https://ntrs.nasa.gov/search.jsp?R=19710010990 2020-03-11T20:40:59+00:00Z ABST RAC T A review of experimental and theoretical work on shock waves i n plasmas where c lass ica l collisions are unimportant i s presented. role of dissipation and dispersion in determining shock s t ruc ture is de- sc r ibed and severa l mechanisms which can provide the requi red dissipa- tion in coll isionless p lasmas a r e discussed. shocks produced i n laboratory experiments as well as the Earth's bow shock occurr ing naturally in space a re described. a shock m a y be considered thin s o as to be t r ea t ed as a discontinuity for flow field calculations. Experimental and theoretical a r e a s where m o r e work is needed are pointed out. The Observations of coll isionless Cr i te r ia a r e given when TABLE OF CONTENTS Abstract I. INTRODUCTION 11. THEORETICAL DISCUSSION 111. EXPERIMENTAL TECHNIQUES IV. COMPARISON O F EXPERIMENT AND THEORY V. EARTH'S BOW SHOCK VI. SUMMARY REFERENCES Page ... 111 1 5 15 19 25 31 33 -V- I. INTRODUCTION Ln plasmas, interactions between fluid elements can occur on a basis other than binary collisions between individual particles. Collective effects lead to plasma turbulence which produces dissipation scale lengths which can be f a r shor te r than the ordinary collision mean free path. Therefore, shock phenomena, which depend on the interaction length, can be thinner than a mean free path. lent scale lengths is important in order t o determine when a shock is thin enough to be t reated as a discontinuity for flow field calculations. addition, shocks which are thinner than a mean f r e e path provide examples for studying mechanisms of collisionle s s dissipation. The study of shock s t ructure and turbu- In The first suggestion of the existence of collisionless shocks was made by Gold (1955) to explain the phenomenon of "sudden commencements" which are small, rapid rises in the Ear th ' s magnetic field occurring a day o r so af ter a solar f lare. protons which a r e emitted as a b las t is estimated at 103 km/sec f r o m the delay t ime of a day. minutes thus indicates that the front of the cloud which was compressing the Ear th ' s field can be no thicker than 105 k m which is f a r l e s s than a collision mean f r e e path of 108 km. The character is t ic velocity of the cloud of The sharp r i s e of the magnetic field in a couple of Research on collisionless shocks was fur ther motivated by the coming of the space age which lead to the accessibil i ty of -- in si tu observa- tions in the steady f l u x of collisionless plasma flowing f rom the sun, the so-called so la r wind (Pa rke r 1958), and by r e sea rch on controlled thermo- nuclear fusion where high temperatures a r e required to achieve a sustained thermonuclear reaction. standing off of the Ea r th ' s dipole field which ac ts as an obstacle to the solar wind was .postulated independently by Zhigulev and Romishevskii (1 960), Axford (1962), and Kellogg (1962) and confirmed by Ness , Scearce and Seek (1964) by direct observations using satell i te probes. control thermonuclear fusion, it has been widely hoped that collisionles s shocks, like their collisional counterparts, would t ransfer flow energy directly into ion thermal energy. The possible existence of a collisionless shock In the attempt to Collective interactions between fluid elements occur in a plasma a s a direct consequence of the fact that the constituents, composed of ions and electrons, a r e charged fluids. Fluctuations in the densities and mean velocities of these charged species ac t as sources for long range electr ic and magnetic fields. F o r nonequilibrium plasmas, i. e. plasmas contain- ing spatial inhomogeneities and/or non- Maxwellian velocity distribution functions, the approach to equilibrium often generates unstable fluctuations thus giving r i s e to turbulent electromagnetic' fields. These turbulent fields cause a random scattering of the plasma particles thus increasing their random velocity and producing dissipation on scale lengths which depend upon the particular instability involved. The study of collisionless shock s t ructure is more complicated than that of collision-dominated shock s t ructure because a plasma contains a r ich spectrum of smal l amplitude waves. P lasma waves with different wavelengths propagate with different wave speeds and thus remain near each other for only a short time giving r i s e to weak interactions. As a result , there exis ts in plasmas a state of weak turbulence which has no analogue in ordinary gasdynamics. sound waves, hence waves of different wavelengths remain near each other for long t imes and produce strong interactions. Turbulence develops rapidly f rom laminar conditions with no distinct interim state a s can exist in a plasma. that there a r e a variety of scale lengths associated with different aspects of the shock s t ructure , such a s the magnetic field change, electron heating, or ion heating. Ln gases there a r e only nondispersive Another consequence of the large number of plasma waves is Before proceeding fur ther , we must define what we mean by a col- l isionless plasma shock. A simple definition of the collisionle ss property is that for a given shock s t ructure the dissipation provided by c lass ica l collisions is l e s s than the dissipation which is required to make the shock. By classical collisions we r e fe r to ordinary momentum transfer collisions, e. g. coulomb collisions, a s well a s charge exchange and ionizing collisions. The principal fo rms of dissipation which will be discussed a r e resist ivity and viscosity. For collisionless plasmas, thermal conductivity appears to be unimportant. Resistivity inhibits e lectr ical cur ren ts which resul t f r o m the relative streaming of electrons and ions whereas viscosity inhibits the relative bulk flow between two plasmas. Generally speaking, the former heats electrons while the la t ter heats ions. these dissipative mechanisms a r e still present, however, they do not resu l t f rom collisions in the c lass ica l sense but ra ther f r o m the interaction of particles with turbulent wave fields. F o r collisionless plasmas In this review we hope to present a picture of the current state of experimental and theoretical understanding of the s t ruc ture of collisionless shocks. This review will not be exhaustive, but will report on the signifi- cant observations which have been made and the theoretical ideas which appear to be relevant for explaining the observations. entirely on shocks in magnetized plasmas where the magnetic field plays a decisive role in determining the shock structure. of electrostatic shocks where only electr ic fields determine the shock structure. We shall concentrate We omit any discussion In Section I1 a theoretical discussion is presented which provides Included a framework for understanding the experimental observations. a r e discussions of nonlinear wave theory, anomalous dissipation in a col- l isionless plasma, and cr i t ical Mach numbers. description of the principal laboratory experiments and their limitations. In Section IV we descr ibe the observed features of laboratory shocks and Section 111 presents a brief -2 - re late the observations to theoret ical ideas. description of the Ea r th ' s bow shock which s tands in the solar wind. Section VI contains a summary of our cu r ren t understanding of collision- l e s s shock s t ruc ture and points out some unresolved questions and a r e a s where future r e s e a r c h will be fruitful. In Section V we present a - 3 - 11. THEORETICAL DISCUSSION The effect of a shock on the flow properties of a collisionless plasma is shown schematically in Fig. 1. The electron (or ion) particle density n, magnetic field B, and temperature T = Te t Ti (subscripts e and i repre- sent electrons and ions respectively) are a l l increased ac ross a fast shock while the flow velocity V is decreased. We a r e concerned here with the possible mechanisms which can lead to the formation of a shock with a thickness Ls which is small compared to a typical flow field scale. The jump conditions a c r o s s the shock (de Hoffman and Tel ler 1950), and thus the shock s t ructure , depend on severa l parameters: the Alfven Mach 1 number MA = V/CA where CA is the Alfven speed [ CA = B/ ( ponmi)Z ] ; the angle 6 between the flow direction and the magnetic field; and the rat io a lso be characterized by an additional parameter , the ratio of electron - 1 gyro to plasma frequency, wce/w [ oce = e B/me; Ope = e (n/Eo me)’] , but in a lmost a l l ca ses of interes!%is ra t io i s less than unity and does not appear to be c r i t i ca l for classifying shock structure. of thermal to magnetic p re s su re (3 = 2ponKT/B 2 . The upstream flow can There is no single theory which descr ibes the whole problem since the parameter range is la rge and the dissipation processes involve turbu- lence which is difficult to t rea t analytically. aspects of this problem separately and a s a resul t , our t reatement will of necessity be somewhat fragmentary. f i r s t half dozen years of r e sea rch since 1958, theoretical ideas were m o r e abundant than observations, but in the l a s t half-dozen years , observations of collisionless shocks occurring naturally i n the solar wind and produced in laboratory experiments have tended to stimulate theoretical approaches. We therfore study various It is interesting to note that in the The shock s t ruc ture for low Mach number and weak dissipation i s These effects lead to a non-monotonic determined by dispersive effects. r i s e a c r o s s the shock a t a scale determined by the wavelength a t which small amplitude waves become dispersive. c lass ical dissipation in the form of turbulence becomes the dominant factor in determining shock s t ructure . of the l inear waves going unstable, and thus the turbulent scale may depend upon the particular instability. turbulent dissipation but only some examples which have been observed o r sugge s ted by observations. F o r higher Mach number non- The turbulence resu l t s f rom one o r more We will not discuss the entire range of An important feature of collisionless shocks is the appearance of cr i t ical values for parameters in the sense that they define a demarcation between different processes . In particular, we shall see that there is an angle (Ic such that precursor waves cannotpropagate ups t reamfor 6 > e,, ,a Mach number above which dissipation dominates over dispersive effects, -5- UPSTREAM I SHOCK I DOWNSTREAM N "I N D E N S I T Y : n, TEMPERATURE :TI n2 T2 c 5357 Fig. 1 Schematic of Collisionless Shock in a Magnetized Plasma - 6- J, and a cr i t ical Mach number M.'' above which resis t ive dissipation i s insuf- ficient to form a shock and a viscous-like dissipation is required. A. Dispersion Effects and Nonlinear Waves Dispersion re la tes to the functional dependence of the frequency 0 on wave nyJmber& fo r small amplitude waves. quency waves below the electron plasma frequency in a collisionless plasma has been derived and discussed by Stringer (1963) and Formisano and Kennel (1969) by treating the electrons and ions a s separate fluids with equal densities and sca la r pressures . regime where fluid equations a r e not usually applicable, there a r e regions in parameter space where certain scale lengths l e s s than a mean f r ee path provide the basis for the use of fluid equations (Chew, Goldberger and Low 1956; Petschek 1958). In other regions of parameter space, e. g. , near particle gyroresonances andwave numbers k such that kXD - 1 (AD is the Debye length Ce/upe where Ce = (KTe /me)z is an electron thermal ve- locity), the fluid equations a r e not adequate and a kinetic description is needed. predicting the general properties of wave behavior in a plasma. The spectrum of low fre- Although we a r e working in a collisionless However, the two fluid description has been quite successful in It i s convenient to view shock waves a s being formed f r o m the steepening of a gradual p re s su re pulse. In ordinary gasdynamics a pres - sure pulse steepens by nonlinear effects until sufficiently steep gradients occur so that dissipative effects become important and a steady monotonic shock s t ructure is achieved. Similarly, the magnetohydrodynamic (MHD) equations with collisional dissipative terms lead to the same phenomena as reviewed by Kantrowitz and Petschek (1966). propagation speed of small amplitude waves i s independent of wavelength and the only change in propagation propert ies that occurs a t shork wave- lengths is dissipation. The two fluid equations a r e equivalent to the MHD equations for wavelengths which a r e long compared to Ri = c/wpi, the speed of light divided by the ion plasma frequency. Hence steepening of a gradual p re s su re pulse i s also to be expected in a collisionless plasma. However, a t short wavelengths dispersion, not dissipation, l imits the fur ther steepening of a p re s su re pulse. erated and c a r r y energy either forward o r backward epending on their dispersive properties. ei ther a leading o r trail ing wave t ra in . A shock i s formed when dissipa- tion damps the wave train. In both of these cases the Small amplitude waves a r e gen- The resul t will be an oscil latory s t ructure with F igures 2 and 3 show specific examples of these effects for the fast wave mode propagating either perpendicular, obliquely o r paral le l to the magnetic f ie ld for a cold plasma. for the fast mode shown in Fig. 2 f o r smal l wave number k and nonparallel propagation resul ts in An expansion of the dispersion relation (1 1 1 2 - 0 k - CA [ 1 t k2 RZi (tan C#I - c2) t 0 (k4) - 7 - I / - 0 I I 0 J w > 'A a I e \ U l u - J V \ I I I I I I Rj-1 IO'R;' Rg' .c- 0.1 R r I c 5395 W AVEN U M B E R k Fig. 2 Phase Velocity of the Fast Mode Propagating (a) Perpendicular , (b) Oblique, and ( c ) Parallel to the Magnetic Field for f3 = 0 i n Hydrogen - 8- ( a 1 Perpendicular C 5361 ( b ) Oblique Fig. 3 Trailing and Leading Wave Tra ins Associated with (a) Per- pendicular and (b ) Oblique Propagation - 9- 1 - where E = (me/mi)2 << 1 and 4 = n / 2 - 8 i s the angle between ,k and the normal to E . As a small amplitude wave steepens and generates shorter scale lengths, the higher order t e r m s in k in Eq. (1) become important leading to waves which propagate either fas te r o r slower than the original pulse depending on whether 4 > E o r $< E respectively. Thus for perpen- dicular propagation ( $ < E ) pulse steepening will lead to trail ing wave t ra ins when scale lengths reach the electron length Re = E R i = c/o while for oblique propagation (4 > €)preceding wave t ra ins can be expected when scale lengths approach $Ri - Ri cos 8 a s indicated schematically in Fig. 3 . The MHD equations, and hence the two fluid equations, can support both fast and slow shock waves. hypothesized to exis t (Petschek 1964; Petschek and Thorne 1967), there have only jus t recently been some repor t s of observations of slow shocks in the laboratory (Bratenahl and Yeates 1970) and in the solar wind (Chao 1970). Therefore we shall confine our attention here to fas t shocks which have been studied extensively. Pe Although slow shocks have long been A s a pulse steepens, l inear wave theory breaks down and one must look f o r nonlinear solutions of the two fluid equations. solutions exis t and have been studied extensively for cold plasmas and per- pendicular (Adlam and Allen 1958; Davis, Liist and Schliiter 1958), paral le l (Saffman 1961a), and oblique (Saffman 1961b; Kellogg 1964; Cordey and Saffman 1967) propagation and more recently for w a r m plasmas (Inoue 1968, 1969; Crevier and Tidman 1970). The qualitative description of nonlinear waves ag rees with the propert ies deduced f rom the l inear waves although there are some quantitative differences. In general the nonlinear wave appears as a pulse of magnetic field (large amplitude) traveling at a velocity fas te r than the fas t wave speed and ei ther preceded by o r followed by a wave train. These nonlinear wave t ra ins have relevance for collisionless shocks since a smal l amount of dissipation will damp the wave t ra ins (lead- ing o r preceding) and a transit ion f rom state one to state two a s indicated in Fig. 1 will have been achieved (Sagdeev 1962a, 1966). These nonlinear As i n the case of l inear waves there is a small angle 4 of order E (for MA << c - l ) which separates the leading f rom trail ing wave t ra ins . The character is t ic length for nonlinear waves propagating perpendicular to the magnetic field is of o rde r Re while for oblique propagation the scale length is of o rde r Ri which is considerably broader . There is an upper limit to the Alfven Mach number above which nonlinear solutions to the two fluid equations have not been found. cussed in Section 11-C. The significance of this l imit will be dis- B. Dissipation F o r la rge Alfven Mach numbers, dissipation increases . If there is enough dissipation, the oscillations in the wave t ra in will be either cri t ically or over-damped resulting in a monotonic shock s t ructure . Dissipation resu l t s f r o m plasma turbulence which is generated by unstable plasma waves. The unstable waves must eventually re turn their energy to the -10- particles, and this process can proceed by severa l routes, e. g. resonant damping directly into particles and mode-mode coupling into waves which a r e then damped into particles. Due to the complexity ar is ing f r o m the large number of possible plasma waves, theorists have attempted to explain collisionless dissipation in t e r m s of particular instabilities and many have been suggested. We will discuss only those which relate to direct observa- tions. F o r perpendicular shocks, l a rge gradients in magnetic field a r e possible due to the smal l scale of the collisionless length Re. It was recognized ear ly (Sagdeev 1962a, 1962b; Kellogg 1964) that the electron cur ren t flowing in the plane of the shock perpendicular to ?iJ and creating the jump in ?iJ would produce unstable electrostatic plasma waves. If the relative s t reaming velocity between ions and electrons, vd , i s grea te r than the electron thermal velocity ( v d > Ce) , a strong two-stream instability can c rea te enough plasma turbulence to provide a l lcoll isionless" resist ivity and preferentially heat electrons (Buneman 1959). The pr imary effect of this instability is to limit the s t reaming velocity to the o rde r of the electron thermal velocity. Sagdeev (1962a, 1966) pointed out that for a low f3, low Mach number, perpendicular shock, this limitation on v d due to plaqma turbulence increases the leading gradient length to be of order R e p-2. This observation is an example of how dissipation can replace dispersion in determining gradient lengths. A weaker cu r ren t than that required to excite the two-stream insta- bility can a l so produce unstable, electrostatic, plasma wave turbulence if the electron temperature exceeds the ion temperature. An analysis using a kinetic description (Jackson 1960; Stringer 1964) shows that when Te> 5 - 10 Ti, a relative d r i f t between ions and electrons in excess of the ion-acoustic speed C s > KT/mi causes the ion acoustic wave mode to be unstable. up to opi. velocity of electrons, this weaker ion acoustic instability gives r i s e to stochastic e lec t r ic fields Ek which produce an effective resist ivity for electrons with the resu l t that their random velocity and thus their tem- perature i s increased. The collisionless res is t ive dissipation produced by this mode of plasma turbulence has received the mos t theoretical attention. This instability involves ion plasma oscillations and frequencies In contrast to the two-stream instability which l imits the flow A rough est imate can be made of the maximum amount of heating which can occur a s a resu l t of stochastic encounters with turbulent e lectr ic fields Ek. a particle can acquire by interacting with a coherent wave packet is A V, = e E k / m o d where a d = o - kV is the effective wave frequency a s seen by a particle traveling with velocity V. In a succession of stochastic encounters, the particle random walks in velocity space with the total A V increasing a s the square root of the number of encounters. spends a t ime to in the turbulent field, the total number of random changes AV, is to/T where T is the wave packet coherence time. the temperature increase is thus given by . In a single encounter the maximum change in velocity, A V,, If a particle An estimate for -11- The thermal energy density produced can be expressed in t e r m s of the energy density contained in the electr ic field oscillations Wk = E Ek2/2; 3 Equations (2) and (3) a r e general and not specific to the type of particle o r type of turbulence. observations of turbulence. Est imates of this type can be used for comparison with More detailed calculations have been ca r r i ed out by a number of The ear ly work on ion acoustic wave turbulence neglected the authors. presence of the ambient magnetic field. Kadomtsev (1 965) has calculated that a large amplitude of turbulent fluctuations with a wave number spec t ra l density which var ies a s km3 I n (kX production of waves by l inear grow& against nonlinear wave scattering by the ions. Sagdeev (1967) has calculated on the bas i s of a quasilinear theory that this microturbulence could produce an effective collision frequency fo r electrons of v e whichis severa l o rde r s of magnitude l a rge r than would be expected on the%asis of coulomb collisions. Kral l and Book (1969a, 196913) have extended the calculation of Sagdeev by including the effect of the magnetic field. By means of a quasi- l inear theory they a r r ive a t an est imate for the shock thickness ( 3 ~ , / 2 ) 1 n se which seems to agree well with severa l experiments in which the rat io upe/wce ranges over more than an order of magnitude. This theory has recently been refined (Biskamp 1970). Sagdeev (1970) have reconsidered the wave damping by ionsland a r r ive at a different scaling fo r the shock thickness, Ls - ( mi/me)4 R e , which i s a lso consistent with observations. ) - l would resul t f rom balancing the - 10-2 (Te Vd/Ti C s ) w p i - 0. 1 - 1 0 Ls = wce Bekshtein and We now turn our attention to two theoretical suggestions for ion dissipation mechanisms which apply pr imari ly to perpendicular and oblique shocks respectively. The f i r s t (Tidman 1967a) involves an electrostatic ion-ion two-stream instability whose main resu l t i s to heat the upstream ions by interacting directly with the downstream ions traveling a t a smaller velocity. A simple cr i ter ion for this instability to be operative i s A V i < C s where AVi is the relative drift between the cold upstream and hot down- s t r eam ions. since the conditions necessary for this instability, hot electrons and low It is difficult to estimate the amount of ion heating produced - 12- relative ion drift, a r e not easily met; a recent calculation (Forslund and Shonk 1970) indicates that two interpenetrating ion s t r eams do not ther- malize effectively by this mechanism. The second f o r m of ion dissipation i s associated with electro- magnetic wave turbulence. This turbulence is produced by the fast mode (see Fig. 2) propagating obliquely to the magnetic f ie ld and above the ion gyrofrequency, the so-called whistler wave (Storey 1953). An ea r ly col- l isionless shock theory (Fishman, Kantrowitz and Petschek 1960; Camac e t al. 1962) suggested that the whistler waves which had group velocities equal to the upstream flow would remain in the vicinity of the shock a long time and be greatly amplified. waves and hence the upstream flow energy was dissipated into magnetic field turbulence. This assumption has been found to be invalid as the mea- sured amplitude of the wave turbulence is too smal l to account fo r the dis- sipation (Patr ick and Pugh 1969). Instead, the latter authors estimate that these waves provide the required dissipation by d i rec t scattering of the ions by the wave electr ic fields according to the scheme leading to Eq. (2). Although this suggestion i s plausible, no detailed calculation of the amount of ion heating by this mechanism has been performed. This theory assumed no damping of the C. Cri t ical Mach Number We have already mentioned that a t a sufficiently large Mach number, low p, perpendicular shocks undergo a transit ion f r o m an oscil latory to a monotonic s t ructure due to increased turbulence. number experimentally determined to be around 2. 5 < MA < 3. 0, a per- pendicular, low p shock undergoes another change in s t ructure which will be described la ter . The ion acoustic turbulence which has been described previously resul ts in an effective resist ivity and perferentially heats elec- trons. It has been widely suggested that the observed change in s t ructure is related to the fact that there i s a mximum Mach number for which re- sistivity alone can provide the required dissipation. This phenomenon is analagous to the situation in ordinary gasdynamics where there i s a cr i t ical Mach number above which a shock with a continuous s t ructure cannot be formed in a gas of zero viscosity but finite thermal conductivity (Landau and Lifshitz 1959). s ical collisions, some kind of anomalous ion viscosity i s required to achieve higher Mach number shocks. At a higher Mach In a plasma in which the ions do not undergo any clas- The two fluid equations with no dissipation permi t no solution when the flow velocity through a large amplitude s t ruc ture is decreased to the ion acoustic speed C . F o r a cold plasma (Cs = 0 ) and perpendicular Allen 1958; Davis e t al. 1958). F o r high Mach numbers , a significant fraction of the ion s t r e a m would be reflected forward and the assumption of a single ion s t r e a m is violated. breaking" but has not been observed in the laboratory. Instead, for Mach numbers this large, t he re is always sufficient dissipation that a monotonic s t ructure does f o r m with downstream properties specified by the Rankine- Hugoniot conditions. propagation, the ion s low velocity is reduced to ze ro a t MA = 2 (Adlam and This phenomenon is called "wave - 1 3 - .I. The cr i t ical Mach number, M1', above which a shock without ion viscosity cannot be formed is given by the condition that the downstream flow velocity equals the downstream ion acoustic speed, (1969a) and Coffey (1970) have discussed the value of the cr i t ical Mach number for perpendicular shocks and find that fo r y = 5/3, M" = 2. 76. Coroniti (1970) has discussed a possible explanation of why the cr i t ical downstream flow velocity is C s which is a different velocity than the fast wave speed in a collisionless plasma. He showed that in a plasma with resist ivity q, the fast wave speed decreases to Cs for scale lengths small compared to LM = q/p0V2, i. e . , the magnetic field becomes decoupled f r o m the plasma oscill?$.ons for a magnetic Reynolds number of o rde r one. As a resul t , i f MA < M", a pulse f r o m the piston traveling a t the down- s t r eam fast wave speed approaches the shock f r o m behind until it steepens enough to reduce its speed to Cs2 which is less than V2 and l-j,ence energy f r o m the piston is blown downstream. propagating toward the shock is slowed to Cs2 which now exceeds V2. In this case the piston can feed additional energy to the shock, more energy than the shock can dissipate by resist ivity alone. V2 = Cs2. Woods However, fo r MA > M", a pulse It has been hypothesized that above the c r i t i ca l Mach number a It was sug- viscous subshock forms (Marshall 1955; Kantrowitz and Petschek 1966; Coroniti 1970) in order to dissipate the additional energy. gested that this subshock should have a smal le r scale length over which viscous dissipation occurs , and this dissipation should interact directly with the ions allowing the formation of higher Mach number shocks. In- deed numerical integration of the two fluid equations including ion viscosity introduced artificially indicates that the shock s t ructure can separate into a thin layer where viscous ion heating takes place imbedded within a broad- er magnetic s t ructure (Macmahon 1968). Regardless of the details of the viscous dissipation region a broad s t ructure i s expected due to the gyration of the heated ions in the magnetic field. Woods (1969b) has car r ied out a detailed analysis based on ion orbits and found that the observed width is of o rde r Ri which is identical to an ion gyroradius based upon the Alfven speed (Ri E C A / W ~ ~ ) . The suggestion of the formation of a viscous subshock i s based upon a thickest shock hypothesis. This hypothesis s ta tes that the dominant dis- sipation mechanism is the one which operates on the smallest gradient and can provide the required dissipation (Kantrowitz and Petschek 1966). In a collisionless plasma there i s no a p r i o r i reason why a t higher Mach number a viscous "subshock" should f o r m because the viscous dissipation could have a l a r g e r scale length than that associated with resistivity. At low Mach number this viscous dissipation m a y not be operative due to its in- stability threshold level and hence the thickest shock hypothesis may still be valid. F o r perpendicular shocks, there is no definite observation of the scale length, broad o r thin, of the dissipation mechanism which pro- duces ion heating. that the ion heating may take place in a thin layer of order severa l XD = Ce/mpe at the rear of the magnetic shock structure. to e lectrostat ic turbulence resulting f r o m an ion-ion streaming instability discussed in Section 11-B. There has been a theoretical suggestion (Tidman 1967b) This heating i s due - 14- 111. EXPERIMENTAL TECHNIQUES A number of experimenters have built devices to produce and study coll ision-free shock waves and they have met with varying degrees of suc- cess. Among the problem a r e a s most usually encountered a r e : (1) fa i lure to attain coll ision-free conditions ahead, within, and behind the shock; ( 2 ) not enough t ime o r space for a shock to detach f r o m the piston and achieve a steady s t ruc ture ; ( 3 ) insufficient spatial o r temporal probe resolution to study shock s t ruc ture ; and (4) density gradients in the flow and finite radius of curvature effects which render the shock nonplanar. These effects as well as the following considerations define regions in pa rame te r space within which one can do coll ision-free shock experiments. An upper limit to the ups t r eam plasma density is s e t by' the collisional mean f r e e path. ups t ream flow and those in the shock mus t be l a rge r than the shock thickness. Since this mean f r e e path sca les as the square of the ion kinetic energy, up- s t r e a m velocities g rea t e r than l o 7 cm/sec yield mean f r e e paths g rea t e r than 0. 25 m e t e r s even for relatively high densities equal to 1015/cm3. The mean free path for coulomb collisions between ions in the It is a l so necessa ry to have the c lass ica l conductivity due to electron- ion collisions la rge enough s o that the t r ansve r se cur ren t which c rea tes changes in the magnetic field cannot produce significant dissipation (joule heating). path in the ups t r eam plasma be sufficiently long so that the turbulence i s not damped by classical collisions. shock must be g rea t e r than 1 eV to prevent such damping for laboratory experiments with ups t r eam densities in excess of 1014/cm3. The mos t res t r ic t ive requirement is that the electron m e a n free The electron tempera ture ahead of the A lower limit on the ups t ream density is fixed by requiring that the shock thickness be small compared to the s ize of the device while the l a rges t charac te r i s t ic dimension in the plasma, Ri = c/wpi, be small com- pared to the flow dimension. The length, Ri, is inversely proportional to the square root of the plasma density and is approximately 3 c m when the plasma density is 1014/cm3. Hence, to obtain a relevant collisionless plasma shock experiment, the velocity and density mus t both be large. This either a pulsed o r steady experiment. requirement leads to a power level at least of the o rde r of 10 7 watts for The maximum value of ups t r eam magnetic field is l imited by the requirement that the ups t ream flow velocity be at least equal to the Alfven speed, i. e. MA L 1. To date, workers have achieved maximum flow ve- locit ies of the o rde r 5 x l o 7 cm/sec which limits the maximum magnetic field to a few kilogauss for densities of the o rde r 1014/cm3. limit on the magnetic field comes f r o m requi.ring that the l a rges t possible The lower -15- sca le length, the ion gyro radius, be less than the s ize of the device. Ap- proximately 5.0 gauss is the lower limit for velocities of 5 x lo7 cm/sec and 0. 5 m e t e r scale s ize experiments. The re a r e many compromises that mus t be made i n the choice of pa rame te r s for a given experiment and to date no single experiment satisfies all the requirements for any appreciable range of MA and p. ments can be divided into two categories: steady and unsteady. a super - Alfvenic plasma is blown against an obstacle producing a standing shock in much the same manner as a wind tunnel. accelerated to super Alfvenic velocity into a s ta t ionary plasma producing a shock running out ahead as i n a n ordinary shock tube. take on various fo rms , including rapidly moving e lec t r ic and magnetic field gradients o r a slug of high density plasma. Shock experi- In the fo rmer , In the la t ter , a piston is The piston may A plasma wind tunnel was built by Pa t r ick and Pugh (1967, 1969) i n which a fully ionized plasma was blown against a magnetic dipole cavity producing a standing oblique shock wave analogous to the Ea r th ' s bow shock as shown in Fig. 4a. were approximately 10 eV and 5 x 1013/cm3 corresponding to a n ups t r eam mean f r e e path of 30 c m which is comparable to the scale s ize of the experi- ment. Thus, the high Te ups t r eam makes this experiment t ruly collision- f r ee on both s ides of the shock. la rge enough to resolve shock s t ructure , but it is possible that two dimen- sional effects caused by density gradients in the flow and a finite radius of curvature of the shock may be important. The minimum ups t r eam B is l imited by residual fields f r o m the plasma source limiting the maximum Mach number and p to be below th ree and one, respectively. The Rankine- Hugoniot jump conditions have been verified and detailed spatial turbulence measurements have been obtained. Self-consistency is achieved: the mea- su red turbulence provides sufficient dissipation to satisfy the jump conditions. The ups t ream plasma electron tempera ture and density The shock thickness, - 3 R i - 10 cm, is Unsteady shocks have a l so been produced by rapidly imploding mag- netic field gradients in devices called Z and 8 pinches, In Fig. 4b we show a schematic d iagram of a Z pinch (Paul et al. 1965). The initial plasma is produced by an oscil latory axial cu r ren t which first ionizes then heats the background gas to densit ies and tempera tures of approximately 7 x lOl4/cm and 1 eV, respectively, giving an electron-ion mean f r e e path in the ups t ream plasma of o r d e r 0. 5mm. Thus, the plasma is collisional. The shock is pro- duced by a rapid r i s e in axial cu r ren t which causes the azimuthal magnetic field to implode, driving a cylindrically converging perpendicular shock into the initial plasma. Shock speeds up to 25 cm/p s e c have been achieved with Mach numbers in the range 2 The shocks are thin ( Ls 5 1 cm) compared to flow dimensions and detach f r o m the piston to become steady for about 10 c m before convergence effects a r i se . F o r low Mach number perpendicular shocks, p recu r so r waves cannot propogate and thus the fact that the ups t r eam plasma i s collisional may not be of grea t significance. These experiments at low MA a r e also self-consistent in that measu red turbulence levels (Paul , Daughney and Holmes 1969) can account for the re- quired dissipation. 3 MA 5 6. F o r l a r g e r MA, the shock thickens and, s ince the -16- ELECTRODES BIAS FIELD COILS 0 0 0 0 0 o / ! \ , MAGNETIC PISTON I - I MAGNETIC PISTON . / 150cm / + r 50cm BIAS B PISTON FIELD COILS 0 0 0 o n 0 0 0 0 0 0 0 ( c ) e PINCH , PLASMA , &/;AS F I E L Y O I L S \ b , ACCELERATOR 0 0 0 0 P p . 2 7 0 cm-4 ALUMINUM PLASfflA PISTON ( d ) PLASMA PISTON C5363 Fig. 4 Schematic Diagrams of Laboratory Experiments - 17- u p s t r e a m conductivity is low, the shock s t ruc tu re can be affected by col- l is ional damping. Also, fo r these higher values of MA, the shock thick- ness becomes comparable to the sca le s i ze of the experiment. Another vers ion of this imploding shock device is the 6' pinch shown schematical ly in Fig. 4c and built by many exper imenters (Alikhanov e t al. 1969; Chodura e t al. 1969; Hintz 1969; DeSilva et al. 1969; Zagorodnikov, Smolkin and Sholin 1967). These perpendicular shocks a r e produced by a rapidly imploding axial f ie ld driven by a n azimuthal c u r r e n t as oppose% to the axial cu r ren t of the Z pinch. The ups t r eam plasma i s produced by ei ther e l ec t r i c d i scharges o r photoionization. In the Z pinch, the piston field is perpendicular to the bias f ie ld while for 6' pinch experiments the B fields are ei ther para l le l o r antiparallel . The resu l t s f r o m these numerous 6' pinch exper iments span a wide range of pa rame te r s . detached shocks a re not observed (Chodura et al. 1969; DeSilva et al. 1969) whi le for densit ies of the o r d e r of 1014/cm3 the r e su l t s of Paul e t al. (1965) a r e recovered. F o r high p and Mach number, i. e. p 2 1 and MA--> 4 (Hintz 1969; Chodura e t al. 1969), shocks a r e observed but are considerably thicker ( o r o r d e r Ri = 1 c m ) than the low p, low MA case. to note that many of the r e su l t s of the 6 pinch exper iments a r e cha rac t e r i s t i c of the specific device and may not be assoc ia ted with coll isionless shock waves themselves; e. g. , DeSilva et al. (1969) find no c l ea r shock s t ruc tu re for ant iparal le l piston and bias fields while Hintz (1969) observes a m o r e pronounced shock s t r u c t u r e for the ant iparal le l case. F o r ups t r eam densi t ies below 1013/cm3, s teady, - It i s impor tan t An interest ing var ia t ion of the perpendicular 6' pinch is that built by Robson and Sheffield (1969) i n which a shor t 6' pinch coil i s u sed and a curved piston with oblique propagation is produced. The u p s t r e a m plasma i s coll isional and a l a r g e amplitude damped whis t le r wave i s observed out ahead of a s h a r p rise i n the magnetic field, the width of the total s t ruc tu re approaching Ri - 1 cm. No turbulence is observed and it is unclear i f any col l is ionless dissipation mechanisms are involved except c lose to the cen te r - line where the shock is perpendicular and the r e su l t s of Paul e t al. (1965) are again recovered. Another unsteady experiment is shown i n Fig. 4d which produces an oblique dis turbance (F r i edman and Pa t r i ck 1968, 1970). gun ionizes and acce le ra t e s a mass of a luminum w i r e and injects it into a background, magnet ized p lasma at 107 cm/sec. by ultraviolet radiation f r o m the gun. 1015/cm3 and thus, fo r an es t imated e lec t ron t empera tu re between 1 and 10 eV, the u p s t r e a m plasma i s highly coll isional ( A 1 mm). Alfven Mach numbers range f r o m 30 up t o s e v e r a l hundred and the ups t r eam p is o r d e r 1 - 10. It i s not clear whether o r not a shock f o r m s ; however, a momentum t r a n s f e r mechan i sm on a sca l e less than 1 cm, which is of the o r d e r of Ri, is p re sen t fo r MA^ 100. Here a coaxial Preionizat ion is achieved The u p s t r e a m density i s approximately At higher MA, little coupling is observed. - 18- IV. COMPARISON O F EXPERIMENT AND THEORY F o r perpendicular propagation and low (3, the cha rac t e r of the shock s t ruc tu re changes significantly as the Mach number i s inc reased from%nity. In Section IV A we first d i scuss this sequence and then discuss high (3 shocks. In Section IV B we present the experimental r e su l t s fo r oblique shocks. We shal l compare experimental resu l t s with the theoret ical work d iscussed in Section 11. A. Perpendicular Shocks 1 < M < 2, low (3. - In this Mach number range, the shock s t ruc tu re -A is dominated by dispers ion r a t h e r than dissipation result ing in the shock s t ruc tu re shown schematical ly in Fig. 5a. 1969). The inital gradient length is of o r d e r Re = c / o pe followed by a damped wave t r a i n of wavelength - 10 Re. the earlier theore t ica l discussion in that both l inear and non-linear wave theor ies pred ic t f o r these conditions ini t ia l gradient lengths of o r d e r Re followed by a t ra i l ing wave t r a i n with wavelength of the o r d e r a few times Re. (Hintz 1969; Alikhanov et al. These observations a g r e e with Although dissipation is not dominant it is manifest i n the damping < p < 10-1, a of the wave t ra in . Since fo r these low (3 exper iments , Mach number of 1. 5 exceeds the two stream instabil i ty threshold (Sagdeev 1962a, 1966), the init ial shock gradient should be given by Ls - Re(3-1/2= 3 - 10 Re as d iscussed i n Section I1 B. No direct evidence exis ts at this time to distinguish coll isional f r o m turbulent dissipation. .l. 2 < MA. < M-., low (3. - At higher Alfven Mach numbers , dissipation is s t rong enough to completely damp out the t ra i l ing wave t r a i n and shock s t ruc tu res become monotonic with thicknesses of the o r d e r 10 Re as shown in Fig. 5b. (Alikhanov e t al. 1969; Hintz 1969; Paul e t al. 1965; Goldenbaum 1967). by the Thompson sca t te r ing of l a s e r light (Paul e t al. Stamper 1967; Chodura e t al. Te alone is sufficient to account for the inc rease in (3 requi red by the Rankine- Hugoniot conditions. heated (by joule diss ipat ion) and that no ion heating o r viscosity is present . The e lec t ron t empera tu re jump a c r o s s the shock has been m e a s u r e d 1967; DeSilva and 1969) and the r e su l t s show that the r i s e in It is concluded therefore that only the electrons a r e The shock width is l a r g e r than would be expected i f no e lec t ron col- l ions w e r e p re sen t and it is concluded that an anomalous res i s t iv i ty exists which has broadened the shock front beyond the coll isionless skin depth Re. An es t imate of the amount of res i s t iv i ty o r equivalently, the effective col- l is ion frequency, v e, is obtained by assuming the shock has a magnetic Reynolds number Rm, of the o r d e r of unity, - 1 9 - e c5359 Fig. 5 i 2 Typical Perpendicular Shock St ruc tures for (a) MA - 1. 5 , (b ) M A > 2 p << 1 and - 20- - R = P o 0- Ls v1 = 1 m where o = e L n ( m e v e )-l. , Ls - 10 R e ff 14 7 F o r n = 7 x 10 /cm3 and V1 = 2. 4 x 10 cm/sec (Paul et al. 1965), the effective coll ision frequency is of the o r d e r w p i . This value exceeds the c lass ica l value (Spitzer and H a r m 1953) based upon coulomb collisions for measu red e lec t ron t empera tu res in the shock front by at l ea s t two o r d e r s of magnitude. (Alikhanov e t al. 1969) . coherent density fluctuations to be between two and th ree o r d e r s of mag- nitude g r e a t e r than that expected f r o m random fluctuations. culations show that this turbulence gives r i s e to an anomalous res i s t iv i ty which can provide the requi red dissipation. Frequencies of this o r d e r have been detected by probes 1969) and by l a s e r sca t te r ing (Paul , Daughney and Holmes The l a t t e r au thors have m e a s u r e d enhanced laser sca t te r ing f r o m Thei r cal- .b F o r Alfven Mach numbers in the range, 2 < M A < M"', the e lec t ron drift velocity assoc ia ted with the t r a n s v e r s e cu r ren t exceeds C s and if Te << Ti i n the u p s t r e a m plasma, the c u r r e n t is unstable as d iscussed in Section I1 B. 1969), it is thought that some other mechanism, possibily the two stream instabil i ty prehea ts the e lec t rons until Te becomes g rea t e r than T i i n the shock front and then the cu r ren t is unstable. The frequencies observed a r e of the o r d e r w wave number spec t rum predicted by Kadomtsev (1965) has been exper i - mentally ver i f ied by Daughney, Holmes and Paul (1970). again that K r a l l and Book's (1 96 9b) theory yields a shock thickness Even i n the one experiment where Te < T i (Chodura et al. i n ag reemen t with theory; fu r the rmore , the turbulent W e a l so mention P i 3 S 2 e L = - R Q n which ag rees well with s e v e r a l exper iments i n ranges over m o r e than a n o r d e r of magnitude. c e which the ra t io w pe/w ce .L The conclusions to be reached fo r 2 < MA < M." a r e : ( a ) The sca l e length of the shock thickness is of o r d e r 10 Re; (b) Perpendicular shocks a r e res i s t ive with the p r i m a r y dissipation mechanism being joule d is - sipation; ( c ) Theory and experiment appear to be i n substant ia l agreement that ion acoust ic wave turbulence produces the requi red anomalously l a r g e r e s is t ivity, .L M". < MA < 4, low f3. - As the Mach number is inc reased beyond a cr i t ica l value experimental ly determined to be about 2. 7 , the cha rac t e r of -21- the shock changes as indicated schematical ly in Fig. 5b. a broad foot at the leading edge while at the r e a r the s teep gradients present at lower Mach numbers a r e still observed (Paul e t al. 1965; Alikhanov et al. 1969; Hintz 1969). a n effective ion thermalizat ion (Alinovskii et al. 1970) whereas f o r MA < M" no ion heating was observed. The sca l e length of the broad foot i s of o r d e r Ri as t o be expected when hot ions a r e present (Woods 1969b). No detailed turbulence measu remen t s have been made on the broad foot p r t i o n of these shocks but the fluctuation leve l detected by l a s e r sca t te r ing does not appear to be as g rea t as that assoc ia ted with the s teep gradient shocks at lower Mach number (Paul , Daughney and Holmes 1969). The p resence of t he rma l i zed ions violates the single s t reaming hypothesis of the two fluid descr ipt ion and either a full kinetic theory o r a multi-ion fluid descr ip- tion is necessary. The shock grows Measurements of the ion velocity distribution now show The appearance of the broad foot is re la ted to the fac t that these Mach numbers exceed the c r i t i ca l Mach number d iscussed i n the theoret ical sect ion and therefore joule heating is insufficient t o provide the r equ i r ed dissipation. The ions m u s t acquire some randomization in velocity pro- ducing counter s t reaming ions in the broad foot. This configuration m a y be unstable (Sagd ev 1966) leading to fluctuations with frequencies of the o r d e r (0 ci)1?2 fo r which t h e r e is s o m e labora tory evidence (Zagorodnikov et al. 1967). However, the mechanism leading to ion viscosi ty and pro- ducing ion heating i s still unclear. T idman 's (1967b) suggestion of a thin subshock has not been ver i f ied; the spa t ia l resolut ion of shock exper iments is not g r e a t enough to e i ther conf i rm o r deny even the existence of a thin subshock. A theory based upon the orbit ing motion of ions ref lected f r o m the shock by the e lec t ros ta t ic potential and hot ions diffusing u p s t r e a m (Woods 1969b) is in agreement with observations (Pau l 1965); however, it m u s t be emphas ized that the u p s t r e a m plasma is coll isional and thus the m e a s u r e d sca l e lengths m a y not be totally determined by col l is ionless p ro - cesses . 4 < M, low p. - F o r the low p, high Mach number case , the total shock s t ruc tu re thickens to R i and the double s t ruc tu re d isappears as shown in Fig. 5b. (Paul et al. 1965; Hintz 1969; Chodura et al. 1969). Per- pendicular shocks at low p have not been produced much beyond MA : 6. High p. - F o r modera t e p 'I- 1 and low MA < M", the shock is similar to the low p case (Hintz 1969) with Ls - 10 Re. number the shock thickens as a whole and no double s t ruc tu re is observed. At high MA, the shock thickness approaches Ri. Above the c r i t i ca l Mach F o r high p of o r d e r 5 and modera te Mach number of o r d e r 4, the shock is th icker than the low p case , of o r d e r Ri (Chodura 1969). ion heating is observed both above and below the c r i t i ca l Mach number and ion heating general ly inc reases with increasing Mach number. s ince these a r e thicker shocks and the ups t r eam p la sma is coll isional, the coll isionles s turbulent p rocesses m a y be altered. Significant However, -22- B. Oblique Shocks When the shock is not perpendicular to g , col l is ionless shock s t ruc tu re changes significantly and p r e c u r s o r wave t r a ins with shock thicknesses of the o r d e r Ri a r e expected. and Pugh 1967, Robson and Sheffield 1969) although significant differences exis t between these two experiments . ing shock produced by the interact ion of a super Alfvenic flow and a mag- netic dipole in a simulation experiment of the Earth's bow shock. shock thickness i s of o r d e r a few Ri for Mach numbers in the range 1, 5 5 MA 5 3 and the Rankine-Hugoniot jump conditions a r e obeyed. A broad spec t rum of turbulent whis t le r waves with frequencies up to a few t imes w c i is observed ahead of the shock. and the re is ample t ime f o r t rans ien ts to decay and turbulence to develop. The turbulence level is not l a rge enough to account for the dissipation purely in t e r m s of wave energy in cont ras t to the ear ly theory of Camac et al. (1962), however, a n es t imate of ion thermalizat ion due to ion scat ter ing off the waves (Pa t r i ck and Pugh 1969) as obtained f r o m Eq. ( 2 ) i n Section I1 B m a y be sufficient to account fo r the r equ i r ed dissipation. It is to be noted (Pa t r i ck and Pugh 1969) that when one compares the labora tory turbulence data with E a r t h bow shock data, the turbulence levels a r e quite similar in- dicating that whis t le r wave turbulence is an impor tan t feature of oblique col l is ionless shocks, both in the labora tory and space as d iscussed i n the next s e ction. Waves of this type have been observed (Pa t r i ck Pa t r i ck and Pugh (1967) s e e a s$.and- The The ups t r eam plasma is coll isionless A moving oblique shock has been generated f r o m a curved magnetic piston in a modified 6 pinch apparatus (Robson and Sheffield 1969). single damped whis t le r wave t r a i n out ahead of a magnet ic field compress ion is observed with overa l l s t ruc tu re of the o rde r R;. is coll isional and it is not c l ea r that turbulence has had a chance to develop. The dispers ion cha rac t e r i s t i c s and magnetic field polarizations of the wave t r a i n have been m e a s u r e d and these observations a g r e e wel l with the con- cept of a whist ler with sufficient group velocity to s tand in the ups t r eam flow. A The u p s t r e a m plasma A dis turbance propagating obliquely to the magnetic field at v e r y high number Mach number (30 < MA < l o 3 ) h a s been produced by injecting a highly ionized aluminum plasma at speeds of o r d e r l o 7 cm/sec into a weakly magnet ized p l a sma (F r i edman and Pa t r i ck 1968). velocity CA i s achieved by totally ionizing the background gas with the ultraviolet pulse f r o m the acce le ra to r discharge. stantly decelerating, and the u p s t r e a m plasma is coll isional, it is not ce r t a in that a shock has formed. that the momentum coupling of the piston with the background p lasma de- pends on the value of ambient magnetic field. Wi th no magnetic f ie ld ap- plied, the two p lasmas in te rpenet ra te f r ee ly with l i t t le coupling. When a weak magnetic f ie ld is applied ups t ream, the aluminum plasma dece ler - a tes rapidly effectively exchanging its momentum with a l l the background p lasma which it encounters. a significant magnetic p r e s s u r e o r to affect ' the motion of the aluminum ions. The nature of the momentum t r ans fe r mechan i sm is not understood. The low Alfven Since the piston is con- The main r e su l t of this experiment is This magnet ic field is too sma l l to provide -23- In t e r f e romet r i c measu remen t s indicate that the aluminim plasma- background p lasma in te r face is no thicker than the resolution capability of the diagnostic which is of o r d e r Ri - 0. 5 c m (F r i edman and Pa t r i ck 1970). The Mach number below which coupling s e e m s to take place is around one hundred. The turbulence spec t rum of the magnetic f ie ld o r density fluctuations has not been completely m e a s u r e d to date. laser sca t te r ing exper iments indicate a l a rge level of p lasma density However, fluctuations is present . .# -24- V. EARTH! S BOW SHOCK The so la r corona expands forming a radially outward flow of, part ic les , predominantly fully ionized hydrogen, which has been called the so la r wind. The so la r wind was f i r s t predicted ( P a r k e r 1958) and la te r confirmed by d i rec t measurement (Neugebauer and Snyder 1962) to have a flow velocity in excess of the Alfven speed in the vicinity of the Earth. sca le s izes l a r g e r than an ion gyroradius ( - 102 km) and a shock has been observed (Ness et al . 1964) to stand in the flow ahead of the Earth. The E a r t h ' s dipole magnetic field f o r m s an impenetrable body f o r A schematic d i ag ram of the interaction of the so l a r wind with the Earth! s dipole magnetic field in the ecliptic plane is shown in Fig. 6. The propert ies of the u p s t r e a m so lar wind va ry widely due to so l a r activity, but typical values a r e indicated i n F i g . 6; under unusual con- ditions actual values can differ by m o r e than a n o rde r of magnitude f r o m those quoted. The collision mean free path is of order the distance f r o m the E a r t h to the sun and hence the s t ruc tu re of the flow by the E a r t h is t ru ly coll isionless. the flow direction i s 450 on the average, but has a la rge var iance, cause of the wide var ia t ion in the magnetic field direct ion the shock is generally oblique, The angle between the magnetic field direction and Be- The shock sepa ra t e s the relatively quiet interplanetary magnetic The shape of the shock surface, the Mach angle and the field f ro in the m o r e compressed and turbulent magnetic field in the magnetosheath. standoff distance (Ness et al. 1964) are given v e r y well by MHD fluid equations (Sprei ter and Jones 1963; Spre i te r , Summers and Alksne 1966; Spre i te r and Alksne 1970). the presence of sca le s izes such a s the ion gyroradius (- 102km) which a r e much sma l l e r than the flow sca le s i ze ( - lO5km). ion and electron iner t ia l sca le lengths a r e Ri - 50 - 100 k m and Re - 1 - 2 km, length ( - 2 0 me te r s ) are resolvable by satel l i te measu remen t s , The justification for this fact is attr ibuted to Typical values of the Hence a l l relevant coll isionless sca le lengths including the Debye An important feature of the E a r t h ' s bow shock is that it is con- stantly in motion; this fact complicates interpretat ion of data. a r e taken in the satel l i te reference f r a m e , appropriate to a par t icular physical p rocess often depend upon the relat ive velocity between the satel l i te and the shock and/or par t ic les , occasions (Holzer, McCleodandSmith 1966; Heppner e t al. 1967) the shock appears to oscil late with maximum excursion of the order an E a r t h radius and an average shock velocity of approximately 10 km/sec , occasions (F rede r i cks et al , and might be explained by a corrugat ion traveling along the shock surface. Since data sca le lengths and frequencies On many On other 1970) the shock motion i s quite different -25- n - 3-10cm-~ IBI - 4- 7 y ( i y = I O - ~ ~ O U S S ) v - 300-700 km/sec T, - io5 OK -1oev Ti- 2 - IOev MA- 5-10 /3 - 0.3-3 ffE= 6.4~ io3 km c 5358 Fig. 6 SUN 9-- v MAG N ETOPAUSE Schematic Diagram of the Earth's Bow Shock -26- Since the typical shock velocity is in excess of the satel l i te velocity, of order 1 - 2 km/sec, meaningful data reduction requi res knowledge of the shock velocity. An average s t ruc tu re compiled f r o m many observations (Ness e t al . 1964, Heppner e t al. 1967) taken with magnetometers having a low frequency cutoff of order 5 Hz i s shown i n Fig. 7. The init ial B field gradient is observed to change within 1 - 2 seconds, the maxim- compress ion is typically reached in about t en seconds, and the field re laxes in another 30 - 60 seconds. By assuming an average velocity of 10 km/sec, es t imates for these sca le lengths a r e as shown. The init ial r i s e in B i s of the order Ri and the full shock width is 5 - 10 Ri in agreement with laboratory simulations (Pa t r i ck and Pugh 1967) and the previous theore t ica l discussion of obliquely propagating waves, The above description i s a n overal l view of the shock s t ruc tu re ; in actuality f iner scale s t ruc tu re has been observed. A broad s p e c t r u m of whist ler wave turbulence (Holzer et al. Russel, Olson and Holzer 1968; Olson, Holzer and Smith 1969) has been observed. s ince the magnetosheath is charac te r ized by fluctuating magnetic fields, As one c r o s s e s the shock f r o m the so l a r s ide, not only the amplitude of the turbulence inc reases but the shape of the spec t r a changes a s well (Olson et a l , 1969). nitude inc rease in the power spec t ra l density) cannot be explained a s just a compress ion of the interplanetary field but indicates that waves a r e being generated in the shock. a s one p rogres ses into the magnetosheath. In general , above the ion whereas in the shock and magnetosheath, the fall off i s f a s t e r , (frequency ) - 3. 1966; Smith et a l , 1967; The high turbulence level extends well into the shock s t ruc tu re Thus the increased amplitude (2-3 o rde r s of mag- The shape of the spec t r a continues t o change gyrofrequency (- 0, 2 cps) the interplanetary s p e c t r u m fa l l s off a s (frequency)' 1 roughly a s The appropriately normalized s p e c t r u m of whist ler wave turbulence measu red i n the bow shock has been compared to the s p e c t r u m measured in a laboratory simulation (Pa t r ick and Pugh 1969). power spec t r a l density of the magnetic field oscillations multiplied by the frequency and normalized to the average value of magnetic field as a function of frequency normalized to the ion gyrofrequency. The spec t rum is essent ia l ly flat below a few t imes the ion gyrofrequency. t ion is an indication that whist ler waves with a group velocity sufficiently la rge to stand in the shock front a r e present , is insufficient to provide the dissipation on the bas i s of wave energy a s i n the theory of Camac et a l . (1962), ion scat ter ing may account for much of the turbulence a s discussed in the theoret ical section. The recent higher frequency data f r o m OGO-111 (Olson et al. 1969) indicate a higher level of turbulence and this discrepancy is not understood, In Fig. 8 we show the This observa- Although the turbulence level At higher Mach numbers , much higher frequency oscillations (- l o 3 Hz) have a l so been detected on occasion (F rede r i cks et al . 1968, 1970) which a r e e lectrostat ic in nature, i. e . , t he re a r e no corresponding - 2 7 - C 5362 Fig. 7 G r o s s Magnetic S t ruc ture of the Bow Shock -28- I O7 X x \ \ X X X - X X SOLAR WIND uti= I sec-1 MA h, 8 - x PLASMA WIND TUNNEL wci= IO6 sec" MA 2.6 X \U\ OGO- I x x X X X X X B 2627 I 6' I I O IO2 _____e c i Fig. 8 Whist ler Wave Turbulence Spec t ra Obtained in the Plasma Wind Tunnel (Pa t r i ck and Pugh 1969) and the E a r t h ' s Bow Shock - 29- magnetic oscillations accompanying them, noise a r e associated with the s teep gradients of the lower frequency whist ler waves. t o be a s sma l l a s Re suggesting that the electrostat ic noise resu l t s f r o m ion acoustic instabil i t ies driven by t r a n s v e r s e cu r ren t s in much the s a m e manner as observed in low Mach number perpendicular shocks. These bu r s t s of e lectrostat ic The sca le length of these gradients in B a r e observed The observations obtained thus f a r indicate a grea t var ie ty d s t ruc tu res present in the bow shock. At a v e r y low Mach number of MA = 1, 5 (Heppner e t al. 1967), a smooth shock s t ruc tu re with l i t t le turbulence has been observed, while at higher Mach numbers (MA = 2-5) whistler wave turbulence s e e m s to dominate the shock s t ruc ture , At very high Mach numbers , MA > 5, both whist ler wave and electrostat ic wave turbulence may be present , The subject of turbulence frequency spec t r a is somewhat obscured by the fac t that none of the data has been Doppler shifted which requi res a knowledge of both the phase velocity and the wave vector , F o r example, i f the waves present in the shock front spec t ra a r e whist lers according t o Camac et al. (1962), they must have a group velocity e ual and opposite t o the so l a r wind velocity which i s of the order 400 k m P sec . F r o m the dispers ion relation for these waves, the i r wavelength i s s eve ra l t imes Ri (kRi- 1) o r of the o rde r 100 km. Thus, the Doppler shift, &-I, i s of the s a m e o rde r a s the frequencies measu red in the satel l i te f r ame . The Doppler shift can change the detailed shape of the spec t ra , but can neither shift the spec t ra t o different frequency r eg imes nor fundamentally a l t e r the physical conclusions, F o r the high frequency electrostat ic waves, the Doppler shift will likewise be comparable t o the measu red frequencies , Detailed compar ison of theory with the observations must wait until m o r e information on this problem of Doppler shifting is available. Another interesting feature of the shock is the observation of a flux of protons outward f r o m the shock (Asbridge, Bame and Strong 1968). It appears that par t of the so l a r wind is acce lera ted to higher velocities a t the bow shock and then re-emit ted forward. This forward reflection of ions i s reminiscent of the existence of a c r i t i ca l Mach number fo r per- pendicular laboratory shocks above which ion viscosi ty effects a r e im- portant a s discussed for perpendicular shocks. In conclusion, the Earth! s bow shock provides a laboratory fo r making detailed measu remen t s of coll isionless shock s t ruc ture . whist ler wave turbulence seen a t modera te Mach number for oblique shocks is similar t o that observed in the laboratory. the par t ic le velocity distribution function a r e now becoming available which will allow a n investigation of the ion thermalizat ion process , The Measurements of - 3 0 - VI. SUMMARY We have seen that shock waves exis t in an ionized fluid flowingd at super-Alfvenic speeds with a scale length much shor t e r than the collisional mean f r ee path. mine for what conditions shocks a r e thin and thus appear a s a discontinuity in the fluid description of the flow. F o r low Mach number , low p , perpen- dicular shocks, the shocks a r e very thin, of the o r d e r 10 Re , whereas for a l l other conditions, i. e . , higher Mach number , p , and/or oblique propaga- tion, the shock is thicker , on the o rde r of Ri. Since the ra t io of R i to Re i s just the square root of the mass ra t io , the thicker shocks a r e only four t imes the thinner shocks for hydrogen. F o r modera te Mach number , the quantity Ri i s comparable to the ion gyro radius (based upon flow velocity) since the ion gyro radius = MA Ri. Thus pract ical ly speaking, the thickness of coll isionless shocks is always comparable to o r l e s s than the ion gyro radius . a fluid description i s valid for the overal l flow and the shock can be t rea ted a s a discontinuity in the coll isionless fluid. s t ruc ture on the other hand, requi res a full kinetic description taking into account var ious turbulent dissipation mechanisms. A p r i m a r y goal of coll isionless shock r e s e a r c h is to de t e r - F o r flow sca l e s which a r e l a rge compared to the ion gyro radius , Investigation of the shock The dissipation which mus t be present in the shock s t ruc tu re in o rde r to satisfy the Rankine-Hugoniot conditions a r i s e s f r o m the interactions of the par t ic les with turbulent waves which a r e generated and amplified in the shock. F o r low Mach numbers , dispers ion effects determine the init ial gradients in the shock while dissipation fixes the total shock thickness. At higher Mach numbers , dispersion i s unimportant and dissipation alone de te r - mines the shock s t ruc ture . T h r e e types of turbulent dissipation have been observed and p rogres s has been made on understanding two of them. shocks below the c r i t i ca l Mach number M", ion acoustic waves a r e driven unstable by t r a n s v e r s e cur ren ts and the resul t ing turbulence sca t t e r s and heats e lec t rons . required dissipation. At Mach numbers grea te r than M', ions a r e heated a s a r e su l t of a viscous-like mechanism which i s neither well documented nor wel l understood. F o r oblique shocks, a turbulent spec t rum of whist ler waves i s observed in the shock front which appears to have a sufficient amplitude to sca t t e r the ions an amount n e c e s s a r y t o account for the required dissipation. Hence, for oblique and high Mach number perpendicular shocks, ions a r e heated direct ly in the shock. F o r perpendicular, low p This anomalous resis t ivi ty is sufficien: to account for the More work i s needed to obtain a bet ter understanding of the s t ruc ture of coll isionless shocks. measu remen t s of the turbulence which occurs in the shock front. Laboratory experiments could provide detailed Theoretical -31- work, which has been guided in the pas t by experimental observation, should p rogres s to the point where c r i t i ca l tests of theoret ical ideas can be sugges- ted. thickness and s t ruc ture on various p a r a m e t e r s e . g. , me/mi and Wce/Wpe, would be useful. and theory which should become m o r e prevalent in the future we cite the comparison made by Daughney et al. (1970) with the predicted spec t rum of Kadomtsev (1 965), and the comparison of the theoretically predictecbshock thickness made by Kra l l and Book (1969b) with the resu l t s obtained in seve ra l different experiments . As a resu l t , experiments which explore the dependence of the shock As examples of m o r e detailed comparisons of experiment The most significant advance in the study of the Ear th ' s bow shock wil l come f r o m multiple satel l i te probes which allow the separat ion of spat ia l f r o m t empora l variations and the determination of the polarizations of the turbulent f ields and wave vectors . be made of the shock thickness and turbulent wavelengths so that Doppler shifts can be inferred. Only then can meaningful measu remen t s The mechanism which leads to the anomalous ion viscosity in perpen- dicular shocks for Mach number g rea t e r than M" still r ema ins to be ex- plained. the ion heating which occurs in oblique shocks due to whist ler wave turbulence. At high Mach number (MA - 5 - 10) the relat ive importance of (electromag- net ic) whist ler wave turbulence and fine scale (e lectrostat ic) ion acoustic wave turbulence in providing dissipation needs to be cr i t ical ly assessed . F u r t h e r m o r e , the a r e a of very high Alfven Mach number where the energy contained in the magnet ic field is insignificant a s compared to kinetic energy is becoming m o r e relevant as the possibil i ty of forming e lec t ros ta t ic shocks, a topic not discussed in this review, is being actively investigated. next few y e a r s should provide a considerable increase in our understanding of the above phenomena. In addition, m o r e theoret ical work is needed in o rde r to explain The - 3 2 - REFERENCES Adlam, J. H . , Allen, J. E . , Phil. Mag . , 3 , 448-455 (1958) Alikhanov, S. G . , e t a l . , Plasma Phys. and Contr. Nucl. Fus. R e s . , I, - # 47-68 (IAEA, Vienna, 1969) Alinovskii, N. I. , Eselevich, V. G. , Koshilev, N. 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R E P O R T DATE 7 6 . T O T A L N O . O F P A G E S 7 6 . NO. O F R E F S December 1970 37 8 0 Be. C O N T R A C T O R G R A N T NO. 98. O R I G I N A T O R ' S R E P O R T NUMBER(S) F4462 0 - 70- C- 002 3 9752 - 02 b. P R O J E C T NO. '' 6 1 1 0 2 F R e s e a r c h Report 363 9 6 . O T H E R R E P O R T NO(S> ( A n y other number8 that may be ensigned I this report) d, 681308 10. A V A I L ABIL ITY /L IMITATION NOTICES 1 1 . SUPPLEMENTARY NOTES 12. SPONSORING M I L I T A R Y A C T I V I T Y A F Office of Scientific Resea rch , (SREP) 1400 Wilson Boulevard, ArlinEton. Vir- 7,2209 . . . 13. ABSTRACT A review of exper imenta l and theoret ical work on shock waves in p lasmas where c l a s s i ca l coll isions are unimportant is presented. ro le of dissipation and dispers ion i n determining shock s t ruc tu re i s de- s c r ibed and seve ra l mechanisms which can provide the requi red diss ipa- tion in col l is ionless p lasmas are discussed. shocks produced in labora tory experiments as wel l as the E a r t h ' s bow shock occurr ing natural ly in space are described. a shock m a y be considered thin s o as to be t r ea t ed as a discontinuity for flow field calculations. Experimental and theoret ical areas where m o r e work i s needed a r e pointed out. The Observations of col l is ionless C r i t e r i a a r e given when - _ Security Classification Security Classification 14. K E Y WORDS 1. Collisionless shocks 2. Shocks 3 . MHD shocks 4. P l a s m a turbulence 5. Collisionles s dissipation INSTRUCTIONS 1. 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Collisionless shocks in plasmas

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Collisionless shocks in plasmas

Abstract

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