Numerical calculations relevant to the initial expansion of the polar wind

JOURNAL OF GEOPHYSICAL RESEARCH , VOL. 87, NO. All, PAGES 9154-9170, NOVEMBER 1, 1982

Numerical Calculations Relevant to the Initial Expansion of the Polar Wind

NAGENDRA SINGH AND R. W. SCHUNK

Center for Atmospheric and Space Sciences, Utah State University, Logan, Utah 84322

The collisionless expansion of an H+-O + electron plasma into a vacuum was studied because of the relevance to the polar wind. A systematic parameter study was conducted by varying the ion composition, the initial ion-electron temperature ratio, and the scale length of the density gradient at the plasma-vacuum interface. The effect of gravity on O + was also simulated. The character of the expansion is different depending on whether H + is the major or minor ion. When H + is the major ion, the density profile in the expansion region is always concave. On the other hand, when H + is a minor ion, a distinct plateau forms in the H + density, H + drift velocity, and electrostatic potential profiles. The longer the density-gradient scale length at the plasma-vacuum interface, the longer it took for plateau formation. However, after plateau formation, the expansion proceeds nearly in an identical fashion regardless of the initial density-gradient scale length. When O + was fixed to simulate gravity, the lower end of the plateau region remained fixed at the location of the initial plasma-vacuum interface, whereas when O + was allowed to expand the H + plateau region moved along with the moving O + density front. For all cases, energetic H + and O + ions were observed in the expansion region. This suggests that the energization of ionospheric ions through the process of plasma expansion could be one of the mechanisms for creating the energetic ion population of ionospheric origin in the magnetosphere. Such a process would operate over the entire high-latitude region, not just on auroral field lines. Since the largest density-gradient scale length at the plasma-vacuum interface that was considered is much less than typical ionospheric scale lengths, the numerical results cannot be directly applied to the polar wind. However, they illustrate the basic physical processes occuring in the polar wind expansion.

1. INTRODUCTION

Numerous theoretical models have been developed over the last decade to describe the polar wind, including hydrodynamic models [Banks and Holzer, 1968, 1969a, b; Maru- bashi, 1970; Banks, 1973; Bailey and Moffett, 1974; Banks et al., 1974, 1976; Strobel and Weber, 1972; Raitt et al., 1975, 1977, 1978a, b; Schunk et al., 1978; Ottley and Schunk, 1980], hydromagnetic models [Holzer e! al., 1971], kinetic models [Lemaire and Scherer, 1970, 1971, 1972, 1973; Le- maire, 1972] and models based on generalized transport equations [Schunk and Watkins, 1979, 1981, 1982]. On the basis of these model studies, the plasma flowing up from the polar ionosphere into the magnetosphere should have the following characteristics: (1) The composition is dominated by the light ions H + and He+; (2) the H + flux should vary from 107-5 x 108 cm -2 s -1 and the He + flux from 105-107 cm -2 s -1, depending on the geophysical conditions; (3) The ion temperatures should be less than 8000øK, i.e., less than 1 eV; (4) The H + temperature distribution should be anisotropic with T• > Tñ for supersonic H + outflow and T• < Tñ for subsonic outflow; (5) The H + velocity distribution should be asymmetric, with an elongated tail along the magnetic field in the upward direction; and (6) The electron temperature distribution should be anisotropic if the H + outlfow is supersonic.

Although significant progress has been made in under- standing the dynamics and energetics of the polar wind, to date there have been no time dependent model studies of the polar wind. However, during the last two decades, studies involving the expansion of a plasma into a vacuum have been

Copyright 1982 by the American Geophysical Union.

Paper number 2A1059. 0148-0227/82/002A- 1059505.00

conducted, and these studies have a direct bearing on the polar wind.

Guterich et al. [1966, 1968] were the first to show that such an expansion can create very energetic ions. Since these initial studies, the problem has been widely studied, particularly in connection with laser-fusion research [Dena- vit, 1979; Guterich et al., 1973, 1979; Allen and Andrews, 1970; Widner et al., 1971; Crow et al., 1975; Wt'ckens et al., 1978; Bezzerides et al., 1978; Wickins and Allen, 1981; Mora and Pellat, 1979; True et al., 1981] Some of these studies considered the expansion of a plasma with multi-ion species and a two-temperature electron population, which is relevant to the plasma expansion from a laser-irradiated pellet (see, for example, Wickens and Allen [1981] and True et al. [ 1981 ] and references therein).

Despite the fact that Guterich et al. [ 1968, 1973] noted that the production of energetic ions in expanding plasmas could be relevant to both space physics and astrophysics, the energization of magnetospheric ions of ionospheric Origin has not been considered in connection with the expansion of the ionospheric plasma along geomagnetic field lines. How- ever, the polar and trough winds are related to the same physical mechanism occurring in a plasma expansion. In the previous polar wind studies, steady state conditions were assumed so that the continual acceleration of ions, as suggested by Gurevich et al. [1966], was not seen.

In this investigation, we studied the expansion of an H +- O + electron plasma into a vacuum in order to simulate flow conditions in the polar wind. We varied the relative concen- tration of O + and H+, considering cases where H + was both a major and minor ion. We also varied the initial electron-ion temperature ratio, and we considered different scale lengths for the density gradient at the plasma-vacuum interface. In addition, we simulated a gravitationally bound O + plasma. However, it should be emphasized that our results cannot be

9154

SINGH AND SCHUNK.' INITIAL EXPANSION OF THE POLAR WIND 9155

No

VACUUM

X=-L I X=O X= L 2

Fig. 1. The numerical simulation region. At t = 0 the region -L• -< x -< 0 is filled with a plasma, while the region 0 -< x -< L2 is empty. The boundary at x = L2 is completely absorbing, while at x = -L• the plasma density is kept fixed at all times to simulate the effect of a plasma reservoir.

directly applied to the polar wind owing to the small spatial extent of the plasma simulations. Nevertheless, our results indicate that there are physical processes operative in a plasma expansion that are not taken into account in the existing steady state models of the polar wind, and yet they could be important.

Satellite observations indicate that the ions in the magnetosphere of ionospheric origin are much more energetic than those in the ionosphere [Baugher et al., 1980; Horwitz and Chappell, 1979; Singh et al., 1982]. The energization of these ionospheric ions can be explained in terms of wave-particle interactions [Sharp et al., 1977; Ungstrup et al., 1979; Whalen et al., 1978; Klumpar, 1979; Papadopoulos et al., 1980; Singh et al., 1981; Chang and Coppi, 1981]. However, in this paper we suggest an additional mechanism for the energization of ionospheric ions that is connected with the outward expansion of the topside, high-latitude ionospheric plasma along open geomagnetic field lines.

The plan of the paper is as follows. In section 2 we review some theoretical work on the expansion of a plasma into a vacuum. The numerical technique used in our simulations is described in section 3. Section 4 deals with the presentation of numerical results. The discussion of the results and

conclusions are given in section 5.

2. A REVIEW OF PLASMA EXPANSION INTO A VACUUM

Figure 1 shows a schematic of the initial plasma density configuration that has been frequently used in studies involving expanding plasmas. At t = 0, the half-space x < 0 is filled with a semi-infinite, electrically neutral collisionless plasma. For t > 0, the plasma is allowed to expand into the vacuum and the subsequent temporal evolution is followed. To date, a number of theoretical studies, both numerical and analytical, have been conducted in order to elucidate the basic physical mechanisms involved in the plasma expansion. Plasma expansions have been carried out for a range of conditions, including both single and multi-ion plasmas, both hot and cold ion temperature distributions, and two-electron temperature distributions. In the paragraphs that follow, we briefly review those studies that are relevant to our work.

2.1. Single Ion Plasma

The simplest treatment of the plasma expansion is obtained by assuming that the electrons always stay in an equilibrium with the developed electric fields; thus they

obey the Boltzmann distribution,

Ne(x) = ZiNo exp [ecb(x)/kBTe] (1)

where Ne denotes the electron density, No is the ion density in the unperturbed plasma, Te is the electron temperature, kB is the Boltzmann constant, Zi is the ion charge number, and & is the electrostatic potential.

Assuming that the ions are cold, the ion continuity and momentum equations reduce to

ONi O • -I- • (NiVi) = 0 (2)

at Ox

o vi o vi Ze ocb + Vi = (3)

at Ox Mi Ox

where Vi is the ion flow velocity, Mi is the ion mass, and Ni is its density.

If quasi-neutrality is assumed, the set of equations (2) and (3) allow self-similar solutions, which depend only on the ratio (x/t) of the independent variables x and t;

Ne = ZiNi = ZiNo exp [- (• + 1)] (4a)

Vi = Cs(• + 1) (4b)

qb = - (kBTe/e)(• + 1) (4c)

for (s c + 1) >- 0, where s c = (x/Cst) is the self-similar variable and Cs = (keTe/Mi) •/2. For s c + 1 < 0, the plasma remains unperturbed.

The most fascinating aspect of the above solution is the possibility of ion acceleration, because Vi "• (x/t). This acceleration is caused by the polarization electric field,

d4• •sTe • E= - •m(•+ 1) (5)

dx e Cst

where H is the step function. Note that at a given time E is a constant for x > -Cst.

Figure 2 shows the characteristics of the self-similar solution for the collisionless expansion of a single ion plasma

N

R'AEEFACTION WAVE NDING PLASMA

Cst !

-%t o

- x

Fig. 2. Self-similar (S-S) solution for the expansion of a plasma into vacuum. (top) S-S density profile. For t > 0 a rarefaction wave propagates into the region x < 0 as the plasma expands into the vacuum. (middle) S-S electric field, which is uniform in the region -Cst -• x < o•. (bottom) S-S ion drift velocity profile. Note the linear increase in Vi with x. At a given x, Vi decreases as t-•

9156 SINGH AND SCHUNKi INITIAL EXPANSION OF THE POLAR WIND

into a vacuum. As the expansion proceeds, a rarefaction wave propagates into the plasma at the ion-acoustic speed. The density profile in the expansion region is concave at all times, and the associated polarization electrostatic field does not vary with position, but its magnitude decreases inversely with time (see equation (5)). Because of this electric field ion acceleration occurs, and some very energetic ions exist far from the expanding plasma front.

The basic assumptions in deriving the self-similar solutions given by equations (4a)-(4c) are (1) the ions are cold, (2) the electrons obey the Boltzmann distribution, and (3) charge neutrality prevails (Ne = ZiNi). The first assumption was examined by Gurevich et al. [1968], who obtained self- similar solutions by using the ion Vlasov equation in place of the continuity and momentum equations. The major modifi- cation due to the nonzero ion temperature is introduced near s • = -1; the discontinuity at x = -Cat is smoothed out.

In the work of Denavit [1979] and Mora and Pellat [1979], the assumption that the electrons obey the Boltzmann distribution has been examined. This work showed that the self-

similar solutions are valid as long as

x(Mi) 1/2 • = Cat << ZMe (6) Physically, this condition implies that as long as the electron transit time across an expansion region with the electron thermal speed is much shorter than the expansion time with a speed Ca, the Boltzmann distribution is justified.

We now examine the third assumption about charge neutrality, which is valid when the scale length for the variation of the potential L >> hal, where hd is the Debye length. The scale length L from the self-similar solution (4c) is easily found to be

L--- Cst (7)

Since ha --• (re/Ne) 1/2 and Te remains nearly constant while Ne varies as given by (4a), we have

V,0 hd= exp [(s e + 1)/2] (8)

O)e0

where V• is the electron thermal speed and OOeO is the electron plasma frequency in the undisturbed plasma. The condition for validity of the self-similar solution can be found by equating (7) and (8),

V,0 Cd = •exp [(sea + 1)/2] (9a)

O)e0

which gives

•:a + 1 = 2 In [wiot] (9b)

where O)i0 is the ion plasma frequency in the undisturbed plasma. Note that (9b) is the same as (36) of Mora and Pellat [ 1979], except for a correction term due to the deviation from the Boltzmann law (1). We note that for values of s • approximately in excess of s•a given by (9b), the potential given by the self-similar solution (4c) is not valid. Owing to the limitations imposed by equations (6) and (9b), the self-similar solutions have a spatial-temporal restriction that can be expressed as -Cat -< x -< [21n (•oiot) - t] Cat. Therefore, the self-similar solutions are valid over a finite range of x for t •> •oi0-1. Physically, this is the time it takes the ions to respond

to the rapid electron expansion and produce a quasi-neutral plasma flow. It is only after this overall quasi-neutrality has been achieved that the self-similar solutions are valid. Later

we show that our numerical calculations based on the

Poisson equation, which do not require the assumption of quasi-neutrality, support this result.

2.2. Multi-Ion Plasma

The expansion of a collisionless plasma with several ion species was studied for the first time by Gurevich et al. [ 1973]. The desired self-similar solutions for the expansion of such plasmas were obtained for various approximations. Since their work has a direct bearing on our calculations, we briefly summarize their important results.

These authors have considered specific examples, in which the plasma contains two types of ions. The major ion has a mass Ml, charge number Z1, and number density in the unperturbed plasma Nl0. Similarly, the minor ion has the corresponding parameters M2, Z2, and N20. Note that N20 < Nlo.

When the minor ion density is sufficiently low, the electrostatic potential is determined by the major ion (equation (4c)) and the trace of a few impurity ions are accelerated in the potential. Then, the density and flow velocity of the minor ion are given by

O•1(•1)

N2(s•1)- N20 (V/2- Z _ 1)(Z- cq(S•l)) (10)

1 $2(•l ) = 'V/•Cal •' + Z • O•1(•1 ) (11)

where el(S•l) is the solution of equation

q- 011 -- Z In (el) + Z In

(12)

where the upper and lower signs in (11) and (12) depend upon whether Z > 1/V• or Z < 1/V•. s•l is the self-similar parameter for the major ion,

•1 = (13) Call

Z = Z2M1/21/2ZiM2 (14)

and Cal is the acoustic speed Ca for the major ion. Note that •1 differs from the self-similar parameter r of Gurevich el al. [1973] by a factor of V• (s e = V•r).

For very large values of Sel >> Z, el -• 0 and the asymptotic results for N2 and U2 are given by

Z1 M2 ] N2 = N20 2 ZI M2 exp --•1 1 (15) Z2 Mi Z2 Mi

U2 = Cal(•l+g2 MI) Z--• M2 (16) Equation (15) shows that when ZiM2/Z2M1 > 1, the density of the minor ion falls faster with increasing Sel than that of the

SINGH AND SCHUNK.' INITIAL EXPANSION OF THE POLAI• WIND 9157

major ion, which is given by

N1=N10exp[-scl- 1] (17)

which is the same as (48). Thus, the impurity ion stays minor. An example of such a situation is when the major ion is H + and the impurity ion is O +.

On the other hand, when Z1M2/Z2M1 < 1, the minor ion density decays slower than that of the major ion, so for sufficiently large values of •1, N2 > N1, and hence, the minor ion becomes the major ion at some distance in the expansion region. As a matter of fact, at that distance the plasma essentially becomes a single ion plasma and further expansion proceeds as discussed earlier according to (1)-(3). However, the boundary conditions for the further expansion are not the same as those imposed by the undisturbed ambient plasma. Gurevich et al. [1973] used the solutions (15) and (16) as boundary conditions for the continued expansion and showed that

N2 N2k = N20 N/• 1 = m exp [- 1 - •ll/V•Z] (18)

U2 = S2k = (•1/ q- •/•Z) Csl •ll • •1 • •lu and

N2 = N2k exp L[ Z2M1 (sol- •lu) •1 > •lu

U2 = Csl •1 + [Z1M2 •l > •lu where

(19)

(20)

(21)

•ll = V'2 [y- In [N20Z2 (1 - 2-1/2Z-1)] (228)

y = 2In [(N10Z1) 1/2 + (NloZ1 + N20Z2(1 - 2-1/2Z-1)) 1/2]

+ 2 NloZ1 q- N20Z2(1 - 2-1/2Z-1) -- 2-1/2 (22b) •lu = •ll + V•Z (1 - 2-1/4Z-1/2) (23)

Note that •ll is obtained by imposing the condition that N1 --• 0 and SClu is obtained by equating (19) and (21).

We now briefly point out some salient features of the above solutions given by (18)-(23). Equations (18) and (19) show that the minor ion density and flow velocity are constant over the range •ll < •1 • •lu. Thus, a plateau forms in the spatial distributions of the physical quantities. The width Ascl of the plateau region can be derived from (23);

A•I = •lu -- •ll = X/2-Z (1 - 2-1/4Z-1/2) (24)

Thus, the width depends only on the charge to mass ratios (see equation (14)).

For so1 > so1,, the expansion occurs as though the plasma is a single-species ion plasma. For example, (21) can be easily reduced to

+ (25)

where sc2 is the self-similar parameter for the ions of type 2; •2 = x/Cs2t. When Z2 = Z1, (25) reduces to the form in (4b).

In section 4, while presenting our numerical results, we shall check the validity of the above analytical results.

3. SIMULATION TECHNIQUE

The numerical calculations in this paper were carded out by solving the Vlasov equations for the different ion species by the method of characteristics [Singh, 1980; Sakanaka et al., 1971]. If f, is the velocity distribution function for ions of species o•, this equation takes the following form:

Of. Of. q. • + V + E = 0 (26) at Ox M. O V

where t is time, V is velocity, x is the spatial coordinate, E is the electric field, q. and M. are the charge and mass of the ions. We assumed that the electrons obey the Boltzmann distribution. The electric field is given by the Poisson equation,

OE 1 (xz• q.N. _ eNe ) (27) where N. is the density of ion species a, Ne is the electron density, e is the magnitude of the electronic charge, and e0 is the permittivity offree space. The ion density N. is given by

N.= f-•o• f. dV (28) Using (1) and (27), it is possible to eliminate the electron density from (27). Differentiating (27) once and eliminating ONe/OX from (1), we obtain

g = kBTe • Eø Ox2 J/l N,- e0•x (29) e a Ox This equation is the same as that given by Mason [1971], except for the summation over the ion species. Note that (29) gives a relation between N, and (b (E = -&b/dx) similar to that given by self-similar solutions [Gurevich et al., 1966, 1968, 1973] only when

dx • = -•x << • N./eo (30) which implies the assumption of charge neutrality. Our calculations show that this assumption breaks down whenever steep density gradients occur.

We solved (26) and (29) over the region -L1 -< x -< L2 (see Figure 1). At t = 0 part of this space, -L1 -< x -< 0, was filled with a charge neutral plasma with known distribution functions f,, while the rest of it was empty. Equation (29) was solved by iterations with the boundary conditions,

E(x = -L1) = E(x = L2)= 0 (31)

The boundary conditions on the particles are as follows. At the boundary x = -L1, we kept the electron and ion distribution functions Maxwellian with a constant density and temperature for all times t -> 0. The boundary at x = L2 was purely absorbing; any particle reaching there was lost. Note that due to the boundary conditions in (31), our calculations are strictly valid up to the time either when the rarefaction shock front reaches the boundary at x = -L1 or when the expansion of plasma creates a sufficiently large plasma density at x = L2 to produce sufficiently large electric

9158 SINGH AND SCHUNK: INITIAL EXPANSION OF THE POLAR WIND

TABLE 1. Summary of Parameters in Various Computer Runs Identified by the Case Numbers

No(H +) No(O +) T(H +) T(O +) Ax/ Case No No d/XDi Te Te •Di L I/XDi L2/XDi

1 0.9 0.1 4 1 1 4 1.6 x 103 2 x 103 2 0.1 0.9 4 1 1 4 1.6 x 103 2 x 103 3 0.1 0.9 4 0.1 0.1 4 1.6 x 103 2 x 103 4 0.1 0.9 2 X 10 3 1 1 10 5 X 10 3 10 4 5 0.1 0.9 400 1 1 8 4 X 10 3 6.4 X 10 3 6 O. 1 0.9 800 1 fixed 8 4 x 103 8 x 103

Note that d indicates the distance over which the initial density profiles of the ions fall from the high densities No(H+) and No(O +) to zero near the interface.

fields. However, with a suitable choice of the simulation- plasma lengths L• and L2, the rarefaction and expansion of plasma can be established without boundary effects.

The solutions of the ion Vlasov equations were carded out over a grid-space in the x-V plane. In most of our calculations, the grid spacing Ax = 4XDi, where }kDi = Vt(H+)/tOpi, Vt(H +) = [kBT(H+)/M(H+)] •/2, and %,i is the plasma frequency for a purely H + ion plasma. In some calculations, as discussed at appropriate places, we used Ax = 8 and 10 hDi. The velocity grid for H + ions was chosen to be A V = 0.25Vt(H+), while for O + ions it was AV = 0.06Vt(H+). The time step in advancing the solution was At = 0.5 %,i-• for H + ions and At = %,•-• for O + ions. The time steps and grid spacings used in our calculations are similar to those used by others [Sakanaka et al., 1971; Mason, 1971] in related calculations.

We have used the following definitions and normalizations; distance • = x/}kDi , velocity 12 = V/Vt(H+), time t = t%,i, potential $ = e&/kBT(H+), and electric field/• = eEhDi/ keT(H+). The normalized drift velocities for O + and H + are denoted by 12D(O +) and 12D(H +) and they are defined by

lYo(a ) = lYf• d f• dV (32)

Any physical quantity with the subscript '0' refers to the undisturbed plasma.

A summary of the parameters for the various computer runs that we conducted is given in Table 1.

4. PLASMA EXPANSION

Results from the computer runs corresponding to the cases in Table 1 are presented below. Cases 1-4 deal with the situation in which both ion species are allowed to evolve self-consistently, while in cases 5 and 6 the heavy O + ions are assumed to be immobile. The motivation for this assumption is to simulate the effect of gravity on the expansion. Cases 4, 5 and 6 deal with the effect that the initial density- gradient scale length has on the expansion. Typically, the results are presented by showing the temporal evolution of the ion density, ion drift velocity, and electrostatic potential profiles. However, phase-space plots illustrating the evolution of the ion distribution functions are also shown for some

cases. The numerical results are compared with the self- similar solutions.

4.1. Plasma Expansion with H + Major

This case deals with the situation in which the light ion is the major ion (case 1). Figures 3a, 3b, and 3c show the temporal evolution of the H + density, H + drift velocity, and

the electrostatic potential profiles. The density profiles in Figure 3a show that the H + ions expand rapidly as time increases. The density profiles in the expansion region (x > 0) always remain concave, as predicted by the self-similar solution. However, as we shall show later, this concave feature of the density profiles occurs only when H + is the major ion; when H + is minor, a plateau formation in the density profile occurs over a limited range for x > 0. We find that in the expansion region the N(H +) values are slightly larger than those given by the self-similar solution equation (4a). On the other hand, in the rarefaction region x < 0, they are lower than those given by the self-similar solution. For example, the self-similar solution for cold ions, as given by (4a), predicts that the rarefaction shock front at t = 320 should reach g = -320; where a discontinuity in the slope of the density profile should occur. However, our calculations for warm ions show that at t = 320 the rarefaction distur-

bance has reached as far as • = -1000, and there is no discontinuity in the slope. Such features were seen in the numerical simulations of a plasma with warm ions by Dena- vit [1979]. The disappearance of the discontinuity at the rarefaction front is a warm-ion effect, as pointed out by Gurevich et al. [1966, 1968].

The temporal evolution of the potential profile shown in Figure 3b has the following noteworthy features. First, we find that the self-similar and numerical results for the potential agree well over a limited range. For example, at ? = 180 the two results compare well until • = 1400 or • -< 8. This value of • is close to the •s value given by equation (9); •s = 21n(?) - 1 = 9.4. When • >- •s, the plasma Debye length becomes comparable to the scale length for the variation of the potential, making space charge effects significant. Next, we note that the numerical results give zero electric fieM for • •> •. Denavit [1979] obtained a similar result from his numerical simulations in which both electrons and ions were

treated kinetically. Also, we note that the self-consistent potential profiles for ? = 60 and 180 in Figure 3b show that in the expansion region, where the potential profiles become concave, the net space charge is negative (d2cb/dx 2 > 0). Thus, the electrons lead the expansion, as previously pointed out by Crow et al. [1975]. Finally, we note that the self- consistent potential profiles are similar to a double-layer type of potential distribution.

The ion drift velocity profiles are shown in Figure 3c. VD(H +) is shown for ? = 60, 180, and 480, while VD(O +) is shown only for ? = 480. For H+, the broken lines show the self-similar solutions and the solid lines show the self- consistent drift velocities. Note that the two solutions are

very close in the expansion region. An intriguing feature of the drift velocity profiles is that at

SINGH AND SCHUNK: INITIAL EXPANSION OF THE POLAR WIND 9159

I.O

0.9

0-8

0.7

0.6

... 0.5 '•Z

P- 0.4

• 0.3

0.2

o.i

• =64 .................. /• =128

--.-- •: 192

• }:256

• :320

0

-1200 -800 -4.00 0 400 800 1200 1600 2000

Fig. 3a. Temporal evolution of the H + density profiles for the case when H + is the major ion [N(H+)/No = 0.9; T(H +) = T(O +) = Tel. Note that the profiles are always concave in the expansion region.

any instant of time VD increases nearly linearly with x, giving values of VD larger than those corresponding to the potential drop at that instant of time. A careful examination of the temporal evolution of the potential and electric field profiles reveals the explanation. The potential profiles in Figure 3b show that the electric field produced in the expansion region continually occupies a larger region; however, its magnitude decreases. This is schematically shown in the inset of Figure 3b. Thus, the ions in the expansion region are continually accelerated for a longer length of time as they move ahead in the expansion region. It turns out that the net result of such an evolution of the electric field is nearly a linear profile for the drift velocity V D.

It is worth mentioning that in the locations where E = 0 in the expansion region, the number density of ions is extreme- ly small; typically N < 10 -4. However, these relatively few ions are highly energetic.

Figure 3c also shows the drift velocity, VD(O+), for the minor O + ions at ? = 480. The dotted line shows the result of

the numerical calculations, while the dashed line marked s - s(O +) is the drift velocity that occurs for the expansion of a pure O + plasma. This dashed line is described by the equation

Note that this equation is obtained from (4b) by using the normalizations described in section 3. It is surprising that the drift velocity of the minor O + ion is so closely described by the self-similar solution for a pure O + plasma expansion.

We now compare our numerical results for the drift velocity of the minor O + ion with that predicted by the

theoretical work of Gurevich et al. [1973], as briefly discussed in section 2. Since Z] = Z2 = 1, the parameter Z = 1/16X/• < 1/X/• (see equation (14)). Therefore, we use the lower set of signs in (11) and (12). Table 2 shows the solutions a• versus • = x/Cs(H+)t. We see that for •(H +) = 0, a• •- 0.0625; thus (11) gives

02(• = 0) = 0.21[Te/T(H+)] •/2 (34)

When •(H +) •> 1, a• makes an insignificant contribution to 02, thus

[92(•]) = • + 16 [Te/T(H+)]•a (35) Note that (Y2(•) is the normalized velocity of the minor O + ion. A comparison of (34) and (35) with (33) shows that the theory of Gurevich et al. [ 1973] for the expansion of impurity heavy ions yields results very close to that given by the self- similar solution (33) for the expansion of a pure heavy ion plasma. Therefore, the flow of heavy ions is not affected by the light ions even if the light ions are major owing to the inertia of the heavy ions. In this light, the agreement between our numerical results and equation (33) involving VD(O +) is not surprising after all.

We have already noted that VD(H +) is also closely de-

I I

12 --

w >

• -

-1200 -800 -400

=e,o' (c)

/ 480 • /,,''" // // •-.-- .,.•-.•:+ -•,_

•.. •.•.•.•.< ß ß _ •' I I I • I

O 400 •OO 12OO I•OO 2OOO

Fig. 3b, c. (b) Temporal evolution of the electrostatic potential profiles for the case when H + is the major ion. The solid curves correspond to the self-consistent numerical solutions, while the dashed curves are the S-S solutions. The inset shows the temporal evolution of the electric field. (c) Temporal evolution of the H + drift velocity profiles for the self-consistent (solid curves) and S-S (dashed curves) solutions. Also shown are the self-consistent (dot- te{ curve) and S-S (dashed curve) solutions for the O + drift velocity at t = 480.

9160 $1NGH AND $CHUNK: INITIAL EXPANSION OF THE POLAR WIND

•000

80

813.oo ,iso.oo :•1•o.0o • NORMALIZED DISTRNCE X •lO

!0.00

320

8b.oo tbo.oo ::,tm.oo 3 •o.oo NJ)RMRLIZED DISTANCE X

ubo.oo

o o

_1

z=;. i

160

oo 8b.oo t•o.oo :,1•o. oo 3! NORMRLIZED DISTANCE X •10

0.00

o

=' 2qo

•oi

z i.

ß DO oh. DO •:•-' •.o. DO 320 NORHRLIZ[[I DIST•NCF X • 0 ]

q00

..oo ,o.oo o.oo NORMALIZED DISTANCE X •

•o. oo qbo. oo 0'

Fig. 3d. H + phase-space plots at selected times for the case when H + is the major ion. In each plot, contours of f (H +) are shown as a function of velocity and position. Note the change in the x axis.

scribed by the self-similar solution, given by

IYD(H +) +)],/2 = •- + [Te/T(H Equations (33)-(35) show that when M? >> 1,

VD(H + )/VD(O +) • 1

(36)

(37)

This is an interesting result. Later, in section 5, we shall discuss that (37) could probably explai n some observations of energetic H + and O + ions streaming at nearly equal speeds in the distant magnetotail [Frank et al., 1977; Hardy et al., 1977].

Figure 3d shows the phase-space plots for H + ions at selected times. The curves shown are the contours of

$INGH AND $CHUNK.' INITIAL EXPANSION OF THE POLAR WIND 9161

TABLE 2. Solutions to Equation (12)

H + Major 0 + Major 0 + Minor H + Minor

•(H +) a• •(0 +) a• - 1 -0.663 - 1 10.6 -0.752 -0.5 -0.174 8 -0.355 -0.25 1.6 6 -0.086 -0.1 5.2 4

0 -0.0625 13.5 2 0.0283 -0.05 23.2 1 0.185 -0.01 0.34 -0.001

constant f(H+). At t = 0, the contours are limited to x -< 1600, and they are symmetric about V = 0 owing to our assumption of a Maxwellian distribution at t = 0. As time elapses, the expansion of the H + ions is clearly visible. The expansion is associated with an acceleration of the ions; the contours bunch around the self-similar solution (4b) for the flow velocity of cold ions. This latter solution is shown by the solid slant line at ? = 240 and 480. The entire phase-space can be divided into three distinct regions' an undisturbed plasma on the extreme left, a rarefaction region, and an expansion region in which ions are accelerated.

4.2. Plasma Expansion with H + Minor

In this case, the lighter ion (H+) is the minor ion, while the heavy ion (O+) is the major ion (case 2). We now show that the expansion of H + when it is a minor ion is very different from when it is a major ion. Figures 4a, 4b, and 4c show the temporal evolution of N(H+), •x), and VD(H+). The most noteworthy feature of the density, potential and drift velocity profiles is the development of plateaus over a finite distance in the expansion region. Figures 4a, 4b, and 4c

Fig. 4b, c. (b) Temporal evolution of the electrostatic potential profiles for the case when H + is a minor ion. (c) Temporal evolution of the H + drift velocity profiles. Also shown are the self-consistent (solid curve) and S-S (dashed curve) solutions for the O + drift velocity at t = 480. Note the formation of a plateau in the & and VD(H +) profiles.

O.I

' I ' I ' I ' I ' I ' I '

- • -

0.02 -- t!!.• .•.....

o.o,- - .,

0 • • • .Xxl ...... • .... l'•• • , -• -• 0 400 •00

0.09

0.07'

O .05

+

3:: 0.04 <Z

0.0:•

Fig. 4a. Temporal evolution of the H + density profiles for the case when H + is a minor ion [N(H+)/No = 0.1' T(H +) = T(O +) = Te]. Note the presence of a plateau in the density profiles.

clearly show that the plateau region moves as the expansion proceeds. The key to the formation of a plateau is the different expansion properties of the heavy and light ions. In order to demonstrate this, we show in Figure 5 N(H +) and N(O +) profiles for one instant of time, ? = 240. The corresponding potential profile is shown in Figure 5b. Note that in and beyond the plateau region H + is the dominant ion [N(H +) >> N(O+)]. Also, in the plateau region the potential & is nearly constant, giving a nearly zero electric field. Thus, in this region charge neutrality prevails,

/Q(H +) =/Qe = exp (•b•(H+)/Te) (38)

Since •(H +) = 0.011 in the middle of the plateau, (38) gives the potential there

Te

•]•pl = T(H +-•• In [/•p/(H +)] = -4.5 (39) which agrees with the value shown in Figure 5b. The self- similar solution for the potential for a pure O + expanding plasma is shown by the dashed curve marked S-S(O +) in Figure 5b. Note that the potential profile from the undisturbed plasma to the plateau region is closely given by this curve. Thus it turns out that the heavy major ions set up a potential in which the light minor ions are accelerated. The accelerated ions move ahead of the O + ions and neutralize

the negative space charge, creating a plateau in the potential


o

o o

g

*G .OD

o

8b.oo tb0.00 :,)4o.oo 3•o,oo ubo.oo NORMALIZED DISTRNCE X •10'

128

eb. oo t bo. oo 2t•o. oo •o, oo who. oo NORMALIZED OlSTRNCE X wlO'

192

o

o o

258

o

7o oo

eb 0o tho•oo •ho.oo s;,o.oo u_o.oo NOF{MALIZED DISTANCE X •,10'

, i .

eb.oo ,bo.oo 2tm.oo 3'•o.oo u•oo.oo NORMF{LIZED DISTANCE X xlO'

38q

o o

.oo eb.oo ,bo'1oo 2•o.oo 3'•0.00 ubo.oo NORMALIZED DIST•qNCE X •10' NOF{MRLIZED DISTRNCE X •10'

Fig. 4d. H + phase-space plots at selected times for the case when H + is a minor ion. In each plot, contours of f(H+) are shown as a function of velocity and position. Note the change in the x axis.

distribution, and hence, in the density and the drift velocity profiles (see Figure 4c). Note that the drift velocity Vo(H +) in the plateau region is given by

12o½(H +) = •pl •* 3 (40) In and beyond the plateau region, the plasma is described by the expansion of a relatively low density, nearly pure, H + ion plasma.

We note that the formation of the plateau in the expansion

region results from self-consistent numerical calculations. Once it is recognized that a plateau forms, where the physical quantities are nearly constant over a range of space, a quantitative estimate of the physical properties, such as Nee, •pl, and Vo•,(H+), can be made using simplified equations of continuity and momentum. The continuity of flux dictates that

]•pl•Dp = •r (41)

$INGH AND $CHUNK: INITIAL EXPANSION OF THE POLAR WIND 9163

<z

o.e (c) --

• = 240 0.6 --

0.4 --

0.2 --

o I

<-0.

io

I i - 0 400 800 1200 1600 2000

x

Fig. 5. Plasma exEansion when H + is a minor ion. (a) N(O +) and N(H+) profiles at t - 240 for the case shown in Figures 4a-d. N(H+) is only shown for x >- 0. The plateau appears when N(H+) •> N(O +) in the expansion region. (b) The corresponding potential profile at • - 240. Two straight lines, marked by S-S (O +) and S-S (H+), show the self-similar solutions for the expansion of pure O + and pure H + plasmas. Note that the plateau region lies between the two lines. The potential drop from the undisturbed plasma to the plateau is closely l•iven by S-S (O+). (c) The self-consistent H + drift velocity profile at t - 240 (solid curve labeled Vo(H+)). Also shown are the self-similar solution for the expansion of a pure H + plasma (dashed line labeled S-S(H+)), the H + drift velocity obtained from the theory of Guterich et al. [1973] (dotted curve labeled U2(H+)), and the asymptotic result of equation (16) (dashed line).

where Fr is the random flux crossing a planar surface in the undisturbed plasma. This flux is given by

Fr =Nø(H+•)I'2kBT(H+)]I/2 2 •rM(H +)

or

Pr = Fr/(No[keT(H +)/M(H +)] 1/2)

= 0.1 x 0.4 =0.04

The momentum equation for H + ions yields

(42)

d d6 (43)

Integrating (43) from the undisturbed plasma to the plateau, we obtain

- T(H+- •- In (]Qp/2) (44) Combining equations (42) and (44) gives an equation for Npl ,

Npl 2 In (]Qp/2) T(H +) ^ -- •'r 2 (45) re

The above equation provides a straightforward method for the determination of the density in the plateau region. Once the density is known, the potential and H + drift velocity are easily determined.

As noted earlier, beyond the region of the plateau Ix > xu, see Figure 5b] the expansion occurs as for a pure H + ion plasma. In order to illustrate this point, we show that the maximum electric field in this region is given by the self- similar solution of an H + plasma. Using J = 240 in (5) we obtain

• ehD 1 = • E = r-= 4.17 x 10 -3 (46) kaT(H +) t

The electric field at 2 = 10 3 in Figure 5b can be obtained by the slope of the potential profile; this slope is 4.16 x 10 -3, which agrees with the value in (46).

Knowing that beyond the plateau region the electric field is described accurately by the self-similar solution, it is easy to show that the potential in this region is given by

• = (•pl + (•u- OTe/T(H +) (47)

where •, - x,/C,(H+)t. Knowing the potential distribution, the density distribution in this region is given by

]Q(H +) = ]Qe = exp [-s • + & + q•plr(H+)/re] (48)

The validity of the above relations can be determined using arguments similar to those used in deriving (9); we obtain

C(H+)t = Vtøex p [{s•.- s•.- $plT(H+)/Te}/2] %0

or

•. = •,• + •pt[T(H+)/Te] + 2/n(J) (49) Since beyond the plateau region VD(H +) is also closely described by the self-similar solution (36). s% can be obtained by equating the drift velocity in the plateau region. given by (40). to the self-similar solution;

• = X/2 (•pl -- [Te/T(H+)] •/2 (50)


In the present case Te = T(H +) and •pl • -4.5, so that

7 =2 (51)

Thus, at • = 240 (49) gives

8.5 (52)

and then the self-similar solutions should be valid up to g < 240 x 8.5 = 2030. Figure 5b shows that for g >• 1600 the potential becomes nearly constant, giving almost a zero electric field.

Before comparing our results, which are based on a self- consistent numerical solution, with those obtained by Gure- vich et al. [1973] it is useful to summarize our findings. We emphasize that when the light ions are minor, a plateau forms in the spatial distribution of the physical quantities. The upper end of the plateau is given by (50), while the lower end is determined by the evolution of the heavy major ion. In the plateau region E •- 0, and then the density /•p/(H+), potential •pl and drift velocity are given by (45), (39), and (40), respectively. Beyond the plateau region (x > xu), these quantities are described by the set of self-similar solutions (47), (48), and (36) up to a distance x -• Xs = Cst.

We now compare the results of Gurevich et al. [1973] on plateau formation and expansion beyond the plateau, which are based on a self-similar solution of fluid equations, with those from our numerical calculations. In the present situation the parameter Z = 16/X/• (see equation (14)). The solution of (12) for a versus • = •(O +) are given in Table 2. The behavior of the drift velocity in (11) is plotted in Figure 5b as indicated by U2(H+). We have also plotted the asymptotic formula (16), which at some values of •(O +) gives the drift velocity in the plateau region, as indicated by (19). This value of •(O +) is •ll given by (22). Since ]•2oZ2 -- 0.1 and •0Z• = 0.9, we obtain

•11=6.9 or .•ll/•= 1.73[ re ] 1/2 T(H+ ) (53) Then, from (23)

T(H+ ) (54) According to above relations, the plateau region in Figure 5c should extend from • = 415 to 1130, as shown by the horizontal solid line marked G; this line is obtained by the intersection of the vertical lines (53) and (54) with the lines marked (16) and s-s(H+), respectively. Comparing the properties of the plateau region, as predicted by Gurevich et al. [1973], with our numerical results, we conclude that the simple analytical theory of these authors does not predict the correct nature of the plateau. The most striking difference is that these authors overestimate the drift velocity in the plateau region. The reason for this discrepancy is because the asymptotic formulae (15) and (16) are not valid near •l. Thus, their use of these formulas as a boundary condition at the lower end of the plateau region is invalid.

We note that the variations of the physical quantities near the lower end of the plateau region are not very sharp; therefore, it is not possible to define an exact lower end, and hence an exact width of the plateau region.

o.i

.o4

.03

.02

.Ol

o

-8oo -4oo o 400 800 12oo 16oo 2ooo

Fig. 6a. Temporal evolution of the H + density profiles for H + a minor ion and hot electrons [Te/T(H +) = 10; T(H +) = T(O+)].

Figure 4d shows the phase-space plots for the minor ion H +. Note the sudden acceleration ofH + for x > 1600 and the appearance of the plateau region, in which the velocity distribution function does not change with x. Beyond the plateau region, a further acceleration of H + is also evident. The propagation of the rarefaction wave in the negative-x direction is also clearly visible. Thus, the entire phase space can be divided into five regions; on the extreme left is the undisturbed plasma, then the rarefaction zone comes, followed by the zone of acceleration of the minor H + ions by the electric field set up by the major O + ions. This acceleration zone is followed by the plateau region, in which the physical quantities are nearly constant. The last zone is the expansion zone of a nearly pure H + plasma.

4.3. Effect of Hot Electrons on the Plasma Expansion

As far as the ion composition is concerned, this case is the same as case 2, except that the electron temperature is 10 times the ion temperature. The temporal evolution of N(H+), qb and VD(H +) are shown in Figures 6a, 6b, and 6c, respectively. A comparison of these results with those shown in Figures 5a, 5b, and 5c indicates that the qualitative features of the temporal evolution for the present case are the same as for case 2, in which the electrons and ions have the same initial temperature. However, quantitative differences occur. The larger electron temperature leads to the development of a larger potential, and hence, the H + ions are accelerated more for hot electrons than for cold elec-

trons. Furthermore, the H + density in the plateau region is smaller in this case than in case 2. Using (45), we obtain

l•pl21tl(•pl 2) = -- 1.6 x 10 -4 (55)

The solution of this equation is •el -• 4 x 10 -3. This is in


o • • • • plateau region. For x > Xl, N(O +) decays rapidly, giving N (b) - (H +) >> N(O+). Corresponding to-•l = 600, the self-similar

-20 parameter •(O+) at • = 180 is given by

•(O +) -• 4.2 (58) -40

l'• 18o This value is slightly smaller than that predicted by (53) from

x • the theory of Gurevich et al. [1973]. The asymptotic formu- -60 xxx lae (15) and (16) of Gurevich et al. [1973], which determine x _ the properties of the plateau region, are valid only when (see \

-80 -ioo I I I I I

//S-S (H+)•=180

•;. 1

//

-800 -400 0 400 800 1200 1600 2000

Fig. 6b, c. (b) Electrostatic potential profile at • = 180 for H + a minor ion and hot electrons. The dashed line S-S (0 +) is the self- similar solution for the expansion of a pure 0 + plasma. (c) Temporal evolution of the H + drift velqcity profiles. The dashed line S-S(H+) is the self-similar solution at t - 180 for the expansion of a pure H + plasma.

good agreement with the density of about 4.5 x 10 -3 in the plateau region shown in Figure 6a. This density gives a potential drop between the undisturbed plasma and the plateau region, of magnitude d)•,l-• -101n (4.5 x 10 -3) = 53, which also is in good agreement with the numerical calculations (Figure 6b). Correspondingly, the drift velocity in the plateau region is much larger for hot electrons than for cold electrons.

Figures 6b and 6c show that when the electron temperature Te >> Ti, the plateau region is relatively more well defined than for the case with Te •- Ti. In the present case, the slopes near the ends of the plateau region change very abruptly. Since the potential for x < Xl, the lower end of the plateau (see Figure 6b), is remarkably close to the self- similar solution determined by O + ions, the O + density can be calculated by such a solution;

/•(O*) = 0.9 exp [-s•(O *) - 1] (56)

192

At • = .•'l = 600

•(O*) •- 4.86 x 10 -3 (57)

This density is nearly equal to the density of H + ions in the

z= C

'o'.oo go. oo t'.•o. oo 2'ao. oo 3•o. oo ubo. oo NORMRLIZED DISTRNCE X •10'

Fig. 6d. H + phase-space plots at selected times for H + a minor ion and hot electrons. Note the change in the x axis.


their discussion leading to their equation (18)),

e(O +) 16

+ 1 >> 1 (59)

Thus, the value of •(O +) in (58) certainly does not satisfy this criterion. Hence the theory of Gurevich et al. [1973] is inadequate to describe the plateau. For example, if we use this theory to estimate the drift velocity in the plateau region at } = 180, we get 12D(H+) = 18 and the plateau extends from • = 980 to 2700. The corresponding plateau region, according to our calculation, extends from ;•l = 600 to gu = 1300 given by (50), and the H + drift velocity is 12D(H +) = 11.

Figure 6d shows the phase-space plots for this case at selected times. This figure is similar to Figure 4(d); however, we note that the H + acceleration is much greater for hot electrons than for cold electrons. Also, the transitions between the various regions are more sharply defined.

4.4. Effect of Initial Density Gradient at Plasma-Vacuum Interface

Case 4 is similar to the case 2, but with an important difference; the initial density profiles of N(H +) and N(O +) have large density gradient scale lengths at the plasma- vacuum interface. In case 2 the ion densities fell from their

values in the unperturbed plasma to zero over the grid spacing Ax, while in the present case the density decrease occurs over a distance d = 2000 Xo. In order to simulate such a density gradient, we had to increase the length of the simulation region; L• = 5 x 103 h o and L2 = 104 }kD. The temporal evolution of the density profiles for O + and H + ions are shown in Figures 7a and 7b, respectively, while that of the electrostatic potential qb is shown in Figure 7c.

Figures 7b and 7c reveal the important result that as far as the expansion properties of the light minor ion are concerned, they are qualitatively similar to those shown in Figures 4a-4c, for which the initial density profiles near the interface were abrupt. As for that case, we observe the formation of a plateau region in the potential and H + density profiles. Beyond the plateau region, the expansion occurs as though the plasma is a pure H + plasma, as discussed in detail for case 2.

The important difference between cases 2 and 4 concerns the time scale of the plasma expansion. For example, in case 2 a well-defined plateau formation took place as early as t = 64, while in the present case a plateau formation is seen only at about J = 2400. This slow temporal evolution is related to the development of a relatively smaller electric field due to the relatively smaller density gradients. This could be in- ferred from (12), which shows that E -• Oln l•10k, where 02E/Ox 2 and OE/Ox are small.

For this case we also examined the expansion of O + ions. Figure 7a shows the profiles of N(O +) at • = 1200 and 2400 and the O + drift velocity VD(O+). In this figure we also show the self-similar solution for VD(O+), assuming the expansion of a pure O + plasma. We find that at relatively large times, • •> 2000, the density profile in the expansion region (x > 0) can be fairy well approximated by the self-similar solution. The drift velocity curves in Figure 7a show that the self- consistent values of VD(O +) are slightly below the S-S(O +) line. A similar behavior of VD(O +) can also be seen in Figure 4C.

4.5. Effect of Gravity on the Plasma Expansion

We now discuss cases 5 and 6, in which the heavy ions are not allowed to expand into the vacuum. The motivation for these cases is that in the polar wind gravity has a much larger effect on O + than on H +. Therefore, to simulate this effect we have considered plasma expansions with O + fixed. The two cases 5 and 6 differ in that in case 6 the scale length d of the O + density profile at the plasma-vacuum interface is double that in case 5, for which d = 400 Xo.

Figures 8a, 8b, and 8c show the temporal evolution of the N(H+), &, and Vo(H +) profiles, respectively, for case 5. For the sake of clarity in the presentation, curves are shown only for two instants of time; t = 704 and t = 1408. In Figure 8c, we have also shown the self-similar solution for Vo(H +) when a pure H + plasma expands. We observe that, as for cases 2-4, the major features of the temporal evolution are the development of a plateau in the profiles of N(H+), &, and Vo(H+), and beyond the plateau the expansion occurs in accordance with a pure H + plasma. The electric field beyond the plateau region, when evaluated as -d$/dg, is found to be fairy well approximated by the self-similar solution (5) for a pure H + plasma; the slopes dcb/dx are indicated in Figure 5b, and it is easy to verify that they scale as 1/•. Similarly, the drift velocity Vo(H +) asymptotically joins the self-similar solution (Figure 8c). The values of N(H +), Vo(H +), and qb in the plateau region are approximately given by (45), (40), and (39), respectively.

The results of the calculation for case 6 are shown in

Figures 9a, 9b, and 9c. Note that despite the fact that there are different density gradients in the O + profiles for cases 5 and 6, the results in Figures 9a, 9b, and 9c have nearly the same qualitative features as in Figures 8a, 8b, and 8c, respectively. The only noticeable difference is in the location of the slight peak in the density; in case 5 at • = 1408 the peak occurs at a larger distance from x = 0 than that in case 6. However, it tums out that the distances of the peaks from the location of the uniform O + density, as indicated in Figures 8a and 9c by Ax e, are nearly the same.

The important difference between the cases when the O + ions are immobile and when they expand is in the extent of the plateau region. When the O + ions are immobile, the plateau region starts near the initial interface (x = 0) and extends to large x as time increases. On the other hand, when O + is allowed to expand, the plateau occupies a limited region of space and it moves as a whole as the O + and H + ions expand.

5. SUMMARY AND DISCUSSION

We studied the expansion of a collisionless, electrically neutral, H+-O+-electron plasma into a vacuum. At t = 0, we assumed that a region of space -L• -< x -< 0 was filled with plasma, while the region 0 -< x _< L2 was empty. For t > 0, the plasma was allowed to expand into the vacuum and its evolution was followed by solving the Vlasov equation for each ion species, the Boltzmann relation for the electrons, and the Poisson equation for the electrostatic potential. In order to elucidate the basic physics involved in the plasma expansion, we studied the effect of varying the ion composition, the initial ion-electron temperature ratio, and the scale length of the density gradient at the plasma-vacuum interface. We also simulated the effect of gravity on O+.

As is well known for a single-ion species plasma, the

$INGH AND $CHUNK: INITIAL EXPANSION OF THE POLAR WIND 9167

0.9

0.8

0.7

0.6-- 6

•0.• s •' \ '. ø-.

-I

0.2 • •• q 2 O.I • I o i I I o

.05

<z .03

-• - •1 I (c) - -2 m \ --

-3- II \\. -

-'• .... 3?.oo - '5 m %•

<"t)- -6 -- • --

-- -•o I I , . I I I ,[

-4 -2 0 2 4 6 8 IOxlO

x

Fig. 7. Plasma expansion for a relatively small initial density gradient at the plasma-vacuum inte•acc. (a) N(O+) at selected times (left axis), and VD(O +) at • = 24• (fight •is). Also shown is the co•csponding sclf-simil• solution for VD(O+). (b) N(H +) at selected times. (c) Potenti• profiles at selected times. Note that the formation of the plateau in the N(H+) and & profiles takes much longer than for the cases shown in Figures 4-6.

evolution of the plasma consists of a rarefaction wave propagating into the plasma-filled region and a plasma expansion into the initially empty region. Out main contribution has been to determine the effect that another ion species has on the temporal evolution of the plasma expansion. We have found that there are significant differences depending on whether H + is the major or minor ion.

When H + is the major ion, the plasma expansion has the following characteristics:

1. The H + density profile in the expansion region is always concave.

2. The heavy, minor O + ion has a very small effect on the H + expansion.

3. The H + phase-space is divided into three regions: an

9168 SINGH AND SCHUNK.' INITIAL EXPANSION OF THE POLAR WIND

undisturbed plasma, a rarefaction region, and an expansion region in which ions are accelerated.

4. The expansion of both H + and O + can be described reasonably well by self-similar solutions.

5. Both H + and O + ion acceleration occurs as a result of

the plasma expansion. Far from the initial plasma-vacuum interface, the accelerated H + and O + ions attain the same drift velocity.

When H + is the minor ion, the plasma expansion has the following characteristics:

1. The major O + ions have an important effect on the H +

,,

I • • o•' • • i o., -"'-'"'-.--.7 -

.O• , ', ii _ '\ I

-

<•,

............ 14OB

O I I I•'•l I' '"

<'9-

-2

-4

-6

-B

-IO

<>n4

I I I I I _ (c)/•/_

i i i

-3200 0 3200 6400

Fig. 8. Plasma expansion with immobile O + ions. The initial O + density near the plasma-vacuum interface decreases from 0.9 to 0 over a distance of d = 400hD. (a) N(H+) profiles at selected times. (b) Corresponding potential profiles. (c) VD(H +) profiles. Also shown are the self-similar VD(H +) profiles for a pure H + plasma (dashed lines).

O.I I I I I I I

(a)

I I

• (b)

.OB

.06

+

v

<Z

.02

-2

-4 <•

-6

-8

v

<>o

0 • I.," I I I I I / -3200 -1600 0 1600 3200 4,B00 6400 8000

Fig. 9. Plasma expansion with immobile 0 + ions. This case is the same as for Figure 8, except that d = 800XD.

expansion. Since O + is more massive than H +, the O + expansion lags behind the H + expansion. In the O + expansion region, the polarization electrostatic field set up by the O + ions and electrons acts to accelerate the minor H + ions to velocities much larger than their thermal speed.

2. Beyond the O + expansion region, the plasma becomes predominantly an H + ion plasma and a distinct plateau region forms in which the H + density, H + drift velocity, and electrostatic potential are constant.

3. Beyond the plateau region, the H + plasma expands like a single-ion-species plasma.

4. The H + phase-space is divided into five regions; the undisturbed plasma, a rarefaction zone, an H + acceleration region owing to the O + electron electric field, the plateau region, and the expansion zone of a nearly pure H + plasma.

5. The H + acceleration due to the expansion process is larger when H + is a minor ion than when it is a major ion.

6. Increased electron temperatures produced enhanced electrostatic potentials and H + drifts in the expansion region.

7. The smaller the density gradient at the plasma-vacuum interface, the longer it took for the plateau to form. However, after plateau formation, the expansion proceeds nearly in an identical fashion regardless of the initial density gradient.

8. When O + was fixed to simulate the effect of gravity, the lower end of the plateau region remained fixed at the location of the initial plasma-vacuum interface, whereas when O + was allowed to expand the plateau region moved along with the moving O + density front.


Since our numerical calculations were for a one-dimen-

sional plasma of a relatively short length, we cannot directly apply our results to the polar wind. Nevertheless, these calculations were useful in elucidating the basic physics involved in the initial expansion of a neutral plasma. As far as the polar wind is concerned, such an expansion should occur whenever there are plasma density enhancements in the high-latitude topside ionosphere. Such density enhancements, in turn, are connected with the horizontal motion of the plasma field tubes that is associated with magnetospheric convection. The basic convection pattern is a two-cell pattern with antisunward flow over the polar cap and return flow at lower latitudes. As the plasma field tubes follow the various convection trajectories they are subjected to a variety of chemical and physical processes, which lead to both plasma density enhancements and depletions. Density enhancements at high altitudes could arise from the direct action of ionization sources or from electric field heating at low altitudes followed by the upward diffusion of the hot plasma to higher altitudes.

Associated with plasma density enhancements in the topside ionosphere will be an expansion of the plasma in the manner described in this paper. If the initial plasma expansion occurs predominately at an altitude where H + is the major ion, the temporal evolution of the expansion should lead to H + density profiles that are always concave in the expansion region. Also, an energization process is connected with the plasma expansion, and both H + and O + ion acceleration occurs. At altitudes far above the initial expansion region, the accelerated H + and O + ions should be strongly field aligned and drifting at about the same speed. This latter result might explain the observations of energetic H + and O + ions streaming at nearly equal speeds in the distant magnetotail [Frank e! al., 1977; Hardy et al., 1977].

On the other hand, if the initial plasma expansion occurs predominately at an altitude where H + is a minor ion the temporal evolution of the expansion will be different from that described above. In this case, an expanding H + density shock front should form and propagate up and out of the topside ionosphere along the geomagnetic field lines. A plateau region should also form behind this shock front in which the H + density, H + drift velocity, and the electrostatic potential are constant. The length of this plateau region depends on the extent to which the major O + ion participates in the initial plasma expansion; if O + expands the lower end of the plateau region will move along with the O + shock front. The plateau region, as described in this paper, may only be observed a distance of from 1-2 Re above the initial expansion region because at greater distances the diverging magnetic field geometry may distort its characteristics.

When H + is a minor ion, the plasma expansion process produces energetic H + and O + ions just as it does when H + is the major ion, but the H + acceleration is greater when H + is a minor ion. The typical energy gained by H + ions during the expansion process is about 10kTe. Since Te probably varies from about 1000 ø to 10,000øK in the high-latitude topside ionosphere, the energy of the accelerated H + ions should lie in the range of from 1 to 10 eV. It should be emphasized that this ion acceleration will occur at any location in the high-latitude ionosphere where plasma expansion is taking place. It may occur at different locations at the same time or at different times at the same location, depending on the ionospheric conditions. It should also be empha-

sized that the production of energetic ions is inherent to the process of plasma expansion and should occur anywhere in space plasmas where a local density enhancement occurs.

We note that the polar wind processes that we have described relate to the initial expansion of the plasma due to a density (or pressure) enhancement. For long times, the steady state solutions obtained from the hydrodynamic, kinetic, and generalized transport equations should ade- quately describe the characteristics of the polar wind.

Finally, we discuss some of the more important limitations of our study. First, we assumed that the plasma expansion was collisionless. This restricts the application of our results to altitudes above approximately 1000 km [cf. Raitt et al., 1975; Schunk and Watkins, 1982]. At this altitude, the H + outflow is subsonic and H + is a minor ion. At higher altitudes, where the transition to supersonic flow occurs, the polar wind is collisionless. At still higher altitudes, H + becomes the dominant ion. Therefore, the discussion of how our numerical results relate to the polar wind is not affected by our assumption of a collisionless plasma expansion.

In all of our calculations, we assumed that the plasma expansion started from an initially stationary plasma. If a density enhancement occurs in an initially flowing H + plasma, the results should be qualitatively similar to those obtained when O + was allowed to expand, as compared with the case when O + was held fixed. That is, the ionospheric side of any plateau that forms in response to the H + density enhancement would simply move along with the already flowing plasma.

The most serious limitation of our study is connected with the relatively short length of the simulated plasma. Although we varied the density gradient scale length at the plasma- vacuum interface by two orders of magnitude, it is still much smaller than that typically found at polar wind altitudes. Unfortunately, the current state of the art computers are still much too small to simulate the plasma expansion along a significant segment of a field line. Nevertheless, our numerical calculations have indicated that there are physical processes operative in a plasma expansion that are not taken into account in the existing steady state models of the polar wind.

Acknowledgments. This research was supported by NASA grant NAGW-77 and NSF grant ATM-8015497 to Utah State Uni- versity.

The Editor thanks J. A. Fedder, J. Lemaire, and T. E. Holzer for their assistance in evaluating this paper.

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(Received April 26, 1982; revised July 12, 1982;

accepted July 13, 1982.)

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Numerical calculations relevant to the initial expansion of the polar wind

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