Collisionless Shock Waves

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By: Roald Z. Sagdeev

Charles F. Kennel

Reprinted without permission from Scientific American, April 1991

Collisionless Shock Waves

Shock waves resonate through the solar system, much like the reverberating boom from a supersonic jet. In the latter case, the disturbance is caused by an aerodynamic shock, an abrupt change in gas properties that propagates faster than the speed of sound. It had long been recognized that in a neutral gas, such as the earth's atmosphere, particles must collide if shocks are to form.

Beginning in the 1950s, we and our colleagues theorized that, contrary to the expectations of many scientists, similar shock waves could form even in the near vacuum of outer space, where particle collisions are extremely rare. If so, shocks could play a significant role in shaping space environments.

"Collisionless" shocks cannot occur naturally on the earth, because nearly all matter here consists of electrically neutral atoms and molecules. In space, however, high temperatures and ultra-violet radiation from hot stars decompose atoms into their constituent nuclei and electrons, producing a soup of electrically charged particles known as a plasma. Plasma physicists proposed that the collective electrical and magnetic properties of plasmas could produce interactions that take the place of collisions and permit shocks to form.

In 1964 the theoretical work found its first experimental confirmation. Norman F. Ness and his colleagues at the Goddard Space Flight Center, using data collected from the IMP-1 spacecraft, detected clear signs that a collisionless shock exists where the solar wind encounters the earth's magnetic field. (Solar wind is the continuous flow of charged particles outward from the sun.)

More recent research has demonstrated that collisionless shocks appear in a dazzling array of astronomical settings. For example, shocks have been found in the solar wind upstream (sunward) of all the planets and comets that have been visited by spacecraft. Violent flares on the sun generate shocks that propagate to the far reaches of the solar system; tremendous galactic outbursts create disruptions in the intergalactic medium that are trillions of times larger. In addition, many astrophysicists think that shocks from supernova explosions in our galaxy accelerate cosmic rays, a class of extraordinarily energetic elementary particles and atomic nuclei that rain down on the earth from all directions.

The study of plasmas began in the 19th century, when Michael Faraday investigated electrical discharges through gases. Modern plasma research dates from 1957 and 1958. During those years, Soviet Sputnik and American Explorer spacecrafts discovered that space near the earth is filled with plasma. At the same time, till then secret research on controlled thermonuclear fusion conducted by the U.S., Soviet Union and Europe was revealed at the Atoms for Peace Conference in Geneva, greatly increasing the freely available information on plasmas.

Fusion research focuses on producing extremely hot plasmas and confining them in magnetic "bottles," to create the conditions necessary for energy-producing nuclear reactions to occur. In 1957, while searching for a method to heat fusion plasmas, one of us (Sagdeev) realized that an instantaneous magnetic compression could propagate through a collisionless plasma, much as a shock moves through an ordinary fluid.

Magnetic fields that thread through plasmas make them behave somewhat like such a fluid. A magnetic field exerts a force (the Lorentz force) on a moving electrically charged particle. The field can be thought of as a series of magnetic lines through the plasma, like the field lines around a bar magnet that can be made visible with iron filings.

The Lorentz force always acts perpendicular both to the direction of the magnetic field line and to the direction in which a particle is moving. If the particle moves perpendicular to the field, the force acts like a rubber band, pulling the particle back and constraining it to move in small circles about the magnetic field line. The particle can, however, move freely in the direction of the magnetic field line. The combination of the free motion along and constrained, circular rotation across the magnetic field shapes the particle's trajectory into a helix that winds around a magnetic field line.

The Lorentz force makes it difficult to disperse the plasma in the direction perpendicular to the magnetic field. The maximum distance over which particles can move away from the field, called the Larmor radius, is inversely proportional to the field strength. In the weak interplanetary magnetic field, the Larmor radius amounts to several kilometers for electrons and several hundred kilometers for more massive ions. These distances may seem large, but they are tiny compared with the size of the region where the solar wind encounters the earth's magnetic field.

The shock that forms there, called a bow shock, has the same parabolic shape as the waves that pile up ahead of a speedboat. It stretches more than 100,000 kilometers across. When the scale is larger than the Larmor radius for ions, the collective motion of plasma particles across the magnetic field actually drags the field lines along with it. The magnetic field thus becomes "frozen" into the plasma.

In short, a magnetic field endows collisionless plasmas with elastic properties analogous to those of a dense gas, and so a plasma wave crossing a magnetic field behaves somewhat like an ordinary sound wave. The theoretical analysis of collisionless shocks therefore started by following the ideas developed from earlier research on aerodynamic shocks.

Suppose, for example, a sudden compression creates a sound wave in air. As the wave travels, its shape--that is, its profile of pressure and density--changes. Because the most compressed regions of the wave move the fastest, the wave grows stronger and its leading edge becomes sharper. The great German mathematician Bernhard Riemann showed how this phenomenon, called wave steepening, creates shock waves.

Ultimately the faster-moving denser air behind catches up with the slower air ahead. At this point, the sound wave behaves somewhat like an ocean wave heading toward shore. A water wave steepens, overturns and then crashes into foam.

A sound wave reaches an analogous but different climax. As the wave grows so steep that it is about to overturn, individual gas molecules become important in transporting momentum between neighboring points in the gas: molecules from the faster, denser region of the wave rush ahead of the steepening wave front, colliding with molecules in the slower region ahead of the wave and exchanging momentum with them. In this way, the slower molecules are brought up to the speed of the moving wave.

This exchange of momentum is caused by molecular viscosity. In this process, momentum is passed from the overtaking wave crest and imparted to the undisturbed region ahead of it, much as in a relay race a baton is passed from one runner to the next. Molecular viscosity becomes highly efficient when the thickness of the wave front shrinks to the average distance that a particle can travel before it collides with another, a distance known as the collision mean free path. (The mean free path of a molecule in air is about one ten-thousandth of a centimeter long.) At this thickness, steepening and viscosity balance each other, and a steady shock wave forms. The resulting shock represents an almost steplike change in gas velocity, density and pressure.

Before physicists knew of a mechanism that could replace molecular viscosity in plasmas, it made little sense for them to talk of collisionless shocks. Consequently, the topic lay fairly dormant for many years. Then, in the late 1950s, one of us (Sagdeev) and, independently, Arthur R. Kantrowitz and Harry E. Petschek, then at the Avco-Everett Research Laboratory near Boston, suggested that a similar sort of momentum relay race could take place in a tenuous plasma. They theorized that in a plasma, waves rather than individual particles pass along the baton.

The plasma relay race depends on the fact that the speed of a plasma wave changes with wavelength, an effect called dispersion. Indeed, whereas in ordinary gases the speed of a sound wave is practically independent of wavelength, in collisionless plasma a wave is very dispersive. That is, its speed may either increase or decrease as its wavelength shortens, depending on the angle between the direction of propagation of the wave and orientation of the magnetic field.

According to Fourier's theorem, a fundamental theorem of mathematics, any wave profile consists of many superimposed waves, or harmonics, of different wavelengths. (By analogy, white light is composed of many distinct colors, each of a different wavelength.) If the wave profile steepens, it excites harmonics of ever shorter wavelength.

For wave propagation that is not exactly perpendicular to the magnetic field, dispersion causes shorter-wavelength harmonics to travel faster than the longer-wavelength ones (negative disperson). The effects of dispersion become significant when a steepening shock front becomes about as thin as the Larmor radius for ions.

At this point, the shorter-wavelength harmonics race ahead of the front into the undisturbed plasma upstream. These harmonics carry along the momentum, like the fast molecules in a sound wave.

The competing actions of steepening and dispersion yield a series of wave pulses that propagate in the direction of the shock. As a result, the front acquires the shape of a "wave train." The weakest (smaller-amplitude) waves announce the arrival of the train, and successively stronger oscillations build up until the full shock transition arrives. The length of the train (in other words, the thickness of the shock front) depends on how rapidly the energy of the waves dissipates.

For waves propagating exactly perpendicular to the magnetic field, dispersion causes the harmonic wave speed to decrease at shorter wavelengths. Short-wavelength harmonics now trail behind the shock front, and so they cannot affect steepening of the overall wave. In this case, the shock passes the momentum baton to a series of compressional pulses called solitons.

Solitons in perpendicular shocks are approximately the thickness of an electron's Larmor radius, and they are created when the wave profile steepens to that scale. The steepening front radiates an ordered sequence of solitons, led by the largest (highest-amplitude) one and trailed by successively smaller ones that ultimately blend into the smooth state behind the shock. The length of the soliton train depends on how fast the soliton energy is dissipated into heat.

Waves on the surface of shallow water behave very much like dispersive waves in collisionless plasma. The theory of shallow water waves was developed in the late 19th century, culminating in the classic work of Diederik J. Korteweg and G. DeVries that first described the solitons that occasionally propagate down Dutch canals. The seemingly recondite analogy between shallow water solitons and plasma solitons expresses a general physical truth: solitons can form whenever wave steepening and dispersion compete.

One implication of this fact is that solitons form even in shocks that do not propagate exactly perpendicular to the magnetic field. The wave pulses mentioned earlier can also be thought of as solitons, the difference being that these solitons are rarefactive (low density) rather than compressive. In this case, short- wavelength harmonics travel relatively slowly (positive dispersion), and the greater the amplitude of the rarefactive soliton, the more slowly it propagates.

As a result, the wave train terminates with the strongest soliton. Surface tension in water creates small waves that have positive dispersion and rarefactive solitons. The physics of water waves therefore provides an analogy to both types of dispersion found in collisionless plasma.

The elegant theory of solitons is an impressive achievement of modern mathematical physics. In 1967 Martin Kruskal and his colleagues at Princeton University proved that any wave profile in a dispersive medium that can support steepening evolves into a sequence of solitons. By relating soliton theory to the problem of elementary particle collisions, which has been studied in depth in quantum physics since the 1920's, they showed that solitons preserve their identities when they collide, just as particles do.

The understanding of dispersive shocks remains incomplete without a knowledge of how to dissipate the energy of waves or solitons into heat. If not for the effect of dissipation, the train of wave structures making up the shock front would be infinitely long. In effect, the fundamental question of how collisionless shock waves transport energy and momentum has reappeared, but in a new guise.

In 1945 the great Soviet physicist Lev D. Landau discovered a dissipation mechanism that requires no collisions between particles. Among the randomly moving particles in a plasma, a few happen to travel at a velocity that matches the velocity of the plasma wave. These particles are said to be in resonance with the wave. An intense exchange of energy can take place between a wave and the particles resonant with it.

In the early 1970s one of us (Sagdeev) and Vitaly Shapiro, also at the Institute of Space Research in Moscow, showed that Landau's mechanism damps solitons by accelerating resonant ions.

Consider, for example, a train of compressive solitons propagating perpendicular to the magnetic field. Each soliton generates an electric field parallel to its direction of motion. Ions traveling close to the resonant velocity move slowly compared with the solitons, and the soliton electric field is able to stop and reverse the motion of these ions. The soliton loses part of its energy to the ions resonant with it during the interaction.

The process does not end here, because the magnetic Lorentz force curves the path of the reflected ion so that it returns again and again to the same soliton. Each encounter adds to the energy of the particle. The Lorentz force, which grows stronger as the particle velocity increases, eventually throws the ion over the top of the first soliton. The acceleration continues as the ion encounters the remaining solitons in the wave train.

The resonant ions gain energy much as surfers gain speed by riding ocean waves. This analogy inspired John M. Dowson of the University of California at Los Angeles to design a new kind of charged particle accelerator, which he dubbed the SURFATRON.

The heating of ions by solitons can form a shock if the number of ions in resonance is great enough. Such is the case if the ions are hot. If not, the solitons find another way to dissipate energy: they themselves generate microscopic plasma waves that heat the plasma.

Plasma electrons flow over ions, thereby creating the electric current responsible for the characteristic soliton magnetic field profile. If the ions are cold, the electrons can easily move at supersonic velocities relative to the ions, in which case the electrons amplify extremely small scale electric field oscillations called ion acoustic waves. These waves, which do not affect the magnetic field, grow in an avalanche-like fashion.

The plasma particles collide not with one another but with these ion acoustic waves. After the waves develop, the plasma enters a microturbulent state.

In 1968 Robert W. Fredericks and his colleagues at TRW in Los Angeles were the first to detect ion acoustic waves in shocks. They made this discovery using instruments on the OGO-5 spacecraft that were designed specifically to study plasma waves in space. Since then, plasmawave detectors have been included on most space mission concerned with solar system plasmas, notably the International Sun- Earth Explorers (ISEE 1, 2 and 3) in earth orbit and the Voyager 1 and 2 missions to the outer planets. The late Fred Scarf of TRW and his collaborators often played back the microturbulent-wave electric fields recorded by the ISEE and Voyager spacecraft through an ordinary loudspeaker. To most listeners, shocks would sound cacophonous; to our ears, however, they were a symphony of space.

Although easy to record, microturbulence has proved difficult to understand completely. Theorists turned to numerical computations to help elucidate the behavior of a strongly microturbulent plasma.

By solving millions of equations of motion for the particles, computer simulation shows how ion acoustic waves grow and heat the plasma. Today's supercomputers are just beginning to give scientists comprehensive understanding of many different kinds of microturbulence.

Even without knowing the detailed nature of microturbulent plasma, physicists can deduce its general behavior. Electrons in the plasma transfer their momentum to ion acoustic waves, which in turn transfer it to ions. This process retards the motion of the electrons in the plasma and so creates resistance to the electric current. In some shocks, ion acoustic-wave resistance grows sufficiently intense to suppress the generation of solitons. When this happens, no wave train forms, and the shock is called resistive.

Although both simple dispersive and resistive shocks have been found in space, most shocks observed there have entirely different characteristics from those discussed so far. Most shocks are sufficiently powerful that neither dispersion nor resistance can prevent steepening from causing the waves to overturn. Overturning then leads to a host of new shock phenomena.

A consideration of shallow water waves, once more, helps to illustrate the process of overturning. When a shallow ocean wave grows sufficiently high, the tip of its wave crest swings forward through an arc and ultimately collapses under gravity. The water stream from behind the crest collides with that ahead, giving rise to the foam on whitecaps. Thus, a large wave crashing toward shore repeatedly overturns, or "breaks."

A plasma wave also develops overlapping velocity streams as it overturns. The fastest stream, which comes from the wave crest, invades the plasma ahead of the shock front. The Lorentz force turns the ions in this stream back into the shock. These reflected ions ultimately mix with those behind the front. If the shock is weak, its structure will remain steady. If the shock is strong, ion reflection will temporarily overwhelm steepening; however, the shock soon steepens again, and the cycle repeats.

Recent numerical simulations by Kevin B. Quest and his colleagues at Los Alamos National Laboratory confirm the idea that very strong shock waves consist of a repeated cycle of steepening, overturning and ion reflection.

The interactions between reflected and flowing ions can also lead to microturbulence. The Voyager spacecraft detected ion acoustic waves, this time generated by ions reflected by Jupiter's bow shock [see top illustration on page 110]. Near the earth, reflected ions generate waves in the solar wind at the geometric mean of the frequencies of rotation of the ions and electrons about the earth's magnetic field; this mean is called the lower hybrid-resonance frequency.

In 1985 the Soviet-Czech Intershock spacecraft made the first definitive measurements of lower hybrid turbulence in the earth's bow shock. Around both planets, the ion acoustic waves take energy from ions and give it to electrons. Some heated electrons escape forward into the solar-wind flow, others back into the shock zone.

So far we have concentrated on those shocks propagating more or less at right angles to the magnetic field, those physicists call quasiperpendicular. Plasma turbulence is even more important when the shock propagates almost parallel to the magnetic field. The field no longer holds back the fast particles that rush ahead of a quasiparallel shock. These particles are a major source of turbulent instability.

The ability of the magnetic field to channel particle motion along field lines creates a situation analogous to a fire hose left spraying water on the ground. Bends in the hose become increasingly curved by the centrifugal force of the flowing water; eventually the hose wriggles uncontrollably on the ground.

The magnetic field channeling the overlapping plasma streams ahead of a quasiparallel shock experiences a similar instability, often called the fire-hose instability. The centrifugal force that bends the magnetic field lines is proportional to the density of energy in plasma motion along the magnetic field. Instability occurs when this energy density exceeds that of the magnetic field. Many physicists conceived of the fire-hose instability independently, but the version invented in 1961 by Eugene N. Parker of the University of Chicago was tailored specifically to quasiparallel shocks.

The plasma fire-hose instability leads to a random flexing of the magnetic field lines. This kind of magnetic turbulence can be regarded as a chaotic ensemble of "torsional" waves, that is, ones that twist the magnetic field lines. They are known as Alfven waves, after Hannes Alfven of the Royal Institute of Technology in Stockholm, who first described them.

Alfven waves, like ion acoustic waves, can exchange energy and momentum with ions in resonance with them. As far as the ions are concerned, the interaction with Alfven waves mimics the effect of collisions. Thus, Alfven waves limit how far ions escaping the shock can penetrate upstream and determine the thickness of the quasiparallel shock.

Theory predicts that collisions between ions and Alfven waves should be nearly elastic, that is, they should involve only slight changes in energy despite a large change in momentum (for example, when a rubber ball bounces off a hard wall, its momentum reverses, but its energy remains essentially the same). As a result, the Alfven turbulence inside the shock front should disintegrate relatively slowly. This notion led us to conclude in 1967 that quasiparallel shocks could be much thicker than quasiperpendicular ones.

The very first measurements of the earth's bow shock by the IMP-1 spacecraft in 1964 hinted at the substantial differences between parallel and perpendicular shocks. The data returned by IMP-1 were somewhat puzzling at first because sometimes the shock appeared thin and other times it appeared thick. Three years later we suggested that shock structure could depend on the orientation of the interplanetary magnetic field.

In 1971 Eugene W. Greenstadt of TRW and his colleagues assembled the first evidence that the thickness of the earth's bow shock does indeed vary with the direction of the solar-wind magnetic field. Since this field constantly changes direction, the regions where the bow shock is locally quasiperpendicular and where it is quasiparallel are always moving, even if the shock itself remains fairly stationary. Wherever the shock is quasiperpendicular, it is thin; where it is quasiparallel, it is thick [see illustration on page 107].

In the early 1970s spacecraft began to detect small fluxes of energetic particles, ion acoustic waves and Alfven waves far upstream of where the earth's bow shock was understood to be. The ISEE program, which started in 1977, established that all the upstream activity is actually part of the extended quasiparallel shock. The shock is so thick that it dwarfs the earth, and therefore earth-orbiting satellites cannot really measure its size.

Another, larger class of shocks does lend itself to investigation by spacecraft, however. Flares in the solar corona occasionally launch gigantic shock waves that propagate through the interplanetary medium to the far reaches of the solar system. These can be observed as they sweep by instrumented spacecraft.

One of us (Kennel), along with colleagues in the ISEE project, found that the region of Alfven and ion acoustic turbulence upstream of quasi-parallel interplanetary shocks can be more than a million kilometers thick.

Alfven waves play a particularly prominent role in the shocks that form ahead of comets as they pass through the solar wind in the inner solar system. Cometary nuclei are far too small to cause any detectable physical disturbance in the flow of the solar wind (the nucleus of Halley's comet, for instance, measures only about 15 kilometers across), and the nuclei possess a negligible magnetic field. Because of these properties, comets cannot generate shocks in the way that planets do. Nevertheless, scientists have found that when comets approach the sun, they create large collisionless shocks.

Sunlight evaporates atoms and molecules from the surface of a comet's nucleus. Most of the liberated gas is ionized by solar ultraviolet light and forms a plasma cloud similar to the earth's ionosphere. The solar wind never penetrates the cometary ionosphere, and it is not the ionosphere that forms the shock wave. The key players in producing cometary shocks are the few neutral atoms and molecules that manage to escape the comet's ionosphere. These, too, are ultimately ionized, but farther out, where they have entered the solar wind.

The newly ionized particles respond to the electric and magnetic fields of the solar wind by joining the flow. They increase the mass density of the solar wind, which, according to the law of conservation of momentum, decreases the wind speed. Because cometary ions are much heavier than the protons of the solar wind, a number of cometary ions can slow the wind appreciably.

More than 20 years ago Ludwig Biermann of the Max Planck Institute for Astophysics in Munich suggested that such a decelerating solar- wind flow should produce a shock similar to a planetary bow shock. During its 1986 encounter with Comet Halley, the Soviet spacecraft Vega-1 heard the plasma wave cacophony that signaled the existence of a shock wave about one million kilometers from the nucleus, the distance predicted by Biermann's theory.

The Soviety Vega, Japanese Suisei and the European Giotto spacecraft encountered both quasiperpendicular and quasiparallel shocks at Comet Halley. The quasiparallel shocks were similar to those at the planets. Heavy ions upstream of the quasiperpendicular cometary shocks generated intense Alfven-wave turbulence, however, something that does not happen around the planets.

Shocks that generate Alfven waves can also accelerate a small group of particles to high energies. The "collisions" of particles with Alfven waves return escaping particles back to the shock front. Each time they recross the shock, the particles increase their energy. This acceleration mechanism is based on one proposed by Enrico Fermi in 1954.

In 1986 one of us (Kennel) and his ISEE collaborators found that a theory of Fermi acceleration developed for interplanetary shocks by Martin A. Lee of the University of New Hampshire successfully passed the test of observations. Yet the Fermi process develops so slowly that the protons accelerated by quasiparallel interplanetary shocks only reach energies of a few hundred thousand electron volts in the one day it takes the shock to travel from the sun to the earth. In comparison, cosmic rays--energetic subatomic particles and atomic nuclei from deep space--have energies up to 100 trillion electron volts.

Exploding stars--supernovas--create very strong shocks that speed into the interstellar plasma at tens of thousands of kilometers per second. We cannot put a space probe ahead of a supernova shock, so we cannot say for sure whether the shock generates Alfven waves and accelerates interstellar ions. We can, however, apply to supernova shocks the theory of particle acceleration that is being tested today using solar system shocks.

Since supernova shocks last about a million years before dying out, particles have time to reach extremely high energies via the Fermi process. Working independently, Germogen F. Krymskii of the Institute of Space Physics Research and Aeronomy in Yakutsk, U.S.S.R., Roger D. Blandford of the California Institute of Technology and Ian W. Axford of the Max Planck Institute for Aeronomy in Katlenburg-Lindau, together with their colleagues, showed in 1977 that the distribution in energy of the particles accelerated by collision-less shocks is virtually identical to that of cosmic rays.

The origin of cosmic rays has long been a puzzle. Many astophysicists now believe that they are created when supernova shocks accelerate particles, although it is still not understood how the particles reach the highest energies observed.

Collisionless shocks probably exist even around remote galaxies. Dynamic processes in the centers of some active galaxies (possibly involving a massive black hole) create supersonic jets hundreds of thousands of light-years long. Shocks are likely to occur when the jets interact with the plasma surrounding the galaxy. Radio emissions from the jets indicate that electrons are accelerated to extremely high energies. Albert A. Galeev, director of the Soviet Institute of Space Research, suggests that a theory he and his colleagues developed to explain how lower hybrid waves accelerate electrons in the earth's bow shock may also clarify how electrons are accelerated in galactic jets.

Contemporary collisionless shock research encompasses phenomena that vary tremendously in scale and origin. The concepts that we and others developed 20 years ago have turned out to be a reasonable basis for understanding collisionless shocks. Spacecraft have found individual examples of most of the shock types predicted by theory. Still to come are refined measurements and numerical calculations that simulate in detail the impressive variety of shocks found in nature.

In most cases, the fairly simple mechanisms we have described here are intertwined in fascinating ways. Yet even now collisionless shock theory has enabled physicists to speculate with some confidence on the physical processes underlying some of the grandest and most violent phenomena in the universe.

ROALD Z. SAGDEEV and CHARLES F. KENNEL have been friends and colleagues since they met at the International Centre for Theoretical Physics in Trieste in 1965. Sagdeev heads the theory division of the Soviet Institute of Space Research and is professor of physics at Moscow Physico-TEchnical Institute. Last year he joined the physics department of the University of Maryland at College Park. In addition to his astronomical and physical research, Sagdeev has been active in the areas of arms control, science policy and global environment protection. Kennel is professor of physics at the University of California, Los Angeles, as well as consultant to TRW Systems Group, where he participates in space plasma experiments. He is also a distinguished visiting scientist at the Geophysical Institute of the University of Alaska, Fairbanks, and a collector of native Alaskan art.


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UPSTREAM WAVES AND PARTICLES. Journal of Geophysical Research, Vol. 86, No. A6, pages 4319-4529; June 1, 1981.

HANDBOOK OF PLASMA PHYSICS. Edited by M. N. Rosenbluth and R. Z. Sagdeev. North-Holland Publishing Company, 1983.

COLLISIONLESS SHOCKS IN THE HELIOSPHERE: REVIEW OF CURRENT RESEARCH. Edited by Bruce T. Tsurutani and Robert G. Stone. American Geo-physical Union, 1985.

NONLINEAR PHYSICS: FROM THE PENDULUM TO TURBULENCE AND CHAOS. R. Z. Sagdeev, D. A. Usikov and G. M. Zaslavsky. Translated from the Russian by Igor R. Sagdeev. Harwood Academic Publishers, 1988.