WHAT DO WE REALLY KNOW ABOUT COLLISIONLESS SHOCKS?
N. A. Krall
Krall Associates, 1070 America Way, Del Mar, CA 92014, USA, E-mail: Nkrall@aol.com
The things that we really know about collisionless shocks are summarized; examples are given of measurements which confirm the theoretical foundations. Typical expected and observed phenomena is described. Some shock theory which is apparently not incorporated into current analysis is also presented.
This paper is organized strictly along the lines of the title, namely, what do we really know about collisionless shocks. Thus we will emphasize the basic theoretical ideas underlying this phenomenon (Tidman and Krall, 1971; C. T. Russell, 1995), as well as the unambiguous experimental measurements related to shocks. We will give little discussion of the cutting edge ideas for explaining some of the observations; perhaps these ideas will soon be part of "what we really know," but perhaps not. Finally, we will note that there are things we really do know about collisionless shocks that do not seem to be incorporated into current analysis, and give as an example the ballistic transport of turbulence out of a shock by particles which pass through the shock layer (Krall and Tidman, 1969a and 1969b).
The shock structures discussed are those that occur in hot low-density fully ionized plasmas (Krall and Trivelpiece, 1973), in which collision times are long compared to transit times through the shock structure. We define a shock in a very broad sense as a transition layer which causes a change in the state of the plasma, and which is stationary in time in some reference frame. In laboratory plasmas, the transition layer generally propagates through the plasma, changing the plasma state as it flows (Iskul'dskii et al., 1965; Stamper and DeSilva, 1969; Paul et al., 1965; Davis et al., 1971; Chin-Fatt and Griem, 1970); in the bow shock of the earth and planets, the transition layer is stationary in the planet rest frame, and a flowing plasma (the solar wind) passes through this layer, reducing its flow and increasing its temperature (Formisano, 1977).
In collision dominated gases, the transition across a shock occurs in a distance of the order of a few collision mean free paths, and the shock structure can be determined relatively simply, by taking into account the collisional dissipation coefficients. In a collision-free plasma the mechanisms by which the plasma state is changed by passage through the shock layer are far more complex. Energy and momentum can be transferred from plasma particles into electric and magnetic field oscillations by instabilities, and these collective motions must be taken into account when conservation laws are applied to relate the preshocked state to the post shocked state. Similarly, the ions and electrons are affected differently by instabilities, so there is no reason for them to have the same temperature, or even that the temperature be isotropic. This alters the jump conditions across the shock, and adds to the richness and variety of possible shock wave phenomena (Sagdeev, 1966). At the larger Mach numbers allowed by the turbulence, thermal spread in the plasma flow allows particle reflection, which provides an additional dissipation mechanism. This emphasizes the need for a selfconsistent approach in developing a shock theory (Berdotti et al., 1966).
2. COLLISIONLESS SHOCKS ARE BASED ON LINEAR PLASMA WAVES
Nonlinear waves which have a constant profile in their wave frame are a first step towards construction of a shock wave model. It is generally possible to relate the nonlinear waves to the more familiar linear ones, though the more highly nonlinear modes of the plasma may bear little resemblance to the linear modes.
A heuristic feeling for what should be expected in the way of nonlinear waves can be obtained by calculating the evolution of an initially specified disturbance in an infinite plasma. The plasma fluid equations can be solved by expanding the field amplitudes F = F1 + F2 + ... leading to a set of equations
D F1 = 0
In Eq. (1) D is an operator which acts on ei(t - kr) to give the dispersion relation for linear plasma waves D(k,) = 0, whose roots give the plasma wave spectrum of frequency and wave number k. Figure 1 shows two types of plasma waves. Both waves have the property that they are dispersionless (/k = constant) in some region and become dispersive at shorter wavelengths. Consider a wavepacket in the dispersion-free region of the curve. All waves in this packet propagate at the same velocity. The next order equations have a driving term Q(F1F1), which produces components 1 + 2, k1 + k2. Since these components lie on the curve D = 0, the coupling is resonant, leading to a rapid spreading of the wave packet to higher wavenumbers. This represents a steepening of the wave. The steepening stops when the shorter wavelengths no longer lie on the D = 0 curve. So the existence of constant profile waves is a consequence of both nonlinearity and dispersion in the plasma equations.
Fig. 1. Typical dispersion curves for sound-like waves in plasma.
As can be seen in Figure 1, the steepening can stop because higher frequency waves go either slower (b) or faster (a) than the waves in the constant speed frequency region. In case (b) the higher frequency waves will fall behind the main pulse, and damp out, while in case (a) the higher frequency waves outrun the main pulse. This produces the two possibilities for nonlinear wave structure shown in Figure 1. These nonlinear extensions of linear wave theory are the basis of shock structure. Mathematically, nonlinear solutions are found in the plasma wave equations, both of isolated waves, called solitons, and of wave trains. The solitons typically have a velocity equal to that of a specific dispersionless plasma wave, and a width equal to the wavelength at which that wave becomes dispersive. The dotted line in Figure 2 shows a soliton. In the absence of dissipation, solitons propagate through but do not interact with a background plasma. When dissipation is present, the solitons interact as they propagate, so that plasma which passes through a soliton changes state, shown by the solid line in Figure 2. In this way solitons become the basis of collisionless shocks.
Fig. 2. Variation of the magnetic field through a magnetosonic shock and soliton.
Some measure of the occurrence of turbulence in a plasma shock transition (at least in principal) can be obtained by determining the distribution functions or their moments ahead and behind the shock. The distributions up and downstream can be non-Maxwellian if the time scale of the shock front is shorter than the two-body collision time, so that the usual three conservation laws from mass, momemtum, and energy flow through the shock leave the distributions undetermined. For collisionless shocks an infinite number of conservation equations in fact exist. For example, dv v<f>2 is constant across a laminar shock but suffers a jump in a turbulent shock. Similarly, the average energy flux, for each species, dv v3<f> - (2e/m) dv v<f>, which is constant across a laminar pulse, changes across a turbulent pulse, reflecting the diversion of energy into fluctuations or "heating."
3. COLLISIONLESS SHOCK WAVES CAN BE CLASSIFIED BY WAVE TYPE AND BY DISSIPATION STRENGTH
Because of their basis on linear waves, collisionless shocks can be classified by their geometry, since the geometry often determines the wave properties. Thus shocks are classified by the angle the plasma flow makes with the magnetic field. This leads to perpendicular, parallel, and oblique shocks, as shown in Figure 3. Perpendicular shocks are based on magnetosonic waves, shown in Figure 4, and parallel shocks are based on whistler waves or ion waves, shown in Figure 5. When the plasma becomes so hot that the magnetic field can be ignored, both these waves degenerate into ion acoustic waves, = kcs, where cs is the ion sound speed = (Te/Mi)1/2. At low Mach number, the parallel shock is based on the lower frequency Alfven waves, while at high M it is based on whistler waves. The oblique shocks are more complicated, because there are many distinct types of wave structure (see Tidman and Krall, 1971, p. 88), and shock structures can be based on many of these branches.
Fig. 3. Geometry of perpendicular, parallel, and oblique shocks.
Fig. 4. Dispersion curves for the magnetosonic mode in a zero-temperature plasma.
Fig. 5. Dispersion curves for the whistler and ion modes in a zero-temperature plasma.
In addition to wave type, shocks are classified according to the level
of turbulence which causes the dissipation. The turbulence itself of course
depends on the shock structure, since turbulence is driven by the nonequilibrium
features in the transition layer. For example, currents in the shock front
required by the jump in B-field,
x B = 4j, can drive instabilities even in weak shocks, while counterstreaming ions characteristic of strong shocks which reflect some plasma can drive ion beam modes unstable. In general therefore there is a correlation between Mach number and strength and type of turbulence. The strength of turbulence is measured by the size of <nE>, a dissipation coefficient which arises from the quasilinear treatment of plasma turbulence. The average distribution function of plasma particle velocities, f(v), evolves according to
where the terms E and f are the fluctuations from linear instability. These average out to zero individually, but their products do not vanish. Taking velocity moments gives the fluid equations, with <En> determining the turbulent drag rate dV/dt due to turbulence.
With this preamble, we observe that low Mach number shocks are classified as laminar, with smooth profiles of plasma parameters and fields, and a low level of <En> providing dissipation but not disrupting the profiles. At higher Mach number, the increasing level of <En> broadens the B and E field profiles, but not necessarily the plasma profiles. This means that some plasma can be reflected from the shock front, leading to counterstreaming plasma, and trapped orbits. This gives an ordered shock structure, which may not be smooth. This type of shock is termed quasilaminar. Finally, at high Mach number, irregular and possibly oscillating shock structures form, which may show little evidence of the wave properties on which they are based. These are called turbulent shocks. Figure 6 is a sketch of these classifications of shock wave.
Fig. 6. Sketch of the variation of fields through laminar or turbulent shock transitions.
4. WE KNOW THAT COLLISIONLESS SHOCKS ARE UBIQUITOUS
Collisionless shocks have been produced in many laboratory experiments (Iskul'dskii et al., 1965; Stamper and DeSilva, 1969; Paul et al., 1965; Davis et al., 1971; Chin-Fatt and Griem, 1970), where they have been a test bed for theories of plasma turbulence (Davidson and Krall, 1977). However, it is in naturally occurring plasmas that the greatest variety and parameter range for these structures is found. Whereas in laboratory experiments parameters vary by factors of 2, shock waves in space have parameters varying over several orders of magnitude, even in the same location at different times. Examples of naturally occurring collisionless shocks include the shocks produced when the solar wind interacts with planetary objects (Formisano, 1977; Spreiter and Stahara, 1995), as well as the shocks produced in interplanetary situations where plasma streams near the sun are either compressed or are superimposed on coronal ejections (Luhmann, 1995). Other examples in the literature include shock waves inferred in supernova remnants (Drury, 1995), in the rotating plasma near pulsars e.g., the Crab Nebula (Rose et al., 1974), in the solar corona associated with radio bursts (Mann and Claben, 1995), and in the plasma environment of comets (Coates, 1995).
4.1 Laboratory-produced Shock Waves
The most frequent laboratory example of the collisionless shock is the -pinch experiment (Iskul'dskii et al., 1965; Stamper and DeSilva, 1969; Paul et al., 1965; Davis et al., 1971; Chin-Fatt and Griem, 1970). In this experiment a cylinder of plasma confined by a magnetic field Bo is subjected to a pulsed B field at its boundary. The pulsed field creates a magnetic disturbance which propagates across the magnetic field, at Alfven-like speeds. A magnetic shock often forms. Figure 7 shows a plot of a magnetic profile at one instant of time in an experiment. The case shown had B antiparallel to Bo. The figure lists some of the plasma processes going on during the magnetic field implosion. These have been confirmed by numerical simulation and in some cases directly by measurement. Results of the experiments included emission of plasma radiation at p and 2p, ion acceleration of V = 2 Vshock, generation of other plasma waves, electron heating to 1-10 keV, and generation of ion waves.
Many of these features are directly reflected in satellite measurements of collisionless shocks in space plasma.
Fig. 7. Magnetic profile during magnetic implosion in a -pinch experiment.
4.2 Solar Wind-produced Shock Waves
The solar wind is a tenuous plasma with a frozen-in magnetic field, which flows radially out of the solar corona, and is composed of electrons, ions, alpha particles, and occasionally other heavy ions. The low density (few cm-3) and huge space scale (of order 100's of km) are obviously vastly different than laboratory experience, but dimensionless constants such as nT/B2 are similar to experiments, as are dimensional quantities such as Alfven speed, plasma temperature and Mach number. In contrast to experiments, however, these parameters may change by factors of 100 or more over a period of months. Figure 8 shows some of the phenomena observed in the bow shock. Clearly, much of the phenomena observed in the laboratory is echoed in the bow shock. In particular, the shock produces a number of significant effects upstream, including the following:
Energetic electrons stream away from the shock along B-field lines; electron properties are functions of the local B-geomery, fe (B), acceleration takes place in the shock front, and electrons propagate along B.
Ion beams and hot ion populations are produced in the shock layer and propagate upstream; energetic ions are produced by acceleration in the shock, or by wave-particle interaction upstream.
Plasma waves propagate upstream; these waves are electron plasma oscillations near the tangent field line, and ion sound waves at greater depth.
Fig. 8. The earth's bow shock. The magnetic field structure and nearby regions are not to scale.
Clearly the dependence on B reflects the change in character of the shock as the geometry goes from perpendicular to oblique to parallel. A wide spectrum of waves are generated from whistlers at 1 Hz, to ion waves at a fraction of a Hz, and at frequencies in between. Some of these are attributed to waves born in the shock, but some may be emitted by plasma reflected from the shock front and streaming through the incoming solar wind. Counterstreaming particles are well known to be a source of plasma waves (Papadopoulos, 1973; Tidman and Northrup, 1968), as is the turbulent shock itself.
In addition to the earth's bow shock, the magnetic field of other planets is also observed to stand off the solar wind, forming a shock front. Even weakly magnetized or unmagnetized planets (Venus and Mars) can deflect the wind, because of the density of the planetary ionosphere itself (Spreiter and Stahara, 1995). Figure 9 hows the geometry of these various situations. The scales of these structures vary from the 4,900 km for Mercury to 6 x 106 km for Jupiter. Space plasmas are clearly a rich laboratory for the study of the collisionless shock.
Fig. 9. Sketches of planetary bow shock and magneto/ionopause obstacle shapes.
5. BALLISTIC EFFECTS IN TURBULENT SHOCKS
The theory and experimental base for collisionless shocks is far too broad to be covered in a paper of this short length. To partly compensate for its many omissions, the remainder of the paper will focus on an aspect of shocks which is really known, but does not seem to be generally used in analysis, and might help explain some data. This aspect is the effect of ballistic transport of information from the shock to other regions, both before and behind the shock layer (Krall and Tidman, 1969a and 1969b).
In the linear approximation, turbulent fields in a stable plasma evolve spatially in two ways from their distributions specified on some boundary. A part of the field decays by Landau damping. A second part f1 exp (ix/v)fo(v) does not decay at large x, but becomes highly oscillatory, so that the charge density e dv exp (ix/v)fo(v) vanishes (Montgomery and Tidman, 1964). This second part is termed "ballistic," since it consists of a memory of the initial or boundary conditions carried by streaming particles.
The second order ballistic spatial decay of electrostatic turbulence behind a shock occurs much more slowly than would be the case if Landau damping alone controlled the wake. The phenomenon can be visualized as follows.
Consider two Fourier components 1 and 2 (1 2) of the first-order fields in the wake immediately behind the shock. Each of these waves modulates the ballistic part of the other wave giving rise to a second-order term ~ exp [i(1 - 2)t - i(1 - 2)x/v] in the distribution function. This produces an electric field (after v integration) that phase mixes away in a length ~ V/ | 1 - 2| (V being a thermal velocity), which can be longer than the Landau damping length (O'Neil and Gould, 1968). The electrons can carry the ballistic memory of the shock turbulence farther downstream than the ions due to their higher thermal velocity, although this is partially offset because they are more easily scattered by particles or other waves.
Another consequence of the ballistic effect is that the use of conventional quasilinear theory (Krall and Book, 1969; Tidman, 1967) (which neglects these effects) will often be inadequate to describe even weak turbulence in a shock wave. The weak turbulence in the leading-edge region is likely to be greatly influenced by ballistic contributions carried back upstream through the shock by protons or electrons which have traversed the large amplitude turbulence in the mid-shock region. Also, for turbulence generated in the leading edge, these terms will influence its propagation into the shock. There is not sufficient path length inside a shock wave for these ballistic fields to phase mix away. Although the ballistic effects of second-order fields persist the farthest distance, it will be first-order ballistic terms which dominate at small distances, and will compete strongly with quasilinear effects inside the shock region.
The idea of the ballistic wake can be extended to explain turbulence upstream of the shock layer, carried not as a wave but as a modulation of a plasma electron distribution which had earlier passed through the shock (Krall and Tidman, 1969b). In one example, the magnetic field in the vicinity of the moon was found to have fluctuations upstream of the lunar wake region (Figure 10) (Ness and Schatten, 1969). These fluctuations were only observed on magnetic field lines that cross the moon's wake, they tended to be transverse, i.e., B B|| where the subscripts denote components perpendicular and parallel to the average magnetic field Bo, their frequencies were broad based, 0 to 5 cps (typically i/2 ~ 0.1 cps, i = eBo/Mc), and the fluctuations extended along Bo both upstream and downstream for a distance of several 1000 km and have a maximum amplitude and highest frequency at the edge of the plasma cavity in the moon's wake. The wake appeared to be the source of these fluctuations.
Fig. 10. Typical spatial distribution of field fluctuations near the moon.
At first sight one is tempted to interpret these field fluctuations as due to waves emanating from the wake region and traveling parallel to Bo, such as whistler, Alfven, or other low frequency waves. However, over most of the above frequency range such waves have group velocities well below the solar wind speed; they would be convected downstream and could not propagate against the solar wind.
When ballistic effects are included, a different picture emerges. At the edge of the plasma cavity behind the moon (Figure 10) the large gradients in plasma density generate low-frequency drift instabilities. Instabilities arise because gradients in plasma and field energy density represent situations that are far from thermodynamic equilibrium, and the plasma finds ways to relax towards equilibrium with a release of free energy.
Electrons moving across the wake along the magnetic field lines can carry a "memory" eix/v fo of this localized turbulence well away from the turbulent region, a distance x ~ Ve/ where Ve is the electron thermal speed. Beyond this distance the electrons phase mix away the memory of the turbulent region since electrons perturbed at many different times overtake each other and the fields produced by averaging over all these electrons become small. Thus we may expect to observe turbulence well away from the unstable sheath.
This is a mechanism entirely distinct from wave propagation. It gives local fluctuations whose frequency and wave numbers are determined not by the local dielectric properties of the plasma but by the fluctuation spectrum in the turbulent region. A direct calculation (Krall and Tidman, 1969b) related those ballistic effects to the Ness and Schatten (1969) experiment.
There are two basic facts here. The first is a very general one, that field fluctuations can be due not to local instability but to particle distributions that arrive at a point x with a memory of an initial perturbation at x = 0; the fields due to these ballistic effects may sometimes dominate fields due to plasma waves, which also transmit energy away from a turbulent boundary. The second fact is that these ballistic effects were confirmed by the measurement of magnetic field fluctuations near the moon.
We conclude that there are indeed many things which we really know about collisionless shock waves.
- We know in detail the source and structure of collisionless shocks
- We know many of the dissipation mechanisms which dominate collisionless shocks
- We know many of the effects of collisionless shocks, up and downstream.
Equally true is that we do not firmly know all the mechanisms for many of these effects. Finally, there are some things we know about collisionless shocks which are not incorporated into present analysis.
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