock case with over the pre-shock levels of quasi-perpendicular shocks. This is seen most impressively in the energetic ion density, which is nearly constant over this distance/time scale at quasi-parallel shocks and much higher than that in quasi-perpendicular shocks, signifying on the one hand the importance of energetic particles in quasi-parallel shock dynamics, on the other hand the capability of quasi-parallel shocks to accelerate particles to substantial energies. The presence of energetic ions (particles) far in front of the quasi-parallel shocks and the enhanced pre-shock levels indicate the importance of foreshocks in quasi-parallel shock dynamics. In the following we will therefore ?rst concentrate on the foreshock. The physics of quasi-parallel shocks cannot be understood without reference to the foreshock. The foreshock is that part of the upstream shock region that is occupied with re?ected particles. At a curved shock, like the Earth’s bow shock, the foreshock starts on the shock surface at the location where the upstream magnetic ?eld shock-normal angle exceeds ΘBn 45? . From that point on electrons and ions escape along the magnetic ?eld in upstream direction. Since electrons generally move at a larger parallel velocity than ions they are less vulnerable to the convective motion of the upstream magnetic ?eld line to which they are tied, and so there is generally a region closer to the foreshockboundary magnetic ?eld line where only upstream electrons are found. This region is con?ned approximately between the line that marks the electron foreshock boundary and the more inclined line line that marks the ion foreshock boundary. An example of this geometry was depicted in Figure ?? for the bow shock of the Earth. More schematically this is shown in a simpli?ed version in Figure 3 for the particular case that the upstream magnetic ?eld forms an angle of 45? with the symmetry axis of the shock. In this case half of the shock is quasiperpendicular and the other half is quasi-parallel. The ?gure also shows the directions of three shock normals, the narrow foot region in front of the quasi-perpendicular shock, and the two (electron and ion) foreshocks. Particles escape from the quasi-parallel shock along the upstream magnetic ?eld. The magnetic ?eld is convected toward the shock by the perpendicular upstream velocity component V⊥ as shown in the ?gure. This component adds to the velocity of the upstream particles leading to an inclined foreshock boundary. Since the ions have much smaller speed than the electrons, the ion foreshock boundary is more inclined than the electron foreshock boundary. In discussing the properties of the foreshock one thus has to distinguish of which foreshock is the talk. However, the properties of the electron foreshock are not as decisive for the formation of a quasi-parallel shock as are the properties of the ion foreshock. Because of this reason we will, in the following, refer to the ion foreshock as the foreshock. The electron foreshock properties we will mention only later.

A.

Ion foreshock

The ion foreshock is not a homogeneous and uniform region. The re?ected ion component evolves across the ion foreshock from the ion foreshock boundary to the centre of the ion foreshock and from there towards the shock. Speaking of a re?ected ion component that can unambiguously identi?ed as being re?ected, i.e. streaming into the upward direction, makes sense only in the immediate vicinity of the ion foreshock boundary. Here the re?ected ions appear as a fast ion beam the source of which can be traced back to the shock. Deeper in the foreshock the beam component cannot be identi?ed anymore.

The ion foreshock boundary region

First identi?cations of re?ected beam protons in space in the magnetic ?ux tube connected to the Earth’s bow shock wave were reported by Gosling et al. [40] and Paschmann et al. [96] who distinguished those beams by their poorly resolved distribution functions from more di?use protons deeper in the foreshock. Interestingly, observations in the foreshock of interplanetary travelling shocks did not show any indication of such beams but only the di?use ion component. Figure 4 gives an observational example of such a re?ected ion beam that propagates very close to the

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FIG. 3: Schematic of the the relation between a curved shock and its foreshock in dependence on the direction of the upstream magnetic ?eld B, shock-normal n, and shock-normal angle ΘBn for the special case when the magnetic ?eld is inclined at 45? with resepct to the symmetry axis of the shock. In this case the upper half of the shock becomes quasi-parallel (ΘBn < 45? ), the lower half is quasi-perpendicular (ΘBn > 45? ). The velocity of re?ected particles is along the magnetic ?eld. However, seeing the ?ow the ?eld-line to which they are attached displaces with perpendicular velocity. This velocity shifts the foreshock boundary toward the shock as shown for electrons (light shading) and ions (darker shading). The ion foreshock is closer to the shock because of the lower velocity of the ions than the electrons. For the electrons the displacement of the electron foreshock boundary is felt only at large distances from the shock.

FIG. 4: ISEE 1 observation of a re?ected ion beam on November 19, 1977 propagating along a magnetic ?eld line that was connected
to the quasi-parallel Earth’s bow shock. Left: A two-minute average pseudo-threedimensional ion velocity-space pro?le in the (vx , vy )-plane showing the undisturbed and cold (narrow) plasma in?ow in negative vx -direction, and the fast and warm (broad) beam of re?ected ions propagating in positive vx -direction and spreading in vy . This beam is quite anisotropic in temperature. Velocities are in km s?1 . The scale on the right is count rates, and background count rates were suppressed by choosing only values above 50 s?1 . Right: Contour plot of a similar beam a little earlier showing that the beam is centred on the magnetic ?eld that connects to the shock, is quite narrow along the ?eld and about 2-3 times as broad perpendicular to the ?eld [after 96]. The cross indicates the origin (zero velocity), the dot the bulk ?ow centre. The 10?25 s3 cm?6 level ?ux contour has been marked.

foreshock boundary upstream away from the shock. The bulk ?ow is the narrow cold beam in the left part of the ?gure which is displaced in negative vx -direction (note that in this ?gure the positive direction points away from the

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FIG. 5: Ion phase space at three locations along the shock-connected magnetic ?eld line in a high-Mach number supercritical quasi-perpendicular shock the magnetic ?eld of which is shown in the top panel. The shock-normal angle is ΘBn = 74.5? . The three lower plots show downstream, shock ramp/shock foot, and distant upstream phase-space plots. The ramp/foot plot shows the presence of the inconing ?ow (SW) and gyrating ions in the foot. The upstream plot shows the usptream ?eld-aligned relatively hot (large velocity spread) beam well separated from the in?ow (SW) [from 64].

shock). The re?ected beam is less dense (lower count rates) but much more energetic. It is displaced in +vx -direction, i.e. streaming away from the shock, and has also a ?vy -component, i.e. it constitutes a gyrating bunch of ions moving away from the shock. In the right part of the ?gure it is seen that the beam is moving away along the magnetic ?eld line that is connected to the shock, while the bulk of the plasma ?ows in positive direction. These beams along the foreshock boundary play some role in the foreshock dynamics as they seem to represent a source population for the entire ion foreshock. Whether and why this is really so is not yet been fully understood as the shock should re?ect ions at almost every place in its quasi-parallel state. However, it seems as that only the group of ions that escape from the shock along the foreshock boundary can form such beams. This points on a further interesting relation between quasi-perpendicular and quasi-parallel shocks at a curved shock surface with a smooth transition from quasi-perpendicular to quasi-parallel as sketched in Figure 3 and realised in space, for instance, at planetary bow-shocks. It seems as so these beams escape from the quasi-perpendicular region of the shock along the nearly tangential ?eld lines. This would also be in agreement with the observation [41] that the foreshocks of extended interplanetary shocks do not show any signs of re?ected ion beams. They are only very weakly curved being nearly planar, and not possessing a recognisable quasi-perpendicular area on the surface. [64] analysed ion distributions along magnetic ?eld lines that were connected to the quasi-perpendicular area of Earth’s bow shock. Investigating the origin of those beams these authors found that, indeed, the observed ion beams at the foreshock boundary result from re?ection at the quasi-perpendicular shock and that, without the presence of a quasi-perpendicular region, there would presumably be no distinct foreshock boundary and no ion beams escaping into the foreshock. If this is really the case, then the population in the quasi-parallel ion foreshock is indeed provided by two di?erent ion sources, the beams from the quasi-perpendicular shock region and the genuine foreshock ion population. The latter has no beam character but is rather a di?use ion component [see also 82]. [64] estimate that roughly 2% of the ion in?ow leaves the shock ramp upstream in the form of a beam along the magnetic ?eld. They argue that the ions which escape along the magnetic ?eld, are re?ected from the very ramp/overshoot region where they have been in resonance with low-frequency plasma waves, which they assume to be large amplitude Alfv?n-whistler waves. These ions experience pitch-angle scattering and pitch-angle di?usion e towards small pitch angles, and subsequently can escape along the magnetic ?eld in the upstream direction. Since the conditions for escape depend in the ?rst place on the pitch-angle scattering process, the beams should be highly variable in time and location. The mechanism might still sound a bit speculative as long as no simulation proves its reality, but any mechanism which is able to pitch-angle scatter ions along the magnetic ?eld in a quasi-perpendicular

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FIG. 6: Left: The reduced ?eld-aligned ion distribution functions in the foreshock region showing the ?eld-aligned re?ected ion beam (FAB) at 1220-1225 UT on February 18, 2003 which arrives at the spacecraft location along the magnetic ?eld line that is connected to the quasi-perpendicular region of the Mach number MA ? 8 supercritical shock (smooth solid line). The upstream bulk velocity was V1 660 km s?1 . Also shown is the reduced parallel distribution function (dashed) for the di?use ion distribution observed at 1130-1135 UT the same day (for the geometry see Figure 7). It is clearly seen that the latter is about symmetric to the magnetic ?eld direction indicating the symmetry of the ring of di?use ions, while the foreshock-boundary ?eld-aligned beam is ?owing in the direction away from the shock into the upstream medium (negative velocities). Note also that the di?use distribution appears as a smooth tail on the full ion distribution. The small gap on the left is uncertain as it dips into the 1-count level. Right: The two-dimensional phase space plot for the time interval 1220-1225 UT when the ?eld-alighned beam was observed. Indicated are the bulk upstream ?ow, ?eld-ligned ion beam, and the gyro-phase bunched residue of the di?use upstream ions [after 57].

shock will naturally cause ion beams to escape from the ramp both in the upstream and in the downstream directions. Such simulations require a three-dimensional treatment which is not in reach yet. A measured example of the ion foreshock-boundary ?eld-aligned ion beam distribution [57] is shown in Figure 6 for an upstream ?ow velocity V1 660 km s?1 , Mach number MA = 8, and average shock-normal angle ΘBn 15? [2], at an upstream distance from the shock (in this case again the Earth’s bow shock). This distribution is a so-called reduced distribution; it is the integrated over pitch-angle φ and perpendicular velocity v⊥ magnetic ?eld-aligned 2 phase space distribution function f (v ) ? 2π v⊥ dv⊥ f (v⊥ , v ) that has been appropriately binned and smoothed. The information to be taken out of this ?gure is that the reduced ion-beam distribution (solid line) is narrow in velocity, maximising at a speed (in absolute terms |vb | 800 km/s) that is only slightly larger than the parallel ?ow velocity (V 1 640 km/s, when taking into account the shock-normal angle), i.e. |vb | 1.25 V1 . It is directed opposite to the ?ow. For comparison a reduced parallel di?use ion distribution is shown in the same plot taken deeper in the foreshock. This distribution is about symmetric to the magnetic ?eld direction, indicating the about circular phase-space distribution of the di?use ion component which appears as an energetic tail on the main ion distribution (note that the small gap on the left of the dashed di?use-ion distribution curve is questionable as it dips below the 1-count level). [82] reported CLUSTER observations of foreshock-boundary ion beams simultaneously with di?use ions. They found that the nominal ion-beam velocity Vb 1.7 V1 had no relation to any known shock-re?ection mechanism like specular re?ection, a conclusion which supports pitch-angle scattering as the beam-injection process as this is independent of the speed of the in?ow. [57] have carefully analysed the relation between the observed ion distribution at the upstream spacecraft position and the calculated shock-normal angle ΘBn , determined from the local upstream magnetic ?eld direction and the predicted shape of the bow shock. The result is shown in Figure 7 for three successive times on February 18, 2003 when the CLUSTER spacecraft was outside the bow shock. At 1130 UT and 1355 UT the shock normal angles ΘBn ? 15? and shock-spacecraft distances ? 7.5 resp. ? 6.7 RE were similar. Despite this similarity the observed ion phasespace distributions were completely di?erent. At 1130 UT no low-energy gyrating ions were observed, while they ˙ were present at the later time 1355UT. Hence, at 1130 UT the spacecraft must have been closer to the ion foreshock boundary, i.e. the foreshock boundary was more inclined than at the later time such that beam particles scattered from the foreshock boundary have not arrived at the location of the spacecraft. Due to the velocity ?lter e?ect they

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FIG. 7: Reconstruction of the shape and location of the ion-foreshock boundary from the measured upstream plasma properties (direction of magnetic ?eld, speed and density) for three successive times in order to explain the ion-phase space observations in Figure 6 [from 57]. The cold magnetic ?eld-aligned ion beam is observed when the spacecraft is located in the vicinity of the ion-foreshock boundary at 1220 UT, while at 1130 UT and 1355 UT no beam was detected. It is assumed that the beam is generated along the foreshock boundary at the position where at the quasi-perpendicular shock surface the shock-normal angle is roughly about ΘBn ? 60? .

have been separated out at spacecraft distance. The reconstruction of the position of the ion-foreshock boundary using the measured upstream conditions and shape of the shock for this period is shown in Figure 7. Indeed, at 1220 UT the spacecraft was close to the foreshock boundary and, as expected, detected the ion beam (as seen in Figure 6). This observation supports the above advocated view that the ion-foreshock beam is generated in the transition region from quasi-perpendicular to quasi-parallel shock. The remaining questions to be answered are: what pitchangle scattering mechanism is responsible for the generation of such a beam, and what is the fate of the foreshockboundary beam-ions during their propagation along the foreshock boundary? Do they contribute to the foreshock ion population and if, in what way? Currently we are not able to answer either of these questions de?nitely. In particular, the pitch-angle scattering mechanism is unknown or at least uncertain. When discussing wave generation, we will touch on the problem of the fate of the beam. Below we present evidence for the scattering of the ion-beam ions and merging into the upstream foreshock di?use-ion population.

Di?use ions

Re?ected upstream magnetic ?eld-aligned ion beams are observed at the foreshock boundary only. The second (and main) ion component encountered in the foreshock is the di?use ion population which is detected there as the energetic extension of the in?ow plasma. It is widely assumed that the origin of this component is also at the shock as there is no other energetic particle source available. However, close investigation of the di?use ion component in the foreshock of the Earth’s bow shock wave has demonstrated that these ions are not produced in a specular re?ection process at the shock. Rather their origin is of di?usive nature. So far it has not been possible to identify the source of these ions though some models have been proposed which we will brie?y discuss below in relation to the appearance of foreshock waves. We will also return to these particles in the next chapter on shock particle acceleration. Here, we merely discuss some of their properties and provide evidence that they are indeed coming from the shock, proving that the shock is an energetic ion source. The foreshock-boundary ion beams also merge into the foreshock particle distribution by scattering from the foreshock boundary o? their self-generated wave spectrum and subsequent being convected downstream by the bulk ?ow. [96] have mapped this merging process by following the evolution of the foreshock-boundary beam distribution shown in Figure 4 during the convection. In that ?gure the beam was detected close to the foreshock boundary. Figure 8 show its form deeper in the foreshock when it has spread substantially in angle evolving into half of a ring distribution

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FIG. 8: The evolution of the ion foreshock-boundary ion beam during its convection downstream into the foreshock as has been seen
by ISEE 2 on 04 November 1977. Left: A one-minute average pseudo-threedimensional ion velocity-space pro?le in the (vx , vy )-plane showing the spreading of the beam in angle around the bulk ?ow without merging into the bulk ?ow. Velocities are in km s?1 . The scale on the right is count rates, and background count rates were suppressed by choosing only values above 50 s?1 . Right: Contour plot of the partial ring distribution. The direction of the magnetic ?eld is also shown [after 96]. The 10?26 and 10?27 s3 cm?6 level ?ux contours have been marked.

FIG. 9: ISEE 1 observation of di?use ion beam on November 19, 1977 propagating deep inside the foreshock away from the ion foreshock boundary. Left: A two-minute average pseudo-threedimensional ion velocity-space pro?le in the (vx , vy )-plane showing the undisturbed and cold (narrow) plasma in?ow in negative vx -direction about centred and surrounded by a ring distribution of fast and warm (broad) di?use ions that have been re?ected from the shock but have been processed in the foreshock region when propagating from the foreshock boundary into the foreshock. These ions are hollow in the sense that they separate from the bulk distribution but have a nearly isotropic distribution function. Velocities are in km s?1 . The scale on the right is count rates, and background count rates were suppressed by choosing only values above 50 s?1 . Right: Contour plot of a similar ring a little earlier showing that the ring centre (star) is slightly displaced from the bulk ?ow (dot) on the magnetic ?eld that connects to the shock and from the centre (cross) of the phase space frame [after 96]. The 10?26 and 10?27 s3 cm?6 level ?ux contours have been marked. Note the near isotropy of the ring distribution.

already. Even deeper inside the foreshock the re?ected ion distribution assumes the shape of a full ring around the bulk distribution as is shown in Figure 9. The three observations depicted in Figures 4, 8 and 9 are from di?erent times; it has, however, been checked that they are at distances corresponding to increasing distance from the foreshock boundary such that the assumption of the convectively processed beam evolution is well founded (or at least reasonable) even though it has not been directly proven. It is interesting to note that a gap remains between the original ion-phase space beam distribution

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FIG. 10: Partial-density gradient of the di?use foreshock-ion component along the shock-connected magnetic ?eld, and density e-folding lengths, determined from CLUSTER observations in the Earth’s bow shock foreshock [after 56]. Left: The parallel partial di?use-ion density as function of distance from the shock along the magnetic ?eld ?ux tube connecting the spacecraft to the shock. The di?use ion density falls o? exponentially with distance suggesting an ion-energy dependent di?usive process being responsible for transport of the ions upstream of the shock. Right: The e-folding distance of di?use ions along the magnetic ?eld as determined as function of energy from the exponential decay of the ion density. The e-folding distance increases linearly with ion energy.

and the bulk-?ow distribution, which is another indication that the evolving distribution is part of the evolution of the ion-foreshock beam. Of a distribution that does not evolve out of a beam one expects a less regular behaviour and, generally, no such well expressed gap between bulk ?ow and beam in velocity space. In fact, the main energetic ion component in the foreshock is irregular and lacks a well expressed gap. The discrepancy between the smooth no-gap foreshock distributions [131] and the gap-observations of [96] has, in fact, been noted much earlier [112, 115] without giving an explanation but suggesting a continuous ion source at the parallel shock. Below we provide further arguments for the two-source, foreshock-boundary beam and continuous extended shock-surface source hypothesis. Nonetheless, this conclusion must be taken with care because Figure 9 shows the gap progressively closing. Being su?ciently far, i.e. even farther away from the ion foreshock boundary then in this ?gure, it will not anymore be possible to distinguish between beam-evolved and genuine di?usive-ion distributions. Ultimately, both distributions will have merged indistinguishably. Still there is no agreement whether and where, i.e. at what distance from the ion foreshock boundary this merging of the two populations occurs. [151], using AMPTE IRM measurements of di?use ion densities upstream of the quasi-parallel (bow) shock found that the di?use ion density decreases exponentially with shock distance. This investigation was substantially improved by [56] in an attempt to infer about the source of the upstream di?use energetic ions. These authors determined the partial-density gradient of di?use ions in the energy range from 10 to 32 keV as a function of distance from the (bow) shock. This investigation was made possible due to the availability of the CLUSTER spacecraft, a four identical-spacecraft mission which during this measuring period had an inter-spacecraft separation distance between ? 1 and ? 1.5 RE . They used a nominal bow shock model [98], based on the measured upstream ?ow parameters (basically the dynamic pressure of the ?ow), providing the shock-spacecraft distances along the magnetic ?eld ?ux tube for the individual CLUSTER spacecraft and the local shock-normal angles ΘBn . The average ΘBn over the whole 10 hours of observation time was 20? ± 8? proving that CLUSTER was in front of the quasi-parallel shock far away from the ion-foreshock boundary, and the Mach number was MA ? 8. Di?use ion partial densities were determined as function of distance from the shock and in four consecutive energy bands every 32 s at two spacecraft. This allowed to determine the partial di?use ion-density gradients along the magnetic ?eld as function of energy from the density di?erences between two spacecraft and the di?erences of the spacecraft distances along the magnetic ?eld to the shock intersection point. Perpendicular density gradients were neglected. The obtained parallel gradients were attributed to the average CLUSTER location. These gradients were then used to ?nd the e-folding distance of the density variation. The results of this investigation are shown in Figure 10. It is learned from this ?gure that in the deep foreshock, i.e. at distances far away from the ion-foreshock boundary, the di?use ion component is densest close to the shock with density decaying exponentially with increasing upstream distance from the shock along the magnetic ?eld. This spatial decay of the di?use partial ion density Ni (E, z) ? exp[?z/L(E)] is di?erent for particles of di?erent energy E.

11 The e-folding distance L(E) ? E turns out to increase linearly with energy, i.e. low energy particles are con?ned to the shock. The higher the ion energy the deeper can the ions penetrate into the upstream plasma. The proportionality constant determined from these data under the special conditions of the Earth’s bow shock is ? 0.14 RE /keV. This behaviour of the energetic foreshock ions provides indisputable evidence for the extended parallel shock-surface origin of the di?use ion component. The source of the di?use ions lies at the quasi-parallel shock. In order to be found at a distance upstream of the shock the ions undergo a di?usion process along the magnetic ?eld. These ions are thus completely di?erent from the beam ions found at the ion-foreshock boundary. The e-folding distance for the di?use ions is given by L(E) = κ (E)/V1 , with spatial di?usion coe?cient κ (E) = 1 v (E), where is the di?usion length (parallel ion mean free path) and v the particle velocity (note that the 3 di?usion coe?cient has the correct dimension [κ ] = m2 s?1 ; justi?cation of the di?usion assumption will be given in the chapter on particle acceleration). From balance between convective in?ow and di?usion into upstream direction, √ one can write (E) = 3L(E) E1 /E ? E · E 1 . The di?usion length increases as the root of the product of particle energy E and upstream ?ow energy E1 . In the solar wind the ?ow energy is a few keV, and a 20-keV di?use ion will have a typical parallel di?usion length (or mean free path) of ? (1 ? 2) RE . This is a rather short distance, orders of magnitude shorter than the collisional mean free path of an ion. Hence, the di?usion estimate suggests that strong wave-particle interactions can be held responsible for the scattering and acceleration of the di?use particle component, which enables it to di?use and escape upstream from the shock and populate the foreshock. The di?usion process is energy dependent with the most energetic ions di?using fastest. These interactions should take place in the quasi-parallel shock transition because, as we have shown above, the di?use upstream-ion density maximises closest to the shock. It is interesting to estimate the corresponding upstreamion collision frequency νc,ui v/ . For the 20 keV-upstream ions this yields νc,ui ? 0.2 Hz. This value is comparable to the ion cyclotron frequency ωci /2π = (0.1 ? 0.3) Hz in the B 8 nT upstream to B 30 nT shock ramp magnetic ?eld [56] during the time of observation. It seems that waves, electromagnetic and/or electrostatic, related to the ion-cyclotron frequency are involved into the process of upstream ion di?usion. Since this di?usion is energy dependent, this process is not a simple pitch-angle di?usion as in the case of the generation of the ion beam that propagates along the foreshock boundary. The di?use ion component must have experienced a substantial acceleration in this process, and this acceleration is located at or around the shock transition and contrasts with the ion-beam acceleration which is a scattering process followed up by pick-up acceleration when the upstream-propagating beam ions are subject to the e?ect of the main-bulk-stream convection-electric ?eld in which they become accelerated in the direction perpendicular to the magnetic ?eld to roughly four times the energy of the bulk ?ow, thereby evolving into the ring distribution that characterises their phase space distribution.

B.

Low-frequency upstream waves

In the frame of the upstream bulk ?ow the two ion components that populate the ion foreshock carry a substantial amount of free energy which is subject to dissipation. Since this dissipation is collisionless it can proceed only through the excitation of waves and wave turbulence through instability upstream of the quasi-parallel supercritical shock. On the other hand it is obvious that the presence of neither of the components can be understood without complete knowledge of the waves in the foreshock and their interaction with the particles. Since their ?rst detection by [91], [110] and [31], observation of shock-upstream waves has been a long-standing issue. Their existence was predicted by [149], followed by hydromagnetic [6] and kinetic [46] theories of electromagnetic wave excitation and propagation upstream of a collisionless quasi-parallel supercritical shock. [163] suggested that they might develop into discrete wave-packets as had been inferred from observation by [110]. The ISEE 1-3 spacecraft allowed for a more elaborate investigation of the properties of upstream waves [cf., e.g., 48, 49, 50, 67, 80, 107, 108, 109, 131, 132, 133, 144, among others]. More recently, CLUSTER measurements have been used to investigate the temporal and spatial structure of upstream waves and wave turbulence [cf., e.g., 23, 24, 77, 78, 82, 87, 88, 90, among others]. We will brie?y review the properties of the upstream waves in the ion-beam and di?use ion region in view of the observations and mechanisms of their generation. Wave generation is coupled to particle-energy loss and to particle scattering both playing a substantial role in particle acceleration. It will therefore be quite natural that in the next chapter on particle acceleration at shocks we will return to the upstream-wave problem. [11] gave a comprehensive review of the various types of waves encountered upstream of quasi-parallel shocks in the ion foreshock. In his words, “upstream particles cause upstream waves .... Once a wave is created .., it then propagates, and so its continued existence relies on it remaining in a region where it is undamped. Its properties might even change as it propagates. Observationally, the wave propagation is superimposed on the convection of the plasma frame, which introduces Doppler shifts in frequency, and possible reversal of polarisation sense.... And that is not the end of the story, since one must take account of the feedback of the waves on the particle distribution

12 function..., and even the possibility that the ... shock injection of particles into the foreshock os modulated either by intrinsic processes or even by the foreshock waves themselves.” Burgess’ review was organised by the observed wave frequencies. He distinguishes between Low Frequency Waves (5 mHz- few 100 Hz) and High Frequency Waves (> 1 kHz), the latter covering the electrostatic waves from ionsound to electron plasma waves, as well as radiation. Radiation generation will be discussed in detail later. Here, we only note the almost continuous presence of waves in the ion-acoustic band which have been known since [105] to populate the complete foreshock region. These spectra might be composed of several di?erent modes, ion-sound, electron-acoustic, Buneman modes, electron-cyclotron harmonics, and others. Their generation mechanism is not clear yet. They might, via a number of di?erent instabilities, be the result of the presence of the hot foreshock-electron component, which also invades the ion foreshock, or they are excited by unresolved narrow electron beamlets that emanate from the quasi-parallel shock. They might also be excited by plasma inhomogeneities, spatial inhomogeneities in the electron distribution, or they are the result of nonlinear wave-wave interaction which is expected to take place in the foreshock. Currently these questions are di?cult to answer and await further observation, simulation and theory. Little has changed so far since Burgess’ remarks concerning high frequency waves. However, there has been substantial progress in the understanding of the low frequency waves and their role in quasi-parallel shock dynamics. The waves that are most important in shock formation propagate in the ultra-low frequency range < 0.1 Hz. Usually they have large (magnetic) wave amplitudes, around |b|/B ? 0.2 ? 1.0, which identi?es them as highly nonlinear. These large amplitude waves had already been observed by [110] to have wave forms from monochromatic to solitary waves, frequently with steep edges resembling shocklets and suggesting that the waves have experienced nonlinear steepening during their evolution and propagation. They sometimes show the typical signs of ?uctuations that are connected to these edges and obviously propagate in the whistler mode. Thus these forms are indeed little shock-like structures. In the same frequency window, large-amplitude pulsations have been identi?ed. These are very typical for quasiparallel shocks. [126] and [125], identifying them in the AMPTE magnetic ?eld measurements, coined the (somewhat ugly as in German it means mud) name SLAMS for them, which stands for ‘Short, Large Amplitude Magnetic Structures’. We prefer to call them upstream pulsations, here. Their duration is 10-20 s; they have very large amplitudes |b|/B ? 5, indeed, but appear as a more coherent structure that is embedded in the ultra-low frequency wave turbulence. Like the ultra-low frequency waves, they propagate in upstream direction in the plasma rest frame while being swept toward the shock by the convective ?ow. Their polarisation is mixed with – possibly – left-hand polarisation (in the plasma rest frame) slightly dominating, suggesting their ultra-low-frequency wave origin. Sometimes the polarisation is di?erent on both sides of the upstream pulsation, indicating that they have evolved by some process which produces both kinds of polarisation, which is similar to a solitary wave. The propagation velocity of the upstream pulsations exhibits an interesting amplitude dependence. The upstream directed pulsation speed increases with amplitude, which also is a solitary wave-like property. Moreover, they grow when approaching the shock and entering the increasing density gradient of di?use ions, and they play an important role in shock reformation. [125] suspect that these pulsations are the ‘building blocks’ [an expression used by 126] of quasi-parallel shocks. Another interesting property is that upstream pulsations contain thermal plasma with properties of the upstream ?ow, while being surrounded by the hot foreshock plasma. One would therefore believe that their source region is located at the ion-foreshock boundary. Being created there by an ion-ion beam plasma interaction they might grow nonlinearly until reaching quasi-equilibrium like solitary structures, having captured the upstream plasma, and afterwards being convected toward the shock into the heart of the ion foreshock. We will return to these interesting structures when discussing simulations below.

Ion-beam waves

Each of the two di?erent upstream-ion populations is responsible for the excitation of its own instabilities. In this section we deal only with those waves which are excited by the foreshock-boundary ion beam. From Figure 6 we obtain that the situation is that of an ion-ion beam (when for the moment neglecting the gyrophase-bunched di?use-ion component). In the frame of the upstream ?ow the re?ected foreshock-boundary ion beam propagates upstream along the magnetic ?eld at parallel speed vb ?(2 ? 3)V1 , where V1 600 km s?1 . The upstream ion temperature (in energy units) for this case was Ti (1 ? 2) keV, yielding roughly a thermal ion velocity of vi 200 km s?1 . Moreover, the beam can be taken as warm with thermal speed vb,th vb . Hence, vb > vi with magnetised background and beam ions. On the other hand, the electrons are hot with Te 100 eV. a. The expected wave modes. Since the large parallel speed of the beam corresponds to a large parallel temperature anisotropy, it is clear that the beam can excite long-wavelength negative-helicity Alfv?n waves via the ?rehose e instability. In addition, because the beam is mildly warm, it can excite the resonant left-hand ion-ion beam instability,

13

FIG. 11: Reduced parallel distribution functions along the ion-foreshock boundary showing the cold main ?ow ion distribution Fi (v ),
the hot main ?ow electron distribution Fe (v ) shifted to the left into the direction of the foreshock ion beam for keeping zero-current conditions, and the ion-foreshock ion beam distribution Fib (v ). This con?guration is unstable with respect to ion-ion beam instabilities and the ion-beam driven ion-acoustic instability.

which is possible for vb > VA ? 100 km s?1 . And under conditions, when the beam thermal speed can be considered to be small, it excites the right-hand resonant ion-ion beam mode. Both waves propagate with the beam upstream along the magnetic ?eld on the background of the upstream ?ow. They are not as fast as the beam, however, and are thus subject to downstream convection with the ?ow towards the shock. The ?rehose mode, at the contrary, moves against the beam and thus by itself moves downstream in the direction of the shock position, when excited. In all three cases the foreshock-boundary beam will lead to the excitation of low-frequency Alfv?n and ion cyclotron e waves which in the shock frame approach the shock while having their source on the foreshock-boundary ?eld line. During this shock-directed convection and/or propagation they populate the foreshock region with low frequency electromagnetic ?uctuations, which might further interact with the di?use foreshock-ion component. On the other hand, since the phase and group velocities ω/k ? (?ω/?k ) ? VA vb of these waves are of the order of the Alfv?n e velocity and are, thus, much less than the beam and ?ow velocities, downstream convection will quickly remove them from the foreshock-boundary source region. Hence, their further evolution in the foreshock is determined by the competition between nonlinear wave steepening and interaction with the di?use foreshock-ion component. Including the electrons (while so far neglecting the electron foreshock component) leaves us with an ion-acoustic unstable phase-space con?guration. In the upstream ion-plasma frame a relatively dense foreshock-boundary ionbeam is propagating upstream on a cold ion-hot electron plasma. In order to keep the plasma current-free the electron component is slightly retarded creating conditions under that ion-acoustic waves can be excited. On the other hand, the con?guration is not able to excite neither the Buneman-two stream instability nor – because the beam propagates solely parallel to the magnetic ?eld – the modi?ed-two stream instability. This is shown in Figure 11. The instability is excited by the velocity di?erence between the ion foreshock-boundary ion beam Fib (v ) and the slightly shifted to the left hot (Te ? 100 eV) electron distribution because the velocity di?erence ?V between ion beam and electron component ve > ?V ? 103 km s?1 > cia ? 100 km s?1 exceeds the ion acoustic speed while being less than the electron thermal velocity ve . These ion acoustic waves occupy a relatively broad spectrum with downstream parallel phase velocities < vib and become swept towards the shock. Since they quickly leave the foreshock boundary and since the beam is not hot, ion-Landau damping in the source region plays no role. However, when entering the di?use-ion foreshock region these waves encounter the hot di?use ion component, will interact with it, and will thereby experience Landau damping. b. Observations. Observationally, it is di?cult to distinguish between foreshock-boundary waves and di?useion generated waves. Two types of waves that can be related to the presence of the upstream beams have been reported in the vicinity of the bow shock. [31] found monochromatic large-amplitude discrete wave packets in the frequency range ω/2π ? 0.4 Hz which [47] could show to be right-hand polarised whistlers at ω 10 ωci propagating on the plasma rest frame and being unable to escape far upstream. The other class of waves is of smaller amplitude |b|/B ? 0.1 and frequency of the order of ω/2π ? 1 Hz. These waves form trains which are directly tied to the upstream foreshock-boundary ion beams [50]. In fact, as [50] demonstrated they occur only in the presence of the observed re?ected 2 ? 5 keV foreshock-boundary ion beams [22] which [57] have shown to evolve from a beam into a gyrating particle component that in the deep foreshock is superimposed on the di?use ion component before it merges into it. However, as [? ] have shown from measuring the electron distribution function, it is not the ion beam who

14

FIG. 12: Left: The combined power spectral density of magnetic ?uctuations excited by the ion foreshock-boundary ion beam with
central frequency around ? 1 Hz [lower part, after 50] and at < 0.1 Hz [left upper part, after 67] as measured by the ISEE spacecraft. This spectrum is obtained in the ISEE spacecraft frame. The low frequency spectrum is shown at three di?erent times corresponding to (from below upward) increasing distance from inside the foreshock to the foreshock boundary. It is about ?ve orders of magnitude more intense than the high-frequency waves. It is worth noting that the high frequency spectrum of [50] toward lower frequencies in the overlap with the [67] spectrum does not show any indication of the lower frequency peak. As indicated by the thin straight line, it would smoothly continue into the 1930 UT branch at 0.01 Hz. This lack of the low frequency peak is probably accidental and due to the particular conditions at the time of measurement. Right: The wave dispersion relation for the high frequency ? 1 Hz whistler-mode waves determined from a tentative estimate of the relavant wave numbers and transformed into the plasma rest frame. It is seen that these waves in the observation range have about linear dispersion and frequencies ω ? (20 ? 100) ω ci , far above the ion-cyclotron frequency ω ci [after 50].

excites these whistlers. The probability that they are driven by the particular electron distribution in the foreshock is rather higher. This is a very favourable case as these waves carry information about the generation mechanism. The waves propagated obliquely (< 60? ) with respect to the magnetic ?eld, at an average angle of ? 45? . The observed spectrum and the dispersion relation determined from the measurements are shown in the high-frequency part of Figure 12. This ?gure has been combined from the high-frequency observations of [50] and the low-frequency observations of [67], both obtained from the ISEE spacecraft. Tentative wave numbers have been determined for the high-frequenc waves from measuring the time delay of the wave front arrivals at the two spacecraft ISEE 1 and ISEE 2 spacecraft yielding (surprisingly) short wavelengths < 100 km. Knowing the wave number, the frequency has been back-Doppler shifted ω = ω ISEE ? k · V1 into the plasma frame yielding unusually high frequencies ω (20 ? 100)ωci . This procedure determines the dispersion relation (naturally with large errors as indicated by the bars), which is found to be about linear in the narrow high frequency range of the waves in the rest frame of the plasma. The waves that have been left-hand polarised in the spacecraft frame turn out to become right-hand polarised in the plasma rest frame. From these properties it was initially concluded that the waves propagate in the whistler branch and have been excited by the cool ion-ion beam instability, which would be consistent with our initial discussion. This conclusion is, however, questionable. The high rest-frame frequencies are not in agreement with model calculation [38, 131] using the realistic observed beam properties. These yield wave frequencies of the order of ω ? 0.1 ωci for both the ion-ion beam and ?rehose modes. In fact, [27] report CLUSTER observations of similar waves but with much lower frequency ω/ωci ? 0.1 and wavelength the order of ? 1 RE ? 6000 km, which is in excellent agreement with the theoretical predictions. These waves are cold ion beam-excited fast-kinetic (magnetosonic) whistlers, in the terminology of [38] and [63]. The high-frequency waves ? 1 Hz whistlers are instead most probably driven by the particular electron

15

FIG. 13: Left: The excellent correlation between the density and magnetic ?eld variations in the ion-beam excited foreshock-boundary low-frequency waves as measured by CLUSTER [after 24]. The variations are practically in phase thus identifying the ?uctuations as fast magnetosonic.

distribution in the foreshock [34] rather than by the cold-ion beam instability. This claim is also supported by the fact that deeper in the ion foreshock, where the electron distribution becomes more isotropic, these waves do not occur separate from smaller magnetic ?eld structures (large-amplitude magnetic pulsations of the SLAMS type or shocklets). In quite good agreement with the CLUSTER measurements are the ISEE foreshock waves analysed by [67] who found the spectral peak at frequencies ω ? 10?2 Hz with the spectrum broadening with increasing distance from the ionforeshock boundary. This is shown in the upper left part of the spectrum in Figure 12 where the measurements of [50] and [67] have been combined. Even though the conditions on the two observation times were not identical, one sees that the high-frequency whistlers identi?ed by [50] occur on the approximate high-frequency extension of the [67] 19302030 UT spectrum which is closest to the ion foreshock boundary. Note, however, that the low-frequency part of the [50]-spectrum (not shown here other but indicated by a thin straight line in the ?gure) was ?atter and in the (relatively short) overlap region with the [67]-spectrum did not show any indication of the magnetosonic foreshock-boundary waves, which must be due to the particular conditions prevailing during this observation period. Thus the [50] waves are a (high frequency) wave species that is di?erent from these more common low-frequency/long-wavelength fastmagnetosonic waves. These latter waves are also of much higher (? 5 orders of magnitude) spectral density as is seen from Figure 12 and has been con?rmed by the later CLUSTER measurements. Moreover, the magnetosonic waves are left-hand polarised in the plasma frame, and their compressive fast-magnetosonic character is proved from the in-phase variation of the density and magnetic ?eld ?uctuations shown in Figure 13. The presence of these low-frequency ion-beam generated waves implies that the ion beams interact with the ?uctuating electromagnetic ?eld. In this process they become scattered and di?use in phase-space. This is the reason for the spreading of the ion beam in phase space and the ?nal merging into the di?use background distribution. [57] have followed this evolution of the beam as we have described above. It is responsible also for the gradual spreading of the spectrum in the upper left part of Figure 12. A spectrum similar to that given there can be found in the paper of [57]. [2], from CLUSTER observations, inferred the nature of these ultra-low frequency (ULF) waves and showed that their correlation lengths along the wave vector direction k is of the order of 1 ? 3 RE , while it can be a factor of three larger in the direction perpendicular to k, rendering these waves oblate though nearly planar. During times of hot beams [23] observed upstream propagating left-hand low-frequency waves which have been excited on the Alfv?n-ion-cyclotron branch of the kinetic dispersion relation. These waves are kinetic Alfv?n waves e e which have been excited by the hot upstream propagating beam similar to the one shown on the right in Figure 6. In the spacecraft frame these waves because they are swept downstream by the ?ow, appear as right-handed waves. [23] report that with onset of the waves the beam gets more di?use. This can either be interpreted as the reaction of the waves on the beam or that the spacecraft enters the region where the initially cold beam enters the foreshock, spreads in velocity and after becoming hot enough generates the observed waves.

16

FIG. 14: Left: The dispersion relation of low-frequency waves as measured by CLUSTER in the foreshock away from the foreshock
boundary [after 85], determined from the “wave-telescope” analysis method. The upper panel shows the dispersion relation ω(k) consisting of several parts and distinguished by the sense of polarisation as right-hand, left-hand and linear. The di?erent branches cluster. There are forward (positive frequency parallel to the magnetic ?eld) and backward (negative frequency antiparallel) branches. The lower panel shows the wave propagation angle for the shorter wavelengths being mostly oblique ear θ ≈ (30 ? 25)? , for longer wavelengths being more perpendicular to the magnetic ?eld. Right: The ?ve low-frequency modes in a cold beam plasma system of similar conditions as the foreshock plasma and for the propagation angle θ = 24? showing similarity to the observed dispersion relation [89].

The question arises whether there are simulation studies available of the evolution of the re?ected ion beam and the generation of upstream waves. This question cannot be de?nitely answered at present. It is clear that a simulation of this kind should reconstruct the region of the foreshock boundary which requires that a curved shock surface must be assumed from the beginning. In other word, this problem can be investigated only in a two-dimensional simulation. Two-dimensional simulations are available in hybrid form but have have not been applied to curved shocks. Hence, this problem remains an open simulation problem. On the other hand, in the following we will extensively discuss the role di?use particles play in shock formation and upstream wave generation for planar shocks in one and two dimensions and hybrid as well as PIC simulations.

Di?use ion waves

Long-period ? 30 s waves are the rule in the foreshock. They occur together with the di?use foreshock-ion component [cf., e.g., 80, 113] which is located deeper inside the foreshock. Because of this reason, any waves that are excited by the di?use ion component are restricted to the interior of the foreshock. One even has de?ned some fuzzy boundary of these waves called ULF-wave boundary [42, 109], which is even more inclined against the upstream magnetic ?eld than the ion-foreshock boundary and outside of which the ULF-wave activity should be weak. The reason is that ULF waves, if propagating upstream, can move at most with fast-magnetosonic velocity which for a supercritical shock is smaller than the stream and also less than the re?ected ion beam velocities. The advection by the ?ow will blow them downstream towards the shock and con?ne them to a region relatively close to the shock [81] bounded to upstream by the ULF-boundary. [85] determined the dispersion relation of the low-frequency waves in the foreshock from the three-dimensional observations of CLUSTER. Their result is shown in Figure 14. The left part of the ?gure shows the dispersion relation in the plasma rest frame and the angle of wave propagation with respect to the local average upstream magnetic ?eld. The scatter is quite large. Nevertheless it is surprising to ?nd a well expressed nearly linear part on the dispersion relation even though the dispersion relation is composed from the contributions of several distinct modes with di?erent polarisation. The representation is linear, which, on the k-axis, emphasises the short wavelengths (larger k values).

17

FIG. 15: Left: Short wavelength spectra for the magnetic ?uctuations measured by CLUSTER in the foreshock away from the foreshock boundary. These spectra are obtained by application of the “wave-telescope” analysis method. The perpendicular ?uctuations are more than one of magnitude more intense than the parallel ?uctuations while the spectra decay about according to the Kolmogorov 5 ? 3 -law of stationary inertial turbulence. Indication of a cut-o? is seen fat wave numbers larger than the inertial wave number kin where strong dissipation sets on. From the maximum at k ? 10?3 one concludes that in this wavelength range the spectral energy is injected by instability of fast and Alfv?nic magnetosonic waves which cascade nonlinearly forward towards shorter wavelengths. Right: The e probability distributions of the parallel and perpendicular magnetic ?uctuations. The dashed distributions are log-normal of same maximum. These distributions exhibit extended tails and thus indicate non-stationary and probably intermittent not fully developed magnetic turbulence [90].

From linear theory one expects that the long wavelengths (small k) should be more pronounced. Unfortunately, this region could not be resolve su?ciently at the available CLUSTER separation distances at that time [89]. These low frequency waves, in the plasma frame, seem to propagate close to perpendicular to the magnetic ?eld and, therefore, are probably in the fast magnetosonic mode. Moreover, it seems that a straight beam dispersion relation contributes to the perpendicular propagating waves. It also seems that in the low-frequency (magnetosonic) waves have negative frequency. Linear dispersion theory for the parameters during the observation time yields the curves on the right in the ?gure showing the coupling between the presumable beam mode and the four plasma modes at wave numbers kVA /ωci < 0.3. The experimentally determined dispersion relation resembles this clustering of couplings at small k and very low frequencies ω. However, application of linear theory is dangerous if not questionable because the foreshock plasma is highly disturbed, the low frequency waves have rather large amplitudes and can barely be related linearly, and the plasma is very inhomogeneous exhibiting steep gradients in density and ?eld. For an Alfv?n speed e VA 30 km s?1 and an ion-cyclotron frequency ω/2π 1 Hz, wavelengths around the linear wave coupling are of the order of k/2π 100 km and should thus be a?ected by the plasma inhomogeneities (note that decreasing the reference cyclotron frequency decreases the wavelength even further). Nevertheless, at the very low frequencies and very long waves the waves are probably in the fast magnetosonic wave band. More mysterious are the higher frequency-short wavelength waves. From comparison with the linear dispersion relation they seem to ?t on the right-handed whistler branch (high frequency magnetosonic whistler R+ ). However, inspection of the dispersion relation indicated that all kinds of polarisation are scattered along the dispersion curve. The curve itself is very irregular even though the propagation angle of the waves seems to be about constant at weakly oblique angles. This wave composition suggests that we are not dealing here with one single wave mode but rather with the short wavelength part of a turbulent spectrum which has generated all kinds of short wavelength ?uctuations with di?erent almost randomly distributed polarisation in a forward cascading process from long to short wavelengths. This idea has been elaborated in more detail in [90] who found that the shorter wavelength spectrum is indeed about featureless and power-law, close to a ? 5 -Kolmogorov-spectrum of stationary turbulence with most of the power in 3 the perpendicular (non-compressive) magnetic component providing another argument for Alfv?nic and magnetosonic e turbulence. This is shown on the left in Figure 15. The ?gure also indicates the respective gyro- and inertial wave

18 numbers, kci = vi /ωci and kin = c/ωpi . There is indication that the spectra change slope at around these number due to onset of ion viscosity and inertia. The large maximum on the perpendicular power curve near k ? 10?3 km?1 corresponds to a wavelength of λ ? 6000 km reported earlier [e.g., 23] and is at the right position for energy injection by instability. However, the conclusion that the deep foreshock is subject to fully developed fast magnetosonic/Alfv?nic turbulence e is not fully justi?ed. This becomes obvious from considering the probability distribution of the ?eld ?uctuations as given on the right in Figure 15. Would the ?uctuations be normally distributed then their distribution functions would have the shape of the dashed curves. Instead the distributions are skewed into tails. In the case of the parallel component the distribution lacks symmetry. Such tails may indicate that the turbulence is non-stationary, intermittent or inhomogeneous. All three cases might hold in the foreshock, in particular as the foreshock is a rather limited spatial region which is bounded from two sides and is subject to plasma injections and plasma losses. Nevertheless, the spectra determined contain signs of strongly nonlinear and turbulent interactions, and the waves, particularly the low-frequency waves, are of large amplitude and interact with the plasma ion component as also with other waves. These waves a?ect the upstream backstreaming ion component scattering, heating, and accelerating it. On the other hand the upstream ion component is responsible for the existence of the waves as it is the ultimate energy source of the waves. And, to close the cycle, the waves cannot escape upstream very far from the foreshock since in a supercritical shock the Mach number of the ?ow is higher than any Mach number based on the wave speed and is thus high enough for the ?ow to advect the wave spectrum towards the shock. As long as the waves do not completely dissipate their energy during this convection, the electromagnetic wave energy accumulates at the location where they arrive at the shock. The consequence of this accumulation is that the waves a?ect the shock, cause instability of the shock surface, and reorganise the shock front. This makes quasi-parallel shocks non-stationary and subject to some kind of irregular reformation. One may thus expect that a supercritical quasi-parallel shock does not represent a solid shock surface over a large area. It consists of a more or less dense patchwork of areas which together form a shock but which also appear and disappear in an irregular manner, come and go, and organise the shock in a certain volume where the entropy increases but where a multitude of very di?cult to handle processes takes place that can be investigated only experimentally or with the help of properly designed numerical simulations. These are, however, more di?cult to design than in the quasi-perpendicular case, because of the greater variability of the conditions at a quasi-parallel shock. Since waves and particles are tied to each other, simulations cannot be discussed separately for waves and for particles. We therefore delay the discussion of the simulation results to the later section on quasi-parallel shock reformation. There it will become obvious what the upstream waves are good for and what their role is in the quasi-parallel shock process. An important question that has not been addressed anywhere in the investigation of the ion-foreshock dynamics concerns the role of the foreshock electron component which, in the large ion-foreshock domain, consists of two populations, the ? 100 eV upstream electron population which belongs to the quasi-neutral upstream ?ow, and the hot ? several kev-electron population which is the product of the shock re?ected electrons we are going to discuss in the next paragraphs. This component is isotropic but irregular and might contribute to ion-foreshock instabilities thereby a?ecting the foreshock turbulence and shock formation. Since these questions are di?cult to be treated theoretically one has to wait until numerical simulations will become capable of including them into full particle codes in a similar way as has been done [cf. 75, 76] for the electron-ion populations in the feet of quasi-perpendicular shocks.

C.

Electron foreshock

Even though the ion-foreshock occupies the larger part of the foreshock and, which is more important, plays the decisive role in the dynamics of the quasi-parallel supercritical shock, it would be an ignorant attitude not to mention the part that aside of the main ?ow is populated solely by the re?ected higher energy electron component, i.e. the electron-foreshock region. Schematically its extent and relation to the ion-foreshock is shown in Figures 3 and 16. The electron foreshock has been identi?ed already by [114] in the OGO 5 satellite observations. [114] noted the occurrence of enhanced electron ?uxes near the bow shock, which were related to observations of high-frequency electric-?eld spikes at 30 kHz. They concluded that these spikes resulted from Langmuir waves that had been excited by electron beams arriving from the bow shock along shock-connected magnetic ?eld lines. (A detailed overview of the early observations can be read in [58].) It could be con?rmed that these electron ?uxes were magnetic-?eld aligned and of higher energy than the bulk electrons in the main ?ow. This was interpreted as electron beams emitted into upstream direction from the shock, even though no mechanism was known that could provide the required shockelectron acceleration – in fact, a de Ho?man-Teller-frame shock-re?ection mechanism had already been proposed by [138] and was reinvented and elaborated on much later by [164].

19

FIG. 16: Synoptical view of the two foreshocks: the electron and ion foreshocks, respectively.

Electron beams

In analogy to the ion-foreshock boundary the electron-foreshock boundary (which is the ultimate upstream boundary of the foreshock) carries narrow bursts of electron beams, which escape into upstream direction from the shock along the magnetic ?eld and which are slightly displaced by the convective ?ow into downstream direction. An example of a narrow energy spectrum of such a beam measured by the IMP-8 satellite is shown on the left in Figure 17. The spacecraft was in the electron foreshock boundary for a relatively short time only. Being near apogee, the spacecraft was about standing. Thus the short contact time with the beam most probably indicates that the electron-foreshock boundary is fairly narrow. The electron beam also occupies only a small volume in velocity space corresponding to a narrow bump on the distribution [1]. Only one energetic electron-gyroradius deeper in the electron foreshock the electron beam becomes depleted and just forms an energetic tail of hot halo electrons on the electron distribution function of the ?ow. Figure 18 shows electron phase-space observations from the electron instruments on ISEE 1 & 2 during crossings of the electron-foreshock boundary. The left part of the ?gure [taken from 36] is a much higher time and velocity (respectively energy) resolution plot than that in the former ?gure. It is nicely seen how the electron distribution evolved from a ?eld-aligned (nearly) Maxwellian distribution prior to contact with the foreshock boundary, through a ?eld-aligned-beam-like distribution at contact, into a distribution with an energetic tail along the magnetic ?eld, where the electron beam has been completely washed out. This transition takes place within a time interval of 15 s, just allowing to obtain the three measurements. The pseudo-threedimensional plot of the electron distribution function on

20

FIG. 17: Left: The electron energy distribution as seen by IMP-8 at three times when crossing the foreshock boundary. Right: The
high-frequency wave spectrum of the electromagnetic radiation as measured some distance away from the shock near the foreshock boundary. Two types of emissions are visible, the so-calle ‘harmonic’ emission f = 2fpe at twice the electron plasma frequency fpe = ωpe /2π, and the weaker so-called ‘fundamental’ emission at fpe . [after 44].

the right [taken from 1] was obtained on another day when the spacecraft passed the electron-foreshock boundary and a series of distributions were recorded from upstream, across the foreshock boundary, and into the electron foreshock. Here only the foreshock-boundary distribution is shown when the narrow electron beam has evolved ?owing into upstream anti-shockward direction along the magnetic ?eld. Note the narrow angular extension of the beam, its weakness, and its comparably large velocity which allows to identify it just outside the bulk of the hot ? 100 eV background electrons which are much hotter than the beam (indicated by the small beam width in v ). The conclusion that can be drawn from these observations, which in the follow-up time had also been con?rmed by measurements of other spacecraft like AMPTE IRM and AMPTE UKS, is that similar to the ion foreshock, the electron foreshock consists of a very narrow region along about the shock-tangential upstream-magnetic ?eld line (or ?ux tube) which is populated by a cold, weak, and fast electron beam, and a broad electron region where the electron distribution exhibits tails and in addition contains a hot, fairly isotropic electron component. The electron beam (like the ion beam at the ion-foreshock boundary) has its origin near the shock-tangential magnetic ?eld line which is connected to the quasi-perpendicular part of the shock. The mechanism of its generation has not yet been unambiguously clari?ed. Presumably if consists of a combination of the de Ho?man-Teller frame mechanism of [138] and the bending of the shock surface plus some kind of stochastic acceleration in the electric ?elds which evolve in the shock transition, ramp and overshoot. Such ?elds have indeed been reported recently on scales below the electron skin depth λe = c/ωpe to be very large, of the order of 100 mV m?1 parallel to the magnetic ?eld and 100 mV m?1 perpendicular to the magnetic ?eld [5]. The presence of the hot electron component in the foreshock, on the other hand, requires another mechanism that heats and accelerates electrons at a quasi-parallel shock over a large area of its surface.
Langmuir waves

So-called ‘gentle electron beams’, which are just those beams one observes in the electron-foreshock boundary – fast, but not too fast (Vb few times V1 ), parallel (Vb ≡ Vb , Vb⊥ = 0), weak (Nb Ne ), cool (veb < ve Vb ), are know to be the drivers of Langmuir waves with dispersion relation √ 2 2 ω 2 (k) = ωpe + 3ve k 2 , vres > 3 ve (1) via the resonant kinetic gentle-electron beam instability. The Langmuir resonance conditions is ω ? k · vres = 0, where vres ? Vb is the resonant electron velocity. Combining it with the dispersion relation, it is easy to see that in order for the wave number k to be real the resonant velocity must satisfy the condition on the right in the above equation. Below we will discuss the properties of these waves in more detail in connection with observations.

21

FIG. 18: Left: Three successive reconstructions of the reduced parallel electron distribution function during crossing of the electron
foreshock boundary. The distributions are only seconds apart, at 1650:37 UT before touching the foreshock boundary, at 1650:43 UT just crossing it, and at 1650:52 UT being behind it in the electron foreshock. The beam is visible only during the short crossing time. Afterwards the electron distribution shown a heated non-symmetric tail, indicating that the spacecraft sees the depleted beam in the hot electron-foreshock plasma [after 36]. Right: The full two-dimensional electron distribution function during another crossing of the electron-foreshock boundary at the moment when the beam is visible as a narrow enhancement of the distribution in the negative parallel velocities (upstream along the magnetic ?eld). An instant later, when the spacecraft entered into the electron foreshock the beam had disappeared [after 1].

c. Stability of the electron beam. However, there are a number of severe limitations to this instability which theoretically would inhibit its application to the electron-foreshock boundary beam. The most stringent of these limitations is that it is believed that quasi-linear saturation of the gentle-beam instability should quickly, i.e. within a time as short as about ?t ωpe ? 10, deplete the beam and transforming it into a plateau on the distribution function. For an upstream plasma density Ne 5 × 106 m?3 the plasma frequency is ωpe /2π 20 kHz. Hence, the beam should become depleted within ?t 0.08 s. A beam of velocity Vb 104 km s?1 (which satis?es the above condition on the resonant velocity in an in?ow of Te ? 100 eV) would thus propagate just 800 km upstream of the shock along the magnetic ?eld before being depleted, with the implication being that the beam could not escape from its shock-source region. This distance is much less than the distances from the shock at which both the electron-foreshock boundary beams and the beam-excited plasma wave spectra have been observed. Either quasilinear saturation of the Langmuir instability does not work, or the observed plasma waves are no Langmuir waves. The latter is probably not true. So the question arises of how the beam can avoid becoming depleted due to quasilinear e?ects. There are several possibilities of inhibiting beam depletion. It has for long time been believed that collapse of Langmuir waves resulting from modulation instability (also known as oscillating two-stream instability) could shift the beam-excited Langmuir waves out of resonance and remove them from reaction onto the beam. This is a very elegant mechanism which is based on the ponderomotive force the Langmuir waves exert on the plasma. The ponderomotive force is the gradient of the wave pressure which is strongest right in the place where the Langmuir waves are most intense, i.e. in the beam region. For the inert ions of the plasma background the intensity gradient |e|2 of the fast large-amplitude oscillating Langmuir electric ?eld presents a pressure force which pushes them out of place thus decreasing the plasma density locally. Since the electrons react immediately and follow the expelled ions in order to maintain quasi-neutrality, the wave-ponderomotive-pressure force drills holes into the plasma which are ?lled with Langmuir waves. The density variation produced δN ≈ ?
0

4mi c2 ia

|e|2

(2)

is nothing else but an ion-acoustic wave. The Langmuir waves, when becoming intense enough, can drive ion-acoustic waves of speed cia , which exist only due to the presence of the Langmuir wave pressure gradient. The broad Langmuir waves in this way organise into a large number of density cavities, and the waves are removed from resonance with the

22 beam electrons. Enhancement of the outside pressure then forces the holes to shrink. This shrinkage of the hole size shortens the Langmuir wavelength, increases the wave number, reduces the phase velocity and, in this way, shifts the waves into the main electron distribution where they can undergo Landau damping and dissipation. Unfortunately, this kind of modulation instability and collapse of Langmuir waves could never be experimentally approved at shocks. It is therefore quite unlikely (though not impossible) that it occurs and stabilises the beam. Another possibility of stabilising the beam by pushing the Langmuir waves out of resonance is scattering them on thermal ions. This process reads + i → + i . For purely elastic scattering only the momenta k of the Langmuir wave and pi of the ion i are changed according to the momentum conservation equation k + pi → k + pi (3)

where the primed quantities are after collision of wave and ion. Loss of momentum decreases the wave number k and shifts the wave phase velocity to higher values out of resonance with the beam; gain of momentum increases k and brings the wave phase velocity down into the bulk of the electron distribution where it is again dissipated. The e?ciency of this process is, however, low and presumably insu?cient for complete stabilisation of the beam. In addition it strongly depends on the available particle number; it is thus most probable to work in the immediate vicinity of the shock only. It does, however, produce a new Langmuir wave that plays a role in the generation of radio-radiation from a shock. Because of this capacity it nevertheless remains to be of interest in the physics of collisionless shocks. [84] investigated this nonlinear wave scattering o? ions including the polarisation cloud which accompanies each ion. They found that of the ion distribution the ions with velocity v 2vi are the most e?ective scatterers of Langmuir waves. The inclusion of the polarisation cloud increases the scattering e?ciency by a factor of three which does not change the above conclusion. However, the wave number of the scattered wave, its direction of propagation and other wave properties change substantially. Nevertheless, the e?ects are not strong enough to alone explain the long survival of the electron beam over distances very far from the shock. A review of all the available theories has been given by [83] who performed simulations of a spatially bounded beam. It seems that many e?ects act together in order to allow the beam to survive. Weak scattering o? ions removes some waves from the beam. The remaining waves are only partially reabsorbed because the fast beam front runs them out such that they stay in the trail of the beam only. In this way only the trail part of the beam is depleted and retarded, which a?ects only the low velocity component of the beam distribution, assuming that the beam has a small but su?ciently large natural beam spread ? veb ve , as is suggested by the observations shown, for instance, in Figure 18. If injection is continuous, this implies that the beam starts pulsating, which may be a reason for the observed strong Langmuir wave variations along the electron foreshock-boundary ?eld line. Finally, the Langmuir 2 waves, riding on the beam, are generally slower than√ beam; since v 2 > 3ve , their group velocity, which carries the res the energy that can be absorbed, is at most v g < ve 3. So they are more vulnerable to the downstream convection than the beam electrons. The convective V -motion of the upstream ?ow thus sweeps them readily out of the beam ⊥ region at the electron foreshock-boundary ?eld line. All these arguments taken together explain why the beam can survive over long distances and, in addition, why the region across the electron foreshock boundary where the beam is detected is as narrow as found in the observations. d. Electron foreshock-boundary waves. Figure 19, taken from [18], shows a beautiful double passage of the CLUSTER-spacecraft quartet through the Earth’s bow shock on December 22, 2000 in the light of the CLUSTER plasma wave spectrum recorded by the WHISPER plasma wave instrument aboard the four CLUSTER spacecraft. Figure 20 shows an example of these high-resolution measurements CLUSTER came from the magnetosphere, entered the downstream region of the shock, passed the shock to upstream, re-entered the shock, and escaped downstream again. The relevant signatures of each region are seen in all four spacecraft. The uppermost light blue trace is the electron plasma frequency fpe = ωpe /2π which maps the local plasma density. The shock appears as the intense broadband emission with maximum intensity in the low-frequency waves. In the density it is mapped as a steep drop in fpe to the low upstream density values. Close to the shock a number of intensi?cations in fpe can be recognised. These occur at the time of contact with the electron foreshock boundary when intense plasma waves are excited in the Langmuir mode. It is interesting to note that the intensi?cation occurs in spots and is not necessarily narrow-band. It may exceed fpe , or it may also drop below it: the plasma frequency may have ‘hair’ or ‘beards’. Moreover, sometimes intense lower-frequency emissions are detected half way between the plasma frequency and the low frequency emissions in the foreshock. The most intense of these emissions belong to the region closer to the shock where the density gradient has not yet settled to the upstream values, but weaker emissions of the same kind occur farther away in the upstream low density domain. A detailed analysis of all these waves detected by CLUSTER has not yet been undertaken and thus is not yet available. It is, however, highly suggestive that the observed intense spots in the plasma frequency, with the ‘hair’ parts exceeding and the ‘beard’ parts hanging down from the local fpe , are related to magnetic ?eld-aligned electron beams emanating from the shock and propagating upstream along the electron foreshock-boundary magnetic ?eld as these are the same

23

FIG. 19: A CLUSTER spacecraft passage across the Earth’s bow shock region on December 22, 2000. The spacecraft are coming from
the magnetosphere, pass across the downstream magnetosheath region with its enhanced density, cross the shock into the upstream region and back again and remain in the downstream magnetosheath. The hole region is seen in the light of the broadband plasma wave electric ?eld spectrum from about 0 kHz up to 80 kHz. The ?gure shows the four panels of all four CLUSTER spacecraft (with their somewhat childish names Rumba, Salsa, Samba, Tango). The uppermost light blue emissions in each panel belong to the electron plasma frequency fpe = ωpe /2π which on most of the path is thermal noise mapping the local plasma density. The passage of the shock is signalled by the appearance of intense broadband waves starting at low frequencies and being correlated with a fairly steep drop in the plasma frequency respectively the plasma density. Outside the shock in the low density upstream region the strong spots of intensi?cation of the plasma frequency indicate contact with the electron foreshock boundary beam and excitation of Langmuir waves. the intensi?cation is highly structured, sometimes stretching above, sometime hanging down below fpe . Also seen are intermediate frequency emissions near the shock. The lowest panel shown the estimated from fpe plasma density for all four spacecraft [after 18].

signatures as those observed with the ISEE and AMPTE spacecraft, though with much better frequency resolution and higher sensitivity here. For instance, the latter two spacecraft were unable to resolve the plasma frequency which in the CLUSTER data is nicely distinguishable even though it represents just thermal noise. It is most interesting that these contacts with the electron foreshock boundary occur quite irregularly throughout the hole period when CLUSTER was upstream of the shock and that they do not seem to exhibit a one-to-one correlation between the four

24

FIG. 20: An example of ISEE 1 high frequency wave observations during contact and passage of the electron foreshock boundary. The
thin faint spotty line around 30-40 kHz is the local plasma frequency fpe = ωpe /2π ? 30 kHz, seen here only as thermal noise with lesser instrumental sensitivity than in the CLUSTER observations. The occasional intense dark spots as the one marked by the white arrow are brief contacts with the foreshock boundary ?eld line when the beam occurs. deeper in the foreshock the spectrum broadens non-symmetrically, evolving ‘hair’ and ‘beards’. Some moving structures can be identi?ed in the hair. The emissions deeper inside the foreshock are correlated with the occurrence of low frequency emissions below a few kHz which are probably ion-sound waves [after 30].

spacecraft. Taking into account that the upstream plasma frequency away from the shock is fairly constant this implies that the shock-normal angle ΘBn changes on a irregular and fast time scale, i.e. that the direction of the upstream magnetic ?eld is highly variable even under conditions of apparent quiescence in the upstream medium. The conclusion drawn from this would not be that the upstream magnetic ?eld is subject to violent variations as in the electron foreshock there is no reason for the magnetic ?eld to be strongly a?ected. Rather this variability points on the temporal variation of the shock area near the tangential ?eld line. Even a small change in the shock normal there will cause a large variation in the location of the tangential magnetic ?ux tube at a distance upstream of the shock. Reasons for such a variability have been given in the section on quasi-perpendicular supercritical shocks, but to repeat here, the main reason is that the super-critical shock is neither in thermal nor in thermodynamic equilibrium and is thus by its very nature subject to changes in all its physical parameters. In addition to the plasma frequency emissions, which hint on contact of the spacecraft with the shock-tangential ?eld line that makes up the electron foreshock boundary, there are a number of most interesting features seen in the wave spectra of Figure 19. One of them we merely note at this place because we will return it farther below. It is the broadband signals that are occasionally seen in many places, in the plasma frequency as signals covering a frequency band of up to ? 20 kHz, in the lower frequencies near the shock, an in particular in the shock transition itself where the frequency range of the emission ?f > 80 kHz in all cases exceeds the whole range of the instrument. There are no known plasma waves of a comparable bandwidth reaching from ? 0 kHz deep into the range of free space radiation frequencies. Such spectra are, however, known to be produced from narrow spatially localised electric ?eld structures. The broader the spectrum the narrower will the structures be. We therefore conclude that in all the places where such broadband emissions are encountered, and in particular in the shock transition, we are dealing with very narrow localised and intense electric ?elds which have been generated by nonlinear processes. Such structures are presumably solitons and/or phase space holes either of electronic or ionic nature. The so far most elaborate investigation of the width of the electron-foreshock boundary beam and its relation to the excitation of plasma waves has been performed by [30] and subsequently re?ned and supported by theory by [65]. An example of their ISEE 1observations is shown in Figure 20. Since the instrument was much less sensitive than WHISPER the thermal noise at the plasma frequency around fpe ? 30 kHz appears only as a very faint spotty line. Occasional strong narrowband intensi?cation of this line has been identi?ed as contacts with the electron foreshock-boundary beam by inspecting the simultaneous electron measurements > 300 eV. These emissions were indeed found to be very narrow band, just 1-2 kHz wide at fpe . Deeper inside the foreshock the bandwidth of the emissions broadens substantially in a non-symmetrical way. First, it becomes very noisy, consisting of short emissions clumped together in groups. Second, long ‘hair’ emissions evolve of roughly up to 10 kHz bandwidth, while ‘beard’ emissions are also found being generally weaker, but sometimes they extend to low frequencies. Low frequency waves below a few kHz

25 are also detected in connection with these deeper-foreshock emission and were found to smoothly merge from below into the beards. [30] and subsequently [65] could show that the spectra changed from very narrowband in the foreshock boundary ?ux tube to broadband in the foreshock, just a few electron gyroradii away from the foreshock boundary. They also showed that the electron beam distribution ?attened over this distance. The waveform of the waves at the plasma frequency in the broadband region away from the foreshock-boundary exhibited modulations, which group the waves into groups of length of a few 10 ms, a frequency roughly corresponding to the frequency of the low frequency waves seen in the dynamic spectrum of Figure 20. e. The nature of electron-foreshock waves. The narrowband electron foreshock-boundary waves detected at the foreshock-boundary ?ux tube are clearly electron-beam driven Langmuir waves of the surprisingly large amplitude of a few mV m?1 at the edge of the foreshock and a few % bandwidth in frequency. It is considerably more di?cult to infer about the nature of the waves deeper in inside the electron foreshock. These waves reach large amplitudes of a few 10 mV m?1 to a few 100 mV m?1 and bandwidths of 30%. (Note that the stationary convection electric ?eld in a ? 500 km s?1 ?ow in a ? 5 nT magnetic ?eld is just E ? 2.5 mV m?1 !) The wave electric ?eld is practically parallel to the upstream magnetic ?eld, and the wavelength is long below and short above fpe . Moreover, these waves are modulated and seem spectrally to connect to the intense low frequency waves. ˙ For plasma frequencies of ? 30 kHz the ion plasma frequency is 0.7kHz. Thus, accounting for the Doppler-broadening of the low-frequency wave spectrum in the fast ?ow, the low frequency waves can be tentatively identi?ed with ionacoustic waves, which accompany the high frequency waves at the plasma frequency. Obviously these wave modulate the latter, it is however not clear whether or not they are created by the high-frequency waves via the modulation instability or whether they are generated in a di?erent interaction between the depleted beam and the upstream ion ?ow via an ion-acoustic instability. We have already argued that the modulation instability is unlikely under the conditions of the electron-foreshock boundary beam. Moreover, the weak modulation of the wave form of the high -frequency wave noted above does not argue in favour of the modulation instability and caviton formation as the waves are not really bundled in localised groups and the wavelength is not changed appreciably. On the other hand, the spiky broadband nature provides a weak argument for some localisation which can, however, be objected owing to the pronounced asymmetry of the spectrum with respect to fpe . [65], ignoring the correlation with the low-frequency waves, interpret the high-frequency waves around fpe on the basis of the kinetic Langmuir wave instability including the beam plasma, i.e. referring to a non-gentle beam of ?nite temperature and appreciable density. Their simpli?ed dispersion relation then becomes √ 2 ωpe Nb π ω ? kVb ω 2vbe ω 2 2 1 = 2 (1 + 3k λDe ) ? 1+i 1, , ?1 < (4) ω N 2 kvbe kve kVb Vb with Nb , Vb , vbe the respective beam parameters, density, velocity, and thermal speed, and λDe = ve /ωpe is the Debye length of the upstream plasma. The ?rst two terms in this expression are the ordinary Langmuir wave. The second term on the right is the beam contribution. for small gentle-beam densities the contribution to the real frequency can be neglected. However, for larger beam densities we see that Nb /N adds to the unity on the left. Hence the frequency of the wave should decrease for small k. Thus long wavelength waves will have frequencies ω < ωpe . This agrees nicely with the observation. Short wavelength waves, where the term 3k 2 λ2 comes into play, will have frequencies De ω > ωpe . This is also in accord with observation. Thus, the inclusion of the dense beam into the dispersion relation does reproduce the basic observed spectral properties of the beam excited waves at least qualitatively in a simple linear way without reference to nonlinear e?ects like the modulation instability and collapse. The occurrence of the low-frequency ion-acoustic waves is then probably due to the electron-ion velocity di?erence of the hot electron component and the cold upstream ion ?ow via the electron-ion acoustic instability. In a series of one-dimensional particle simulation papers, [21] has investigated the dynamics of beams at the foreshock boundary in an attempt to quantify the above qualitative theoretical conclusions. He considered a pure electron beam on an otherwise neutralising ion background. As expected, a gentle weak beam evolves basically according to quasilinear theory into a plateau which for long times remain unchanged. Even in extremely long simulation times there is practically no evolution in wave energy, and the distribution function remains stable once the quasilinear plateau is formed. This happens approximately after a few hundred plasma periods. For a plasma frequency fpe 30 kHz this amounts to a relaxation time of the order of just ?1 ms. Hence, gentle beams at the foreshock should be stabilised within this time, and any nonlinear e?ects will evolve only at much later times if at all. ?1 Excluding the ion dynamics, the simulations have been run until ? 3000 ωpe without any susceptible change in the distribution and wave level, in physical times until ? 10 ms. Moreover, the saturation level of the waves is remarkably low, much lower indeed than the theoretical estimates suggested. One does not expect that weak or strong turbulence e?ects will evolve which could change or prevent plateau formation. For this to happen, of course, ion dynamics should be included into the runs and should produce small k waves, while in the plateau formation the wave number increases because the plateau widens and the beam front proceeds to smaller and smaller velocities until the plateau

26

FIG. 21: Results of a one-dimensional electron-electron beam particle simulation [after 21] intended to explain the high-frequency
plasma waves in the electron foreshock. Left: The electron distribution used. Initially (thin line) it consists of a broad (warm) Maxwellian background distribution plus a narrow (cold) electron beam sitting on the main distribution. All quantities are normalised, velocity is normalised to nominal beam velocity. At end time tωpe = 1053 (solid line) the beam has become depleted and heated but has not disappeared. Right: Real dispersion relations ω(k) (solid lines) and growth rates γ(k) (thin lines) at di?erent simulation times tωpe as indicated by the numbers. Only the unstable part of the real dispersion relations (corresponding to positive γ > 0) is plotted. The Langmuir wave is shown separately with its growth rate labelled fpe which is weekly positive only at very long wavelengths. The main unstable waves are shown to propagate in the beam mode (about straight lines starting at origin of ω and k. Maximum unstable frequencies are well below the plasma frequency ωpe .With progressing time the wavelength of the unstable modes increases (decreasing k). The unstable domain in frequency and wave number shrinks. The unstable frequency decreases as a consequence of the decreasing beam velocity which leads to a decrease in the slop of the beam mode.

is completed and Landau damping stops the further evolution. Therefore, based on these simulations, the dynamics of a clod beam cannot explain the richness of the wave observations in the foreshock boundary. For a very cold beam the evolution is a bit more complicated as the cold beam initially is not subject to the Langmuir instability but rather to the reactive beam instability which generates a broad spectrum below fpe as inferred above [65]. The maximum growth is close to but below fpe , and the spectrum is sharply cut o? just above fpe . Afterwards the beam evolves readily into the kinetic Langmuir stage described above, forming a plateau and stabilising. This transition proceeds due to intermediate electron trapping in the wave, which heats the beam to a temperature when the kinetic instability can take over. One might thus conclude, that the waves observed below fpe indicated the passage of the cold narrow beam front. This, however, is in contradiction to the observation that these waves are not observed in contact with the electron-foreshock boundary but deeper in the foreshock where no cold beams are present. Hence, another interpretation for the waves is needed. One solution is to take into account the bulk streaming distribution [21]. This is shown in Figure 21. Then the interaction becomes an electron-electron beam mode interaction with the possibility to destabilise the electron acoustic wave mode. This mode has frequency su?ciently far below the plasma frequency, ω < ωpe . A condition is that the beam velocity spread is small. Initially the beam also excites frequencies substantially above fpe with large growth rate, but these are quickly stabilised. The unstable modes are actually beam modes with resonance condition ω kVb which are destabilised by Landau damping from bulk electrons, i.e. the beam speed must enter the range of bulk velocities for Landau damping and should thus not be displaced far from the bulk distribution like in the gentle beam case. Plateau formation takes very long time in this case such that the wave can reach quite large intensity of the order of W/N Te 10?3 . For an upstream density of N ? 5 × 106 m?3 and electron temperature of Te ? 100 eV this yields an rms electric ?eld amplitude |e|rms 1 V m?1 in very good agreement with observation. For larger beam temperatures the general trend as in Figure 21 remains valid. It is interesting that the unstable frequency readily drops from higher than ωpe to ω < ωpe while the wave number shrinks, i.e. the wavelength of the maximum unstable waves increases. One should thus observe falling tones in the frequency of the emission. It is also interesting that

27

FIG. 22: Parallel distribution functions in the deep electron foreshock consist of the cold bulk ion distribution and the hot bulk electron distribution on top of which the parallel part of the hot di?use, i.e. warm upstream electron distribution. This distribution in the upstream frame moves away from the shock. The upstream electron distribution adjusts to the current-free condition by being retarded by the small amount |Ve ? Vi | = Vb Nb /N . Its maximum is shifted in the direction of the upstream di?use beam distribution. Such a con?guration should be unstable with respect to electron-acoustic waves (dense hot electrons, less dense cool beam electrons).

during the evolution of the instability the slope of the beam mode decreases which simply re?ects the retardation of the beam during depletion. The ?nal state is low frequency-long wavelength. It seems that this calculation explains the observation of electron foreshock wave emissions. However, the problem about this model is that it should work only right at the electron-foreshock boundary where only emission at the plasma frequency is seen like in the classical case of a gentle warm low density beam. The broadband waves below fpe are observed inside the electron foreshock, where no electron beams have ever been detected. Moreover, it does neither explain the high intensities and the broadband nature of the waves above fpe , nor does it explain the connection of the waves below fpe to the low frequency ion-acoustic waves. Therefore we conclude, that probably a two-temperature counter streaming electron-component plasma with both components warm will be more appropriate to the interior of the foreshock, and probably the cold ion component of the bulk upstream ?ow must also be included. On the other hand, taking these simulations for serious, the deep electron foreshock region should be ?lled with many cold and not overwhelmingly fast electron beams propagating along the magnetic ?eld upstream. Possibly the resolution of the current instrumentation is still unable to resolve them. The observations on the right in Figure 18 might indeed indicate the presence of such small beamlets that are distributed over the gyro-angle. So far, however, they can only been regarded as measurement ?uctuations. The parallel con?guration of the distribution in the foreshock is shown in Figure 22 for a warm beam that is the ?eldaligned part of the di?use electron component in the electron foreshock. Because of the vanishing-current condition the bulk electron component will be slightly retarded at the small amount |?Ve | = V1 Nb /N . This con?guration consists of a hot dense bulk electron component and the warm dilute beam component. In addition to the beam instability it 1 should be unstable with respect to electron-acoustic waves at frequency ω kve (Nb /N ) 2 . Electron-acoustic waves are long-wavelength waves below the electron plasma frequency. Thus the low frequency waves ?nd several explanations, none of them completely satisfying though. For the high frequency waves which exceed the plasma frequency the only available interpretation is that they result from localised electric ?elds. In frequency space these localised waves have a broadband signature. The distributions in the perpendicular direction are non-symmetric halo distributions with the backstreaming electrons populating the halo tails. They still await an in-depth treatment for inferring their contribution to the wave spectra, particle scattering and plasma heating. We close this section by presenting in Figure 23 an average synoptic spectral view of the waves detected in the shock transition as was provided by [105] and [45] from consideration of a large number of shock spectra. There is no new information in this ?gure except that it summarises at a glance the main features in the higher frequency electric and magnetic ?eld wave spectra. The cut-o? of the magnetic spectra at the electron-cyclotron frequency is no surprise. In the electric spectra there is a large temporal variability as can be seen from a comparison of the average and peak values (measured within 1 s measuring time). The variations cover up to two orders in magnitude. Of interest is also that the electric spectra exhibit a maximum at the lower-hybrid frequency, an absorption at the electron cyclotron frequency, and show a broad maximum at the ion plasma/Buneman two-stream frequency before being steeply cut o? towards higher frequencies. This cut-o? occurs due to Landau damping at the Doppler-shifted frequency of ionacoustic waves when the waves are shifted into Landau resonance with bulk electrons. In this frequency range all

28

FIG. 23: A representative distribution of electric shock spectra. Left: Average (1 s-averages) spectral values, showing the broad peak
in the electric ?eld spectra and the exponential decay of the spectrum toward high frequencies. For intermediate spectral intensities thepeak is well developed, while when the spectral intensity is very high it smoothes out. Centre: The same spectra but in the peak values (30 ms resolution) and not in the averages The spectra are similar but much more variable and up to two orders of magnitude more intense. This points to the high variability of the electric wave emissions at shocks. In addition indications of the plasma frequency are seen at the high frequency end, suggesting that the emissions in fpe are highly time variable [after 105]. Right: A schematic summary of shock spectra showing the magnetic spectra being cut o? at the electron cyclotron frequency fce , and the electric spectra containing several maxima at the lower hybrid frequency fLH at its low frequency end, around the ion plasma fpi and Buneman two-stream fB frequencies, with an absorption dip at the electron cyclotron frequency, a steep exponential decay caused by Doppler-shifted Landau damping of the waves in the Buneman and ion-acoustic modes at frequency fpe V1 /ve = V1 /2πλD , and the little bump at the electron plasma frequency [after 45].

kinds of waves overlap, reaching from ion-acoustic waves and electron-cyclotron harmonics to Buneman waves, localised structures like Bernstein-Green-Kruskal modes and solitons, while the modi?ed two-stream instability provides the large maximum at the lower hybrid frequency. The high variability in the peak values results from the presence of these localised structures which recently have been observed [5] in situ but could have been concluded also just from the high variability of these spectra in the frequency range of the Buneman-modes respectively Doppler-shifted ion-acoustic waves. These are nothing but the signatures of many microscopic phase-space holes that obviously accumulate in the shock transition region. Support to such an interpretation has been given long ago by the ISEE measurements of the electron distribution function during shock transition [34] which [45] made responsible already for the high variability of the peak spectral values. In these measurements the electron phase-space distribution function transforms from the upstream streaming Maxwellian to the shock-ramp and downstream ?at-topped heated electron distribution, which just in the short time interval of crossing the shock -ramp magnetic-overshoot exhibited a clear signature of an electron beam that was sitting on the upstream edge of the ?at top of the distribution (as depicted in Figure ??). The electron beam in that ?gure is caused by acceleration in the shock potential and has an upstream directed velocity of a few 1000 km s?1 . Compared to the bulk electron temperature this beam is cool. Thus it may excite electron-beam waves and electron-acoustic waves. However, in combination with the bulk ion ?ow it is also capable of exciting the Buneman two-stream instability which will readily evolve into electron holes and plays a major role in heating the plasma and making the main electron distribution ?at-top. One should also remember that the scale of the overshoot is narrow, of the order of a skin depth, and is most probably due to an intense electron current ?owing in the ramp or ramp-transition region which, presumably, is also related to the observed beam. At high frequencies the spectra also exhibit the small peak caused by the beam excited Langmuir electron plasma waves, while the lower frequency foreshock waves are buried in the fat bump of the Doppler-shifted ion-acoustic waves. Of course, more cannot be concluded from a picture like this. The more detailed discussion of the spectra requires much higher spectral resolution. To the extent as it was available at the time of writing, we have given it here. But much more work is to be done until the various modes which can be excited in the foreshock and shock transition will be understood better.

29
Radiation

Shocks are frequently referred to as sources of radiation. Famous examples are supernova shocks, which are visible in almost all wavelengths, from radio through visible light up to x-rays [e.g., 19], and solar type II shocks (cf. Chapter 8) with their main radiation signatures seen in the radio waves. Supernova shocks are relativistic shocks which are not treated here. Their Mach numbers range from weakly relativistic to highly relativistic, but the energy per particle in them remains to be less than the rest energy of an electron, me c2 = 0.511 MeV, which allows to treat them classically. This does not hold anymore for the central supernova engine which drives the ?ow and which in some cases results in the generation of ultra-relativistic jets. There the shocks become non-classical, and not only radiation losses but also particle generation must be taken into account in their description. f. Observations. The radiation that is occasionally emitted from nonrelativistic shocks is restricted to the radio wave range. They do not generate x-rays or visible emissions because, ?rst, of their comparably low energy per particle, which is less than the rest energy of an electron me c2 , second, because of their low energy transmission rate (particles 2 are not retarded from ?ow speed to rest) and, third, because of the low ‘emission measure’ EM = dr3 ?Ne (r), i.e. the number of radiating particles of density ?Ne in the volume that contributes to the emission of x-rays is too low for providing any measurable intensity. Moreover, since magnetic ?elds are weak and the ratio ωpe /ωec > 1 of electron plasma to electron cyclotron frequency is larger than one, gyro-synchrotron emission is unimportant. Thus, the only means of how free-space radiation of frequency ω ωpe can be produced is via plasma wave emission. Radio emission from collisionless shocks in the heliosphere is a widely studied ?eld including type II solar radio burst, interplanetary type II burst, CME-driven radio bursts and radio emissions from planetary bow shocks. While the solar bursts, because of their high frequency can be observed from Earth, most of the other emissions have been discovered only from aboard spacecraft. Of the enormous wealth of such observations made by the ISEE, AMPTE, Polar, WIND, GEOTAIL and other spacecraft, many of them never published, we pick here just a more recent observation from CLUSTER [154]. Figure 24 shows two cases of such observations on December 22, 2000 and three months later on March 13, 2001, when CLUSTER was crossing the bow shock and moved into the electron foreshock. On December 22, 2000 the upstream density was relatively high with plasma frequency near 40 kHz. The intense spots in fpe between 1200-1300 UT indicate touching of the electron foreshock boundary ?eld line as has been discussed above. Broadband 1 wave modes of around ? 2 fpe frequency indicate beam modes though no frequency drift is detectable. However, the density during this time is low enough for a faint emission to occur at about ? 80 kHz, just two times fpe . At this frequency emission can be only in the electromagnetic free space mode. We are thus witnessing local generation of radio emission from the foreshock boundary. At later times when the spacecraft moved deeper into the foreshock – as visible from the widening of the plasma wave spectrum to both sides, up and down from fpe – the radio emission becomes more di?use and more broadband shifting to lower frequencies than the second harmonic of the plasma frequency. This suggests that possibly here the lower frequency modes close to fpe do actively participate in the generation of emission. On the other hand, radiation from the shock ramp might also contribute to these emissions. The lower panel in Figure 24, recorded on March 13, 2001 during a day of much lower upstream density, shows the more typical case of Langmuir waves at fpe and radio emission at almost precisely 2fpe . Note that near 80 kHz a faint indication of the presence of an emission at the third harmonic ? 3fpe can be made out. The dashed vertical lines in this panel indicate the times when the magnetic ?eld direction changed abruptly with the changes in magnetic ?eld magnitude and density remaining comparably small. The emissions in the plasma frequency and harmonic are well correlated during the entire event. Obviously CLUSTER was constantly close to the foreshock boundary as only intensi?cation in fpe is seen but no ‘hair’ nor ‘beards’ neither evolve except during a period shortly after 1200 UT in the three yellow spots in fpe when the harmonic emission extends to lower frequencies together with the development of a little beard. Two other interesting features can be read from this panel. At early time in the panel, just before the short active phase of the topside sounder at 1020 UT, splitting of the harmonic emission into two narrow bands is seen, which reminds at the band-splitting in type II bursts. The other interesting feature is the large number of drifting emissions with increasing frequencies, which suddenly evolve right after 1200 UT (following the drop in plasma density at 1140 UT). Both features, the splitting of 2fpe and these drifting bursts are not understood yet. The latter might be related to the abrupt changes in density and plasma frequency seen in this panel. These density changes are accompanied also by changes in the magnetic ?eld, which are not shown here. It is worth noting that these banded drifting emissions cannot come from remote simply because the low frequencies arrive ?rst. Radiation from any remote source should be visible ?rst at high frequency. The drifting emissions must be related to a nearby source, most probably the shock or foreshock. Understanding its generation mechanism should provide valuable information about the radiation source. g. Interpretation. Theory of shock-emitted radiation is based on plasma processes which under the prevailing collisionless conditions in the shock and foreshock plasmas refer to wave-wave coupling as the main generation mechanism. Direct emission from particles is unimportant, because the energy losses a particle experiences when becoming

30

FIG. 24: CLUSTER observations of electromagnetic radiation from the electron foreshock, following the multiple shock crossings between 1120-1210 UT that is marked by the intense low frequency noise and its broadband extension through the entire frequency range from 0-80 kHz. Upper panel: Near electron-foreshock boundary typical fpe and lower frequency emissions from 1210-1250 UT. These are accompanied by a 2fpe radiation band near 80 kHz and another weak emission band just above fpe the upper cut-o? of which is decreasing in frequency. At later times the spacecraft is deeper in the shock as indicated by the broadband electrostatic emissions at fpe . But these are correlated with broadening of the harmonic radio radiation towards lower frequencies. Lower panel: A sequence of three hours of low upstream density when the spacecraft remained close to the foreshock boundary. With the exception of a few short periods there is no broadening of the plasma frequency, while all the time a harmonic emission of intensity not much less than at fpe accompanies the plasma frequency. During the time of broadening of the plasma frequency one also observes a downward broadening of the harmonic radiation. At high frequencies near 80 kHz a faint third harmonic band ? 3fpe can be identi?ed. Of interest is the band splitting seen in the harmonic radiation between 1000-1010 UT. Finally, the many upward drifting equally spaced in frequency narrow-band radiation bands after 1140 UT are surprising. They do not have any counterpart in the plasma frequency and must thus be from a remote source that might be related to the small but sharp change in density (drop in fpe ) at 1140 UT. The dashed vertical lines mark changes in the magnetic ?eld (not shown here), when its direction turned abruptly. Some of these are correlated with small variations also in the magnetic ?eld strength [after 154, and Trotignon, private communication].

retarded or re?ected at a shock, are not transformed into radiation. In all non-relativistic cases radiation losses can completely be neglected compared with all other energy losses. Nevertheless, the observed radiation is of interest because in many cases, where no measurements are possible to be performed in situ, radiation is the only direct and presumably identi?able signature a shock leaves, when seen from remote. The other signature is the generation of energetic particles, which will be treated in the next chapter, but energetic particles are a more di?use indicator of a shock, because their propagation is vulnerable to scattering from other particles, obstacles and, in the ?rst place, scattering by magnetic ?elds. They, thus, do not provide an image of the shock as clear as radiation does. Just because of this reason, investigation and understanding of the mechanisms of emission of radiation from shocks enjoys – and deserves – the large amount of attention it receives.

31 Expecting that – presumably – direct particle involvement into radiation is improbable (maybe with two exemptions, which we will note later) we are left with a small number of possible mechanisms, which all belong to the class of wave-wave interaction in weak plasma turbulence. The most probable of these are three-wave processes. These can be understood as ‘collisions’ between three ‘quasi-particles’. Since only these three are involved, the interaction conserves both, energy and momentum, and can symbolically be written as L + L → T, L ≡ {ω L (k L ), k L } , L ≡ {ω L (k L ), k L } , T ≡ {ω(k), k} (5)

Here L stands for longitudinal, and T for transverse – as the emitted radiation of frequency ω and wave number k propagates in the free space mode and thus is a transverse electromagnetic wave while its two mother waves are assumed to be longitudinal (i.e. electrostatic) waves of su?ciently high frequency. Moreover, these frequencies ω L (k L ) etc. depend on wave number through the real parts of the electrostatic dispersion relation DL (ω L , k L ) = 0 and may become quite complicated expressions. In the presence of a plasma there are two free space modes, the ordinary and the extraordinary mode. Naturally, in order to leave the plasma and escape in the form of radiation their frequencies must exceed some lower cut-o? frequency. for the ordinary mode
2 ω 2 = ωpe + k 2 c2

(6)

this is the plasma frequency reached at very long wave lengths k = 0. However, since the speed of light c ve is so large, the radiated wave length is much longer than any of the wavelengths of the longitudinal waves involved. This is immediately recognised when comparing the above ordinary wave dispersion relation with the Langmuir wave relation, with L ≡ ,
2 2 ω 2 = ωpe + 3k 2 ve

(7)

√ which is of exactly the same structure. Hence, as long as ve c/ 3 we will have k k, and the wave number of the radiated mode is practically zero. Momentum conservation of the three interacting quasi-particles becomes simply k L ≈ k L , implying that the interaction selects counter-streaming electrostatic waves. As for an example, any process that is capable of generating Langmuir waves of comparable wavelengths, propagating in both directions along and opposite to the magnetic ?eld, can in principle contribute to generating escaping radiation. From energy conservation ω + ω = ω of the three ‘quasi-particles’ involved one immediately ?nds that ω ≈ 2ωpe . This is the origin of and the simplest mechanism for the generation of the 2fpe -second plasma harmonic radiation and has been proposed more than half a century ago by [39] as an explanation for the observation of solar type II and type III radio bursts. In this simple reasoning we have completely neglected not only the contribution of the Langmuir wave number (which turns out not to be important in magnitude, it just shifts the emitted frequency a tiny amount up in frequency) but also the fact that the electric ?eld of Langmuir waves is polarised along k = ±k B/B and thus along the ambient magnetic ?eld B, while the electric ?eld of the transverse emitted electromagnetic radiation must necessarily be polarised perpendicular to k (because of the absence of space charges at frequencies su?ciently higher than fpe ). Since the electric ?eld will, after collision and annihilation of the two Langmuir waves involved, remain to oscillate along the magnetic ?eld, the emission is preferably directed perpendicular to the magnetic ?eld. It turns out, then, that it is easier to radiate in the extraordinary than in the ordinary free space mode. The extraordinary mode has a slightly more complicated dispersion relation; in a dense plasma with ωpe ωce (as is encountered in near Earth space where supercritical collisionless shocks evolve) it has a slightly higher cut-o? frequency. But the argument about the smallness of k k holds also in this case. Radiation at the second harmonic ω ≈ 2ωpe should therefore be polarised perpendicular to the magnetic ?eld in the extraordinary mode. Unfortunately, foreshock emission has not been found to show any preference in polarisation [103]. Moreover, emission is not only in the second harmonic but has also been detected close to ωpe and at the third harmonic ? 3ωpe . These emissions require di?erent waves to be involved for which a number of mechanisms have been proposed [cf., e.g., 14]. So far none of them could be ultimately veri?ed or even agreed upon, each having its merits and pitfalls. Since radiation is energetically negligible, as we have mentioned above, the whole problem could be put aside. However, since the assumptions made in every radiation mechanism contain important information about the source region, the problem of radiation production in collisionless shocks remains to be tantalisingly urgent and awaits resolution. Since there is no agreement yet about the radiation mechanism, we merely note here some of the di?erent proposals. The ?rst is the above mentioned merging of two counter streaming Langmuir waves. The problems about this simple though suggestive mechanism are numerous. First, Langmuir waves are assumed to be generated by the gentle-beam instability. Ignoring the problem of beam survival during its propagation along the shock-tangential ?eld line, which we have discussed already in detail, gentle beams excite only forward Langmuir waves, which requires some mechanism

32 that backscatters a substantial percentage of waves and inverts the direction of their wave numbers. There are three known elegant processes that are capable of doing this: modulation instability respectively collapse, scattering of Langmuir waves o? thermal ions, and scattering o? ion-sound waves, all three proposed long ago [for an early review of the latter two mechanisms cf., e.g., 156]. 2 Modulation instability generates ion-sound waves via the ponderomotive pressure force Fpmf = ?(e2 /me ωpe ) |e |2 of the high-frequency Langmuir wave, e (r, t). These waves, when becoming locally large amplitude, structure the plasma into a chain of cavities in which the Langmuir waves become trapped. This process generates long wave lengths. It is described by the Zakharov equations for the combined evolution of the Langmuir wave ?eld and the density variation δN , respectively, ? 2 δN ? c2 ia ?t2
2

δN =

0

2

mi N

|e|2 ,

?e 3ωpe 2 + λ ?t 2 e

2

e=

ωpe eδN 2N

(8)

The ?rst of these equations is a driven wave equation for the density variation which for slow time variations, when the derivative with respect to time is neglected, just gives pressure balance between the ponderomotive pressure on the right and plasma pressure on the left, i.e. proportionality δN ? ?|e|2 . In other words, the density variation anti-correlates with the ?eld pressure, which corresponds to caviton formation. The second equation is a nonlinar Schr¨dinger equation for the evolution of the wave amplitude. o The Langmuir waves trapped in the cavitons must bounce back and forth, which naturally creates counter-streaming waves of equal intensity with opposite wave numbers. During collapse the cavities shrink in size, the wave numbers and momenta of the waves increase, and the wave energy density increases as well because of the shrinking volume. This yields both, the counter streaming Langmuir waves being localised in the same region and, in addition, a large radio emissivity. Unfortunately, we have already noted it, this process – as beautiful as it might be – has not been con?rmed experimentally, neither in the observations nor in the full particle simulations. Observed wave intensities are too low in the electron-foreshock boundary and electron foreshock, and the density variations did not indicate the presence of the expected cavities. Simulations, on the other hand support quasilinear evolution and wave scattering o? thermal ions. We note, however, that the most recent detection of the very strong electric ?elds in the shock ramp [5] might indicate that it is not the electron foreshock where one should expect caviton and collapse to work and cause the most intense radiation, rather it might be the very shock transition where shock radiation is generated by such processes. It is, in this respect, most interesting to remind of the strange radiative behaviour reproduced in Figure 24 that was detected by CLUSTER. We also note that similar observations had been made much earlier with the wave experiment on AMPTE IRM in the Earth foreshock (R. A. Treumann & J. LaBelle, unpublished 1986). The observed band splitting and high intensities might have been caused by Langmuir caviton collapse [153]. Other possibilities to produce counter streaming Langmuir waves are scattering of Langmuir waves o? thermal ions [84, investigated this process in full detail numerically including the ion polarisation cloud], a mechanism known since the early sixties. The process reads symbolically L + i → L + i? , where the primed quantities are after the collision, and the star on the ion indicates excitation of the ion as it is too heavy for changing momentum during the collision with the Langmuir wave. It is merely excited while the scattered Langmuir wave has changed direction and lost some of its momentum, i.e. attains a longer wave number and lower frequency. The same process does also work with ion-sound waves as L + IS → L . The scattered Langmuir waves then also change direction by absorbing the ion sound. Both processes have been used for radiation generation [165]. Radiation at higher frequency, e.g. radiation at the third plasma harmonic can be generated by a four-wave process. This is also favoured by caviton formation and collapse since the waves in this case are all con?ned to one and the same volume. However, other mechanisms have also been proposed. All these processes are of the kind of wave-wave interactions and thus their e?ciencies are proportional to the product of the involved relative wave intensities. Since the latter are usually low, the e?ciencies are very small as a rule. An attempt to increase the growth of Langmuir waves has in the recent past been the idea to consider a statistical theory of growth called ‘stochastic growth’ [60, 104]. This attempt takes advantage of the statistically distributed density ?uctuations in the foreshock region like in a random medium. Since the Langmuir-wave growth rate is proportional to δN/N , an average growth rate over the volume of occupation by the Langmuir wave can be calculated. This might be more realistic than using the linear growth rate. Regions of decreased density contribute strongly to the average Langmuir-wave amplitude. In this way an in the average larger emission e?ciency is obtained. Moreover, the averaging procedure introduces a statistical element which supports the incoherence of the relation between the detected Langmuir waves and radiation, which is in partial agreement with the observation. Generation of radiation at the fundamental ω ωpe in a three wave process requires the presence of a low frequency wave. Ion-acoustic waves are one possibility, other possibilities are lower-hybrid waves, Buneman waves, the modi?edtwo stream instability, various kinds of drift waves, and also electron acoustic waves or electron beam waves. In particular the latter are present in the foreshock region and thus can combine with Langmuir waves to generate fundamental radiation slightly above the plasma frequency.

33 The two exemptions when particles become involved are the above mentioned scattering of Langmuir waves o? ions, and the so-called electron-cyclotron maser instability [for a contemporary review see, e.g., 152]. Its advantage is that it operates directly on the free space mode avoiding any intermediate step like three-wave processes or particle scattering. However, it requires a particular form of the electron distribution with a velocity space gradient in the perpendicular direction ?Fe (v , v⊥ )/?v⊥ > 0, a hot electron distribution and low cold electron density. It is barely known whether such distributions are realised at the shock. However, if they are in some place, then the cyclotron maser instability will outrun all other mechanisms and directly feed the free-space electromagnetic radiation modes. Radiation will then be at a harmonic of the electron cyclotron frequency which is a severe restriction if the magnetic ?eld is low and the density high. Therefore, regions of low density and stronger converging magnetic ?elds are the best candidates for this radiation source.

III.

QUASI-PARALLEL SHOCK REFORMATION

In quasi-parallel supercritical shocks there is not such a stringent distinction between the region upstream of the shock and the shock itself like in quasi-perpendicular shocks. The foreshock, which we have discussed in some detail in the previous section, and the shock itself cannot be considered separately. This is due to the presence of the re?ected and di?use particle components in the foreshock. These, as we have seen are the source of a large number of waves. The interaction of these waves with the shock is one of the main issues in quasi-parallel shock physics. In the present section this will become clear when we will be dealing with the formation, behaviour and structure of quasi-parallel shocks as it has been inferred less from observation than from numerical simulations. The reason is that the real observations in space do not allow to separate the particles and waves from the shock. They all occur simultaneously and are interrelated and can never be observed in their initial state. The observations to which we will nevertheless occasionally refer will leave the impression of large-amplitude noisy ?uctuations. In simulations, on the other hand, it is at least to some degree possible to prepare the system in such a way that a single e?ect can be studied. For instance, in one-dimensional simulations the direction of wave propagation can be prescribed which allows studying just waves in one direction and their e?ect on the shock and particles. Moreover, treating the electrons as a neutralising Boltzmannian ?uid suppresses their e?ect on the ion motion and wave genereation. Treating them as an active ?uid allows taking account of electron-ion ?uid instabilities. Finally, full particle PIC simulations can be performed with low or realistic mass ratios in order to investigate di?erent time and spatial scale dependence and the excitation or coupling to higher frequency waves. Most simulations that have been performed in the past have taken advantage of these possibilities.

A.

Low-Mach number quasi-parallel shocks

It is usually assumed that low-Mach number shocks are stable, i.e. show no substantial time variation or reformation. It is not completely transparent why this should generally be so. Firstly, the critical Mach number has been shown by [54] to become small at narrow shock normal angles ΘBn → 0, in which case even low-Mach number quasi-parallel shocks should become supercritical and re?ect ions. Secondly, any fast ions of parallel velocity v > V1 that have been heated in the shock can in principle escape from the quasi-parallel shock upstream along the magnetic ?eld and should appear in the foreshock where they contribute to wave generation. Therefore, it makes sense to investigate the state of quasi-parallel shocks in view of their stability and wave generation even for low Mach numbers. In addition, any waves that are generated in the shock ramp or transition with upstream directed k and fast enough parallel phase or group velocities could also escape from the shock in upstream direction. This could, in particular, be possible just for low Mach number quasi-parallel shocks. To check this possibility [92] have performed one-dimensional hybrid simulations ?nding that initially the quasiparallel shock consisted of phase-standing dispersive (magnetosonic) whistler waves with the last whistler wave cycle constituting the shock ramp. As expected, the wave vectors of these phase-locked magnetosonic whistlers are aligned with the shock normal. At later times, backstreaming ions along the upstream magnetic ?eld excite a long-wavelength whistler wave packet upstream of the shock. In the one-dimensional simulation the wave vector is restricted to the shock normal while it is known from theory that the growth rate is largest along the magnetic ?eld. These oblique whistler waves should thus show up in two-dimensional simulation and may be visible at larger distance for su?ciently large upstream phase velocites. Such two-dimensional (hybrid) simulations with a non-inertial electron ?uid have been performed by [120] for a Mach number of MA = 2.2 and angles ΘBn = 20? , 30? , 45? and by [20] for an angle ΘBn = 30? and high Mach number MA = 5 (Figure 25). The lower Mach number simulations show the presence of a substantial number of backstreaming ions which cause an ion-ion instability in the upstream region. However, the excitation and properties

34

FIG. 25: Two-dimensional hybrid simulations of the evolution of upstream waves. Left column: Upstream wave in a low-Mach number
MA = 2.2 quasi-parallel shock [after 120]. The nominal shock is in the (y, z)-plane. Ions escaping to upstream generate the oblique upstream magnetosonic waves. The contour plot of the two normalised to the upstream magnetic ?eld components of the magnetic ?uctuations shown is taken at relatively early times ωci t = 68. It shows the nearly plane magnetic wave fronts inclined against the shock in direction x and having wavelengths of ? 10/ωpi in z while being much shorter in x. In the vicinity of the shock the wave fronts turn ? more parallel to the shock and produce a non-coplanar magnetic component |by | which is of same order as the |bz |. Moreover, even though the shock has relatively low Mach number, it is not completely stable but shows structure in z direction which is produced by the presence of the re?ected upstream particles and the upstream waves. At places it is impossible to identify one single shock ramp. Right column: Two-dimensional hybrid simulations of the evolution of giant magnetic pulsations (SLAMS) in front of a quasi-parallel supercritical shock [after 20]. Top: The simulation plane showing the structure of the (normalised) magnetic ?uctuation ?eld |b| at time tωci = 45 and shock normal angle ΘBn = 30? . The accumulation of the growing wave fronts at the shock transition, their increasing amplitudes, and their turning towards becoming parallel to the shock is clearly visible from the rotation of the two wave fronts and their k vectors shown in white. Away from the shock transition the angle between k and the shock normal n is large. Close to the shock the two vectors are about parallel. The magnetic ?eld is in the wave front, so ΘBn is close to 90? here. Bottom: Pulsation amplitude and ion phase space. The ?uctuations evolve into large amplitude pulsations when approaching (and making up) the shock. The strong retardation of the upstream ?ow by the pulsations is visible in the shock-normal velocity component (Mach number MA ). In the hybrid simulations this slowing done is accompanied by some ion heating.

of the waves depend strongly on the shock normal angle. Initially, as in the case of [92] phase-locked short-wavelength whistlers appear which are replaced at later times by upstream long-wavelength whistlers with phase velocity directed and amplitude growing towards the shock ramp but upstream directed group velocity, i.e. the shock radiates energy away towards upstream, as one would naively expect, because the shock being supercritical must reject the excess in?ow of energy which it can do by both, re?ecting particles and emitting waves into upstream direction. These waves are excited by the backstreaming ion component in a strongly nonlinear interaction process because of the evolving steep ion-density gradient, which is of the same scale as the whistler wavelength. The k-vector turns away from the magnetic ?eld having comparable components parallel to B and parallel to the shock normal n. For small ΘBn a remnant of the initial phase-locked whistlers survives but disappears at ΘBn = 30? . Close to the shock, where the backstreaming ion density is high, the waves have short wavelengths, and k is almost parallel to n. In the high Mach number simulations no shock is produced but instead re?ected ions were arti?cially injected with same Mach number as the incoming ?ow but with much higher temperature vi = 14.1 VA , forming a spatially uniform ion beam. The intention was to investigate the e?ect of the hot re?ected ions. This is shown on the right in the above ?gure. The result resembles the former one where a shock was generated by re?ection at a wall, but the e?ect in the injected beam case is stronger because of the higher Mach number. Hence it is the hot re?ected ion component that is responsible for the wave dynamics and the shock dynamics. All this can be seen from the two-dimensional intensity contours of these waves in the foreshock, which are plotted at a relatively early time in the shock evolution in the simulations in Figure 25. On the right in this ?gure the geometry

35

FIG. 26: The evolution of the shock normal angle ΘBn on distance from the shock in two-dimensional hybrid simulations for two initial quasi-parallel shock-normal angles ΘBn0 = 2? and ΘBn0 = 20? , respectively [simulation results taken from 120]. The horizontal line at 45? is the division between quasi-perpendicular and quasi-parallel shock normal angles. In both cases thetabn evolves from quasi-parallel direction into quasi-perpendicular direction. The shaded areas identify the quasi-perpendicular domains.

is given, with the magnetic ?eld ?uctuation vector b in the (y, z)-plane. The bottom panel on the left shows contours of the bz ?uctuations in the (x, z)-plane. The nominal shock ramp is at x ≈ 145 λi ion inertial lengths λi = c/ωpi at this time. The upper panel shows the non-coplanar component by -contours in the same representation. Behind the shock the ?uctuations are irregular and disorganised. However, in front of the shock a clear wave structure is visible with strongly inclined wave fronts and of roughly ? 10 λi wavelengths in z parallel to the nominal shock surface. The wavelength in x is about three times as short. These waves are seen in both components, bz , by , are low amplitude at large distance from the shock but reach very large amplitudes simultaneously in both components during shock approach while, at the same time, bending and assuming structure in z-direction that is di?erent from the regular elongated shape at large distance. This deformation of wave front may be due to the residual whistlers near to the shock, but it implies that the shock has structure on the surface in both directions x and z and is not anymore as planar as was initially assumed. The shock becomes locally curved on the scale of the shock-tangential wavelength. The waves deform the shock and, in addition, being themselves of same amplitude as the shock ramp, become increasingly indistinguishable from the shock itself. The shock is, so to say, the last of the large-amplitude magnetic wave pulsations in downstream direction, and the shock-magnetic ?eld is not anymore coplanar, because the waves have contributed a substantial component by that points out of the coplanarity plane. We have emphasised this phrase, because it expresses the importance of the low-frequency upstream magnetic waves in quasi-parallel shock physics. Contrary to quasi-perpendicular shocks where the re?ected gyrating ions in combination with the re?ected-ion excited modi?ed-two stream instability were responsible for the shock dynamics and di?erent kinds and phases of shock reformation, quasi-parallel shock reformation and much of its physics is predominantly due to the presence of large-amplitude and spatially distinct upstream waves. These are the generators of the shock and, due to their presence, the shock changes its character. It is highly variable in time and position along the shock surface and is – close to the shock transition on a smaller scale – ‘less quasi-parallel’ (or more perpendicular, i.e. the shock-normal angle ΘBn has increased on the scale of the upstream waves). The latter is due to the out-

36 of coplanarity-plane component of the upstream waves. In spite of concluding this from a hybrid simulation, this conclusion remains basically valid also in full particle simulations. It had been suggested already earlier on the basis of ISEE 1& 2 observation of the evolution of the upstream ultra-low frequency wave component [43]. The gradual evolution of the shock normal angle ΘBn has been demonstrated in other hybrid simulations by [120] and [20] who investigated the evolution of the shock normal angle in dependence on distance from the shock. This is shown in Figure 26 for two-dimensional hybrid simulations with initial shock-normal angles ΘBn0 = 2? and ΘBn0 = 20? , respectively, which we anticipate here. In both cases ΘBn evolves from quasi-parallel to quasi-perpendicular angles. Qualitatively there is little di?erence between the two cases. At the shock ramp ΘBn is deep in the domain of quasiperpendicular shocks. The only di?erence is that for the nearly parallel case the angle jumps to quasi-perpendicular quite suddenly, just before approaching the shock ramp, while the evolution is more gradual for the larger initial ΘBn . In both cases the evolution is not smooth, however, which is due to the presence of large-amplitude foreshock waves. Transition to quasi-perpendicular occurs for the initially nearly parallel case at the nominal shock ramp while for the initially quasi-parallel case it occurs at an upstream distance of about 100λi from the shock. One notices that this transition is on the ion scale, implying that in the region close to the shock the ions experience the shock occasionally (because of the large ?uctuations in ΘBn ) – and when ultimately arriving at the shock – as quasi-perpendicular. It is thus not clear, whether the electrons do also see a quasi-perpendicular shock, here. However, the ISEE measurements of the electron distribution function by [34] at the shock do not show a di?erence between quasi-perpendicular and quasi-parallel shocks. This fact suggests, in addition that, close to the shock transition, quasi-parallel shocks behave like quasi-perpendicular shocks as well on the electron scale, which is just what we have claimed. We ?nally note that the behaviour of the shock normal angle gives a rather clear identi?cation of the location of the shock transition in the quasi-parallel case, as indicated in Figure 26 by shading. Three distinctions can be noticed: ? ?rst, at larger initial shock-normal angles the transition to quasi-perpendicular angles occurs earlier, i.e. farther upstream than for nearly parallel shocks. This is due to the stronger e?ect of the large amplitude upstream waves in this case; ? second, at larger initial shock normal angles the quasi-perpendicular shock transition is considerably broader than for nearly parallel shocks, i.e. it extends farther downstream before the main quasi-parallel direction of the magnetic ?eld in the downstream region takes over again and dominates the direction of the magnetic ?eld: ? third, at an initial shock-normal angle of 20? , this region is roughly ? 150λi wide, implying that the magnetic ?eld direction behind a quasi-parallel shock remains to be quasi-perpendicular over quite a long downstream distance measured from the shock ramp. For the nearly parallel shock this volume is only about ? 50λi wide. This observation must have interesting implications for the physics downstream of quasi-parallel shocks. For instance, applied to the Earth’s bow shock, where λi ? 103 km. both distances correspond to regions wider than the order of > 5 RE which is larger than the nominal width of the magnetosheath! Thus, behind the bow shock a substantial part of the magnetosheath plasma should behave as if the bow shock would have been a completely quasi-perpendicular shock.
B. Turbulent reformation

When speaking about turbulent reformation we have in mind that a quasi-parallel supercritical shock is basically a transition from one lower entropy plasma state to another higher entropy plasma state that is madiated ba a substantially broad wave spectrum. Such a transition has been proposed by [126] based on the detection of the large amplitude magnetic pulsations (SLAMS) in the foreshock. Figure 27 on the left shows their model assuming that somewhere upstream in the foreshock magnetic pulsations have been excited which become convected downstream toward the shock by the convective ?ow, grow in amplitude and number and accumulate at the shock transition to give rise to a spatially and temporarily highly variable transition from upstream of the shock to downstream of the shock. An important clue in this argument was the observation that, ?rst, the pulsations grow in amplitude when approaching the shock and that, second, they slow down. this slowing down is e?ectively an increase in their upstream directed velocity on the plasma frame with growing amplitude such that their speed nearly compensates for the downstream convection of the ?ow. On the right of the ?gure, which is suggested by the observations of [57], a larger volume is seen. Here the pulsations are the result of growing ultra-low-frequency waves which are generated in a volume inside the foreshock but relatively close to the ion-foreshock boundary. These waves grow to large amplitudes until evolving into pulsation which the ?ow carries toward the shock. Growth, slowing down, and accumulation then lead to the pile up of the pulsations at the shock location and formation of the turbulent shock structure. Of the magnetic ?eld, in this ?gure on the

37

FIG. 27: The patchwork model of [126] of a quasi-parallel supercritical shock mentioned earlier. Left: Magnetic pulsations (SLAMS)
grow in the ion foreshock and are convected toward the shock where they accumulate, thereby causing formation of an irregular shock structure. Note also the slight turning of the magnetic ?eld into a direction to the shock normal that is more perpendicular, i.e. the magnetic ?eld is more parallel to the shock surface with the shock surface itself becoming very irregular [after 126]. Right: The same model with the pulsations being generated in the relatively broad ULF-wave-unstable region in greater proximity to the ion-foreshock boundary. When the ULF waves evolve to large amplitude and form localised structures these are convected toward the shock, grow, steepen, overlap, accumulate and lead to the build up of the irregular quasi-parallel shock structure which overlaps into the downstream direction.

right, we plotted only the shock-tangential upstream ?eld line. In the left part, several ?eld lines are schematically shown exhibiting the ?uctuations imposed by the background level of ultra-low-frequency ?uctuations. Moreover, a certain bending of the ?eld lines is included here in approaching the shock transition with the ?eld lines turning more perpendicular the closer they come to the shock. This bending is what we claim to be a parallel shock turning quasi-perpendicular at a scale very close to the shock. In this schematic drawing, however, there would be no reason for the magnetic ?eld to turn this way. What closer observations and simulations show is, however, that the turning of the ?eld is the result of the presence of the large amplitude magnetic pulsations. This will become clearer below.

Observations

Of course, the model shown in Figure 27 is a schematic model only which, however, has some merits in explaining the observations. The signature of a quasi-parallel shock in the magnetic ?eld is indeed quite di?erent from that of a quasi-perpendicular shock. We have already seen in the electric recordings reproduced in Figure 24 that the quasi-parallel shock appears in the electric wave spectrum as a broadband emission of highest spectral intensities at the low frequency end. The magnetic signature of a quasi-parallel shock is quite similar in that it lacks a clear location of the shock front. Rather one detects a broad region of very large amplitude compressive oscillations in magnetic magnitude and in the direction of the magnetic ?eld that subsequently is recognised as a passage across the quasi-parallel shock. An example is shown in Figure 28 as measured by the CLUSTER spacecraft. This ?gure shows eight hours of observation by CLUSTER in the immediate vicinity of the quasi-parallel shock. It is di?cult to say where in the ?gure the shock transition is located as the large ?uctuations in the magnetic ?eld magnitude and directions mask

38

FIG. 28: Eight hours of CLUSTER magnetic ?eld data during a long passage near and across the quasi-parallel supercritical (Alfv?nic e
Mach number MA ? 12 ? 13, ion inertial length λi ? 140 km) bow shock. The time resolution was 4 s. The top panel shows the variation in the magnitude of the magnetic ?eld. The two lower panels are the respective elevation and azimuthal angles θ, φ in a GSE coordinate frame [data taken from 71]. Large variation in the magnetic compression and direction can be seen to be associated with this quasi-parallel shock crossing. Buried in these large variations on this highly time-compressed scale are many magnetic pulsations (SLAMS). The compressive large amplitude ?uctuations in the upper panel are typical for a quasi-parallel shock transition.

the various back and forth passages across the shock that are contained in the data. Clearly, at the beginning near 1400 UT the spacecraft was in the downstream region. The ?uctuations show that during almost the entire sequence the magnetic ?eld is exhibits compressive ?uctuations. These belong to the shock transition. At the same time large ?uctuation in the direction of the magnetic ?eld are also observed. In the compressions of the magnetic ?eld buried are also upstream pulsations (SLAMS), and many of the changes in direction belong to the ultra-low-frequency waves present at and near the shock. The changes in direction indicate that the shock does not behave like a stationary ?at surface. Instead, it shows structure with highly ?uctuating local shock normal directions. [71] have checked this expectation by determining the local shock-normal angle ΘBn and comparing it with the prediction for ΘBn estimated from magnetic ?eld measurements by the ACE spacecraft which was located farther out in the upstream ?ow. The interesting result is that during the checked time-interval of passage of the quasi-parallel shock the prediction for the shock normal was around 20 ? 30? , as expected for quasi-parallel shocks. However, this value just set a lower bound on the actually measured shock normal angle. The measured ΘBn was highly ?uctuating around much larger values and, in addition, showed a tendency to be close to 90? . This is a very important observation. It strengthens the claim that quasi-parallel shocks are locally, on the small scale, very close to perpendicular shocks, a property that they borrow from the large magnetic waves by which they are surrounded. In fact, we may even claim that locally, on the small scale, quasi-parallel shocks are quasi-perpendicular since the majority of the local shock normal angles was > 45? . By small-scale a length scale comparable to a few times the ion inertial length or less is meant here. The data suggest that, indeed, the quasi-parallel shock is the result of a build-up from upstream waves which continuously reorganise and reform the shock. Figure 29 shows three representative examples of such upstream waves

39

FIG. 29: CLUSTER magnetic ?eld measurements of magnetic pulsation (SLAMS) near and remote from a quasi-parallel shock on
February 2, 2001 [data from 71]. Only the magnetic ?eld magnitude is shown for all four CLUSTER (colour coded) spacecraft. The spacecraft separation was between a few 100 km and 1000 km. Top: Clustered pulsations in the shock transition. Three events of large amplitudes are shown. These structures are very irregular with steep fronts. Note that in spite of the small spacecraft separation the shapes of the structures di?er strongly from spacecraft to spacecraft. Moreover, determination of the pulsation fronts and normals (not shown) indicates high variability over the spacecraft separation distance. Thus the structures are of relatively small scale and large amplitude. Middle: Isolated pulsation at greater distance from shock outside compression region. The structures are seen almost simultaneously at the spacecraft and thus must be of larger size. Copared to the embedded pulsations the amplitudes are lower, and the structures are more regular. Bottom: A shcklet observed outside the pulsation region in the domain of ultra-low-frequency waves. The steep shock-like front is well expressed with the qattached whistler waves it carries with it. Note the much lower amplitude than the pulsations.

which are far from being continuous wave trains. The upper panel is taken from the large density ?uctuation region in the shock transition. This region turns out to consist of many embedded magnetic pulsations (SLAMS) of very large amplitudes. In the present case amplitudes reach |b| ? 25 nT. These pulsations have steep ?anks and quite irregular shape, exhibit higher frequency oscillations probably propagating in the whistler mode while sitting on the feet or shoulders of the pulsations. It is most interesting that the di?erent CLUSTER spacecraft – at spacecraft separation < 1000 km – do not observe one coherent picture of a particular pulsation. This implies that the pulsations in the shock are of shorter scale than spacecraft separation: the di?erent spacecraft observe di?erent structures respectively di?erent pulsations. In addition, the magnetic ?eld directions (not shown in the ?gure) are very di?erent from spacecraft to spacecraft and thus from pulsation to pulsation, and even for one pulsation at its front edge and trailing edge di?erent magnetic ?eld directions are observed. The directions of the magnetic normal across a pulsation change on very short spatial scales. The quasi-parallel shock front has thus a rather irregular shape, which will be bent locally with changing direction of its normal. The second panel shows an isolated pulsation farther away from the shock transition. This pulsation exhibits a much more coherent way on the four di?erent CLUSTER spacecraft. It seems as if the pulsation is still in evolution as three of the spacecraft see a nearly coherent structure while the fourth which is farther away sees it in a di?erent state. Concluding from this event, isolated pulsations seem to have larger dimensions and lower amplitudes, which would be consistent with the assumed solitary properties of pulsations.

40

FIG. 30: Observation of electric ?eld structures in large magnetic pulsations (SLAMS) in the quasi-parallel shock transition region
[data taken from 7, 8]. Structures on three di?erent time scales are shown, corresponding also to three di?erent spatial scales. Top: CLUSTER passage across on (moderately large amplitude) magnetic pulsation in the shock transition. The (smoothed) magnetic ?eld structure is a slightly steepened magnetic bump. The stationary parallel electric potential ?eld across this structure shows a potential ramp with steep gradient at the leading edge of the pulsation. The potential drop of ? 400 V corresponds to an electric ?eld of ? 0.47, V m?1 . Note that the time scale in this panel is 90 s. Bottom left: Six seconds of a CLUSTER passage through the shock transition. The black dots show the spacecraft potential variation which maps the local density variation. Overlaid is the high frequency WHISPER trace of the plasma frequeny. In the magnetic pulsation regions (white) the plasma frequency exhibits huge excursions to both sides similar to those seen in the overview Figure 24 on wave observations. These excursions trace the BGK (nonsymmetric) modes and (symmetric) solitons. Bottom right: One example of one of the solitons on a 2 ms time scale. It is nicely seen how symmetric the parallel potential trough and the corresponding bipolar parallel electric ?eld shape look like in the solitary wave structure.

The third panel shows a shocklet, i.e. a structure which presumably has little in common with pulsations. It is embedded into long wavelength ultra-low-frequency wave trains, evolves into steep front and drives whistlers attached to this front across the ?ow. These waves were already observed by [110]. Their properties indeed resemble those of sub-critical little shocks which propagate against the ?ow, though with slower speed such that they e?ectively are slowly convected towards the shock. [7] report another interesting property of magnetic pulsations in the shock transition region where they overlap to form the quasi-parallel shock. Measurement of the electric cross-SLAMS potential identify a substantial unipolar drop in the electric potential of several 100 V, corresponding to a potential ramp, when passing from upstream to downstream across the pulsation. Such a drop signi?es the presence of an electric ?eld in one direction across the pulsation. Taking the mean size of a pulsation to be roughly 1000 km, the mean electric ?eld is E 400 mV m?1 . However, this ?eld drops mainly at the leading edge of the pulsation. Such a ?eld presumably corresponds to a steep pressure gradient in the pulsation. It could also be generated by an anomalous collision frequency. This remains to be tested by further observations and data analysis. The measurements of [7] anticipated the more recent report of strong electric ?elds in the shock by [5]. In addition to this observation it was found [8] that the single pulsations were subject to a fairly large number of high frequency/Debye scale structures in the electric ?eld seen in the WHISPER recordings (bottom panel), which belong to electron holes or solitons which form in the pulsation gradient regions as shown in Figure 30. The bipolar electric ?eld and unipolar potential across one – indeed very symmetric – soliton is seen in this ?gure. These observations suggest that the pulsations are indeed the main constituents of a quasi-parallel shock with the dynamics on the micro-scale of a quasi-parallel shock going on mainly in the single pulsations of which the shock transition is built. The occurrence of these intense nonlinear electrostatic electron plasma waves at the quasi-parallel shock transition is intriguing. It forces one to draw another very important main conclusion from these observations (and other related observations like those of [5]): that quasi-parallel shocks are sources of electron acceleration into electron beams, which

41 are capable to move upstream along the magnetic ?eld over a certain distance and excite electron plasma waves at intensity high enough to enter into the nonlinear regime, forming solitons and electron holes (BGK modes). Presumably this is possible only when in the supercritical quasi-parallel shock transition region the magnetic ?eld changes from quasi-parallel to quasi-perpendicular on the electron scale ? λe . Indications of such a change on the ion scale λi have been noted above at a number of occasions, but the detection of solitary structures in the electron plasma waves in relation to quasi-parallel shock transitions provides a very strong argument for this to be true on a scale which is well below the ion scale. Only if this is the case, there will be ample reason for electrons to become re?ected and accelerated into beams from the transition region in a quasi-parallel shock. As we already noted, we may, therefore, expect that quasi-parallel supercritical shocks on the electron scale are not anymore quasi-parallel but change to become locally quasi-perpendicular, while on the larger ion scale they still maintain properties of quasi-parallelity. If this conclusion will turn out to be true and will sustain future more sophisticated experimental tests, it will have important consequences for collisionless shock physics. Supercritical collisionless – nonrelativistic – shocks will, in fact on the electron scale, always behave quasi-perpendicularly – and it may be suspected that this conjecture will also hold for relativistic shocks though probably for other reasons (like the generation of transverse magnetic ?elds by the Weibel instability, which becomes dominant in relativistic shocks [see, e.g., 52, 53]). This implies also that the true quasi-parallel shock physics cannot be properly elucidated when ignoring electron e?ects as is, for instance, done in hybrid simulations.

Simulations of quasi-parallel shock reformation

Nevertheless, before turning to the very few more realistic full particle PIC simulations with di?erent mass ratios from small to about realistic, we are ?rst going to discuss here hybrid simulations in one and two dimensions of the formation and behaviour of quasi-parallel shocks. Initial particle [100, 101, 102] and hybrid simulations [10, 118, 119, 142, 160] did already illuminate some of the particular interrelations between the dynamics of quasi-parallel shocks: energy dissipation by short wavelength whistlers at the shock transition, the presence of a di?use hot ion component upstream of the shock, and the importance of upstream waves. Most of these have been the subject of reviews [the interested reader might be directed to the papers by 11, 13, 68, which more or less systematically discuss selected aspects of these interactions]. The selection of results we will provide here is guided by the progress that has been achieved in the understanding of the quasi-parallel shock physics and in its relation to the observations. The ?rst arising question is, whether nearly parallel shocks are at all capable of re?ecting ions. This question is neither nonsensical nor academic. It makes sense, because the investigation of the critical Mach number by [54] becomes unreliable at small angles, say ΘBn 30? , because it is based on linear whistler dispersion and does not take into account the completely modi?ed plasma conditions near the shock ramp. Thus the simple conclusion that the critical Mach number approaches Mcrit → 1 for ΘBn → 0 might be an unjusti?ed extrapolation. Theoretically, the re?ection of particles at small ΘBn must become entirely due to the electrostatic shock potential drop with the magnetic part of the Lorentz force being obsolete. The potential drop, however, should decrease with M, and thus the re?ection of particles should cease, while being of vital importance for the generation of a collisionless quasi-parallel shock. Hence, the question is also not purely academic. Of course, we do already know from the observations that quasi-parallel shocks exist at small ΘBn with their foreshocks being populated by a di?use ion component that excites upstream waves and mediates the beam-generated upstream ion-foreshock boundary waves. The impossibility for this di?use component of being entirely due to shock re?ection in the quasi-perpendicular part of the shock, immediately proves that the quasi-parallel (or even the nearly parallel shock) must be able to re?ect particles upstream. Hence, either a quasi-parallel shock is capable of generating a large cross-shock potential, or it is capable of stochastically – or nearly stochastically – scattering ions in the shock transition region in pitch angle and energy in such a way that part of the incoming ion distribution can escape upstream, or – on a scale that a?ects the ion motion – a quasi-parallel shock close to the shock transition becomes su?ciently quasi-perpendicular that ions are re?ected in the same way as if they encountered a quasi-perpendicular shock. Observations suggest that the latter is the case, while observations also suggest that large potential drops occur in the large-amplitude magnetic pulsations (SLAMS) where they accumulate in the shock ramp [8]. Hence, re?ection of ions will be due to the combination of both e?ects, the electric potential drop and the magnetic de?ection. In fact, this can be a quite complicated process for an ion passing across a number of magnetic pulsations, in each of which it is being retarded and at the same time de?ected by a small angle until its normal velocity component is decreased su?ciently that a further de?ection in pitch angle su?ces to let it return into the upstream region. h. Hybrid simulations in 1D. In agreement with what is known today, ?rst one-dimensional hybrid simulations in an extended simulation box [10] suggested that the reformation of quasi-parallel shocks is about cyclic and is

42

FIG. 31: Fast mode dispersion relation in simulations in the shock frame. Left: The Doppler shifted fast mode dispersion relation in a
supercritical ?ow in the shock frame. The Dispersion relation assumes negative frequencies corresponding to the downstream convection by the ?ow of Mach number MA = 2.8 and at ΘBn = 30? . Waves at zero group velocity have energy at rest in the shock frame. Negative group velocities imply downstream transport of energy, positive group velocities imply upstream transport. Right: Simulated upstream wave dispersion spectra near the shock and upstream of the shock. Near the shock wave energy accumulated around standing and downstream transport. Away from the shock the wave energy still moves upstream [data from 61].

caused by the impact of large-amplitude upstream waves. [119], using ΘBn = 20? and MA = 3.5 with an upstream ion thermal velocity vi = vA (corresponding to βi = 1) in one-dimensional hybrid simulations (with small numerical resistivity) showed that the re?ected ions are not coming from the core of the incident upstream ion distribution but originate in the shell of this distribution, having initial velocities v 1.7vi . These ions escape from the shock quite far upstream and excite ultra-low frequency waves with upstream directed velocity of ? 1.3VA at distances up to > 300λi , which are convected downstream to reach the shock. In this one-dimensional hybrid simulation the only mode in which they can propagate is the compressive fast magnetosonic mode. These waves are in fact what in observations has been identi?ed as pulsations (SLAMS) but is not yet recognised as such here. During downstream convection the waves grow and slow down in the interaction with the foreshock ion component. When approaching the shock they generate a large amount of new re?ected ions. These slow the incident ion population down and steepen the wave crest, which becomes the new shock front. In the time between the arrival of the compressive waves the shock is about stationary and develops phase-locked upstream whistlers which the arriving next wave crest destroys. From these simulations it could not be concluded what process produced the re?ected ions, however, as one-dimensional simulations among su?ering from other de?ciencies select only one particular direction of wave numbers and are thus not general enough for drawing ?nal conclusions. The nature, generation and e?ects of the large-amplitude upstream waves have been further investigated in more detail in one-dimensional [61, 117, 118, among others], and in two-dimensional hybrid simulations [20, 62, 117]. Since shocks are three-dimensional, it is clear that two-dimensional numerical simulations at same resolution come closer to reality. However, they su?er from restrictions in size of the simulation box and simulation time. Since reality does not confront us with an initial state, large boxes and long times are needed. However, for the investigation of particular questions, simulations have the great advantage of providing the possibility to prepare them for answering just those questions. Concerning the propagation of upstream waves in one direction with respect to the shock normal, one-dimensional simulations are just good enough. In order to identify a particular wave mode, the dispersion of the wave must be investigated. This dispersion relation depends on the frame in which it is taken, because the energy/frequency of a wave is not invariant with respect to coordinate transformations; in a medium moving with velocity V it is Doppler shifted according to ω = ω(k) ? k · V, where ω(k) is the dispersion relation in the rest frame of the ?ow. While the Doppler shift at high frequency is negligible, it completely changes the dispersion of ultra-low frequency waves at large Mach numbers. Figure 31 on its left shows the deformation of the fast mode dispersion relation in the shock frame at large Mach number M > 1 and for waves propagating upstream in the plasma frame. The deformation causes negative frequencies of the waves which imply downstream directed phase velocities, which is nothing else but the intuitive downstream convection of the waves by the ?ow. However, the minimum in the dispersion relation implies that waves of a particular frequency and wave number have zero group velocities. In the shock frame the energy of these waves is stationary. Smaller wave numbers have energy moving downstream, larger wave number have energy moving upstream away from the shock. The right part of the ?gure shows simulations of upstream waves according to one-dimensional hybrid simulations by [61] for ΘBn = 30? and a Mach number MA = 2.8. The entire dispersion of the simulated waves is negative. The

43 waves are all convected toward the shock as their Mach number is less than the streaming Mach number. Near the shock most of the wave energy moves downstream and will cross the shock. Still some shorter wavelength waves (large k) move in energy upstream in the shock frame. Farther away from the shock most of the wave energy encountered is seen to move upstream. [117] investigated these waves further in one-dimensional hybrid simulations performing several numerical experiments on them, taking away the shock and instead injecting a di?use ion component from downstream. The main ?nding is that the large amplitude upstream magnetic pulsations (SLAMS) evolve out of the ultra-low frequency wave spectrum in the interaction with the di?use ion component. In accord with observation the pulsations move upstream in the plasma frame. Thereby their upstream leading edge steepens and is right-hand circularly polarised like required for whistlers. However, dispersion is unimportant; the main cause of the evolution of large pulsations is nonlinearity when the wave interacts with the di?use ion distribution. This distribution has a steeper shock directed density gradient than the pulsation wavelength. Moreover, the ?ow becomes decelerated at the leading edge of the pulsation (as is seen in Figure 25), and here the velocity di?erence between the ?ow and the di?use ion component drastically decreases, which shifts the k vector of the resonant wave to larger values, and the wavelength decreases during convection of the pulsation toward the shock. (Note that no resonant ion beam-whistler interaction exists as the beam is hot and di?use.) The standing whistlers are in the leading edge are simply generated by the current ?owing in the steep edge. It is thus concluded that it is the gradient in the hot di?use ion component over a length of the same order as the length of the wavelength which produces the pulsations. Ultimately these pulsation cause a quasi-periodic reformation of the shock, as we have described earlier. This is thus proved by one-dimensional hybrid simulations. The same result is obtained when the simulation starts right away without a shock but with an injected hot beam (which is no surprise as the generation of the shock, before it was removed in the former simulations in order to keep with the wave ?eld, was due to the plasma ?ow-re?ected ion beam interaction). i. Hybrid simulations in 2D. The two-dimensional evolution of the pulsation (SLAMS) was studied later by [120] and [20] (see Figure 25). It basically con?rmed the conclusions drawn from one-dimensional simulations with the three important modi?cations, ?rst, the wave fronts of the pulsations (SLAMS) rotate into a direction that is more parallel to the shock thereby increasing the shock-normal angle locally to become quasi-perpendicular; second, the pulsations have short wavelength in shock normal direction, but are of substantially longer but ?nite lengths in the direction tangential to the shock, which provides structure to the shock in tangential direction; and third, shock reformation is a result due to the steepening and accumulation of the pulsations and is a quasi-periodic process but the downstream structure of the shock over some distance is caused by the downstream convection of the old shock front, i.e. the bulk of the pulsations that had accumulated at the location of the former shock transition. It is interesting to note that from reformation cycle to the next rather large ?uctuations in the magnetic ?eld and density exist in the transition from upstream to downstream which may be capable of trapping particles. The two-dimensional simulations do also con?rm the conclusions that the di?use upstream ion component is responsible for the growth of the pulsations (SLAMS). The simulations by [20], in particular, followed the same scheme as the one-dimensional simulations, injecting hot di?use ions into upstream in order to control the interaction between the di?use ion component and large amplitude pulsations (SLAMS). j. Full particle PIC simulations. So far we dealt just with hybrid simulations where the ions are macro-particles while the electrons represent a charge-neutralising background of zero mass. Clearly, such simulations are unrealistic if whistlers become involved. This is, however, the case, as we have discussed above, whenever large amplitude pulsations evolve at the leading edge of which phase-locked whistlers are attached. The question what role the electrons play in the evolution of the pulsations can only be answered by full particle simulations. These require large simulation boxes and at the same time high temporal and spatial resolutions. So far they could therefore only be performed in one dimension [94, 122, 155]. [94] used a mass ratio of mi /me = 100. They had to small a box (just ? 30λi in the upstream direction) for following the evolution of upstream waves but stressed the importance of whistlers in shock reformation. One decade later it became possible to substantially enlarge the box and at the same time to switch to a larger mass ratio while staying with one dimension only. [122], using the same mass ratio at ΘBn = 30? and MA 4.7, had an upstream entension of ? 200λi and could follow the shock evolution for a time tωci ? 100. Similar simulations with mass ratio mi /me = 50 have been performed by [155]. Here we discuss in greater detail the calculations of [122]. An overview of their results is shown in Figure 32 in the ?xed lab-frame for the main magnetic ?eld component Bz , electric shock potential Φ, bulk stream velocity along the shock normal V , and density N , all as functions of distance x (measured in electron inertial lengths λe ). Far upstream from the quasi-parallel shock the magnetic ?eld exhibits long wavelength ultra-low frequency waves (4) which, when approaching the shock, start steepening at their leading edges (note that in the plasma frame these waves are moving upstream, as also does the shock, i.e. to the left in the ?gure). The amplitude of the wave increases (3) , and the wave becomes a pulsation (SLAMS). Its amplitude is large enough to already substantially brake the upstream bulk ?ow, which causes a drop in V and in Φ, and an increase in density. The pulsation has slowed down the ?ow to a velocity ? 2.8VA already here. In fact the leading edge of the pulsation behaves like a ‘baby shock’,

44

FIG. 32: Full particle PIC simulations of the evolution of a quasi-parallel shock in one dimension only [after 122]. From top: main magnetic ?eld component Bz , electric potential Φ, bulk plasma ?ow velocity V , density N , all in simulation units. The numbers indicate three pulsations (SLAMS). Pulsation 1 was the old shock. Pulsation 2 is the actual shock coinciding with the drop in V to zero and the steep increase in density and potential. Pulsation 3 is just evolving. It already has a steep leading upstream edge and decelerates the plasma ?ow. It will become the next shock ramp. Number 4 indicates a bump in the upstream waves that will become a pulsation. The actual shock ramp has some phase-locked whistlers attached to it which are not well resolved on the scale shown (see the next ?gure).

which later will become the real ‘adult shock’. The shock itself is formed further downstream at the local position of the leading steep edge of the previous pulsation (2). In front of this edge (i.e. the genuine shock at this instant and location) between it and the trailing edge of pulsation (3) a standing phase-locked large amplitude whistler has evolved. This whistler is spatially damped by the approaching pulsation (3). An indication of such a whistler has already been seen in front of the leading edge of pulsation (3) as well. Pulsation (2) (the instantaneous shock) has a substantial downstream extension. Further downstream of it the ‘old shock’ is seen, which was formed at an earlier time by pulsation (1); and even farther downstream a remainder of earlier shock ramps (pulsations) is recognised in the trace of Bz . The instantaneous shock ramp (pulsation 2) is high enough to completely brake the upstream ?ow, the velocity of which drops to zero while the density steeply increases and forms a dense wall. Figure 33 gives and impression of the shock evolution in higher temporal and spatial resolution. It shows instantaneous magnetic pro?les in a box of lengths 400le including the shock ramp with time axis running upward. In this simulation frame the shock moves upstream to the left. The heavy step-like line indicates the approximate location of the nominal shock ramp. Also shown is an upstream ultra-low frequency wave that at time tωci = 82.5 has begun to evolve into a pulsation while convectively approaching the shock, growing in amplitude and developing a steep leading (upstream) ramp in front of which phase-locked magnetosonic whistlers start growing. When approaching the shock at tωci = 92.5, the pulsation kills the phase-locked whistlers that were waiting in front of the ramp by damping

45

FIG. 33: Full particle PIC simulations of the evolution of a quasi-parallel shock in one dimension [data taken from 122]. The pro?le of
the main magnetic component Bz is shown for subsequent simulation times shifted by ? tωci = 2.5 upward. The representation is in the simulation frame, i.e. the shock moves to the left into the upstream direction. The simulation shows the reformation of the quasi-parallel shock resulting from the exchange of the shock with an incoming upstream wave which has steepened to become a pulsation (SLAMS). The magnetic ?eld trace at time tωci = 100 has been overlaid on the ion phase space at this time. The heavy steps show the location of the nominal shock ramp (where the ?ow is stopped). It moves slowly upstream until a new pulsation arrives and when it suddenly jumps forward by roughly 100λe . Also shown is the fast approach of an upstream pulsation starting at tωci = 82.5 and arriving at the shock at tωci = 92.5 to take over the role of the shock. Note that in the minima of the ‘whistler’ ?eld ?uctuations (at tωci = 100) ions are trapped, oscillating back and forth and forming hole vortices in phase space centred around local minima of the electric potential Φ (not shown). The little boxes indicate where particle (ion or electron) phase space distributions have been determined.

them out. At this time the pulsation takes over the role of the shock, and the nominal shock position jumps ahead to upstream by roughly a distance of the width of the pulsation ? 100λe . This process repeats itself at tωci = 102.5 showing that the quasi-parallel on this time scale shock is not stationary but undergoes nearly periodic reformation which is mediated by the

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