A M E C H A N I S M FOR THE F O R M A T I O N OF P L A S M O I D S AND

KINK WAVES IN THE H E L I O S P H E R I C C U R R E N T S H E E T

S. W A N G * , L. C. LEE, C. Q. W E I , and S.-I . A K A S O F U

Geophysical Institute and Department of Physics, University of Alaska, Fairbanks, Alaska 99775-0800, U.S.A.

(Received 1 February, 1988)

Abstract. Satellite observations of the heliospheric current sheet indicate that the plasma flow velocity is low at the center of the current sheet and high on the two sides of current sheet. In this paper, we investigate the growth rates and eigenmodes of the sausage, kind, and tearing instabilities in the heliospheric current sheet with the observed sheared flow. These instabilities may lead to the formation of the plasmoids and kink waves in the solar wind. The results show that both the sausage and kink modes can be excited in the beliospheric current sheet with a growth time ~ 0.05-5 day. Therefore, these modes can grow during the transit of the solar wind from the Sun to the Earth. The sausage mode grows faster than the kink mode for fl~o < 1.5, while the streaming kink instability has a higher growth rate for flo~ > 1.5. Here flo~ is the ratio between the plasma and magnetic pressures away from the current layer. Ifa finite resistivity is considered, the streaming sausage mode evolves into the streaming tearing mode with the formation of magnetic islands. We suggest that some of the magnetic clouds and plasmoids observed in the solar wind may be associated with the streaming sausage instability. Furthermore, it is found that a large-scale kink wave may develop in the region with a radial distance greater than 0.5-1.5 AU.

I. Introduction

Satellite observations of the transition region of the heliospheric current sheet at 1 AU (Borroni et al., 1981; Gosling et aL, 1981; Zhao, Wilcox, and Scherrer, 1983) indicate that (a) the transition width of the current layer ranges from 106 km to 3 x 107 kin; (b) the solar wind speed is typically 50-150 km s - 1 lower than the speed on the two sides of the current layer; (c)the proton density is high and the proton temperature is low inside the current sheet. Other studies of the three-dimensional structure of solar corona and interplanetary space based on satellite observations and interplanetary radio scintillation measurements (e.g., Zhao and Hundhausen, 1983; Rickett and Coles, 1983; Sime, 1983; Hakamata and Munakata, 1984; Newkirk and Fisk, 1985; Fry and Akasofu, 1987) also demonstrated that the velocity of solar wind in the interplanetary space increases with distance from an interplanetary neutral sheet which separates the two hemispheres of opposite magnetic polarity.

Recently, Lee et aL (1988) and Wang, Lee, and Wei (1988) found that the streaming sausage, kink, and tearing instabilities can be excited in a current sheet with a super-Alfv6nic plasma flow. The streaming sausage and kink instabilities, similar to the Kelvin-Helmholtz instability, are caused by the sheared plasma flow. In the presence of a finite resistivity, magnetic islands can be formed inside the plasmoids associated

* Also at Department of Earth and Space Science, University of Science and Technology of China, Hefei, Anhui 230029, China.

Solar Physics 117 (1988) 157-169. �9 1988 by Kluwer Academic Publishers.

158 S. WANG El" AL.

with the streaming sausage mode, and the streaming sausage mode can evolve into the streaming tearing mode. The purpose of this paper is to point out that the streaming sausage, kink, and tearing instabilities can also be excited in the heliospheric current sheet, where a sheared plasma flow is observed.

It may be mentioned that the sausage and kink instabilities in a cylindrical geometry have been studied by many authors (e.g., Kruskal and Schwarzchild, 1954; Alfv6n and F~tlthammar, 1963; Hasegawa, 1975). However, in the previous studies of the sausage and kink modes, the effects of the sheared flow in a neutral sheet are not considered.

2. Streaming Sausage, Kink, and Tearing Instabilities

In this section, we employ an initial-value method (e.g., Steinolfson, 1984; Lee and Fu, 1986) to study the growth rate and eigenmode structures of streaming sausage, kink, and tearing instabilities in a compressible plasma with a sheared flow. We start from the following magnetohydrodynamic (MHD) equations:

8p - - + 7" (pv) = 0, (1) 8t

p +(v '7 )v = - T P + - - ( 7 x B ) x B , (2) 47r

C 2 8 B _ x / x ( v x B ) _ _ _ 7x(r /VxB) , (3) 8t 4n

a(pp-

8t - - + (v .v) ( e p - = 0 , (4)

where p is the plasma mass density, v the velocity of solar wind, P the plasma pressure, B the interplanetary magnetic field, ~ the resistivity, and 7 = 35- is the ratio of specific heats.

The initial one-dimensional equilibrium configuration of the current sheet is described by

Bo(z) = Bo~ tanh ( L ) (cos q~oix + sin cpoiy),

Vo(z) = [Vo~- Vo., sech2 ( L ) ] ix,

(5)

(6)

po(z)=po~+ B%_ s e c h 2 ( ~ ) , (7) 8 n R T o

FORMATION OF PLASMOIDS AND KINK WAVES 159

where Bo, Vo, Po, To represent the zero-order quantities of the magnetic field, velocity, density, and temperature, respectively; Boo, V~, and Poo are, respectively, the x components of the magnetic field and plasma velocity, and plasma density far from the current layer; Vom is the difference of the flow velocity at z = 0 and at z ~ ~ ; ~Po may be considered as the local azimuthal angle of the spiral magnetic field; a B and a v are the characteristic half-thicknesses of the current sheet and sheared flow layer, respec- tively; R is the gas constant. It is noted in the solar wind, av can be larger than as. In our previous calculations (Lee et al., 1988; Wang, Lee, and Wei, 1988), only cases with av = aB and q~o = 0 are considered. The pressure balance condition has been used to obtain the density profile in Equation (7).

In the initial value approach we follow the development in time of a small perturbation about the equilibrium state. Each of the physical quantities may be written as q(x, y, z, t) = qo(Z) + Re{ql(z, t) exp[i(kxx + kyy)]}, where qo(z) is the zero-order quantity and ql (z, t) is the complex first-order perturbed quantity, and k = (kx, ky, 0) is the wave vector. Eight linear differential equations in variables z and t for the eight complex perturbed quantities can be obtained from Equations (1)-(4). These perturbed differential equations can be numerically solved by a fourth-order Runge-Kutta method. If the initial equilibrium state in Equations (5)-(7) is unstable, the fastest growing mode will eventually dominate the numerical solution. One then obtains the growth rate and the eigenmode profiles of the fastest growing mode.

The computational domain ( - L < z < L) must be much larger than the half-thickness aB or av. In our calculation, the system length used is 2L = 32aB, which ensures that the effect of the boundary is negligible. The boundary conditions at z -- + L can be a fixed (ql = 0) or 'free' (Sql/SZ = 0) boundary condition. The results show that for L = 16a~ the growth rate and eigenmode profiles are not sensitive to the choice of boundary conditions.

In the following computation, the perturbed quantities are written in the dimensionless form. The length is normalized by the half-thickness of current layer (aB), the time by the Alfvdn transit time z A (~A = aB/VAo~ ), the magnetic field by Boo, the density by p~, the pressure by Po~ and the velocity by VAo ~, where VA~ = Bo~/xf4np~p~ and Po~ = p~RTo . The plasma beta far away from the current layer is flo~ = 8 n P o J B 2 . The dimensionless growth rate is defined as 7* = 7aB/VAo~ = 7VA, where 7 is the imaginary part of the complex frequency co. The normalized resistivity is defined as tl* = ~lc2/4naB VA~. The grid size is typically taken to be Az = aB/8. The direction of wave propagation is assumed to be in the direction of the initial plasma flow (kx = k, ky = 0), which has the maximum growth rate. Numerical results are shown in Figures 1-6.

Figure 1 shows the magnetic field configurations for three cases: Case (A) the streaming sausage mode (fi~ = 1, Vom = 2 VAo~, q~0 = 0, t/* = 0), Case (B) the streaming tearing mode (fi~ = 1, Vom = 2VAoo, ~0 = 0, t/* = 0 .0058) , and Case (C)the streaming kink mode (fi~ = 2, Vom = 2VA~, q~0 = 0, t/* = 0). Other parameters used are a v = aB and ka B = 0.45. The corresponding eigenmode profiles of Pl, bx, and b z for the three cases are plotted in Figure 2. The magnetic field configurations in Figure 1 are obtained

60

6

S. WANG ET AL.

(A) Streaming Sausage Mode

-6

6 (B) Streaming Tearing Mode

-6

(C) Streaming Kink Mode

-6 0 28

x/6 Fig. 1. Magnetic field configurations for Case (A) the streaming sausage mode (rid = 1, ~/* = 0), Case (B) the streaming tearing mode (flo~ = 1, t/* = 0,0058), and Case (C) the streaming kink mode (fl~o = 2, */* = 0),

Other parameters used are: av = aB, kaB = 0.45, ~b o = 0, and Vom = 2V,~.

f rom the superpos i t ion of the b a c k g r o u n d magnet ic field B o and the pe r tu rbed fields

ob ta ined from the e igenmode profiles. The m a x i m u m value o f the per tu rbed field bx is

chosen to be 0 . 1 5 B ~ . I t c an be seen f rom Figure 1 tha t r econnec t ions of magnet ic field

F O R M A T I O N O F P L A S M O I D S A N D K I N K W A V E S 161

3,

2,

O.

- | .

It/.

5.

-113.

3.

2,

1.

Sausage Mode

13, - t r .

t - - I t

7 I " -

. . . . . ~ . . . . . . I , . ,

~ " t I - -

-8. o. 8. 16.

z/5

Fig. 2.

O.

12.

6,

O.

-6,

-12,

6.

Tearing Mode 2.

. . . . . . . : -7.

i . . . . i ~ Lt* @" 2.

I.

O.

Kink Mode

i 1

4. / O , g

2, -0,8

0, - 1 . 6 - t6 . -8. O.

z18 zl~

. . . . . . . . . f ~

8. t O ,

Eigenmode profiles of the perturbed plasma density p~, and the x and z components of the perturbed magnetic field, b~ and b~, corresponding to the three cases in Figure 1.

lines do not occur in Case (A), since ~/= 0. However, plasmoids are formed even in the streaming sausage mode. On the other hand, magnetic reeonnections do occur in Case (B) and magnetic islands are formed in the presence of a finite resistivity (t/* = 0.0058). In this case the streaming sausage mode becomes the streaming tearing mode. A magnetic field configuration of kink waves caused by the streaming kink instability is also shown in Figure 1. Figure 2 shows that for the streaming sausage mode and streaming tearing mode the perturbed density @1) and the z component of magnetic field (bz) are symmetric with respect to the z = 0 axis, and bx is antisymmetric about z = 0. In the z = 0 plane, b~(0) is zero in Case (A), which corresponds to the streaming sausage mode with zero resistivity. On the other hand, b~(0) is finite in Case (B), which corresponds to the streaming tearing mode with a finite resistMty. For the streaming kink mode, p~ and b x is antisymmetric, and b x is symmetric about z = 0 axis. The normalized growth rates for Cases (A), (B), and (C) are found to be respectively ~,* = 0.059, 0.075, and 0.046.

162 S. WANG ET AL.

The normalized growth rates 7* ( = 7 ZA) versus the normalized wave number ka B for the streaming sausage mode (fl~ = 1, Vom = 2VAoo, r/* = 0, a v = aB) and the streaming kink mode (floo = 2, Vom = 2VAoo, ?]* = 0, a v = aB) are shown in Figure 3. It can be seen that for the streaming sausage mode the maximum growth rate (Tin - 0.060ZA 1) occurs at k a s "" 0.5, and the growth rate is found to be small for k a B > 1. For the streaming kink mode, the maximum growth rate (7,, - 0.12ZA 1) occurs at k a s ~- 1.0, and the growth rate is very small for k a s < 0.35. Numerical results also show that the phase speeds (cor/k) of streaming sausage and kink modes are (Voo - 1.SVAoo) and (Vo~ - 1.1 V Aoo), respectively.

We also calculate the growth rates and eigenmodes for various values of floo. It is found that the streaming sausage mode grows faster than the streaming kink mode When fl~ < 1.5. When fl > 1.5, the streaming kink instability has a higher growth rate. The instability condition for both the streaming sausage and kink modes is found to be VOm ~ 1.2VAo o (Lee e ta l . , 1988; Wang, Lee, and Wei, 1988).

Figure 4 shows the maximum normalized growth rate 7m ZA as a function of Vom/VAoo

for the streaming sausage mode (floo = 1, Vom = 2VAoo, ~o = 0, /7' = 0, a v = a s ) and

the streaming tearing mode (flo~ = 1, Vom = 2VAoo , ~)0 = 0, ~$ = 0.0063, a v = as ) ,

respectively. It can be seen from Figure 4 that the growth rate for the tearing mode remains nearly constant in the range 0 < Vo,,,/VAoo < 1. A large increase in the growth rates occurs at Vom/VAo o > 1.2. It means that the effect of the sheared plasma flow on

0.15 I I I

0.10

O. 05

~ floo = - 2

/ / \ Z==-I U ~ ~ l a u s a g e M ~ e)

0 . 0 0 0.0 0.% 1.0 1.5 2.0

kaB Fig. 3. The normalized growth rate 7 ZA as a function of the normalized wave number kaB for the streaming

sausage mode (flo~ = 1, Vom = 2VAoo) and the streaming kink mode (rico = 2, Vom = 2VAo~)-

0,15

FORMATION OF PLASMOIDS AND KINK WAVES

[ r I

163

0. t0

O. 05

O. O0 0.0

S = 1 6 0

1.0 2.0

VO, IVAoo Fig. 4. The maximum growth rate as a function of the normalized plasma flow speed Vo,,,IVgo~ for the streaming sausage mode (r/* = 0, S = oo) and the streaming tearing mode (~* = 0.0063, S = 160). The

parameters used are: fl~ = 1, av = as , q~o = 0, and gOm = 2 V A c o ,

the growth rate of tearing instability is not important when Vom is sub-Alfv6nic, and the effect becomes very important when Vom is super-Alfv6nic. The importance of the sheared plasma flow on the streaming sausage and tearing instabilities can also be

understood from the fact that the streaming sausage and tearing modes occur on the Alfv6n time-scale ( ~ 10ZA), while the pure tearing mode occurs on a slower time-scale '~' ('CATD) 1/2, where *D = 4 ~ a ~ / t l c2 is the diffusion time (e.g., Priest, 1984). Since the resistivity ~/is very small in the interplanetary space, it follows that the streaming sausage and tearing instabilities are much more important for the formation of plasmoids than

the pure tearing mode without the presence of a sheared flow. We now consider the case with a v --A a B . Figure 5 shows the normalized growth rate

7* versus the normalized wave number k a B for the streaming sausage mode with a v = 2a B. It can be seen from Figure 5 that for the case with a v = 2aB, the maximum

growth rate of the streaming sausage mode with floo = 1 and Vo, . = 2VAo o is 7m ~ 0"03ZA l" The maximum growth rate occurs at k = k,,-~ 0.25a~ 1. Since the

streaming sausage and kink instabilities, similar to the Kelvin-Helmholtz instability, are caused by the sheared plasma flow, the growth rate is determined mainly by the characteristic thickness of the sheared flow layer ( a v ) . In terms of a v , we have k,, -~ 0.5a v 1 and 7mav/VAoo "~ 0.06, which are similar to the case with a v = a B .

We also calculate several cases with the spiral angle ~Po = 45 ~ The maximum growth rate is found to occur when the wave vector is parallel to the plasma flow (k rJ Vo). The magnitude of the maximum growth rate is found to be nearly the same as the case with

qSo = 0.

164 s. WANG ET AL.

t---

Fig. 5.

0.075

O. 050

O. 025

I I I

a v - - 2 a B

0.000 I I i

0.00 0.25 0.50 0.75 1.00

kaB The normNized growth r ~ e of ~ e streaming sausage mode (7%) as a ~nc t ion of the normNized

wave number kas ~ r the case with av = 2aB.

3. Formation of Plasmoids and Kink Waves in the Heliospheric Current Sheet

Satellite observations (Borroni e t al . , 1981; Gosling et aI., 1981; Zhao, Wilcox, and Scherrer, 1983) show that a transition region exists between two spiral sector regions. The thickness of transition region is on the order of 106-3 • 107 km at 1 AU. In the transition regions, the distributions of magnetic field, plasma velocity, and density are approximately described by the expressions in Equations (5)-(7). The magnetic field and plasma parameters across the transition regions at 1 AU are: Boo = 5 nT, V~ = 450 km s - 1, Vom = 50-150 km s - 1, n~ = 5 cm - 3, T e = 1.5 • 105 K,

T i = 4 • 104 K. Here n~ is the number density of the electron and ion, T e and T e are the temperature of the electron and ion, respectively. We then have VA~ ~ 50 km s - 1,

flo~ = 8 r c [ n o ~ k ( T e + Te)]/BZ~ ~ 1.3. Based on these data, it can be seen that the instability condition Vo, . > 1.2VAo~ can be satisfied for the spiral sector transition regions at 1AU. If we take the half-thickness of transition region a v ~ a s = 5 • 10 5-107 km, VAo~ -- 50 km s - 1, and Vo, ~ - 100 km s- 1, we obtain at r = 1AU, the Alfv6n transit time ZA-----104--105 S--0.1--1 day, the growth time 7- 1 _ 10ZA -- 1--10 days, and the wavelength of the excited waves 2 "~ 4 z t a v ~- 107-108 km. The growth time is shorter in the region with r < 1 AU as discussed later.

In the following we calculate the growth rates of the streaming sausage and kink modes as a function of the radial distance r from the Sun. According to Parker's theory

FORMATION OF PLASMOIDS AND KINK WAVES 165

(Parker, 1963), the interplanetary magnetic field in the equatorial plane can be written,

in the spherical coordinates, as

=

B,(r , = - B E rE , (8) u E r

Bo=O~

where co o = 2.7 x 10- 6 s - 1 is the angular velocity of solar rotation, and B e is the

magnetic field and u E the solar wind speed at r = r e = 1 AU. We assume that the plasma density p(r) and the half-thickness of transition regions

a v ( r ) may be written as

p(r) = PE , (9)

r ) (10) av ( r ) = a v e \ r E ~

where PE is the plasma density and a v e the half-thickness at 1 AU. From Equations (8)-(10), the Alfv6n velocity Vao~, the plasma beta ~ and the Alfv6n time ZA in the solar wind can be represented as

I I ( i ~ 2 ~ 1/2

VAoo(r)= VAE 2 l + ~ j j ' (11)

= 2 E(1 + -1 r2J , (12)

"CA(r ) = "CAE 1 + rZJJ (13)

From the results obtained in Section 2, it can be seen that the growth time of the streaming sausage and kink instabilities is Vg = y- 1 ~ 10VA. Figure 6 shows the growth

time rg -- 10v A versus the normalized heliocentric distance for the streaming sausage and kink modes in the heliospheric current sheet for four cases: (a)fie = 1,

a v e = 106 km, (b) fie = 4, a v e -- 106 km, (c) fie = 1, a v e = 107 kin, (d) flF = 1, a v e = 107 kin. The region with dashed lines in Figure 6 indicates that the growth rate of the sausage mode is higher than the kink mode. The solid lines indicates the region in which the kink mode has a higher growth rate than the sausage mode. At r = 0.2 AU, the growth time ,g _~ 0.05-0.5 day, which is smaller than the value at r = 1 AU by a factor of ~ 20.

166 S. W A N G ET AL.

10 6

10 5

10 4

g. 10 3

10 2

/

/ / /

/ /

- / /

[ / / /

' / /

//

/I !

. t / /

t

/ /

I I

O. 1. 2. 3.

r(AU) Fig. 6. The growth time (Zg) for the streaming sausage and kink modes versus the heliocentric distance r. (a) f e = 1, ave = 106 km, (b) f~r = 4, a r e = 106 km, (c) f e = 1, a r e = 107 km, (d) fie = 4, ave = 107 kin. The region with dashed line is dominated by the sausage mode, while the region with solid line is dominated

by the kink mode.

It can be seen from Figure 6 that for cases (a) and (c) with fie = 1, the streaming

sausage mode grows faster than the streaming kink mode in the region 0.1 AU < r < 1.5 AU. Beyond r = 1.5 AU, the streaming kink instability has a higher growth rate. For the cases (b) and (d) with fie = 4, the streaming sausage instability is the dominant process in the region 0.1 AU < r < 0.5 AU, while the streaming kink mode becomes dominant for r > 0.5 AU. On the other hand, it can be also seen that the growth time Zg increases as the characteristic thickness of the sheared flow layer (av) increases.

In the following, we demonstrate that the growth rate of the pure tearing instability in the absence of a sheared flow is too small to form plasmoids in the solar wind. The maximum growth rate of the pure tearing instability can be written as (e.g., Furth, Killeen, and Rosenbluth, 1963; Lee and Fu, 1986)

( ~ m ) t e a r i n g "~ 0.6(ZD Z A) - 0.5. (14)

The growth time for the pure tearing instability can then be written as

"Ct = ( ~ m ) t e a r i n g - - 1.7S~/2 ZA, ( 1 5 )

where S = ZD/ZA is the magnetic Reynolds number.

FORMATION OF PLASMOIDS AND KINK WAVES 167

The growth times of pure tearing mode as a function of radial distance r for various values of S are plotted in Figure 7. In this case, we choose aVE = 107 km and/~e = 1.

The growth time zt for the pure tearing mode increases as the magnetic Reynolds number S increases. It is seen from Figures 6 and 7 that the growth time for the pure tearing mode is much larger than the growth time zg for the streaming sausage and kink modes for S > 1000. For example, the growth time of the pure tearing mode for S = 1 0 4 is Z t=2 X 105 S--2.3 days at r = 0.2AU, and zt_~3 x 106s_~ 35days at r = 1AU. Since the effective magnetic Reynolds number in the solar wind is very high, we suggest that the pure tearing instability is unlikely to cause the formation of plasmoids in the interplanetary space.

4. Discussions and Summary

Based on observations from five spacecraft, Voyager 1 and 2, Helios 1 and 2, and IMP 8, Burlaga et al. (1981, 1987) and Klein and Burlaga (1982) analyzed the magnetic fields behind shocks in the interplanetary space and identified the presence of magnetic

clouds. The magnetic clouds are a magnetic structure with helical magnetic field lines. Based on satellite observations, Gosling and McComas (1987) suggested the presence

of plasmoids in the interplanetary space. They further suggested that these magnetic

Fig. 7.

10 8

10 7

~ 10 6

~ 10 5

10 4 O.

I I

1. 2. 3.

r(AU) The growth time (vt) for the pure tearing mode versus the heliocentric distance r for the case with

fie = 1 and a w = 107 km.

168 S. WANG ET AL.

clouds or plasmoids are associated with the coronal mass ejections which accompany the solar flares or eruptive prominences.

It is interesting to point out that the streaming sausage mode excited by the sheared plasma flow can also lead to the formation of plasmoids or magnetic clouds in the heliospheric current sheet. After their formation, plasmoids tend to be ejected with a super-Alfv6nic speed as observed and simulated in the magnetotail study (Hones, 1979; Birn, 1980; Lyon etal., 1981; Forbes and Priest, 1983; Lee, Fu, and Akasofu, 1985). A shock can be formed as a consequence of the super-Alfv6nic motion of plasmoids. It is possible that the magnetic clouds and plasmoids associated with interplanetary shocks can also be caused by the streaming sausage and tearing instability.

The above results and discussions are based only on the calculations of linear growth rates. In the nonlinear stage of evolution, the streaming sausage mode may also lead to the formation of several pinched regions of the current sheet as shown in Figure I(A). If the resistivity is finite, an enhanced reconnection of magnetic field lines may occur in these pinched regions, where the thickness of the current layer is highly reduced. This process may also lead to the formation of magnetic islands inside the plasmoids formed by the streaming sausage instability.

In summary, we have investigated the growth rates and eigenmodes of the streaming sausage, kink and tearing instabilities in the heliospheric current sheet with a sheared plasma flow. Our results show that plasmoids and kink waves can be formed in the heliospheric current sheet. The instability condition is found to be Vom > 1.2VA~, which is generally satisfied in the heliospheric current sheet. The streaming sausage mode grows faster than the streaming kink mode when/3~ < 1.5, and the streaming kink instability has a higher growth rate when fl~ > 1.5. If the effect of a finite resistivity is considered, the streaming sausage mode evolves into the streaming tearing mode with the formation of magnetic islands. The maximum growth rate for these modes is found to be ])m ----- 0.1 z A 1. The corresponding growth time in the region with 0.2 AU < r < 1 AU is found to be Tg = 10z A - 0.05-5 days. Therefore, these modes can develop during the transit time of the solar wind from the Sun to the Earth. The most unstable modes occur at k a v a 0 . 5 , corresponding to a wavelength 2 ~ 4ha v ~ 107-108km atr = 1 AU.

Acknowledgements

This work was supported by the NSF grant ATM 85-21115, DOE grant DE-FG06-86ER13530 and Air Force contract F19628-86-K-0030 to the University of Alaska.

References

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FORMATION OF PLASMOIDS AND KINK WAVES 169

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1038. Newkirk, G. R., Jr. and Fisk, L. A.: 1985, J. Geophys. Res. 90, 3391. Parker, E. N.: 1963, Interplanetary Dynamical Processes, Interscience Publishers, New York. Priest, E. R.: 1984, Solar Magnetohydrodynamics, D. Reidel Publ. Co., Dordrecht, Holland. Rickett, B. J. and Coles, W. A.: 1983, in M. Neugebauer (ed.), Solar Wind 11, NASA Conf. Publ. Sime, D. G.: 1983, in M. Neugebauer (ed.), Solar Wind 11, NASA Conf. Publ. Steinolfson, R. S.: 1984, Phys. Fluids 27, 781. Wang, S., Lee, L. C., and Wei, C. Q.: 1988, Phys Fluids (in press). Zhao, X.-P. and Hundhausen, A. J.: 1983, J. Geophys. Res. 88, 451. Zhao, X.-P., Wilcox, J. M., and Scherrer, P. H.: 1983, Chinese J. Space Sci. 3, 1.