AD-A14i 284 ON THE LINEAR STABILITY OF TIO-DIMENSIONALCBARIUM i/ICLOUDS I TH4E INVISCID CASE(U) NAVAL RESEARC LABI WASHINGTON DC S T ZALESAK ET AL. 38 APR 84 NRL-MR-53i2

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S. T. Zalesak and J. D. Huba~ MECVRDT FKPR--7;PG ON13& TYPE OF REPORT 13R1. TIME 8 CIEDO 1 4 AT OF REOT .Y,. Mo.. Day, 37PAECONInterim FROM 10183 TO17'I April 30, 1984 I37

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FI ~ELD GROUP SuB. OR Barium clouds Bifurcation Structure Nuclear cloudsU-shaped curve Plasma structure Plasma clouds

19_1USTRACT 'Conm,. on II t ea.- and deittif, by blogl namborI

We examine the linear stability of the steepened backside of two-dimensional ionospheric barium clouds.We derive expressions for the growth rate of perturbations on these backsides which are functions of (1)the ratio M of the integrated Pedersen conductivity inside the barium cloud to that of the background

,*5;1%ionosphere; (2) the overall shape of the cloud; (3) the plasma density gradient scale length L of thesteepened backside; and (4) the perturbation wavenumber k. For sufficiently small kL, the growth time

-~ as a function of M traces out the "U-shaped curve" observed in previous studies.

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11. TITLE (Include Security Clasification)

ON THE LINEAR STABILITY OF TWO-DIMENSIONAL BARIUM CLOUDSL THE INVISCID CASE

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CONTENTS

I. INTRODUCTION ............................................... 1

II. THEO RY ..................................................... 2

III. APPROXIMATING THE GROWTH RATE IN CASES WHERE THEDENSITY GRADIENT IS FINITE ................................. 5

IV. RELATIONSHIP TO THE U-SHAPED CURVE ......................... 6

V. CONCLUSIONS ................................................. 8

APPENDIX - DERIVATION OF THE E X B INSTABILITY GROWTH RATEFOR A DISCONTINUOUS INTERFACE ............................. 9

ACKNOWLEDGMENTS ............................................... 13

REFERENCES ..................................................... 21

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ON THE LNEAR STABILITY OF TWO-DIMENSIONAL BARIUM CLOUDS1. THE INVISCID CASE

I. Introduction

The study of barium cloud phenomenology has been of interest to

ionospheric physicists for more than two decades. The release of barium in

• +- the ionosphere, followed by its rapid ionization, leads to a complex

S .-

Z4 interaction between the barium plasma and the background ionospheric plasma

° "in the presence of ambient electric and magnetic fields. One of the most

interesting phenomena observed is the structuring of the barium cloud: the

"fingering" or "striating" of the backside of the cloud. it is generally

believed that this structuring is produced by the E x B instability [Linson

and Workman, 1970]. A substantial research effort, both theoretical and

computational, has been devoted to understanding this instability and its

application to barium cloud structuring. The purpose of this paper is to

examine the linear stability of the steepened backside of two-dimfensional

barium clouds. We derive a relatively simple expression for the growth

rate of the perturbations on these backsides which are functions of () the

prmtrM which is the ratio of the inertdPedersen cnutvt

inerae Introduition

inside the barium cloud to that of the background ionosphere; (2) the shape

onof the cloud; (3) the plasma density gradient scale length L of the

steepened backside; and (4) the perturbation wavenumber k. For

sufficiently small kL, the growth time as a function of M traces out the

~"U-shaped curve" observed in previous studies [Linson, 1975; McDonald et

al__l, 1981; Overman and Zabusky, 1980].

The presene f abe remainder of this paper is as follows. In Section

II we derive the growth rate of a two-dimensional barium cloud, where we

approximate the steepened backside of the cloud as a discontinuity. In

Section Ita we relax this constraint and let the density gradient scale

length on the steepened backside be finite. In Section IV we relate our

results to the "U-shaped curve" of critical Reynold's number seen by

McDonald et al. 1981. Finally, in Section V we present our conclusions

and plans for future work.

Manuscrip approved Febrary 9, 1984.

5~'4

II. Theory

We are interested in deriving the approximate linear growth rate for

. perturbations applied to the steepened backside of an ionospheric barium

cloud. To achieve this we need several earlier results. The first is the

equation derived by Ossakow and Chaturvedi [1978] which describes the rise

velocities of plasma depletions in the shape of circular or ellipsoidal

cylinders in the presence of a gravitational field perpendicular to a

magnetic field B. It is simple matter to extend their equation to plasma

enhancements in the presence of an electric field Eo (i.e., barium

clouds). The second result we shall need is the equation for the linear

growth rate of the gradient drift E x B instability when the plasma profile

consists of a single discontinuity in plasma density. This result has been

derived by Huba and Zalesak [1983]. Finally, we shall need to know the

dependence of the growth rate on the plasma density gradient scale length

in the real-world regime between the two extremes of a density

discontinuity on the one hand, and a smooth density distribution, where

"local" theory should hold, on the other. For this information we draw

upon the work of Huba et al. (1982] and a suggestion of J.A. Fedder

[private communication, 1983].

Consider an idealized two-dimensional barium cloud consisting of a

uniform density inside of a "waterbag" of ellipsoidal shape, acted on by a

uniform external electric field E x in the presence of a magnetic field B

2z. Let the ellipse axis in the direction of E be denoted by b and the--0other axis by a. Let the ion density of the barium cloud be denoted by nb

and that of the background ionosphere be denoted by no . This situation is

depicted in Fig. 1. Then the results of Ossakow and Chaturvedi (1978] can

be extended to show that the total electric field E inside the barium

cloud is given by

E E (1)- -ol+MR

where

% R a/b (2).

2

_%,• , **o'* • "* -~~*~ . .. "% - --. ,...... ..-. . . . .. . .- -..- . .. . - % . " . ". ." . ...-, ": , ',, , : .,, . , :, .:. ,.. . 5.-. *,.,., . .5 : .:.A.2.: 5'*,.. .-... ... ........ . ..... :....

*~~~~; -77 '_ 7 * -~ - 7

- -nb + no/no (3)

Equation (1) is the first result we shall need for our analysis.

Now consider a one-dimensional barium cloud, infinite and invariant in

the x direction, which consists of a single jump in electron density from

nb + no to no at y = 0. That is, ne = nb + no for y < 0, and ne = no for

y > 0, where ne is electron density. Again we impose an electric field,

denoted here by E, aligned in the x direction. We then perturb the

* interface at y - 0 such that the position of the perturbed

interface n(x) is given by n - a cos kx where a is a small distance. This

situation is depicted in Fig. 2. It is shown in Huba and Zalesak 11983]

that this interface will grow in amplitude with growth rate y given by

k cE M - 1y. kB M+ (

Equation (4) was derived under the assumption that both Hall currents and

the inertial terms in the ion momentum equation are negligible. M is as

defined in Eq. (3). An alternative derivation of Eq. (4), due to the first

author, is given in the Appendix.

Consider a steepened barium cloud, as depicted in Fig. 3. If we

consider only the tip of the steepened backside, and we further consider

only perturbations of sufficiently long wavelength such that kL is small,

where k is the perturbation wavenumber and L is the characteristic scale

length of the density gradient on the backside of the cloud, then the

growth rate of these perturbations should be well approximated by Eq.

(4). However, note that the electric field E seen by the cloud is not the

externally imposed electric field Eo, but rather some smaller electric

field that is a result of the shielding effects of the two-dimensional

cloud. If we approximate our barium cloud as an ellipse as depicted in

Fig. 1, then Eq. (1) yields precisely this smaller electric field:

E -E I +R (5)- -o +MR(

. ; . ....K j ,. ..-... . .......... .., ..... .. .........-. ' '* * . . ...,.,.. ....... ,.,.. .

Substituting this field into Eq. (4) we get finally

cEk o (M - 1) (R + 1)y k -B (M + 1) (MR + 1) (6)

For the special case of a circular shape (R - 1) we get

2k CE0 M-1 (7)B (M + 1)2

Equation (7) traces out an inverted "U-shaped" dependence of y on M,

which we relate to the "U-shaped curve" of McDonald et al. [1981] in

Section IV. The M dependence of y is given by (M - !)/(M + 1)2, which

maximizes at M - 3. A plot of this quantity, as well as its two factors,

(M - 1)/(M + 1) and (M + 1)-1, which are due to I-D instability and 2-D

shielding respectively, is shown in Fig. 4. Note that the falloff in y for

small M is due to 1-D instability effects alone, while the falloff in Y for

large M is due to 2-D shielding alone. Thus we conclude that the

resistance of 2-D high M clouds to bifurcation (see McDonald et al., 1981)

is due to 2-D shielding.

Equation (7) is very similar, but not identical to, an expression

derived by Overman and Zabusky [1980] for the case of a perfectly circular

"waterbag" model of a barium cloud. Their linearized analysis showed the

results of perturbing a circular cloud of radius ro by making radial

sinusoidal displacements of its boundary

r = r + E a cos m (8)0 m

where ro is the initial radius of the cloud, r is the perturbed radius,

am is the perturbation amplitude, $ is the angle measured from the backside

of the cloud, and m is an integer (mode number). They found that

cE

3 am/at ( (2 r o M -2 am+l (9)0 (M + 1)

Note that if we write our perturbation (8) in terms of arc length ro0

multiplied by a wavenumber k then cos m6 - cos k r0 and

m/ro - k (10)

4

. " . .. - -.. .- ... . . ... . .. . ,. ... ., , . ..

.:.

making Eq. (9) yield the same "growth rate" as given by Eq. (7), except

that there is an am on the left side of (9) and an am+l on the right

side. Thus we have not totally recovered the results of Overman and

Zabusky [1980] for the circular waterbag, but the similarity is both

striking and encouraging. Furthermore, our formula Eq. (6) exhibits

several advantages over the Overman and Zabusky [1980] result. For one

thing, we do not have the ambiguity associated with using the term "growth

rate" to describe the rate at which mode m+1 supplies energy to mode m (see

Eq. 9). Equation (6) is a genuine growth rate in the usual sense of the

word. Also, precisely because we have derived Eq. (6) as the product of a

term involving global effects, i.e., the overall shape of the cloud, and a

term involving local effects, i.e., the densities and electric fields at

the steepened edge of the cloud, it is possible to generalize the results

to elliptical clouds (via the R dependence) and, more importantly, to the

case of a finite density gradient on the backside of the cloud, as we shall

see in Section III.

III. Approximating the Growth Rate in Cases Where the Density Gradient is

Finite

Let us return to Eq. (4). This equation was derived under the

assumption that the edge of the barium cloud represented a true

discontinuity in electron density; but real barium clouds consist of finite

gradients. How accurately does Eq. (4) represent the case of a sharp but

finite density gradient and how are we to treat the case where the density

gradient is not even sharp? The work of Huba et al. [1983) gives

numerically generated growth rates for a wide range of ratios of

perturbation wavelength to gradient scale length. However, we are

interested here in finding a simple, closed form expression. We propose

the following, after a suggestion due to J.A. Fedder [private

communication, 1983]

- k cE M - I kL € M + I (la)B M +1 M- 1

4 cE -L > M + 1y L ki> (11b)M I

5%.

*%

where L- 1 is the maximum value of the quantity - for the geometryn e y

.. . shown in Fig. 2 wherein we have replaced the discontinuity by a finite

transition length. Equation (lib) is the classical result for the gradie t

drift instability from local theory (see Linson and Workman [1970]). The

combined Eq. (11) agrees very well with the actual numerical results shown

in Huba et al. [1983] when allowance is made for the fact that the L used

in that paper differs from the one used here by a factor of 1.45. A

comparison is shown in Fig. 5.

Combining Eq. (11) with Eq. (1), we get our final expressions for the

growth rate of instabilities on the steepened backside of a barium cloud,

in the absence of diffusion:

- cE 1 i R1__Y. - c o M - 1 R + 1i M + I(1aB Mk1kL+1 (12a),,.B T + I M + I ' M - 1

"" " cE

Y . o L-1 R + 1 UL > M + iB MR + l" ' M- I

An obvious caveat to the above is that the perturbation wavelength X =

k/21 must be smaller than the total diameter of the barium cloud to have

the concept of a Fourier mode perturbation to make any sense at all.

We note that we have constructed our final expressions such that they

not only very closely recover the results of Overman and Zabusky [1980]

(for R = 1 and L 0 0, as previously discussed) but also accurately

reproduce the other known case of a one-dimensional plasma distribution (R

-0) for the entire range of kL.

N> :IV. Relationship to the U-Shaped CurveEquation (7), when viewed as an expression for y as a function of Mi,

yields a curve which maximizes at M - 3 and monotonically decreases away

from this value for all values of M for which our barium cloud represents a

true enhancement in electron density (1 4 M < -). In fact the growth rate

y is zero for M - 1 and M - -. The corresponding curve of a growth time

-a. y versus M minimizes at M = 3 and goes to infinity at M - I and M = -.

Curves of y versus . which exhibit this qualitative behavior have been

4F %•

:. 6

r ",. W. V W

dubbed "U-shaped curves" by several researchers. If one views barium cloud

bifurcation as a battle between the gradient drift growth rate y and

diffusive processes, then this U-shaped curve can be related to a similar

U-shaped curve which relates the amount of diffusion D needed to stop the

bifurcation of a cloud of a given radius ro, as a function of M. Let us

assume that the effect of diffusion is to contribute a term - k2D to the

growth rate. Let us further approximate our barium cloud as a cylinder (R

= ), with very steep edges (L/ro < 1). Then our growth rate is

cEy 2k o M k2D (13)

B2M+

We will further assume that bifurcation, if it takes place at all, takes

place at kL I. The bifurcation condition is then found by setting y = 0

in (13) with k - 1:

cEL-2 D 2L- (14)B (M+1 2

(1+)

cE' T0 L i M+1 2

_ 1( M+1) (15)

D 2 (14 - 1)

Noting that the left-hand side of (15) is simply McDonald et al.'s [1981]

Reynolds number Re we get

Re 1 (M + 1) (16)2 (M - 1)

V Note that McDonald et al. [1981] got

(1+ 1)2Re - 75 (M - 1) (17)

so the functional forms are the same but our Eq. (16) seems to state that

it requires 150 times more diffusion to stop bifurcation then was observed

by McDonald et al. [1981]. Whence the discrepancy? Part of the difference

can be removed by noting that the "finger" of McDonald et al. [1981]

corresponds to taking R = -, in which case we would get

Re - M(M +1) (18)_- A-(1e -DW)

d4"

- a

%p

.n-r. rr -, -'r -,,-d . Jr '..r, r- - .'*.... . . . . .s-,. .. .- . . -. .. . . ...- . . -. , ".

For large M this removes half of the discrepancy. Accounting for a larger

share of the difference is McDonald et al.'s [1981] equation for computing

L (their Eq. (5)), whlzh appears to give L's which are larger than the

minimum gradient scale length we have assumed here. However, we do not

believe this effect could account for a factor greater than ten, which

still leaves a discrepancy of more than seven. That is, the model we

propose here predicts a factor of at least seven greater diffusion

necessary to stop bifurcation than is given by McDonald et a!. [19P1]. The

U-shaped dependence on M, however, is the same. We hooe to be able to

resolve these quantitative differences in the sequel to this paper

[Zalesak, 1983], where the full effects of diffusion are considered.

V. Conclusions

We have derived an expression, Eq. 12, that we believe to be a good

approximation to the actual growth rate of perturbations applied to the

steepened backside of a two-dimensional ionospheric barium cloud, for the

case of no electron or ion diffusion. Diffusion plays a large role in the

behavior of two-dimensional barium clouds, however, and affects the growth

rate in two ways:

(i) diffusion will determine the maximum density gradient that can

exist at the backside of the barium cloud;

(ii) diffusion will provide a damping mechanism for any perturbation

which attempts to grow on the backside of the cloud. We have

already treated this effect, at least approximately, in Section

IV.

These two combined effects of diffusion are treated in the sequel to this

Vpaper [Zalesak, 1983].

6.'

6*a ~ 9 h Z ,* .1*.

3;- 4: 46I61

Appendix. Derivation of the E x B Instability Growth Rate for a

Discontinuous Interface

The geometry we shall consider is depicted in Fig. 6. The

differential equations describing this system are [McDonald et al., 1981]:

anate+ V-[ne - 0 (Al)

V-[n e V1 ] E.Vn e (A2)

where v - V x B (A3)B

Here ne is the electron density, which is n' for y > n(x) and n fory < TI(x), c is the speed of light, B is the magnetic field, E is the

imposed electric field, and is the induced electrostatic potential. Note

from Eq. (A3) that we have placed ourselves in a reference frame drifting

with the E x B velocity. Our two-dimensional plasma resides in the x-y

Cartesian plane. B is aligned along z.

If E - E x, and we ignore Hall currents, then the geometry depicted in

Fig. 6 is an equilibrium configuration with n - 0. We now apply a

perturbation to the position of the interface, denoted by n(x), of the form

a a cos kx e (A4)

and ask what the response of the electrostatic potentials in the two

regions will b2. Noting that Vn = Vn' - 0, since the fluid is

incompressible and we have only moved the interface, Eq. (A2) becomes

V2' -0 , y > n(x) (A5)

V2- 0 y < n(x) (A6)

If we assume solutions of the form

iati- G'(y) sin kx eiat, y > n(x) (A7)

9 9~

4

• # ,,,.%.%°.. - .,- A ,. -, - •% % % % % - "% - % % -. %" *- , - ..g .-%- -.. :. , . -. *-.' -. .-. a'. .- . . -, -o

. G(y) sin kx eiat v < r (x) (A8)

(Note we are assuming a specified phase relationship between n and ' which

will later be shown to satisfy our boundary conditions), then demanding

that t' vanish at y - + - and b vanish at y - o yields

€" C" e - k y snxe i a t

=C e sin kx e , y > r(x) (A9)

,;k. -. ",_' : =C e+k sin kx e t , y < -I(x) (AlO)

We shall later see that a is pure imaginary, meaning that the perturbation-. does not propagate in the x direction. Therefore the movement of the

interface is determined by the y-components of the velocity fields at the

interface, which must be the same for the interface to be well defined (no

:"" interpenetration or cavitation):

a- v" - v at y = n (All);.- . t y y

but v _ c 36y B ax

y B ax

so

B a'- kC-e-ky cos kx e t (A12)ce c seit ( )

- kC e k y cos kx e y

Since is assumed to be arbitrarily small we have

C C (A13)

but by Eq. (A4)

-n ia a cos kx e (A14)

% . " le -. '..:..10

so comparing (Al2) and (A14) we get

i-B a kC' kC (A15)c

Now we have one last constraint to satisfy, namely that 7.J 0 0. We

will use the integral form of this constraint, that is

.. s(v)J.n dS = 0 (A16)

for any control volume V bounded by surface S(V). Equation (A16) obviously

holds for any control volume not containing the interface n(y), since n - a

constant and V2€ = 0. Consider our interface and the control volume

depicted in Fig. 7. JL' JR' JB and J are the total currents flowing

* through the left, right, bottom and top surfaces respectively of thecontrol volume. We define n - xR) and

J J.n ds = J J + J - J =0 (A17)R L T B= *". nk~ - k y T 17xL

JT -n - - +nkCe x sin kx dx (AI8)

J n- - nkCek y B R sin kx dx (A19)B 3 y L

J JI + II (A20)R R R

where

J he -. xjdy (A21)

I"" YTJR n ( E -- Jdy (A22)R

Since both ax and the integration path length rYT- n. are perturbation

quantities in Eq. (A22), the integral involving n' aY/3x will be quadratic

in the perturbation and hence negligible. We then get

SI YT n'E dy =(YT nRJn'E (A23)

R IIR

- f'h.. ' :.:. -

. 3

*. Similarly

II RJ M n E dy YB n E (A24)* R a] B (A24)

I + .IH YTnE - YBnE + -, kn n')E (A25)"JR " R JR T"

Using similar reasoning we get

SJL YT n ' E - yBnE + ,L(n - n°)E (A26)

JR JL = (n - n')E(lR -nL) (A27)

S We now let

(xR - XL) +0 (A28)

Then

i''q =R J (n - n')E 22 x(A9J - 3'-,IR - L (A29)R JL antx R L)

but from Eq. (A4)

ax Ak sin kxeit (A30)

so

S-R JL (n - n')E a k sin kx e iattXR X L). (A31)

-3 Using Eq. (A28) and noting that YT and YB are perturbation quantities we

can set

x sin kx dx sin kx(XR - XL) (A32)L

UN -kYT +kYB

e "e = (A33)

12

• " -'.• "" ' ? * o-'o,"" . . "-,, o . .- ' " ".,v . * .*-• .- , .. ,.' - 0.'.. " . , ,,. . . " ." " .. " ,"-. . ,, ,,"

and hence Eqs. (A18) and (A19) become

JT + n'kC sin kx xR - XL) (A34)

J i n kC sin kx(x R -XLJ (A35)

JT -B (n + n) kC sin kxxR XL) (A36)

Combining Eq. (A36), (A31), and (A17) and canceling the common term

ksin kx e iatxR - XL) we get

(n + n)C - (n - n')E a - 0 (A37)

Now substituting Eq. (A15) into (A37)

(n +n)i(n - n )Ea,

- n kE (A38)n+n B

" n -n EB' i-" y - k C (A39), .>n + n B

Defining M n/n"

M + 1 k cE (A40)=M+l B-

which corresponds to the result obtained by Huba and Zalesak [19831.

V

Acknowledgments

This work was supported by the Defense Nuclear Agency.

-a3

"- " -"'.,"'""" - .- 2. '" -." . .V '.. ." " ." .2"; '-"2 i -. 2'" -- .-.i:2 --; -i;'.. i ." 2'."Y 2-- " - -i ." i"--1'32

BACKSIDE OF BARIUM CLOUD

.*.',

! n. =n o -+-nb ne no

A

...,Eo X

lq.4 - .

y eB =13BZ

Fig. 1. Idealized "waterbag" elliptical barium cloud geometry. The totalelectron density, barium cloud ion density, and background

T•ionosphere ion dens ity are denoted by ne t. nb , and n o

" " .'respectively. B is the ambient magnetic field, and E. is the

• .- externally imposed electric field. E is the constant shielded

=Velectric field inside the cloud. a and b are the majlor and minor

axes of the ellipse respectively.

S14

• .". •.-.,,. -. .. ' •. . . .",' " ... " ' '."%'. -"-' ' .'.'-*-.-" . % ." . -. .[." . ." "-. "-" . '. ."."" .'..b. '*.-,. , .. .,".'.'.: . ..j ,'''' , '.," r ; % Xe W % % .. "," :.,. .'.

E:E 1

E =Ex

FLUID INTERFACE y (x) cos kx

I,. •B B

no nol +flb

y

x ..

Fig. 2 Geometry of 1-D barium cloud configuration for stability analysis

of discontinuous interface.

I.155

9* ,

',

--40

Fig. 3 Isodensity contours of a steepened 2-D barium cloud. The box

indicates that region of space over which we will apply our -D

stability analysis by approximating the density gradient there as

a 1-D, slowly evolving structure.

16

%

I ID GROWTH RATE ~M+

ITOTAL GROWTH RATE M 1-I(M +I1)

-%.4

- D HEDNG (M+I

I5 10 15M

Fig. 4 Plot of the M-dependence of the growth rate given by Eq. (7), M -

1/(M + 1)2. Also shown are its two component parts, (M - 1)/(M +

1) and (M + )- I , due to 1-D instability and 2-D shielding

results respectively.

!.t1

j..

17

4..0

0S 0

.00

0.0

0.1 1.0 10 100k L

AFig. 5 Plot of normalized growth rate versus kL for the case of the M

lip 39 cloud considered by Huba et al. [19821. The solid line-e I

depicts the simple model shown here, while the circles represent

numerical results from Huba. et al. f 1982].

I'.v

4..

%'4

J* %J

'S 1

all e

Y 0*

.51

ONTOE -E x

_" " F FLUID INTERFACE Y=, 7 (x)z= a coskx

'"'..P., n%,

I_ I

LCONTROL VOLUME V

~-6

Ay 0B =B z

x

Fig. 6 Geometry of I-D barium cloud configuration for stability

analysis, as in Fig. 2. n and 6 are the plasma density and

electrostatic potential, respectively. Primed and unprimed'p

variables denote quantities above and below the interface

y - n(x), respectively. Shown is the control volume used to

evaluate V.J.

19

" "

" YT < a

INTERFACE y : (x)

nI

'V .,

.., .JL J R

- n

~Ye 2 -a"I A

L J XR

Fig. 7. Blow-up of the control volume shown in Fig. 6. Depicted are the

total currents flowing through the four surfaces of the volume.

~ 20

>,,..v,*l2*

.4,°.,

References

Huba, J.D., S.L. Ossakow, P. Satyanarayana, and P.N. Guzdar, Linear theory

of the E x B instability with an Inhomogeneous Electric Field, J.

Geophys. Res., 88, 425, 1983.

Huba, J.D. and S.T. Zalesak, The long wavelength limit of

the E x B instability, J. Geophys. Res., 88, 10263, 1983.

Linson, L.M. and J.B. Workman, Formation of striations in ionospheric ion

clouds, J. Geophys. Res., 75, 3211, 1970.

Linson, L.M., Slab release and onset time of striations, in Multiple Barium

Release Studies, KMR Ionspheric Monitoring Program, Spring, 1975,

HAPREX Final Report, vol. 3, Stanford Reserch Institute, Menlo Park,

Calif., Oct. 1975.

McDonald, B.E., S.L. Ossakow, S.T. Zalesak, and N.J. Zabusky, Scale sizes

and lifetimes of F region plasma cloud striations as determined by thecondition of marginal stability, J. Geophys. Res., 86, 5775, 1981.

Ossakow, S.L. and P.K. Chaturvedi, Morphological studies of risingequatorial spread F bubbles, J. Geophys. Res., 83, 2085, 1978.

Overman, E.A. and N.J. Zabusky, Stability and nonlinear evolution of plasma

clouds via regularized contour dynamics, Phys. Rev. Lett., 45, 1693,

1980.

Zalesak, S.T., On the linear stability of two-dimensional barium clouds.

II. The effect of diffusion, NRL Memorandum Report in preparation,

1983.

21

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