the expansion of a plasma into a vacuum& basic … the parker [1976, ... theory and measurement...

REVIEWS OF GEOPHYSICS AND SPACE PHYSICS, VOL. 21, NO. 7, PAGES 1631-1646, AUGUST 1983

The Expansion of a Plasma Into a Vacuum' Basic Phenomena and Processes and Applications to Space Plasma Physics

URI SAMIR 1

Space Physics Research Laboratory, University of Michigan Ann Arbor, Michigan 48109

K. H. WRIGHT, JR.

Department of Physics, University of Alabama at Huntsville Huntsville, Alabama 35899

N.H. STONE

Space Science Laboratory, NASA Marshall Space Flight Center Huntsville, Alabama 35812

In this review we call attention to basic phenomena and physical processes involved in the expansion of a plasma into a vacuum, or the expansion of a plasma into a more tenuous plasma, in particular the fact that upon the expansion, ions are accelerated and reach energies well above their thermal energy. Also, in the process of the expansion a rarefaction wave propagates into the ambient plasma, an ion front moves into the expansion volume, and discontinuities in plasma parameters occtlr. We discuss the physical processes which cause the above phenomena and point toward their possible application for the case of the distribution of ions and electrons (hence plasma potential and electric fields) in the wake region behind artificial and natural obstacles moving supersonically in a rarefied space plasma. To illustrate this, some in situ results are reexamined. Directions for future work in this area via the utilization of the Space Shuttle and laboratory work are also mentioned.

1. INTRODUCTION

Phenomena involved in the expansion of a plasma into a vacuum, particularly ion acceleration and rarefaction wave propagation, were studied both theoretically and to a lesser extent experimentally in the last decade. Gurevich et al. [ 1966, 1968] were the first to show theoretically that upon the expansion of a plasma into a vacuum, ions are accelerated to high energies. While this physical process was recognized by laboratory plasma physicists, particularly by those working in laser fusion research, it went unnoticed by space geophys- icists, even though it may be one of the fundamental processes underlying many phenomena in space plasma physics and astrophysics. Recently, $ingh and Schunk [1982] used computer simulation calculations of the expansion of a plasma into a vaccum and the resulting production of energetic ions in order to study the energization of high- latitude ionospheric ions in the context of the expansion of the polar wind. They indicate that there are potentially important physical processes operative in a plasma expansion that are not taken into account by the existing steady state models of the polar wind.

The distribution of charged particles and potential (electric fields) in the wake behind an obstacle moving supersonically in a collisionless plasma is also an example of an expansion

•Now at the Space Science Laboratory, NASA Marshall Space Flight Center, Huntsville, Alabama 35812. On leave from the Department of Geophysics and Planetary Sciences, Tel-Aviv Uni- versity, Ramat-Aviv, Israel.

This paper is not subject to U.S. copyright. Published in 1983 by the American Geophysical Union.

Paper number 3R0866.

of a plasma into a void (vacuum) or into a more tenuous plasma. $amir and Fontheim [1981] performed a compara- tive theory-experiment study of the ion and electron distribution in the wakes of the Atmosphere Explorer C and the Explorer 31 satellites. The theoretical model used was based on the Parker [1976, 1977] wake and sheath steady state model. The latter is probably one of the more sophisticated and elaborate numerical models which exist at the present time. Even so, order of magnitude discrepancies between theory and measurement in the very near wake zone were found. The conclusion of that work was that the discrepancies between theory and experiment are due to the use of a steady state theory and a single ion equation using the mean ion mass (see also $amir et al. [1981]). There can be little doubt that the spatial and temporal evolution of electron and ion velocity and density distributions which take place upon the expansion of a plasma into a vacuum is directly relevant to the filling in of the wake region behind an obstacle moving supersonically in space. It is also possible that the structure (i.e., particle and field spatial and temporal distributions) of the 'dark' or 'antisolar' side region behind Venus, behind our moon, and/or in the wakes of Io and Titan is determined, at least partially, by the basic processes involved in the expansion of a plasma into a vacuum or into another, more tenuous plasma.

Furthermore, investigations relevant to the electrodynamic characteristics of satellites and large space structures may benefit from an examination of the plasma expansion processes in modeling the total current collection [$amir, 1982b].

With the advent of the Space Shuttle, including its wide range of capabilities, it should be possible to perform controlled experiments of body-plasma interactions in a manner

1631

1632 SAMIR ET AL.: EXPANSION OF A PLASMA INTO A VACUUM

not possible in the pre-Shuttle era. The study of the phenomena and the physical processes involved in the expansion of a plasma into a vacuum follows directly from the study of 'plasma-obstacle' interactions. Details of a new experimental philosophy including general outlines for practical modes of experimental operation required to achieve specific scientific objectives are given by Sarnir and Stone [1980] and Sarnir [1982a]. In addition, it would be very valuable to perform complementary experiments in the laboratory. Such experiments, which would differ from those conducted in laser fusion research, could be made more directly applicable to the expansion processes in space plasmas.

In this paper the basic physical processes and phenomena which characterize the expansion of a variety of plasmas into a vacuum are discussed in section 2. Section 3 follows with a

reexamination of some of the available in situ wake data and

a discussion of the relevance of laboratory experiments of body-plasma interactions in light of space plasma expansion processes. In section 4 we speculate on possible interpretations of phenomena observed in the interaction between the solar wind and Venus, the solar wind with the earth's moon, and the wake of Titan in terms of phenomena and processes which characterize the plasma expansion into a vacuum. Finally, in section 5 we summarize the present knowledge of the plasma expansion phenomena and processes based on theoretical studies and point toward the required in situ and laboratory simulation experiments needed to examine the present theoretical predictions.

2. EXPANSION OF A PLASMA INTO A VACUUM:

PHYSICAL PROCESSES AND PHENOMENA

In the past decade an extensive effort by plasma physicists working in the area of laser fusion research was devoted to the study of the electric fields and energy and density distributions of particles created by plasma expansion into a vacuum, in particular, the expansion of laser-created plasma from a target pellet. The study was both theoretical and experimental. However, despite the significant achieve- ments already attained, various aspects are still in a rudi- mentary state of understanding. The theoretical studies include both analytical and numerical methods for a wide range of conditions. The types of plasmas considered include (1) plasmas composed of a single electron temperature and a single ion species (see, for example, Gurevich et al. [1966, 1968] (the pioneering work in this area), Allen and Andrews [1970], Widner et al. [1971], Crow et al. [1975], Bezzerides et al. [1978], Mora and Pellat [1979], Denavit [1979], and Gurevich and Meshcherkin [1981a, b]); (2) plasmas composed of electrons with multiple temperatures and a single ion [e.g., Bezzerides et al., 1978; Denavit, 1979; Wickens and Allen, 1979; True et al., 1981]; (3) plasmas composed of a single electron temperature and multiple ion species [e.g., Gurevich et al., 1973, 1979; Gurevich and Pitaevsky, 1975; Gurevich and Meshcherkin, 1981a; Singh and Schunk, 1982; Decoste and Ripin, 1978; Felber and Decoste, 1978; Ander- son et al., 1978; Begay and Forslund, 1982]; and (4) plasmas composed of electrons with multiple temperatures and multiple ion species [e.g., Wickens and Allen, 1981]. The papers cited can be divided according to whether a fluid and/or a kinetic approach is used, whether ions are taken to be cold and/or hot, and whether overall charge neutrality or charge separation is considered. Among the papers which treat the latter case we cite Widner et al. [1971], Crow et al. [1975],

Denavit [1979], True et al. [1981], Gurevich and Meshcher- kin [1981b], and Singh and Schunk [1982]. Among the papers where quasi-neutrality is assumed throughout the expansion region we cite Gurevich et al. [1966, 1968, 1973, 1979], Allen and Andrews [1970], Gurevich and Pitaevsky [1975], Ander- son et al. [1978], Bezzerides et al. [1978], Decoste and Ripin [1978], Felber and Decoste [1978], Mora and Pellat [1979], Wicken} and Allen [1979, 1981], Gurevich and Meshcherkin [1981a], and Begay and Forslund [1982].

Detailed reviews of studies regarding the plasma expansion into a vacuum are given by Singh and Schunk [1982], Denavit [1979], and Gurevich and Pitaevsky [1975]. Hence we restrict the discussion here to basic phenomena and processes and some of the results. The plasma types to be discussed will follow the above order.

We now discuss the expansion processes by considering a semi-infinite plasma held by a diaphragm at its boundary located at x = 0 (see Figure l a). At a time t = 0 the diaphragm is removed, and the plasma expands into the vacuum. We are interested in the evolution of the velocity and density distributions of the plasma particles filling in the vacuum and the electric field they create. As the expansion begins, the electrons move ahead of the ions because of their greater thermal velocity, and some of the ions are subse- quently accelerated by the space charge electric field. A front of plasma, called the 'expansion front,' moves into the vacuum. The density of ions near this front decreases with time. A region of decreased plasma density, a 'rarefaction wave,' moves into the ambient plasma.

Electron inertia in this process can be neglected as long as the ion streaming velocity is less than the electron thermal velocity. The electric field provides continuous acceleration, although its magnitude decreases with time. As a result, the ions from the ambient (source) plasma that move to replace the ions that move into the vacuum region are exposed to a lower electric field and thus will not reach the velocity of the ions that were initially near the t = 0 plasma-vacuum interface. Indeed, it is the ions originally near this interface that attain the highest velocities.

The electron expansion can be treated as isothermal. Denavit [ 1979] showed that the assumptions of an isothermal electron expansion and the neglect of electron inertia are correct to order (ZMe/Mi) 1/2, where Z is ion charge, Me is electron mass, and Mi is ion mass. The source of the ion translational energy is the electron thermal energy. There- fore if the electron gas does not cool, then heat must flow from the ambient plasma to the expansion region. Mora and Pellat [1979] showed that at the rarefaction wave, qe = dEi/dt, where qe is heat flux and Ei is ion energy.

Some characteristic features of the expansion process can be found by solving for the ion dynamics under the assumption of charge neutrality (Ne = ZiNi). Charge neutrality removes the Debye length (as a relevant characteristic length) from the equations. Thus any functional dependence on x or t will be through the combination (x/t). Solutions of this type are commonly referred to as self-similar [Landau and Lifshitz, 1963]. An analytical solution to the cold ion fluid equations, i.e., continuity and momentum equations, is obtained by assuming a space-time dependence for density N• and velocity Vi through the variable • = x/Sot, where So = (ZTe/Mi) •/2 is the ion acoustic speed and Te is the electron temperature in energy units. The self-similar solutions for a plasma consisting of a single ionic species and an ambient

SAMIR ET AL.' EXIANSION OF A PLASMA INTO A VACUUM 1633

Maxwellian electron distribution are

Ne -- ZiNi -- ZiNo exp [-(• + 1)]

= So( + 1)

cb = -(Te/e)(•: + 1) • + 1 -> 0

The polarization electric field E = -Ock/Ox is proportional to (l/t).

Figures lb and lc show the self-similar density and velocity solutions. The rarefaction wave propagates into the plasma at the ion acoustic speed. Note that for large values ofx this theory predicts high-velocity ions with a density that approaches zero. The quasi-neutrality assumption restricts the validity of the self-similar solution to - 1 -< • < •m, where •m is determined by equating a characteristic length of the expansion, L = Sot, to the local Debye length (Te/Ne) 1/2. The •rn -- [2 In (oopit) -- 1], where t > •opi -• and 60pi is the ion plasma frequency. Namely, for values of • > •m the potential due to the self-similar solution [•b = -(Te/e) (• + 1)] is not valid. The time required for the ions to respond to the polarization electric field and produce a plasma flow with Ne = ZiNi is given by •ovi -•. In other words, only after the ions respond to the rapid electron expansion and create a quasi- neutral plasma flow are the self-similar solutions valid. Singh and Schunk [1982] show through their computer simulation computations, which are based on the Poisson equation, that the above conclusion is indeed correct. The assumed Boltz-

mann distribution for electrons will remain valid as long as the time required for the electrons or ions to cross the expansion region is shorter than the expansion time. The velocity at • = •m can be interpreted as the velocity of an ion front, Vr, moving in the expansion region with Vr = 2S0 In (%,•t). The energy of the ions at the front is then given by

!N {t=O) No (a)

VACUUM

RAREFACTION WAVE

• N

No

EXPANDING PLASMA

(b)

Vi=0 •'/ -Sot

•V i / (c) "•-LINEARLY INCREASING V i

Fig. 1. The expansion of a plasma into a vacuum. (a) Initial condition. (b) The evolution of density according to the self-similar solution. (c) The evolution of ion velocity according to the self- similar solution.

E = « ZTe(•m + 1) 2 = 2ZTe [In topit] 2

The energy per charge is then

E/Z = 2Te[ln toeit] 2

where %i = (4rrZe2No/Mi) v2. All ions with the same mass- to-charge ratio will be accelerated in the same manner.

In the pioneering work of Gurevich et al. [1966, 1968], numerical serf-similar solutions were found using a kinetic approach, i.e., the Vlasov equation for ions (further details are given in the appendix). Comparing the results from the kinetic approach and the results from the cold ion fluid treatment shows that including ion temperature smooths out the weak discontinuity at • = -1 (see Figure lb) and introduces differences in the ion density values in the rarefaction region. As the ion temperature Ti increases, the difference between the two approaches increases. In the expansion region at large • (or large x), changing the Te/T• ratio was shown to introduce small differences between the

two approaches. The reason for these small differences is that at large • the ambient ion distribution function evolves to a streaming delta-function-like distribution. The effective ion temperature was found to vary as exp (-2se). Therefore the ion dynamics in the expansion region for large • can be reasonably well described by using the cold ion fluid equations. This is an important physical conclusion, since it specifies a condition (i.e., distance in space) where the fluid

solutions based on the cold ion momentum equation (see also appendix) can be applied.

A comparison between studies using the self-similar approach and those using numerical computer simulations, which drop the assumption of charge neutrality and use the potential determined from the Poisson equation, was performed more recently by Denavit [1979] and Singh and Schunk [1982]. In Denavit's [1979] study, both single and double electron temperatures were considered. It was found that the effect of charge separation is to produce (among other local effects) an ion front (sometimes called an 'expansion front'). In the region between the expansion front and the rarefaction wave some of the general predictions of the self-similar theories are applicable. In other words, in this region the numerical solutions are in accord with those obtained by the self-similar approach. As mentioned earlier, a simple way to describe the range of applicability (say, in x or t) of the self-similar solution vis-h-vis the solutions obtained by considering the Poisson equation is to say that the self-similar solutions are valid for times t which satisfy t -> o¾• -•. This is the time it takes the ions to respond to the fast expansion of the electrons and create a quasi-neutral plasma flow.

Smaller density gradients existing at t = 0 (as compared to the large gradient shown in Figure la) affect the expansion process by increasing the time required for the expansion to

1634 SAMIR ET AL' EXPANSION OF A PLASMA INTO A VACUUM

become self-similar [Felber and Decoste, 1978; Singh and Schunk, 1982].

The case of an expansion of a dense plasma into a more tenuous plasma was also treated by Gurevich et al. [1968] and Gurevich and Pitaevsky [ 1975]. The boundary condition imposed by the second plasma population existing at t = 0 for x > 0 (see Figure la) adds to the variety of phenomena that occurs in the expansion process. Limited acceleration of ions is a feature of this physical situation. Depending on the properties of the second plasma population, there can be trapping of ions in potential wells, excitation of a two-stream electrostatic instability, and jump discontinuities or shock waves occurring when the two plasmas have highly dissimi- lar ion temperatures [Gurevich and Meshcherkin, 1981b]. These jump discontinuities imply the existence of charged sheets moving with constant velocity.

In describing the expansion of a plasma consisting of one ionic species and two electron temperatures we consider the case where the ambient density of the colder electron population (Neo c) is much greater than the density of the hot electron population (Neon). During the early stages of the expansion the ion acceleration is determined by the cold electron component, because Ne½ > NeH. The ion velocity Vi is given by Vi • Sc(• + 1), where • = x/Sct, Sc = (ZTeC/Mi) 1/2, and Te c is the temperature of the cold electrons. When Ne n > Ne c, the hot electron component will control the ion motion according to Vi • Sie(• + 1), where • = x/Snt, Sn = (ZTen/M•) m, and Te n is the temperature of the hot electrons. The energy spectra of the ions will then have two peaks, one proportional to Te c and the other proportional to Te n . The rarefaction wave will propagate at the acoustic speed determined by the cold electron component.

Now consider the case of a two-ion, one-electron component plasma. From the space plasma point of view, this case is of great practical value for planetary ionospheres and magnetospheres. The dynamics for the two-ion plasma depend on the similarity parameter 3' = Z2M•/Z•M2. For the case where the major ionic constituent is hydrogen (M•) and the minor constituent is oxygen (M2), T • 1, each species is found to behave in a self-similar manner at large s e [Gurevich et al., 1973; Anderson et al., 1978; Singh and Schunk, 1982]. Moreover, Gurevich et al. [1973] and Gurevich and Mesh- chefkin [1981a] point out the existence of oscillations in the expansion region and conclude that they should always be present in a two-ion component plasma. The velocity of M1 (H +) is V• = S0(s e + 1) and that of M2 (O +) is V2 - S0(s e + 3') for s e >> % and the energies of the ions at s e- sero are El/Z1 = «Te(•m + 1) 2 and E2/Z2 = « (Te/T) (•m -{- 7) 2, respectively. Hence E2/Z2 > El/Z1 at • = •m. Gurevich et al. [1973] give an expression for the percentage of the total number of 'impurity ions' (i.e., the minor ionic constituent) passing through a unit surface at the point x = x0 having an energy greater than some reference energy as

where p = (Z22M•/2Z1M2), from which it follows that 0.1% of the ions acquire an energy E > E0 = 50p Te. Ifp becomes too large, i.e., p >> 1, then the assumptions used to derive NE>E0 break down.

If •/> 1, we have a plasma where oxygen (M0 is the major constituent and hydrogen (M2) the minor constituent. The relatively immobile oxygen initially provides an additional electric field to accelerate the hydrogen ions. As a result, the hydrogen ions gain a higher initial velocity than in the •/< 1 case. As one might expect, the hydrogen density becomes comparable to the oxygen density at some value of •. A 'plateau region' in ion density, velocity, and potential is obtained, which implies constant values for density, velocity, and potential near the location where N• (O +) = N2 (H+). Although there are quantitative differences in the description of this plateau region by the quasi-neutral (self-similar) treatment of Gurevich et al. [ 1973] and the charge separation (computer simulation) treatment of Singh and Schunk [1982], the gross qualitative effects remain similar. Beyond the plateau region the hydrogen behaves as in the expansion of a one-ion component plasma. Oscillations are seen behind the ion front. These are more pronounced in hydrogen for Te > Ti. Oscillations have also been seen in the spectra of laser pellet interaction plasma [Decoste and Ripin, 1978]. The hydrogen velocity for values of • greater than the plateau region approach that given by the self-similar solution, V2 = S0(• + ¾m). So the energy is

E2/Z2 = « (re/T) (•m + T1/2) 2 ate= •m

The oxygen velocity is given by V• = So(• + 1) in the expansion region for 1 -< • < •m.

Finally, we consider the expansion of a plasma composed of two ion constituents (Z•, M•; Z2, M2) and two electron temperature populations (Te c, Te n) where the ambient cold electron population (Nco) is much greater than the hot electron population (Nno), Z•Nlo > Z2N20, and ,/> 1. As the expansion begins, the initial electric field is predominantly determined by the cold electrons and the ion constituent with the greater charge density, in this case, Z•N•. So the higher Z/M constituent, i.e., Z2/M2, is preferentially accelerated. As the expansion continues, there will be a spatial region where many of the cold electrons will be reflected from the self-consistent electric field set up by the hot electrons. In this spatial region the charge density of the Z2/M2 constituent begins to exceed the charge density of the Z•/M• constituent. The remainder of the expansion is then determined by the hot electron component and the Z2/M2 constituent. The energy spectra of each ion constituent will then have two peaks, one proportional to Te c and the other proportional to Te n . The rarefaction wave moves toward the ambient plasma at a speed determined by the cold electron temperature and the lower Z/M constituent, i.e., (ZiTeC/MO •/2.

Figure 2 summarizes the ion energy for the four types of plasmas discussed above, and the analytic self-similar solutions are quoted. Figure 3 [after Singh and Schunk, 1982] is introduced in Figure 2 because the velocity solution for the self-similar case [Gurevich et al., 1973] was shown to differ with the charge separation computer simulation result [Singh and Schunk, 1982] for the indicated region of space.

In summary, the self-similar and computer simulation computations have shown the following effects when a plasma expands into a vacuum:

1. Ions are accelerated to high energies. 2. A rarefaction wave is created which propagates into

the ambient plasma.

SAMIR ET AL.' EXPANSION OF A PLASMA INTO A VACUUM 1635

3. An ion front (shock) moves into the vacuum region. 4. Excitation of instabilities and plasma waves over

certain volumes in space take place. 5. Strong (or jump) discontinuities in the plasma occur at

the expansion front.

The quantitative significance and intensity of the above phenomena and processes depend, in part, on the specific ionic constituents of the plasma and the relative concentration of the minor ion in the plasma, on the ambient electron temperature, on the ratio of a characteristic linear dimension

PLASMA TYPE

ONE ION SPECIES (M 1, Zl). (1) ONE ELECTRON DISTRIBUTION

(TeL

ONE ION SPECIES (M 1, Z1). TWO ELECTRON DISTRIBUTIONS

(2) (Tell ' TeC) ' FOR N•o >> Ne• o AND 1.5 < [TeH/TeC] < 9.

TWO ION SPECIES (N•o ' M1, Zl) '

(3) (N2oi, M2, Z2) FOR N1o 2o

ONE ELECTRON DISTRIBUTION

(TeL

(4)

TWO ION SPECIES (N•1o ' M1 ' Zl),

(N• o , Z2) FOR ZiN•o > Z2N•o. M2.

TWO ELECTRON DISTRIBUTIONS

(No, T/). (N: ø , T,")

ION ENERGY EXPRESSION

•x : E1 . «Te(•+l)2 Sot Z 1

X E 1 IF N, c > Nell AND • -Sct : • ' :• TeC (• + 1)2 c x E 1 IFNe H •N e ANDS- SH t : ---- « Te H(•+I) 2 ß Z 1

X

•-•- IF •<1: E 1 «Te(•+l) 2 Z1

E2 - « Te (• +,•)2 Z2

E1 )2 IF'y•.l: •- •Te(•+l Z1

FOR ,• '"" 7 «' E2 ß •-SEEFIG. 3 FOR

Z2 ?

H X IF NeC •N e ANDS- set: E 1 .. • Te C(• + 1)2

Zl

E2 =_ « TeC (• + 1) 2

AND X E 1 - E2 =, « Te H • + 1) 2 z2

WHERE' So = ( ZlTel «= ION ACOUSTIC VELOCITY; X = SPATIAL COORDINATE; M1

t - TIME; M 1, M 2 = IONIC MASSES; Z 1, Z 2 - ION CHARGE NUMBERS;

Ne o, No I - AMBIENT VALUES OF ELECTRON AND ION CONCENTRATIONS;

Z2M1 Sc-( ZlTeC)•, S H = (ZITeH) « T e - ELECTRON TEMPERATURE, ?- ZIM2 , M1 M1 . Fig. 2. Analytical ion energy expressions for self-similar solutions for each of the four plasma types.

1636 SAMIR ET AL.' EXPANSION OF A PLASMA INTO A VACUUM

4110 _ ,,,• -

-800 -400 0 400 800 1200 1600 2000

_ _

Fig. 3. Evolution of the H + and O + drift velocities according to the self-consistent computer simulation computations (solid curves). The dashed curve (s-s) is the self-similar solution for O +. f•o(a) is the normalized ion drift velocity for sp•ecies a (• H +, O +) defined by Po(a) = f_•+•l?f, d•/f_•+•f, dV, where V = V/Vr(H +) and Vr(H +) is the hydrogen thermal velocity; • is the normalized distance, equal to X/XDi(H+), where XDi(H +) is the hydrogen Debye length; ! is the normalized time equal to t%,i(H+), where %,i(H +) is the hydrogen plasma frequency. The computations were made for the case of No(H+)/No - 0.1, No(O+)/No - 0.9, T(H+)/Te - 1, and T(O+)/Te - 1, where No is the total ambient ion density. The figure is after Singh and Schunk [1982].

to the Debye length, and on the density gradient at the plasma-vacuum interface.

3. PLASMA EXPANSION INTO A VACUUM:

THE CASE OF PLASMA WAKES

As mentioned in the introduction, the processes and phenomena involved in the expansion of the plasma are of interest to space and cosmic plasmas research, e.g., flow of plasma out of stars, solar physics (flares), and flow interactions with artificial 'obstacles' orbiting the earth and planets as well as flow interactions with planets and their natural satellites. Even so, this subject went unnoticed by most of the space physics community.

It should be noted that the expansion phenomena and physical processes can be adequately examined via the analysis of the density and energy distributions of electrons, ions, and potentials (or electric fields) in the wake region behind 'artificial obstacles.' For example, the mechanisms responsible for ion acceleration can be studied using spacecraft or 'test bodies' orbiting in the terrestrial (or planetary) environment, as well as in laboratory experiments.

3.1. The Expansion: An Ion Acceleration Mechanism

In the brief discussion given below we show some theoretical results for ion acceleration which in situ and laboratory experiments should attempt to examine. Figure 4 [after Gurevich and Pitaevsky, 1975] shows examples of computed ion distribution functions for a single ion and electron plasma expanding into a vacuum. Here

•= Mi = •-- • So 2 ]/2

and g and u are nondimensional quantities defined by

•Mi/ and

u = V = --2 -1/2 So

where No is ambient density, f is the ion distribution function, Ti and Mi are ion temperature and mass, respectively, and V - Vx is the velocity in the direction of the x axis (see Figure 1). As seen in Figure 4, the distribution functions for the ions (in [g, u] coordinates) differ for different values of ß (see also appendix). For ß < 0 (e.g., ß - -3 or -2 in Figure 4) the distribution function g - F(u) is the unper- turbed distribution, whereas for ß > 1 the distribution narrows to delta-function-like shapes, which physically implies that ions are being accelerated. For a plasma with two ion components and one electron temperature distribution the distribution functions g - F(u) for various values of ß are obtained for the two differing ionic constituents. Figure $ shows examples of distribution functions for an O+-H +- electron plasma where H + is taken to be the minor ion, i.e., N0(H +) << N0(O +), where No is the ambient ion plasma density.

From this figure it follows that the H + ions are accelerated to much higher velocities than the O + ions, which are the main ion component of the plasma. Generally, the maximum acceleration of the minor ion depends on the values of Z, Mi, and Te of the ambient plasma and on the ratio (Ro/ho) (where R0 is the characteristic length and ho is the ambient value of the Debye length) and on the relative ambient concentration of major to minor ion constituent, e.g., No(O+)/No(H+). We note that when 3/ - MiZ2/M2Z• > 1 (where M and Z represent the ionic mass and charge, respectively, and the subscripts 1 and 2 indicate the major and minor ions), the minor ions are initially accelerated much more than the major ions. For this case the concentration N2 decreases with increasing • much slower than for the case with 3/< 1. For even larger values of ß the minor ion concentration exceeds that of the major ions. This can be seen in Figure 6,

g

1.o

0.8

'T = --2

0.4

I

Fig. 4. The variation of the normalized distribution g with the normalized velocity u for different values of ?. The plasma consists of one ionic species and one electron distribution: g = (2,rTi/Mi) 1/2 No-if, u = o(2Te/Mi) -1/2, and r = x/t(2Te/Mi) -1/2, where x, t, and v are distance, time, and velocity, respectively. The figure is after Gurevich and Pitaevski [1975].

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0

SAMIR ET AL.' EXPANSION OF A PLASMA INTO A VACUUM 1637

which depicts different distribution functions for different values of •'. The results of Figure 6 are for ambient conditions (N2oZ2/N•oZl) = 0.1. It appears [Gurevich et al., 1973; Singh and Schunk, 1982] that for •' > •'k (where •-k is a specific critical large value of •') the distribution function hardly varies with increasing •-; i.e., a characteristic plateau region is obtained whose width is a function of the charge-to-mass ratio. Beyond the plateau the plasma expands like a single- ion species plasma. Both Gurevich et al. [1973] and Singh and Schunk [1982] also study the opposite case, where N0(O +) << N0(H +).

3.2. Reexamination of Satellite Wake Measurements

A reexamination of relevant results available at present from in situ satellite wake observations was therefore performed. Unfortunately, we find the available results to be meager, fragmentary, and applicable only to the very near wake zone. Hence theory-experiment comparison is limited. This unfortunate situation stems from the following reasons: (1) No experimental probe complements were ever designed setting the phenomena involved in the plasma expansion as a scientific objective. Hence the available relevant data are basically by-products of traditional geophysical observations. (2) The available data are fragmentary, since it was not always possible to gather sufficient reliable measurements to compose the required ensemble of plasma and body parameters. (3) The available relevant measurements are limited (a priori) in their spatial and temporal extent, because most observations were made by probes which were either flush mounted on the surfaces of the satellites or mounted on

relatively short booms. Hence, at best, only very near wake measurements could be examined.

Despite these shortcomings and limitations some indirect experimental findings can be used for a partial theory- experiment analysis.

A quantity often used as an indicator of filling-in processes in the wake is the ratio (le,i(wake)/le,i(ambient)), where e and i represent electrons and ions, respectively. This quantity

N o (0 +) •N o (H +) TWO IONS

ß =- 1 2 =3

I I -4 -2 0 2 4 6 8 u

gl ß r = 1 r=2

1- -- -2

1-o=

-4 -2 0 2 4 6 8 u

Fig. 5. The variation of the normalized distribution functions g] (O+), g2 (H +) with the normalized velocity u for different values of •-. The plasma consists of two ionic species and one electron distribution for the case N•o >> N2o, where N]o and N:o are the ambient densities of the two ionic species. The figure is after Gurevich et al. [1973].

g2. t= -2 r = 0

-2 0 2 4

I T=0 T=2

--2 0 2

;=2 T=4

8 10 U

I I 6 8 u

No+ (r) • NH+ (r)

Fig. 6. The variation of the normalized distribution functions g• (O+), g2 (H +) with the normalized velocity u for different values of •-. The plasma consists of two ionic species and one electron distribution for the case Z2N2o/Z•N•o = 0.1. The figure is after Gurevich et al. [1973].

was determined experimentally from in situ measurements made by a variety of probes on board the satellites Explorer 8, Ariel 1, Explorer 31, Atmosphere Explorer, and U.S. Air Force S3-2. A comparison with various wake models was attempted. Such comparisons, as mentioned earlier, were limited to the very near wake zone. 6eneral agreement with Gurevich_et al. [1970, 1973] was reached only for specific situations and for the spatial regions located on the edges of the wake zone and when it was assumed that the influence of

the accelerated H + ions becomes dominant even though their relative concentration in the ambient plasma was very small [Sam& et al., 1973, 1975; Gurevich et al., 1973; Al'pert, 1976]. It is in the wake edge regions where the self- similar approach is valid. However, since no relevant energy spectrum information was available from any of the above satellites, it was not possible to examine directly the acceleration of ions. Moreover, since all available in situ measurements are limited to the very near wake region, it is not possible to reexamine the spatial distributions in the overall wake region. There is one exception where the electron angular distribution profile was obtained at a distance of SRo (where S is the average ion acoustic Mach number and R0 is the radius) from the center of the Ariel 1 satellite [Henderson and Sam&, 1967] and compared with results of Gurevich et al. [1970] (see also Al'pert [1976]) and the degree of agreement is very good. The limitation in this case is, of course, that the data were obtained for electrons and not for ions.

It should be noted that for the practical case of the wake of an ionospheric satellite both N0(H +) >• N0(O +) and N0(H +) << No(O +) cases are of interest, since the orbit of a standard ionospheric satellite will pass through a plasma which satis- fies both situations and intermediate ones. Unfortunately, none of these interesting phenomena can be examined via the presently available in situ wake observations. The in situ results given in Figure 7 [Sam& et al., 1979] for (l+(wake)/l+(ambient)] = f[Te, No(O +)/No(H +)] are of interest, since they show the quantitative influence of the ambient electron temperature and relative ionic composition (ambient) on the ion depletion in the wake. While the latter quantity is related to wake filling-in processes, the results of Figure 7 standing on their own cannot yield direct information on the ion distributions as a function of •-for further

distances in the wake or yield unambiguous information regarding the acceleration of ions. The results of Figure 7


0.4

0.3

I

R = [0.1 - 0.2]ß R = [1-1.510 R = [4-5]V R = [lO] © • = [lO 2] ß

-R= [103]1 ß

R = [0.1.-.0.2]

I I

-1.5]

500 1500 2500 3500

T e (AMBIENT) "

Fig. 7. Variation of normalized ion current a with ambient electron temperature Te for various values of [N(O+)/N(H+)] = R. [after Sam& et al., 1979].

could be used together with other in situ results (from the Space Shuttle, for example) when they become available (see also Samir and Stone [1980]).

Gurevich et al. [1973], Gurevich and Pitaevsky [1975], Gurevich and Meshcherkin [1981a, b], and others discussed the excitation of ion plasma waves, ion acoustic instability, two-stream instability, strong or 'jump' discontinuities on the ion expansion front, and the dependence of some of the phenomena on the initial density gradient between the plasma and the vacuum. The overall situation is complicated, and no in-depth analysis will be given here. However, the excitation of ion plasma oscillations caused by the

motion of a satellite through the terrestrial ionosphere was inferred for the first time by Samir and Willmore [1965] through an analysis of Ariel 1 satellite measurements. These oscillations were found to exist at least at the edges of the wake of the satellite. While this result is in accord with the

above mentioned theoretical predictions, this in situ evidence cannot be seen as conclusive. Preliminary results obtained recently from the wave experiment [Shawhan and Murphy, 1982; S. D. Shawhan, personal communication, 1982] from the Plasma Diagnostic Package on board the Space Shuttle flight STS-3 showed the existence of 'electrostatic noise generated in the orbiter wake at frequencies near the ion plasma frequency (50 KHz) and below in the ion acoustic mode' [Shawhan and Murphy, 1982]. If the above preliminary result can be correctly interpreted as indication of excited waves due to the motion of the Space Shuttle, then further support is provided to the earlier result of Samir and Willmore [ 1965] and is in accord with theoretical predictions [Gurevich and Meshcherkin, 1981b; Gurevich et al., 1973, 1970].

Samir and Wrenn [1972] through their analysis of the angular distribution of electron temperature Te around the Explorer 31 satellite found that Te(wake) > Te(ambient). This is shown in Figure 8. The authors speculated that this enhancement is probably due to a heat transfer process which takes place through wave-particle interactions in the potential 'well' which exists in the wake behind the satellite and/or to instabilities. Troy et al. [1975] analyzed in situ measurements from another probe mounted on the same satellite. They confirmed the earlier finding that there exists an electron temperature enhancement in the very near wake region and that the enhancement is not due to some instru- mental effect. We should note that no such enhancement was

found at a distance of Z = 5R0 downstream in the wake of the Ariel 1 satellite [Henderson and Samir, 1967]. A possible interpretation for the latter result is that the temperature enhancement is limited to distances downstream, Z, which satisfy Z < SR0, where S is the ionic Mach number, which was S = 4 for the above experiment. It should be noted that the Te measurements at Z - 5R0 represent average values. Gurevich and Meshcherkin [1981b] attempt to explain this temperature enhancement. They claim that the 'region of

4000 -

3000

2000

510 550 600 650 700 750

ALTITUDE (km)

ANGLE OF ATTACK RANGE

180 + 15 ø , 180 + 30 ø

0+ 30 ø 0 + 60 ø

Fig. 8. Variation of electron temperature with altitude for several angle of attack ranges [after Samir and Wrenn, 1972].

SAMIR ET AL..' EXPANSION OF A PLASMA INTO A VACUUM 1639

maximum rarefaction' in the wake of an obstacle in a

streaming plasma has a sharp and not a diffuse boundary which may lead to sharp discontinuities in the plasma properties, among which is the electron temperature. More- over, behind the body the converging beams of the flowing plasma collide [e.g., Stone et al., 1972; Stone, 1981a, b, and references therein] and can lead to the excitation of ion acoustic waves. Since Landau absorption of such a wave is done mainly via the electrons it is possible that this absorption causes the enhancement in the [Te(wake)]. Despite the above reexamination of the electron temperature data it is clear that more experimental evidence is needed in order to establish the existence of the excitation of plasma oscillations (mainly ion oscillations) and the instabilities in the wake of an obstacle moving supersonically in a rarefied space plasma. If the electron temperature behind an obstacle can indeed serve as an indicator of the existence of waves/in-

stabilities in the wake region, it would be useful to determine (experimentally and theoretically) the exact conditions for the existence of such temperature enhancements.

We have reexamined some of the Ariel 1 results obtained

from measurements made by a guarded planar electron probe which was mounted on a boom [Henderson and Samir, 1967]. The probe measures the angular electron distribution at a distance of about Z = 5Ro (Ro is the satellite radius) from the center of the satellite downstream in the wake. Figure 9 shows the variation of the normalized electron current Ie/Io with angle of attack 0. The result can be interpreted as indicating the existence of a 'trailing shock' or 'propagating rarefaction wave.' This is depicted by the structure of the normalized current at the angles of attack 0 = 120 ø, 240 ø. Details on how this plot was obtained are given by Henderson and Samir [1967]. Of course, it would have been more useful to have ion density and composition measurements even for this single location downstream, but such is unfortunately not available. In order to establish unambiguously the existence of the rarefaction wave, similar angular variation profiles are needed for more than one location downstream. This is not available at the present time. As will be shown in the next section, laboratory experiments do indicate the existence of a rarefaction wave propagating into the ambient plasma, as predicted by theory.

Recently, Samir and Fontheim [ 1981] performed a theory- experiment comparison for the angular distribution of the normalized ion current around the Explorer 31 and the Atmosphere Explorer C satellites (see Figure 10). For the maximum wake zone (0 > 150 ø) a discrepancy develops between theory and experiment, increasing to about 2.5 orders of magnitude at 0 = 160 ø, where [I(Theory)/Io] = 0.97

1.0

0.6 ß MORE THAN ONE

MEASUREMENT

I I I 24O

0

Fig. 9. Variation of normalized electron current [le/10] with angle of attack 0 [after Henderson and Sam&, 1967].

,. ts• I I I I I I (9o ø )

.9

.8

.7

.6

.5

.4

.3

.2

.1

0

90 ø 100 ø 110 ø 120 ø 130 ø 140 ø 150 ø 160 ø

Fig. 10. Variation of normalized ion current [1+(0)/1+(90ø)] with angle of attack 0. The dashed curve represents in situ measurements. Iteration 0 represents the neutral approximation (where ions are treated as neutral particles). Iteration 15 represents the self- consistent solution. The figure is after Samir and Fortrheim [1981].

x 10 -5 and [I(meas)/Io] = 0.58 x 10 -2. Details are given by Samir and Fontheim [1981]. The theoretical model used was that of Parker [1976, 1977], which solved the steady state, Vlasov-Poisson equations for a single ion self-consistently. Although the exact cause of the discrepancy is not clear at present, it was suggested [Samir and Fontheim, 1981] that this discrepancy can most likely be removed by the use of the time-dependent equations written separately for the various ionic species.

In summary, we note that while the amount of presently available in situ information regarding the expansion of a plasma into a vacuum through satellite wake studies is meager, fragmentary, and limited, some results obtained in the past through the parametric analysis of the amount of current depletion in the near wake, electron temperature enhancement in the wake, etc., seem to be in accord with theoretical predictions.

3.3. About the Relevance of Some Laboratory Wake Experiments

Laboratory investigations of the expansion of a plasma into the void downstream from test bodies in collisionless

plasma streams and of the propagation of the corresponding rarefaction wave into the ambient plasma have been carried out by Hester and Sonin [1970], Fournier and Pigache [ 1975], Stone et al. [ 1978], and others. In the investigation by Stone et al. [1978] both spherical and cylindrical test bodies were used. The plasma stream conditions were such that the ion acoustic Mach number S and the Debye ratio Re re- mained constant while the normalized test body potential •b was varied over a wide range of values.

Figure 11 shows tranverse ion current density profiles taken at 2 and 3 radii downstream from the maximum cross-

sectional area of the spherical test body. Similar transverse profiles were obtained at several distances downstream from the various test bodies. In each case the distance that the

leading edge of the rarefaction wave had propagated away


I -13 -4 -2 0 2 4

[X/R o]

I •oo = 2.0

i •

-13 -4 -2 0 2 4

[X/Ro]

Fig. 11. Variation of normalized ion current density (I/1o) with normalized transverse distance (X/Ro) at two normalized distances (Z/Ro) downstream from a conducting sphere [after Stone et at., 1978].

from the wake axis and into the ambient plasma stream, A W, was determined. As seen in Figure 12, the variation of A W(S/Ro) is a linear function of Z/Ro, which shows that the leading edge of the rarefaction wave propagates into the ambient plasma stream at the ion acoustic speed. This result is in agreement with an earlier theoretical treatment of body- plasma electrodynamic interactions by Martin [1974].

The data in Figure 12 were obtained for several different test body geometries and for a wide variety of applied test body potentials. The fact that the data from all of these cases fall on the same line shows that the rarefaction wave is not

affected by body geometry or by the applied body potential. It is apparently dependent only on the characteristics of the plasma stream and is generated by plasma moving into the void region swept out by the test body. This is analogous to the theoretical case treated by Gurevich et al. [1966, 1970], where a plasma occupying a half space was released at time to and allowed to expand into the vacuum of the remaining half space. In experimental studies of the type conducted by Stone et al. [1978] the plasma density gradient is created at time to by the motion of the test body through the plasma, and at all subsequent times the ambient plasma expands into the void left in the wake of the test body. Since in the reference frame of the test body the rarefaction wave propagates away from the wake axis at the ion acoustic Mach angle, 0s = sin -l (l/S); in the reference frame of the plasma stream the rarefaction wave propagates into the ambient plasma from the initial void-plasma interface at the ion acoustic speed, as predicted theoretically.

Another, presumably permanent, feature of the wake under certain conditions is the enhancement of electron

temperature in the very near wake region. This finding was first reported by Samir and Wrenn [ 1972], as discussed in the previous section. In the laboratory, similar results were reported by Oran et al. [1974] and Shuvalov [1979, 1980]. The [Te(wake)/Te(ambient)] values from the laboratory experiments significantly exceed the in situ results. There is not yet a clear physical explanation for the heating process that causes this enhancement. Samir and Wrenn [1972] and Gurevich and Meshcherkin [1981b] speculated that the enhancement is probably due to a heat transfer process which takes place via wave-particle interactions in the potential well which exists [Gurevich et al., 1970; Al'pert, 1976] in the

wake and/or via the excitation of ion acoustic waves which

are absorbed (Landau absorption) by the electrons. Intriligator and Steele [1982] reported interesting results

from experiments performed at the University of Southern California's Astrophysical Plasma Laboratory. Although the experiments are basically similar to those reported by Hester and Sonin [1970], Fournier and Pigache [1975], Stone et al. [1972], Oran et al. [1974], and Stone [1981a, b, c], they differ in that the ionic constituent of the synthesized plasma was H + with an energy of the order of KeV. Hence the Intriliga- tor and Steele [1982] results represent the case of the interaction between an obstacle (sphere) and a high-energy plasma, whereas the earlier experiments represent the case of low-energy, plasma-body interactions. Intriligator and Steele [1982] suggest that their experiments may be more realistically related to the interactions of the high-energy solar system and astrophysical plasmas with planetary, lunar, and astrophysical objects.

Intriligator and Steele [1982] indicate that strong fluctuations in the current occur on the edges ('transition region') of the wake, that these fluctuations occur at a low frequency, and that these phenomena are a direct result of the body- plasma interaction. It is very tempting to claim similarity between the above findings and the in situ results of Samir and Willmore [1965] and Henderson and Samir [1967], related to fluctuations on the edges of the wake of the Ariel 1 satellite and the excitation of ion plasma waves in the frequency range of a few kilohertz. However, we feel that at present such a claim is highly speculative. It should be noted that Intriligator and Steele [1982] do not report on the propagation of a rarefaction wave expanding into the ambient plasma or on the enhancement in electron temperature in the near-wake region.

It should be noted that the laboratory data available at present, with the exception of the work of Eselevich and Fainshtein [1980], do not deal explicitly with the ion acceleration mechanism or with the specifics of discontinuities as discussed by Gurevich and Meshcherkin [1981a, b]. More- over, most of the laboratory work done until now does not

25-

O /' ,• -25

• OPEN - SPHERE • CLOSED - CYLINDER (ñ)

.,•'• •b = -3.8 ESTIMATED ERROR: • I I I I I

0 5 10 15 20 25

[Z/R o]

Fig. 12. Variation of the normalized propagation distance of the ion rarefaction wave away from the wake axis [(AW/Ro)S] with normalized distance downstream (Z/Ro), where $ is the ion acoustic speed [after Stone et at., 1978].


deal with cases where Ro (-- Ro/ho) >> 1, which may be of greater relevance to space applications. However, laboratory studies have shown the creation of a rarefaction wave which propagates at the ion acoustic velocity, as predicted by theoretical treatments of plasma expansion, and clearly, specialized experiments can be designed to study aspects of the ion acceleration process and the theoretically predicted strong discontinuities and oscillations.

4. THE EXPANSION OF A PLASMA: PROCESSES AND

PHENOMENA OF POTENTIAL INTEREST TO SOLAR WIND

INTERACTIONS WITH 'PLANETARY OBSTACLES'

4.1. A Few Comments Regarding the Wake of Venus

The depletion of particles in the boundary layer mentioned in the recent review paper by Russell and Vaisberg [1983] may perhaps be connected with the acceleration of ions into the wake of Venus upon the expansion of the postshock ionosheath/magnetosheath plasma. In any case, fluctuations in velocity [Russell and Vaisberg, 1983] are possible in the rarefaction wave (or rarefaction shock) region. It is also possible that predictions based on viscous interactions [e.g., Perez-de-Tejada, 1980] can be alternatively seen in light of the discussion given in this paper regarding the region which is in the proximity of the plasma-vacuum interace and the location of the onset of the rarefaction wave. In this region the self-similar approach is valid, as shown theoretically by Singh and Shunk [1982] and as could be inferred from theory-experiment comparisons [e.g., Gurevich et al., 1970; Sam& et al., 1975]. The slowing down and cooling (or heating) of ions approaching the center of the wake [Russell and Vaisberg, 1983] should be examined in depth through the processes involved in the 'expansion of the plasma into a vacuum. '

Jumps in flow properties such as density, velocity, and potential are in general accord with some theoretical predictions discussed earlier (see also Gurevich and Meshcherkin [1981a, b]), particularly the flows in the wake with properties which are different from those of the external flow. Jumps in the flow properties at the boundary of the wake are of particular interest and perhaps directly relevant to the phenomena and processes involved in the 'plasma expansion.' Although the planetary origin of these ions may complicate the issue, it is worthwhile examining the findings in light of the latter processes in the wake, ignoring the question of particle origin. From our earlier discussion it follows that higher-energy accelerated ions should exist in the wake, while their concentration varies with location downstream.

Ion acceleration associated with magnetic field fluctuations [Russell and Vaisberg, 1983] may be correlated with the rarefaction wave region. If the clouds observed by Brace et al. [1982b] are created outside the ionopause of Venus (see Figure 13), an examination of the nature of the clouds (overall location, energy, etc.) vis-h-vis the discussion of phenomena involved in the plasma expansion, in particular in the wake edge regions may be worthwhile. Moreover, it may not be unfounded to consider the energization of ions from the Venusian ionospheric 'holes' [Brace et al., 1982a; Grebowsky and Curtis, 1981] through the process of plasma expansion. This may, perhaps, provide another relevant accelerating mechanism. Reports on far-wake measurements

_

of particles and fields are given by Russell et al. [1981] and Mihalov and Barnes [1981, 1982]. Mihalov and Barnes [1982] have surveyed the plasma observations from the Pioneer

-• øo • . •n•Hi•• '•e _• CLOUD ZONE

DUSK SUN--; - • --180• VENUS WAKE•

• 120 IONiAUSE

o• • • e • o

Fig. 13. Location of plasma clouds around Venus [after Brace et al. 1982b].

Venus Orbiter during the first series of orbits that intersected the planet's wake in the region 8-12 Rv (Rv is the Venusian radius) downstream behind the planet. Their results, con- trary to those of Venera 9 and 10, do not point toward a well- defined plasma cavity which narrows with increasing distance from the planet and which terminates at -<3-4 Rv. Overall they find the wake region to vary strongly in space and time and to display turbulence. They also discuss the energy spectra (intensity and shift) for H+-O + in the wake and the origin of the O + ions.

We suggest that in addition to the interpretations given by Mihalov and Barnes [ 1982] it might be useful to examine the energy and shift of the particle spectra in terms of phenomena (particularly acceleration mechanisms, instabilities, and wave-particle interactions) involved in the expansion of a plasma into a vacuum for the case of a two-ion plasma with one and/or two electron distributions discussed in section 2.

One possible explanation of the existence of O + in the Venusian wake is that the neutral oxygen which extends above the ionopause on the dayside is being ionized by photoemission and charge exchange processes and then convected down the Venusian tail. Mihalov and Barnes

[ 1982] state that such an explanation is in accord with plasma measurements in the region near the planet and in the wake. However, they state that the thermal speed of these O + ions is much smaller than the magnetosheath flow speed. Al- though a possible explanation of the latter was given in terms of a cooling process, it is not impossible that the O + ions in the Venusian wake are caused by the plasma expansion processes discussed in this paper.

Recently, Intriligator and Scarf [1982] compared particle and wave measurements in the Venusian ionosheath. They found continuously changing ion distributions and corresponding enhanced plasma wave activity. They also found ion acoustic waves generated by plasma instabilities associated with the changing plasma distributions and predict rarefaction and compression of the ionosheath. The observed enhancements in plasma waves were related to interpenetrating ion beams. More details regarding the ion

1642 SAMIR ET AL..' EXPANSION OF A PLASMA INTO A VACUUM

population in the Venusian wake (at 11.5 Rv) are given by Intriligator [1982]. A point of interest here is the statement by Intriligator and Scarf [ 1982] that the results from Venus and Titan suggest that the interaction of a nonmagnetic object with a streaming plasma may produce high turbulence levels, in agreement with the recent laboratory results of Intriligator and Steele [1982].

The orientation of planetary O + fluxes and magnetic field lines in the Venusian wake were discussed by Perez-de- Tejada et al. [1982]. A result of this study is that the direction of motion of the O + ions is uncorrelated with

changes in the direction of the magnetic field vector. This may indicate that [E x B] pickup processes are not sufficient to account for the acceleration and the direction of motion of

the ions and that wave-particle interactions associated with turbulence processes are called upon.

The discussions here and in section 3 clearly indicate the existence of common signatures in the following interactions between a streaming collisionless plasma and a nonmagnetized obstacle: solar wind-Venus, streaming laboratory plasma-target body, and spacecraft-ionosphere. The understanding of common processes will undoubtedly lead toward a unified approach in treating collisionless space plasma- body interactions.

4.2. A Few Comments Regarding the Wake of Titan

It is difficult at present to comment meaningfully on the direct application of our discussion to the case of Titan's wake. However, speculations pointing toward additional directions of thought in interpreting this part of the Voyager 1 fly-by observations may not be unwarranted. As mentioned recently by Gurnett et al. [1982], Titan can interact either with the magnetosphere of Saturn or with the solar wind depending on its orbital position and the position of the magnetopause. If the interaction is with Saturn's magnetosphere, then the flow regime for the interaction is qualita- tively similar to that of an artificial satellite moving in the terrestrial ionosphere/magnetosphere. On the other hand, differences between these cases are due to plasma corotation and to the fact that Titan has a substantial atmosphere. In this respect there is a similarity with the interaction of Venus with the solar wind or, to a lesser degree, the interaction of a comet with the solar wind. However, it may be possible to consider aspects of our discussion in the interpretation of the wave experiment measurements [Gurnett et al., 1982] for the 'low-frequency noise.' It is also possible that the question of the 'slow-mode shock' mentioned by Gurnett et al [1982] in the context of the low-frequency noise is a signature of a 'propagating wave' or a 'trailing shock.'

It is tempting to speculate that the structure of the electron density observed on the edges of Titan's wake is of the kind known to occur in satellite-ionosphere interactions [Hender- son and Samir, 1967]. At present, nothing more definitive can be said. However, if and when more in situ and laboratory measurements relating to body-plasma interactions become available, it would be possible to support or oppose the above speculation.

4.3. A Few Comments Regarding the Lunar Wake

Another kind of body-plasma interaction which takes place in the solar system is that of the solar wind with the moon. The moon has neither an intrinsic magnetic field nor an atmosphere. Hence the solar wind interacts essentially

with the surface. In the present study, seeking 'model unification' for some wake structure in terms of phenomena typical of the expansion of a plasma into a vacuum, we examined some of the moon's experimental wake results [e.g., Lyon et al., 1967; Ness et al., 1968; Serbu, 1969; $iscoe et al., 1969] and theoretical results [e.g., Michel, 1968; Wolf, 1968; Whang, 1968a, b, 1969; Moskalenko, 1972; Lipatoy, 1976]. We find that the existence of a region depleted of charged particles in the very near wake zone was established in qualitative accord with results from satellite ionosphere interactions [e.g., Samir and Willmore, 1965; Samir, 1981] and from laboratory simulation data [e.g., Stone and Sam&, 1981].

Siscoe et al. [ 1969] investigated the distribution of normalized flux in the near lunar wake. They found the wake to be depleted of charged particles, while the edges of the wake showed fluxes larger than the ambient values. It should be noted that the 'leading edge' of the disturbance as it spreads out downstream from the moon, mentioned by Siscoe et al. [1969], is at the location of the rarefaction wave associated with the plasma expansion as discussed in section 2. Rele- vant laboratory results are given by Podgorny et al. [1975] and Dubinin et al. [1977]. Theoretically, regions of rarefaction, recompression, and the existence of an inner shock were predicted [Wolf, 1968; Michel, 1968].

The semiquantitative theoretical diagrams show the main features of the flow in the lunar wake to be in line with our

knowledge from laboratory work and, to a degree, from in situ work. However, we did not find any in situ measurements or discussion which directly relate to the ion-accelerating mechanisms due to the expansion of the plasma into a vacuum. It should be noted that the approach of Michel [1968], Wolf[1968], and Siscoe et al. [1969] is conceptually similar to that of Gurevich et al. [1966, 1968] for the regions where a self-similar approach holds, including the region between the rarefaction wave and the 'plasma free region' [Gurevich and Pitaevsky, 1969, 1971].

A particle approach rather than a fluid approach (as taken by Wolf[1968], Michel [1968], and Siscoe et al. [1969]) was adopted by Whang [1968a, b, 1969], Moskalenko [1972], and Lipatoy [1976]. However, neither approach provided any significant information regarding the acceleration of ions in the wake. A review of lunar wake theoretical studies is given by Sprieter et al. [1970], and the question of the validity of each approach was discussed by Ness et al. [1968] and Dryer [1968].

We believe that an in-depth reexamination of available lunar wake measurements (particles and fields) is worthwhile, particularly in light of the basic phenomena and processes involved in the expansion of the solar wind into the lunar wake ('dark side'). The results from such a study may undoubtedly help in the understanding of plasma-body interactions in space plasma physics.

5. SUMMARY AND FUTURE STUDIES

The fact that phenomena such as ion acceleration, excitation of plasma oscillations, propagation of rarefaction waves and ion fronts, creation of strong and weak discontinuities in the plasma parameters, plasma instabilities, and turbulence are all caused by processes involved in the expansion of a plasma into a vacuum makes this area of plasma physics very interesting but quite difficult to study. However, we are dealing with processes and phenomena which are of funda-

SAMIR ET AL.: EXPANSION OF A PLASMA INTO A VACUUM 1643

mental scientific interest with relevant applications to both laser fusion and space plasma research. This was recognized by laser fusion researchers, and an extensive effort, both theoretical and experimental (but mainly theoretical), has been devoted to this area in the past decade. Unfortunately, the importance of the complex of phenomena and physical processes involved in the expansion of space plasmas into a vacuum, particularly to solar system plasma phenomena, and the possibility of studying them via the interactions of space plasmas with natural and artificial 'obstacles' in space went almost unnoticed by the space geophysics community.

While the existence of rarefaction waves and possible trailing shocks was discussed in the context of the lunar wake and, to a lesser extent, in the context of the Venusian wake, there was no overall comprehensive and systematic study or discussion along the general lines shown in summary in Figure 14.

We hope that the discussion given in this review will be seen as a step toward a unified approach in dealing with the interaction between an obstacle and a space plasma, particularly the extremely complicated wake region. Specific practical situations may require variability in the significance and intensity of specific processes, but there are undoubtedly basic processes and permanent features which are relevant to a wide range of interactions.

Even the state of in situ investigations of the basic processes relevant to space plasma physics for the practical case of spacecraft-ionosphere interactions is still not well understood. An in-depth, comprehensive reexamination of measurements from spacecraft-ionosphere, solar wind- moon, and solar wind-Venus interactions, together with relevant available results from laboratory studies, should constitute a first stage aimed toward a unified approach to the understanding of plasma-obstacle interactions in space plasma research. The structure of the wake, the more complicated region of the interaction, could be largely understood through the phenomena and processes of the expansion of a plasma into a vacuum.

While the above reexamination is essential, it will not suffice for gaining an overall knowledge and understanding of the spatial and temporal structure of the wake region of the interaction. More in situ and laboratory experiments supported by computer simulations and semianalytic, semiquantitative theoretical work will be needed.

Measurements directly relevant to the study of the expansion of a plasma into a vacuum can be performed partly in laboratory simulation studies and via in situ measurements utilizing the Space Shuttle. It would be very valuable to conduct laboratory experiments (different from those orient- ed toward laser fusion research) suitable as much as possible to realistic situations met in space plasma physics. This can be done through the study of wakes. Although it is often difficult to generate, in the laboratory, synthetic plasmas and conditions which are exactly identical to those which exist in space, it may not always be essential to do so. This depends on the scientific objectives of the study. If the major objective is to seek physical understanding of processes and cause and effect relationships, then there may be no need to seek exact scaling between laboratory and space. There can be no doubt that laboratory studies are of scientific importance and have potential applications to space plasma physics. From our present physical understanding, it is possible to speculate that some features observed in the wakes of

EXPANSION OF A PLASMA INTO A 'VACUUM'

(SUMMARY)

(A) PHENOMENA/PROCESSES

(1) ION ACCELERATION IN THE 'VACUUM' REGION.

(2) RAREFACTION WAVE (SHOCK) PROPAGATION INTO THE AMBIENT

PLASMA REGION.

(3) ION FRONT MOVES IN THE DIRECTION OF EXPANSION

(IN THE VACUUM),

(4) EXCITATION OF PLASMA OSCILLATIONS AND INSTABILITIES, OVER CERTAIN VOLUMES.

(5) STRONG ('JUMP') DISCONTINUITIES IN PLASMA PARAMETERS

AT THE EXPANSION FRONT.

(B) THE ABOVE DEPEND ON:

(A) SPECIFIC IONIC CONSTITUENTS OF THE PLASMA.

(B) RELATIVE CONCENTRATION OF IONS IN THE PLASMA.

(c) AMBIENT ELECTRON TEMPERATURE,

(D) DENSITY GRADIENT AT THE PLASMA-VACUUM INTERFACE.

(E) RATIO OF CHARACTERISTIC LENGTH TO AMBIENT •'D' Fig. 14. Expansion of a plasma into a vacuum: phenomena and

processes.

bodies inserted in laboratory streaming plasmas may be permanent features for body-plasma interactions at large. After all, basic physical processes are not necessarily bound- ed by specific plasma and body properties. Their significance and intensity may vary with specific situations but not necessarily their basic existence. Moreover, employing the principle of 'qu,•litative scaling' [e.g., F•ilthammar, 1974; Samir and Stone, 1980] may be sufficient in many cases [e.g., Podgorny and Sagdeev, 1970; Podgorny et al., 1975; Podgorny and Andrijanov, 1978; Andrijanov and Podgorny, 1975; Dubinin et al., 1979, 1981; Stone and Samir, 1981]. The common belief that the exact Vlasov scaling laws have to be adhered to if we are to reflect from laboratory work to space is not necessarily applicable.

The availability of the Space Shuttle and its extensive capabilities make it possible to study the expansion of a plasma in situ through a series of well-conceived controlled experiments. To achieve this goal, relatively small instru- ment packages (i.e., simple small satellites) could be ejected from the Space Shuttle to measure the rarefaction waves, converging streams, energy spectrum of ions, and the variety of discontinuities in the plasma properties which occur in the interface between the ambient plasma and the plasma in the wake. The wake into which the plasma expands would be generated by 'test bodies' such as inflatable balloons, teth- ered spheres, and spheres and/or cylinders mounted on booms. The test bodies can have different surface properties (e.g., different conductivities), sizes, and geometry. It is possible to examine the characteristics of the plasma in the wake for different ratios of [N(H+)/N(O+)] as well as for different ratios of [Ro/ho] of interest in space plasmas. The target bodies selected can be nonmagnetized, magnetized, and bodies surrounded with an atmosphere/ionosphere. A

1644 SAMIR ET AL' EXPANSION OF A PLASMA INTO A VACUUM

detailed discussion of some possible experimental modes of operation and shuttle flight configurations is given elsewhere [Sam& and Stone, 1980].

Finally, we submit that studies of body-plasma interactions, particularly wake studies, provide an excellent frame- work through which the basic physical phenomena involved in the expansion of a plasma into a vacuum can be investigated and the physical processes examined.

APPENDIX: THE BASIC EQUATION TREATED BY GUREVICH ET AL. [1966, 1968, 1973] AND GUREVICH AND PITAEVSKY [1975] UNDER THE ASSUMPTION OF QUASI-NEUTRALITY

The plasma is described by the kinetic equation for the ion distribution function f:

Of Of e O6 Of --+ v ..... 0 (1) Ot Ox Mi Ox

and by the Poisson equation

024, Ox 2 • = -4rre(Ni- Ne) (2)

Equation (9) can also be written in the form

Og 10g dqbN (u-r) =0

Or 20u dr (10)

= = In = g 1/'•- •--• 4•N re No ( rr/3) ri Equation (9) or (10) is the basic equation treated by Gurevich et al. [1966, 1968, 1973] and Gurevich and Pitaevsky [1975].

For t -< 0, x --> -o• the plasma is not disturbed (ambient), while for x --> +o• the plasma vanishes. If it is assumed that the undisturbed plasma obeys a Maxwellian distribution, then the boundary conditions of the 'basic equation' (9) or (10) are r---> -o•, g ---> exp (-/3u 2) and r---> +o•, g ---> 0.

For large r > 0 (which correspond to large x) the ions are strongly accelerated; hence their thermal motion can be neglected. In this case, one deals with the continuity and momentum equations

ONi O • + • (Nivi) = 0 (11)

Ot Ox

where Mi is ionic mass, •b is electrostatic potential (E = -Ocb/Ox), and

Ni= f•o, fdv (3) Assuming quasi-neutrality, (2) reduces to

Ni = Ne (4)

and Ne is given by

Ne = No exp (ecb/Te) (5)

Hence

e cb = Te In (Ni/No) (6)

Substituting in (1), we obtain

•+ v .... In fdv =0 (7) Ot Ox Oo Mi Ox

As mentioned in the text, after a time t when quasi- neutrality is reached, the motion can be treated in the self- similar approach, and f = f(x/t; v). Note that the element of length ('characteristic length') was eliminated.

Introducing the parameters

r=-- .= V g=[ Mi J fNo-' t

(8)

the equation

...... In g du = 0 (9) (u r) Or 20u dr

is obtained for the nondimensional ion distribution function

g. Note that (9) does not impose any requirement of f. (Note that r and s c (given in the text) satisfy s c = r(2) 1/2.

[Ot Ox /l A discussion of the relation and analogy between the

Riemann solution [Landau and Lifschitz, 1963] for simple waves in ordinary hydrodynamics and the self-similar solutions for the collisionless kinetic equation for the case of a quasi-neutral plasma [Gurevich et al., 1966, 1968, 1973; Gurevich and Pitaevsky, 1975] is given by Gurevich and Pitaevsky [1969]. The breaking of the simple wave in the kinetics of a rarefied plasma is discussed by Gurevich and Pitaevsky [1971]. In the latter paper the stability of the self- similar solutions of the kinetic equation in a rarefied quasi- neutral plasma was investigated, and the existence of un- damped ion acoustic oscillations was demonstrated. Moreover, a class of distribution functions for which solutions (stationary solitary waves) exist, even if the thermal motion of the ions is considered, is shown.

Acknowledgments. U. Samir acknowledges the support of the NRC/NAS Associateship Office, the interest of R. Manka, the Program Administrator, and the hospitality of the Space Science Laboratory at NASA/MSFC. K. H. Wright acknowledges support from NASA under contract NAS8-33982.

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(Received January 28, 1983; accepted May 6, 1983.)

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